Introduction and statement of the result
The Sherrington–Kirkpatrick (SK) model was introduced by Sherrington and Kirkpatrick [22 ] as a mean-field Ising spin glass with independent Gaussian pair interactions. Parisi’s replica-symmetry-breaking description uses a hierarchy of order parameters [20 ] . Guerra’s interpolation established the Parisi upper bound for the pressure [12 ] , and Talagrand proved equality [23 ] .
Farhi, Goldstone, and Gutmann introduced the quantum approximate optimization algorithm (QAOA) as alternating evolution under an objective Hamiltonian and a transverse-field mixer, starting from the uniform superposition [10 ] . For this model, Farhi, Goldstone, Gutmann, and Zhou developed an exact description of its performance at every fixed depth in the infinite-size limit and proved concentration at fixed parameters [11 ] . Basso, Farhi, Marwaha, Villalonga, and Zhou related that limit to QAOA on large-degree trees and conjectured that increasing depth eventually attains the Parisi optimum [4 ] , Section 7, eq:7.2 . Boulebnane et al. [5 ] proved a fixed-depth spin–boson representation and used it to obtain numerical evidence for convergence towards the Parisi value. Their representation provides a useful Hilbert-space language for the limit. The construction here must also realize bounded classical computations and eliminate longitudinal seed controls before selecting one ordinary word.
Sels and Morone [21 ] studied QAOA on the SK model using a semiclassical truncated-Wigner approximation and compared its performance with quantum data. Their numerical results and heuristic analysis give further evidence for convergence to the Parisi value and suggest a depth scaling.
The order of limits distinguishes this question from the fixed-instance optimality proved by Farhi, Goldstone, and Gutmann [10 ] , Section VI, eq:10 : for each fixed instance, increasing the depth approaches its optimum. That result does not provide a finite depth that works as the number of spins tends to infinity. We instead choose a complete finite word for each desired accuracy and then let the number of spins tend to infinity. The quantum construction simulates a finite classical calculation, using temporary Gaussian seed controls and then removing them.
There are two ways to supply the classical calculation. Auffinger, Chen, and Zeng proved that the zero-temperature Parisi order parameter has infinitely many support points [3 ] ; Chen proved full support on [ 0 , 1 ) [0,1) [ 0 , 1 ) [6 ] . Our primary route uses the companion Full support of the zero-temperature Sherrington–Kirkpatrick order parameter [19 ] . Its scalar diffusion yields a finite function of independent Gaussian variables and a coefficient formula for its value. We realize that function directly in the rooted Hilbert space and identify the coefficient formula with its energy. The imported scalar facts are full support of the zero-temperature order parameter, the diffusion’s consistency and value identities, and a finite Gaussian coefficient identity. Section 3 states these inputs before constructing their rooted realization.
We also give an independent positive-temperature route. It follows the Parisi-diffusion approach of Montanari [18 ] , whose optimization theorem assumes interval support of the positive-temperature Parisi measure, and the broader framework of El Alaoui, Montanari, and Sellke [8 ] . Lopatto’s full-support theorem [16 ] supplies that positive-temperature support property. The finite-iteration Gaussian and edge-energy analysis of El Alaoui, Montanari, and Sellke [9 ] then gives a near-optimal descendant formula. This second route has its own quantitative and finite-degree statements and does not use the zero-temperature companion.
Model and quantifiers
Let ( J i j ) 1 ≤ i < j ≤ n (J_{ij})_{1\leq i<j\leq n} ( J ij ) 1 ≤ i < j ≤ n σ ∈ { − 1 , 1 } n \sigma\in\{-1,1\}^{n} σ ∈ { − 1 , 1 } n
h n , J ( σ ) = 1 n ∑ i < j J i j σ i σ j , C n , J = 1 n ∑ i < j J i j Z i Z j , B n = ∑ i = 1 n X i . (1) h_{n,J}(\sigma)=\frac{1}{\sqrt{n}}\sum_{i<j}J_{ij}\sigma_i\sigma_j,\qquad C_{n,J}=\frac{1}{\sqrt{n}}\sum_{i<j}J_{ij}Z_iZ_j,\qquad B_n=\sum_{i=1}^{n}X_i. \tag*{(1)} h n , J ( σ ) = n 1 i < j ∑ J ij σ i σ j , C n , J = n 1 i < j ∑ J ij Z i Z j , B n = i = 1 ∑ n X i . (1) Here X , Y , Z X,Y,Z X , Y , Z
P ∗ = lim n → ∞ 1 n E max σ h n , J ( σ ) . (2) P_*=\lim_{n\to\infty}\frac{1}{n}\mathbb{E}\max_{\sigma}h_{n,J}(\sigma). \tag*{(2)} P ∗ = n → ∞ lim n 1 E σ max h n , J ( σ ) . (2) The ground-state Parisi formula and its attainment are due to Auffinger and Chen [2 ] . We use the covariance convention ξ ( t ) = t 2 / 2 \xi(t)=t^2/2 ξ ( t ) = t 2 /2
For p ≥ 1 p\geq1 p ≥ 1 γ , β ∈ R p \gamma,\beta\in\mathbb{R}^{p} γ , β ∈ R p
∣ ψ n , p , J ( γ , β ) ⟩ = e − i β p B n e − i γ p C n , J ⋯ e − i β 1 B n e − i γ 1 C n , J ∣ + ⟩ ⊗ n . (3) |\psi_{n,p,J}(\gamma,\beta)\rangle=e^{-i\beta_pB_n}e^{-i\gamma_pC_{n,J}}\cdots e^{-i\beta_1B_n}e^{-i\gamma_1C_{n,J}}|+\rangle^{\otimes n}. \tag*{(3)} ∣ ψ n , p , J ( γ , β )⟩ = e − i β p B n e − i γ p C n , J ⋯ e − i β 1 B n e − i γ 1 C n , J ∣ + ⟩ ⊗ n . (3) All parameters are deterministic. Write
v p ( γ , β ) = lim n → ∞ 1 n E ⟨ ψ n , p , J ( γ , β ) ∣ C n , J ∣ ψ n , p , J ( γ , β ) ⟩ , Q p = sup γ , β ∈ R p v p ( γ , β ) . (4) v_p(\gamma,\beta)=\lim_{n\to\infty}\frac{1}{n}\mathbb{E}\langle\psi_{n,p,J}(\gamma,\beta)|C_{n,J}|\psi_{n,p,J}(\gamma,\beta)\rangle,\qquad Q_p=\sup_{\gamma,\beta\in\mathbb{R}^{p}}v_p(\gamma,\beta). \tag*{(4)} v p ( γ , β ) = n → ∞ lim n 1 E ⟨ ψ n , p , J ( γ , β ) ∣ C n , J ∣ ψ n , p , J ( γ , β )⟩ , Q p = γ , β ∈ R p sup v p ( γ , β ) . (4) The fixed-parameter limit exists [11 , 4 ] .
Theorem 1.1. For the zero-field SK model (1) ,
lim p → ∞ Q p = P ∗ . (5) \lim_{p\to\infty}Q_p=P_*. \tag*{(5)} p → ∞ lim Q p = P ∗ . (5) Equivalently, for every ε > 0 \varepsilon>0 ε > 0 p p p γ , β ∈ R p \gamma,\beta\in\mathbb{R}^{p} γ , β ∈ R p n n n J J J v p ( γ , β ) ≥ P ∗ − ε v_p(\gamma,\beta)\geq P_*-\varepsilon v p ( γ , β ) ≥ P ∗ − ε
This proves the conjectured eventual Parisi optimality in the fixed-parameter thermodynamic formulation of [4 ] . Measurement in the computational basis immediately gives Q p ≤ P ∗ Q_p \le P_* Q p ≤ P ∗ Q p + 1 ≥ Q p Q_{p+1} \ge Q_p Q p + 1 ≥ Q p
The theorem is stated in expected energy. The argument gives no quantitative finite-size estimate, depth bound, or efficient procedure for choosing the angles.
How the construction works
The proof separates a classical choice of target from its quantum implementation. The intermediate setting is the large-degree regular tree: each vertex carries an independent Gaussian label, and a finite rooted calculation only reads labels within a fixed radius. Normalized sums of centered child calculations become jointly Gaussian coordinates. Section 2 gives a Hilbert-space realization of these coordinates and a continuous quadratic expression for the limiting energy.
The scalar companion provides a bounded odd function of finitely many independent Gaussians whose value approaches P ∗ P_* P ∗
We next approximate all transmitted functions by bounded smooth ones. A finite label assigns a type to each vertex. Splitting the label space into sufficiently small types lets every aggregation omit the receiver’s type, the types of all its path ancestors, and a small reserved set of helper types, at arbitrarily small energy cost. These exclusions prevent a compiled operation from feeding back into its own parameters. They do not assert independence of calculations using the same descendant subtree. Proposition 3.5 gives the resulting bounded finite formula, with its output written as sin ( 2 θ ) \sin(2\theta) sin ( 2 θ ) θ \theta θ
Theorem 4.1 implements these angles using cost and mixer gates together with a longitudinal field formed from the Gaussian labels. To implement a neighbor sum, commutators combine interactions along a receiver–sender–helper–sender walk. When the walk returns to the same sender, the helper spin, held in its initial state, supplies the desired term. We subdivide the sender type and retain only walks whose two sender visits lie in the same part. This keeps every returning walk, while the different-sender contribution vanishes as the parts become small. The proof also controls the effects inside the lower calculations, which are called repeatedly within a fixed surrounding circuit. Helpers have arbitrarily small total probability; discarding their output is included in the energy estimate.
Theorem 5.1 then removes the longitudinal field from the already selected finite circuit. An imperfect cost echo creates a small component with a large coherent displacement. Selective spin rotations change that component while canceling their leading action on the rest of the state; a long cost echo magnifies the change into a finite longitudinal field. Separating its direction from the circuit’s existing Gaussian coordinates gives a seed that can be reused at every requested seed gate. A separate one-particle calculation shows that this Gaussian identification also preserves the rooted energy. The elementary ensemble-control ideas are related to those of Li and Khaneja [15 ] ; Section 5 proves the estimates for one fixed circuit, including its changing insertion directions and the singular echo.
After all these choices there is one finite ordinary cost–mixer word. Only at this point do we invoke the fixed-word tree-to-SK identity of Basso et al. [4 ] . The exact error allocation and the order of choices are given in Section 6, after the three constructions have been established.
A regular MaxCut consequence
For a finite simple graph G = ( V , E ) G=(V,E) G = ( V , E ) E ≠ ∅ E\ne\varnothing E = ∅
H M C ( G ) = 1 2 ∑ { u , v } ∈ E ( I − Z u Z v ) , B G = ∑ v ∈ V X v . H_{\mathrm{MC}}(G)=\frac{1}{2}\sum_{\{u,v\}\in E}(I-Z_uZ_v),\qquad B_G=\sum_{v\in V}X_v. H MC ( G ) = 2 1 { u , v } ∈ E ∑ ( I − Z u Z v ) , B G = v ∈ V ∑ X v . For a , b ∈ R p a,b\in\mathbb{R}^{p} a , b ∈ R p ⟨ ⋅ ⟩ p , a , b G \langle\cdot\rangle^{G}_{p,a,b} ⟨ ⋅ ⟩ p , a , b G H M C ( G ) H_{\mathrm{MC}}(G) H MC ( G ) B G B_G B G a , b a,b a , b ∣ + ⟩ ⊗ ∣ V ∣ \lvert+\rangle^{\otimes\lvert V\rvert} ∣ + ⟩ ⊗ ∣ V ∣
q p , G ( a , b ) = ⟨ H M C ( G ) ⟩ p , a , b G ∣ E ∣ . q_{p,G}(a,b)=\frac{\langle H_{\mathrm{MC}}(G)\rangle^{G}_{p,a,b}}{\lvert E\rvert}. q p , G ( a , b ) = ∣ E ∣ ⟨ H MC ( G ) ⟩ p , a , b G . Write OPT ( G ) = MaxCut ( G ) / ∣ E ∣ \operatorname{OPT}(G)=\operatorname{MaxCut}(G)/\lvert E\rvert OPT ( G ) = MaxCut ( G ) / ∣ E ∣
Corollary 1.2 (Regular MaxCut in successive limits). For every η > 0 \eta>0 η > 0 p ≥ 1 p\ge1 p ≥ 1 γ , β ∈ R p \gamma,\beta\in\mathbb{R}^{p} γ , β ∈ R p D 0 ≥ 2 D_0\ge2 D 0 ≥ 2 D ≥ D 0 D\ge D_0 D ≥ D 0 ( D + 1 ) (D+1) ( D + 1 ) G G G 2 p + 1 2p+1 2 p + 1
q p , G ( 2 γ / D , β ) ≥ 1 2 + P ∗ − η D . (6) q_{p,G}(2\gamma/\sqrt{D},\beta)\ge\frac{1}{2}+\frac{P_*-\eta}{\sqrt{D}}. \tag*{(6)} q p , G ( 2 γ / D , β ) ≥ 2 1 + D P ∗ − η . (6) The word p , γ , β p,\gamma,\beta p , γ , β D D D G G G γ \gamma γ
Let G n , d G_{n,d} G n , d d d d n n n n n n n d nd n d
Δ D ( G ) = OPT ( G ) − q p , G ( 2 γ / D , β ) . \Delta_D(G)=\operatorname{OPT}(G)-q_{p,G}(2\gamma/\sqrt{D},\beta). Δ D ( G ) = OPT ( G ) − q p , G ( 2 γ / D , β ) . The word may be chosen so that
lim D → ∞ lim sup n → ∞ n ( D + 1 ) e v e n Pr G [ D Δ D ( G n , D + 1 ) > η ] = 0. (7) \lim_{D\to\infty}\limsup_{\substack{n\to\infty\\n(D+1)\ \mathrm{even}}}\Pr_G\left[\sqrt{D}\,\Delta_D(G_{n,D+1})>\eta\right]=0. \tag*{(7)} D → ∞ lim n → ∞ n ( D + 1 ) even lim sup G Pr [ D Δ D ( G n , D + 1 ) > η ] = 0. (7) Here the inner limit keeps D , p , γ , β D,p,\gamma,\beta D , p , γ , β Pr G \Pr_G Pr G q p , G q_{p,G} q p , G
The complete proof is given in Section 7, after the fixed-word tree interface and the SK construction.
The optimality comparison concerns random regular graphs in these successive limits, not all high-girth graphs or fixed-degree optima. No joint size–degree limit is asserted. Here p p p
Reading paths and conventions
Section 2 develops the fixed-word rooted calculus. Section 3 imports the scalar theorems and constructs the bounded excluded formula. Sections 4 and 5 implement it and remove the seeds. Section 6 assembles the ordinary word and proves Theorem 1.1. Section 7 then proves the regular-MaxCut consequence from this theorem and the fixed-word tree interface.
For the positive-temperature reading path, Appendix A replaces the scalar input and its rooted realization. It derives the same classical conclusion independently, using the shared bounded approximation and exclusion lemmas of Section 3. The quantum constructions and final assembly are the same for both paths.
Physical system size is n n n D + 1 D+1 D + 1 p p p N N N β \beta β β 0 \beta_0 β 0 γ ( t ) \gamma(t) γ ( t )
All constants may depend on a fixed finite formula or word. Uniformity in any approximation parameter is stated when required. Degree limits are taken with all words and their angles fixed; further approximation limits concern the resulting tree values.
The fixed-word tree calculus
This section constructs the common space in which we compare ordinary QAOA words and classical formulas. It has three tasks: identify the fixed-word SK value with a tree limit, express that limit as a bounded quadratic form, and justify the controlled operations used in synthesis. The same space will carry the finite Gaussian construction in Section 3.
We first isolate the transfer from an ordinary word to the tree. A word always means a finite word with fixed real coefficients; none of the statements below makes a uniform assertion in its length or its coefficients. The initial vector Ω \Omega Ω
Proposition 2.1 (Fixed-word transfer). Let W W W ( D + 1 ) (D+1) ( D + 1 ) C D = D − 1 / 2 ∑ { i , j } ∈ E Z i Z j C_D=D^{-1/2}\sum_{\{i,j\}\in E} Z_iZ_j C D = D − 1/2 ∑ { i , j } ∈ E Z i Z j
v W = lim D → ∞ D 2 ⟨ W D Ω ∣ Z i Z j ∣ W D Ω ⟩ (8) v_W=\lim_{D\to\infty}\frac{\sqrt{D}}{2}\langle W_D\Omega\lvert Z_iZ_j\rvert W_D\Omega\rangle \tag*{(8)} v W = D → ∞ lim 2 D ⟨ W D Ω ∣ Z i Z j ∣ W D Ω ⟩ (8) Here v W v_W v W
Proof. This is Theorem 1 of [4 ] , together with its normalizations in Equations (2.3), (2.6), and (6.1)–(6.2). Its Equation (3.11) gives the fixed-depth tree value, and Equation (6.3) identifies that value with the SK limit. That paper uses the negative of our tree cost. Conjugation by X X X Z i Z j Z_iZ_j Z i Z j (8) . The SK convention already agrees with the positive Gaussian Hamiltonian in its Equation (6.1). Consecutive gates of the same kind and inverses are included by inserting zero-angle gates. □ \square □
We first construct the rooted space and identify its energy functional. This also gives the Gaussian coordinates used by the classical construction. We then prove the weighted convergence estimates that realize ordinary words in this space and justify controlled gates inside a fixed surrounding circuit.
Pointed spaces and occupation embeddings
The number of children will tend to infinity while the number of circuit layers remains fixed. We therefore organize a symmetric collection of child states by how many differ from the initial state. The limiting space is a symmetric Fock space: its m m m m m m m m m [5 ] ; the occupation, insertion, and surrounding-circuit estimates used below are proved here.
Let ( S , μ , ι ) (S,\mu,\iota) ( S , μ , ι ) ι ( s ) = − s \iota(s)=-s ι ( s ) = − s S S S
K = C 2 ⊗ L 2 ( S , μ ) , ω = ∣ + ⟩ ⊗ 1. \mathcal{K}=\mathbb{C}^{2}\otimes L^{2}(S,\mu),\qquad\omega=|+\rangle\otimes1. K = C 2 ⊗ L 2 ( S , μ ) , ω = ∣ + ⟩ ⊗ 1. For any pointed space below, write ⟨ T ⟩ 0 = ⟨ Ω ∣ T ∣ Ω ⟩ \langle T\rangle_{0}=\langle\Omega|T|\Omega\rangle ⟨ T ⟩ 0 = ⟨ Ω∣ T ∣Ω ⟩ Q = 1 − ∣ Ω ⟩ ⟨ Ω ∣ Q=1-|\Omega\rangle\langle\Omega| Q = 1 − ∣Ω ⟩ ⟨ Ω∣ X ⊗ ( f ↦ f ∘ ι ) X\otimes(f\mapsto f\circ\iota) X ⊗ ( f ↦ f ∘ ι )
For a pointed Hilbert space ( H , Ω ) (H,\Omega) ( H , Ω ) H ∘ = H ⊖ C Ω H^{\circ}=H\ominus\mathbb{C}\Omega H ∘ = H ⊖ C Ω Γ s ( H ∘ ) = ⨁ m ≥ 0 ( H ∘ ) ⊗ s m \Gamma_{s}(H^{\circ})=\bigoplus_{m\geq0}(H^{\circ})^{\otimes_{s}m} Γ s ( H ∘ ) = ⨁ m ≥ 0 ( H ∘ ) ⊗ s m
H 0 = K , H h + 1 = K ⊗ Γ s ( H h ∘ ) , Ω h + 1 = ω ⊗ Ω . (9) \mathcal{H}_{0}=\mathcal{K},\qquad\mathcal{H}_{h+1}=\mathcal{K}\otimes\Gamma_{s}(\mathcal{H}_{h}^{\circ}),\qquad\Omega_{h+1}=\omega\otimes\Omega. \tag*{(9)} H 0 = K , H h + 1 = K ⊗ Γ s ( H h ∘ ) , Ω h + 1 = ω ⊗ Ω. (9) The embedding from height h h h h + 1 h+1 h + 1 ξ ↦ ξ ⊗ Ω \xi\mapsto\xi\otimes\Omega ξ ↦ ξ ⊗ Ω
Write R ι f = f ∘ ι R_{\iota}f=f\circ\iota R ι f = f ∘ ι
P 0 = X ⊗ R ι , P h + 1 = ( X ⊗ R ι ) ⊗ Γ s ( P h ∣ H h ∘ ) . P_{0}=X\otimes R_{\iota},\qquad P_{h+1}=(X\otimes R_{\iota})\otimes\Gamma_{s}(P_{h}|_{\mathcal{H}_{h}^{\circ}}). P 0 = X ⊗ R ι , P h + 1 = ( X ⊗ R ι ) ⊗ Γ s ( P h ∣ H h ∘ ) . These involutions respect the height embeddings and fix the vacuum, so every odd observable has zero vacuum mean. We suppress the height in P P P
The analogous invariant space on a rooted D D D
H D , 0 = K , H D , h + 1 = K ⊗ Sym D ( H D , h ) . \mathcal{H}_{D,0}=\mathcal{K},\qquad\mathcal{H}_{D,h+1}=\mathcal{K}\otimes\operatorname{Sym}^{D}(\mathcal{H}_{D,h}). H D , 0 = K , H D , h + 1 = K ⊗ Sym D ( H D , h ) . The symmetry here permutes children together with their entire subtrees and seed coordinates. It does not discard the seeds. For any pointed space H H H
I D : Sym D ( H ) ⟶ ⨁ m = 0 D ( H ∘ ) ⊗ s m . (10) \mathcal{I}_{D}:\operatorname{Sym}^{D}(H)\longrightarrow\bigoplus_{m=0}^{D}(H^{\circ})^{\otimes_{s}m}. \tag*{(10)} I D : Sym D ( H ) ⟶ m = 0 ⨁ D ( H ∘ ) ⊗ s m . (10) On a symmetric m m m m m m D D D ( D m ) − 1 / 2 \binom{D}{m}^{-1/2} ( m D ) − 1/2 I D , h : H D , h → H h \mathcal{I}_{D,h}:\mathcal{H}_{D,h}\to\mathcal{H}_{h} I D , h : H D , h → H h P h P_{h} P h
Creation is linear in its argument, annihilation is antilinear, and
q ( d ) = a ∗ ( d ) + a ( d ) , [ q ( d ) , q ( e ) ] = 2 i Im ⟨ d , e ⟩ . q(d)=a^{*}(d)+a(d),\qquad[q(d),q(e)]=2i\operatorname{Im}\langle d,e\rangle. q ( d ) = a ∗ ( d ) + a ( d ) , [ q ( d ) , q ( e )] = 2 i Im ⟨ d , e ⟩ . Conjugation by the recursive parity sends the child field q ( d ) q(d) q ( d ) q ( P h d ) q(P_h d) q ( P h d ) q ( d ) q(d) q ( d ) W ( d ) = e i q ( d ) \mathcal{W}(d)=e^{\mathrm{i}q(d)} W ( d ) = e i q ( d )
W ( d ) ∗ q ( v ) W ( d ) = q ( v ) − 2 Im ⟨ v , d ⟩ . (11) \mathcal{W}(d)^{*}q(v)\mathcal{W}(d)=q(v)-2\operatorname{Im}\langle v,d\rangle. \tag*{(11)} W ( d ) ∗ q ( v ) W ( d ) = q ( v ) − 2 Im ⟨ v , d ⟩ . (11) Root insertions and edge energy
At finite degree, a rooted vector records the response of a root observable to a circuit. Let U U U U j U_j U j j j j u u u
U = ( ⨂ j c h i l d U j ) u . U=\left(\bigotimes_{j\ \mathrm{child}}U_j\right)u. U = j child ⨂ U j u . For a root Pauli observable P P P
f P = U ∗ P U Ω = u ∗ P u Ω . f_P=U^{*}PU\Omega=u^{*}Pu\Omega. f P = U ∗ P U Ω = u ∗ P u Ω. because the child operations commute with P P P Z Z Z Z i Z j Z_iZ_j Z i Z j
The insertion vector lives at one root, whereas an edge joins two rooted neighborhoods. The following operator moves one centered vector down to a single child while placing the new root in its initial state. It will turn the edge pairing into a quadratic form on one common space.
Define the centered root shift by
L : H ∘ ⟶ H ∘ , L v = ω ⊗ a ∗ ( v ) Ω . (12) L:\mathcal{H}^{\circ}\longrightarrow\mathcal{H}^{\circ},\qquad Lv=\omega\otimes a^{*}(v)\Omega. \tag*{(12)} L : H ∘ ⟶ H ∘ , Lv = ω ⊗ a ∗ ( v ) Ω. (12) At height h h h h + 1 h+1 h + 1 L L L L ∗ Ω = 0 L^{*}\Omega=0 L ∗ Ω = 0 L ∗ L^{*} L ∗ L ∗ L^{*} L ∗
Proposition 2.2 (Energy as a root shift). Let f D f_D f D ( D + 1 ) (D+1) ( D + 1 ) v v v D D D f D , i f_{D,i} f D , i f D , j f_{D,j} f D , j
lim D → ∞ D 2 ⟨ f D , i , f D , j ⟩ = E ( v ) : = Re ⟨ v , L v ⟩ . (13) \lim_{D\to\infty}\frac{\sqrt{D}}{2}\langle f_{D,i},f_{D,j}\rangle=\mathcal{E}(v):=\operatorname{Re}\langle v,Lv\rangle. \tag*{(13)} D → ∞ lim 2 D ⟨ f D , i , f D , j ⟩ = E ( v ) := Re ⟨ v , Lv ⟩ . (13) For any centered v , w v,w v , w
∣ E ( v ) − E ( w ) ∣ ≤ ( ∥ v ∥ + ∥ w ∥ ) ∥ v − w ∥ . (14) \left|\mathcal{E}(v)-\mathcal{E}(w)\right|\leq\left(\lVert v\rVert+\lVert w\rVert\right)\lVert v-w\rVert. \tag*{(14)} ∣ E ( v ) − E ( w ) ∣ ≤ ( ∥ v ∥ + ∥ w ∥ ) ∥ v − w ∥ . (14) Proof. We give the exact finite-degree decomposition. Put M = D + 1 M = D + 1 M = D + 1 E E E N M N_M N M k ⊗ Sym M ( E ) k \otimes\operatorname{Sym}^M(E) k ⊗ Sym M ( E ) R M R_M R M
J M ξ = ω ⊗ 1 M ∑ j = 1 M Ω ⊗ ( j − 1 ) ⊗ ξ ⊗ Ω ⊗ ( M − j ) . J_M \xi= \omega\otimes\frac{1}{\sqrt{M}} \sum_{j=1}^{M} \Omega^{\otimes(j-1)} \otimes\xi\otimes\Omega^{\otimes(M-j)}. J M ξ = ω ⊗ M 1 j = 1 ∑ M Ω ⊗ ( j − 1 ) ⊗ ξ ⊗ Ω ⊗ ( M − j ) . The latter is an isometry from E ∘ E^\circ E ∘ J M ∗ = L ∗ J_M^* = L^* J M ∗ = L ∗
R M = 1 − N / M , ∥ R M f − f ∥ 2 ≤ M − 1 ⟨ f ∣ N M ∣ f ⟩ . (15) R_M = \sqrt{1 - N/M}, \qquad\lVert R_M f - f \rVert^2 \le M^{-1} \langle f \mid N_M \mid f \rangle. \tag*{(15)} R M = 1 − N / M , ∥ R M f − f ∥ 2 ≤ M − 1 ⟨ f ∣ N M ∣ f ⟩ . (15) Indeed, in the m m m ( M − 1 m ) / ( M m ) = 1 − m / M \binom{M-1}{m}/\binom{M}{m} = 1 - m/M ( m M − 1 ) / ( m M ) = 1 − m / M
Delete the distinguished edge and let P A , P B P_A, P_B P A , P B f D , i f_{D,i} f D , i P B + ( 1 − P B ) P_B + (1 - P_B) P B + ( 1 − P B ) f D , j f_{D,j} f D , j P A + ( 1 − P A ) P_A + (1 - P_A) P A + ( 1 − P A ) s D = J M ∗ f D s_D = J_M^* f_D s D = J M ∗ f D r D = R M f D r_D = R_M f_D r D = R M f D ⟨ s D , r D ⟩ / M \langle s_D, r_D \rangle/\sqrt{M} ⟨ s D , r D ⟩ / M ⟨ f D ∣ N M ∣ f D ⟩ / M \langle f_D \mid N_M \mid f_D \rangle/M ⟨ f D ∣ N M ∣ f D ⟩ / M
⟨ f D , i , f D , j ⟩ = 2 M Re ⟨ s D , r D ⟩ + ε D , ∣ ε D ∣ ≤ ⟨ f D ∣ N M ∣ f D ⟩ M . (16) \langle f_{D,i}, f_{D,j} \rangle= \frac{2}{\sqrt{M}} \operatorname{Re}\langle s_D, r_D \rangle+ \varepsilon_D, \qquad|\varepsilon_D| \le\frac{\langle f_D \mid N_M \mid f_D \rangle}{M}. \tag*{(16)} ⟨ f D , i , f D , j ⟩ = M 2 Re ⟨ s D , r D ⟩ + ε D , ∣ ε D ∣ ≤ M ⟨ f D ∣ N M ∣ f D ⟩ . (16) After multiplication by D / 2 \sqrt{D}/2 D /2 (15) and strong convergence show that the first term tends to Re ⟨ L ∗ v , v ⟩ = Re ⟨ v , L v ⟩ \operatorname{Re}\langle L^*v,v\rangle= \operatorname{Re}\langle v,Lv\rangle Re ⟨ L ∗ v , v ⟩ = Re ⟨ v , Lv ⟩ (13) .
Finally, (14) follows by adding and subtracting ⟨ w , L v ⟩ \langle w,Lv\rangle ⟨ w , Lv ⟩ ∥ L ∥ = 1 \lVert L\rVert= 1 ∥ L ∥ = 1
The occupation hypothesis in Proposition 2.2 is a sufficient hypothesis for the displayed proof; strong convergence alone does not bound the remainder in (16) . Lemma 2.9 and Theorem 2.12 will supply that hypothesis for the quantum gates used here.
Remark 2.3 (Classical rotations). The classical sector consists of vectors whose spin is ∣ + ⟩ |+\rangle ∣ + ⟩ Y Y Y v = ∣ + ⟩ m + ∣ − ⟩ n v = |+\rangle_m + |-\rangle_n v = ∣ + ⟩ m + ∣ − ⟩ n E ( v ) = E ( ∣ + ⟩ m ) \mathcal{E}(v) = \mathcal{E}(|+\rangle_m) E ( v ) = E ( ∣ + ⟩ m ) L v Lv Lv ∣ + ⟩ |+\rangle ∣ + ⟩ L ( ∣ − ⟩ n ) L(|-\rangle_n) L ( ∣ − ⟩ n )
The root-shift functional also applies to classical vectors, before they are compiled into quantum gates. We now give a common Gaussian realization for both classical constructions in this paper. This is a definition on the limiting pointed space; a finite-degree energy limit still requires the hypotheses of Proposition 2.2.
Keep every quantum spin in ∣ + ⟩ |+\rangle ∣ + ⟩ f f f G ( f ) = q ( f ) G(f) = q(f) G ( f ) = q ( f ) f 1 , … , f k f_1,\ldots,f_k f 1 , … , f k ⟨ f i , f j ⟩ \langle f_i,f_j\rangle ⟨ f i , f j ⟩
⟨ e i ∑ j = 1 k t j G ( f j ) ⟩ 0 = exp ( − 1 2 ∥ ∑ j = 1 k t j f j ∥ 2 ) . \left\langle e^{i\sum_{j=1}^{k}t_jG(f_j)}\right\rangle_0=\exp\left(-\frac{1}{2}\left\|\sum_{j=1}^{k}t_jf_j\right\|^2\right). ⟨ e i ∑ j = 1 k t j G ( f j ) ⟩ 0 = exp − 2 1 j = 1 ∑ k t j f j 2 . In particular, orthonormal directions give independent standard normal positions: the variance convention is ∥ f ∥ 2 \|f\|^2 ∥ f ∥ 2
We identify a scalar function of these commuting variables with the vector obtained by applying that function to the vacuum. Under this identification,
∥ ( G ( f ) − G ( g ) ) Ω ∥ = ∥ f − g ∥ . (17) \|(G(f)-G(g))\Omega\|=\|f-g\|. \tag*{(17)} ∥ ( G ( f ) − G ( g )) Ω∥ = ∥ f − g ∥. (17) All fields are realized on the same pointed space, so this equality also specifies their coupling. The empty-descendant embeddings in (9) preserve inner products, commute with passing to the corresponding Fock fields, and keep the local seed factor fixed. Thus directions constructed at different finite heights can first be embedded in one common height without changing their joint Gaussian law.
Definition 2.4 (Finite descendant formula). A finite descendant formula is a finite expression built recursively from a local seed and finitely many Gaussian positions of centered formulas one level below, using pointwise scalar functions. The functions are required to be measurable and to produce square-integrable vectors whenever the formula is used as a vector. Additional empty levels identify formulas of different radii. Its finite-degree version replaces every Gaussian position by the normalized sum of the corresponding independent child variables.
The definition by itself is a limiting-space construction, not a finite-degree convergence theorem. Later approximation arguments impose boundedness or polynomial-growth conditions on the pointwise maps as needed. Because L L L L ∗ L^* L ∗ e 1 , … , e k e_1,\ldots,e_k e 1 , … , e k F F F G ( e 1 ) , … , G ( e k ) G(e_1),\ldots,G(e_k) G ( e 1 ) , … , G ( e k )
L ∗ F = ∑ j = 1 k E [ F G ( e j ) ] e j . (18) L^*F=\sum_{j=1}^{k}\mathbb{E}\left[F G(e_j)\right]e_j. \tag*{(18)} L ∗ F = j = 1 ∑ k E [ F G ( e j ) ] e j . (18) Here F F F L ∗ L^* L ∗
The rooted space, Gaussian positions, and energy functional now give the setting for the classical construction in Section 3. The rest of this section establishes convergence and continuity of the quantum operations that will implement it.
Weights and the topology of convergence
The fields just defined are unbounded. To pass their products and parameter derivatives through limits, we need a common domain and bounds that control both seed moments and excitations at every level. These recursive weights serve that purpose; they are distinct from the top-generation number weight used in the seed-removal argument. If all local-seed coefficients under consideration are uniformly bounded, take W 0 = 1 W_0=1 W 0 = 1 k \mathsf{k} k 1 + ∣ s ∣ 1+|s| 1 + ∣ s ∣ ω \omega ω
W 0 = P 0 + Q 0 ( 1 + ∣ s ∣ ) Q 0 , P 0 = ∣ ω ⟩ ⟨ ω ∣ , Q 0 = 1 − P 0 . W_0=P_0+Q_0(1+|s|)Q_0,\qquad P_0=|\omega\rangle\langle\omega|,\qquad Q_0=1-P_0. W 0 = P 0 + Q 0 ( 1 + ∣ s ∣ ) Q 0 , P 0 = ∣ ω ⟩ ⟨ ω ∣ , Q 0 = 1 − P 0 . Compression is understood on its natural closed domain. This is a positive self-adjoint operator bounded below by one. It differs from multiplication by 1 + ∣ s ∣ 1+|s| 1 + ∣ s ∣ A m − B m = ∑ j = 0 m − 1 A j ( A − B ) B m − 1 − j A^m-B^m=\sum_{j=0}^{m-1}A^j(A-B)B^{m-1-j} A m − B m = ∑ j = 0 m − 1 A j ( A − B ) B m − 1 − j A − B A-B A − B
Since W 0 ω = ω W_0\omega=\omega W 0 ω = ω
Λ h = W h ∣ H ∘ , W h + 1 = W 0 ⊗ 1 + 1 ⊗ d Γ ( Λ h ) . (19) \Lambda_h=W_h|_{\mathcal{H}^{\circ}},\qquad W_{h+1}=W_0\otimes1+1\otimes\mathrm{d}\Gamma(\Lambda_h). \tag*{(19)} Λ h = W h ∣ H ∘ , W h + 1 = W 0 ⊗ 1 + 1 ⊗ d Γ ( Λ h ) . (19) Thus W h Ω h = Ω h W_h\Omega_h=\Omega_h W h Ω h = Ω h W h ≥ 1 W_h\ge1 W h ≥ 1 W D , h + 1 = W 0 + ∑ j = 1 D Q j W D , h ( j ) Q j W_{D,h+1}=W_0+\sum_{j=1}^{D}Q_jW_{D,h}^{(j)}Q_j W D , h + 1 = W 0 + ∑ j = 1 D Q j W D , h ( j ) Q j W h W_h W h
Write
S h = ⋂ m ≥ 0 Dom ( W h m ) , ∥ ξ ∥ m = ∥ W h m ξ ∥ . \mathcal{S}_h=\bigcap_{m\ge0}\operatorname{Dom}(W_h^m),\qquad\|\xi\|_m=\|W_h^m\xi\|. S h = m ≥ 0 ⋂ Dom ( W h m ) , ∥ ξ ∥ m = ∥ W h m ξ ∥. Finite-particle tensors of smooth child vectors and local vectors with all seed moments are dense in every one of these graph norms. One proof first cuts off the spectral variables of W 0 W_0 W 0 Λ h \Lambda_h Λ h
When comparing unitary maps on the varying embedded spaces, extend each embedded unitary by the identity on the orthogonal complement of its range projection. These are genuine unitaries on the common Hilbert space. The projections commute with the weights and converge strongly to one in every fixed graph norm on smooth vectors, so this convention preserves all the bounds and limits. Parameter derivatives of the extensions vanish on that complement.
We will repeatedly use the following precise convention. An operator family F D ( x ) F_D(x) F D ( x ) uniform smooth bounds if, for every nonnegative integer m m m α \alpha α r r r k k k C C C D D D x x x
∥ W D m ∂ x α F D ( x ) ξ ∥ ≤ C ( 1 + ∣ x ∣ ) k ∥ W D r ξ ∥ , ξ ∈ S D . (20) \left\|W_D^m\partial^\alpha_xF_D(x)\xi\right\|\le C(1+|x|)^k\left\|W_D^r\xi\right\|,\qquad\xi\in\mathcal{S}_D. \tag*{(20)} ∥ W D m ∂ x α F D ( x ) ξ ∥ ≤ C ( 1 + ∣ x ∣ ) k ∥ W D r ξ ∥ , ξ ∈ S D . (20) The same condition is required of the adjoint family. Constants may depend on the fixed expression, its finite list of channels, the height, m m m α \alpha α
Two elementary consequences will be useful. Products of a fixed number of such families have the same bounds, with a possibly larger loss of input powers. Also, strong convergence together with bounded higher graph norms gives convergence in every fixed lower graph norm. The latter follows from the spectral inequality
∥ W m ξ ∥ ≤ R m ∥ ξ ∥ + R m − M ∥ W M ξ ∥ ( M > m , R ≥ 1 ) , (21) \lVert W^m \xi\rVert\le R^m\lVert\xi\rVert+ R^{m-M}\lVert W^M \xi\rVert\qquad(M>m,\ R\ge1), \tag*{(21)} ∥ W m ξ ∥ ≤ R m ∥ ξ ∥ + R m − M ∥ W M ξ ∥ ( M > m , R ≥ 1 ) , (21) by taking the strong limit first and R → ∞ R \to\infty R → ∞
Lemma 2.5 (Weyl estimates). Let N N N Γ s ( H ∘ ) \Gamma_{\mathrm{s}}(H^\circ) Γ s ( H ∘ ) m ≥ 0 m \ge0 m ≥ 0 C m C_m C m
∥ ( 1 + N ) m W ( d ) ξ ∥ ≤ C m ( 1 + ∥ d ∥ ) 2 m ∥ ( 1 + N ) m ξ ∥ , (22) \lVert(1+N)^m\mathcal{W}(d)\xi\rVert\le C_m(1+\lVert d\rVert)^{2m}\lVert(1+N)^m\xi\rVert, \tag*{(22)} ∥( 1 + N ) m W ( d ) ξ ∥ ≤ C m ( 1 + ∥ d ∥ ) 2 m ∥( 1 + N ) m ξ ∥ , (22) ∥ ( W ( d ) − 1 ) ξ ∥ ≤ 2 ∥ d ∥ ∥ ( N + 1 ) 1 / 2 ξ ∥ . (23) \lVert(\mathcal{W}(d)-1)\xi\rVert\le2\lVert d\rVert\lVert(N+1)^{1/2}\xi\rVert. \tag*{(23)} ∥( W ( d ) − 1 ) ξ ∥ ≤ 2 ∥ d ∥ ∥( N + 1 ) 1/2 ξ ∥ . (23) For the weight W = W 0 + d Γ ( Λ ) W=W_0+\mathrm{d}\Gamma(\Lambda) W = W 0 + d Γ ( Λ ) ∥ Λ j d ∥ \lVert\Lambda^j d\rVert ∥ Λ j d ∥ ( 1 + ∥ d ∥ ) 2 m (1+\lVert d\rVert)^{2m} ( 1 + ∥ d ∥ ) 2 m
Proof. On finite-particle vectors, ∥ a ( d ) ξ ∥ ≤ ∥ d ∥ ∥ N 1 / 2 ξ ∥ \lVert a(d)\xi\rVert\le\lVert d\rVert\lVert N^{1/2}\xi\rVert ∥ a ( d ) ξ ∥ ≤ ∥ d ∥ ∥ N 1/2 ξ ∥ ∥ a ∗ ( d ) ξ ∥ ≤ ∥ d ∥ ∥ ( N + 1 ) 1 / 2 ξ ∥ \lVert a^*(d)\xi\rVert\le\lVert d\rVert\lVert(N+1)^{1/2}\xi\rVert ∥ a ∗ ( d ) ξ ∥ ≤ ∥ d ∥ ∥( N + 1 ) 1/2 ξ ∥ e i t q ( d ) ξ \mathrm{e}^{\mathrm{i}tq(d)}\xi e i tq ( d ) ξ (23) . Conjugation gives
W ( d ) ∗ N W ( d ) = N + q ( i d ) + ∥ d ∥ 2 . \mathcal{W}(d)^*N\mathcal{W}(d)=N+q(\mathrm{i}d)+\lVert d\rVert^2. W ( d ) ∗ N W ( d ) = N + q ( i d ) + ∥ d ∥ 2 . Commuting N N N m m m (22) . For the weighted version use
W ( d ) ∗ d Γ ( Λ ) W ( d ) = d Γ ( Λ ) + q ( i Λ d ) + ⟨ d , Λ d ⟩ , \mathcal{W}(d)^*\mathrm{d}\Gamma(\Lambda)\mathcal{W}(d)=\mathrm{d}\Gamma(\Lambda)+q(\mathrm{i}\Lambda d)+\langle d,\Lambda d\rangle, W ( d ) ∗ d Γ ( Λ ) W ( d ) = d Γ ( Λ ) + q ( i Λ d ) + ⟨ d , Λ d ⟩ , and [ d Γ ( Λ ) , a ∗ ( d ) ] = a ∗ ( Λ d ) [\mathrm{d}\Gamma(\Lambda),a^*(d)]=a^*(\Lambda d) [ d Γ ( Λ ) , a ∗ ( d )] = a ∗ ( Λ d ) d d d □ \square □
Normalized sums and a unitary central limit
A normalized sum creates or removes one child excitation in the limit. A product of small unitaries has the corresponding field exponential as its limit, but may also retain a scalar second-order term. We prove both statements with the weighted bounds needed for later substitution into a fixed circuit.
Lemma 2.6 (Sums). Let T D T_D T D T D ∗ T_D^* T D ∗ ⟨ Ω , T D Ω ⟩ = 0 \langle\Omega,T_D\Omega\rangle=0 ⟨ Ω , T D Ω ⟩ = 0
D − 1 / 2 ∑ j = 1 D T D ( j ) ⟶ a ∗ ( T Ω ) + a ( T ∗ Ω ) (24) D^{-1/2}\sum_{j=1}^{D}T_D^{(j)}\longrightarrow a^*(T\Omega)+a(T^*\Omega) \tag*{(24)} D − 1/2 j = 1 ∑ D T D ( j ) ⟶ a ∗ ( T Ω ) + a ( T ∗ Ω ) (24) smoothly. Without centering, D − 1 ∑ j T D ( j ) ⟶ ⟨ Ω , T Ω ⟩ 1 D^{-1}\sum_j T_D^{(j)}\longrightarrow\langle\Omega,T\Omega\rangle1 D − 1 ∑ j T D ( j ) ⟶ ⟨ Ω , T Ω ⟩ 1
Proof. Put g = Q T D Ω g=Q T_D\Omega g = Q T D Ω h = Q T D ∗ Ω h=Q T_D^*\Omega h = Q T D ∗ Ω A = Q T D Q A=Q T_D Q A = Q T D Q T D T_D T D
a ∗ ( g ) 1 − N / D + 1 − N / D a ( h ) + D − 1 / 2 d Γ ( A ) , (25) a^*(g)\sqrt{1-N/D}+\sqrt{1-N/D}a(h)+D^{-1/2}\mathrm{d}\Gamma(A), \tag*{(25)} a ∗ ( g ) 1 − N / D + 1 − N / D a ( h ) + D − 1/2 d Γ ( A ) , (25) where factors beyond the truncation are zero. On the m m m D − m D-m D − m D − m + 1 D-m+1 D − m + 1 Λ \Lambda Λ 1 + ∑ Λ j 1+\sum\Lambda_j 1 + ∑ Λ j Λ j ≤ ∑ Λ j \Lambda_j\leq\sum\Lambda_j Λ j ≤ ∑ Λ j (25) . On a fixed-particle smooth tensor the last term tends to zero and the square roots tend to one; density and (21) finish the proof. For an average the same decomposition has off-diagonal factor D − 1 / 2 D^{-1/2} D − 1/2 D − 1 D^{-1} D − 1
The sum lemma determines first-order fields. Controlled gates also require products of D D D D − 1 / 2 D^{-1/2} D − 1/2
Lemma 2.7 (Products of small unitaries). Let K D ( x ) K_D(x) K D ( x ) x ∈ R b x\in\mathbb{R}^b x ∈ R b K D ( 0 ) = 1 K_D(0)=1 K D ( 0 ) = 1 ⟨ Ω , ∂ α K D ( 0 ) Ω ⟩ = 0 \langle\Omega,\partial_\alpha K_D(0)\Omega\rangle=0 ⟨ Ω , ∂ α K D ( 0 ) Ω ⟩ = 0
K α = ∂ α K ( 0 ) = i T α , K α λ = ∂ α ∂ λ K ( 0 ) . K_\alpha=\partial_\alpha K(0)=iT_\alpha,\qquad K_{\alpha\lambda}=\partial_\alpha\partial_\lambda K(0). K α = ∂ α K ( 0 ) = i T α , K α λ = ∂ α ∂ λ K ( 0 ) . The T α T_\alpha T α a ∈ R b a\in\mathbb{R}^b a ∈ R b
K D ( a / D ) ⊗ D ⟶ exp { i ∑ α a α q ( T α Ω ) + 1 2 ∑ α , λ a α a λ ⟨ K α λ + 1 2 { T α , T λ } ⟩ 0 } . (26) K_D(a/\sqrt{D})^{\otimes D}\longrightarrow\exp\left\{i\sum_\alpha a_\alpha q(T_\alpha\Omega)+\frac{1}{2}\sum_{\alpha,\lambda}a_\alpha a_\lambda\left\langle K_{\alpha\lambda}+\frac{1}{2}\{T_\alpha,T_\lambda\}\right\rangle_0\right\}. \tag*{(26)} K D ( a / D ) ⊗ D ⟶ exp ⎩ ⎨ ⎧ i α ∑ a α q ( T α Ω ) + 2 1 α , λ ∑ a α a λ ⟨ K α λ + 2 1 { T α , T λ } ⟩ 0 ⎭ ⎬ ⎫ . (26) The scalar in this exponent is imaginary. The convergence holds smoothly with all a a a a a a a a a D D D θ \theta θ K D ( a / D ; θ ) K_D(a/\sqrt{D};\theta) K D ( a / D ; θ ) K D ( 0 ; θ ) = 1 K_D(0;\theta)=1 K D ( 0 ; θ ) = 1 θ \theta θ θ \theta θ θ \theta θ
Proof. We first identify the candidate limit from exponential vectors; then we prove the bounds needed for smooth convergence. If x , z ⊥ Ω x,z\perp\Omega x , z ⊥ Ω ( Ω + x / D ) ⊗ D (\Omega+x/\sqrt{D})^{\otimes D} ( Ω + x / D ) ⊗ D ε ( x ) = ⨁ m ≥ 0 x ⊗ m / m ! \varepsilon(x)=\bigoplus_{m\geq0}x^{\otimes m}/\sqrt{m!} ε ( x ) = ⨁ m ≥ 0 x ⊗ m / m ! z z z D − 1 D^{-1} D − 1
c = ⟨ z , x ⟩ + ∑ α a α ( ⟨ z , K α Ω ⟩ + ⟨ Ω , K α x ⟩ ) + 1 2 ∑ α , λ a α a λ ⟨ K α λ ⟩ 0 . c=\langle z,x\rangle+\sum_\alpha a_\alpha\bigl(\langle z,K_\alpha\Omega\rangle+\langle\Omega,K_\alpha x\rangle\bigr)+\frac{1}{2}\sum_{\alpha,\lambda}a_\alpha a_\lambda\langle K_{\alpha\lambda}\rangle_0. c = ⟨ z , x ⟩ + α ∑ a α ( ⟨ z , K α Ω ⟩ + ⟨ Ω , K α x ⟩ ) + 2 1 α , λ ∑ a α a λ ⟨ K α λ ⟩ 0 . Taylor’s formula makes the remaining error o ( D − 1 ) o(D^{-1}) o ( D − 1 ) x , z x,z x , z e c e^c e c t = ∑ α a α T α Ω t=\sum_\alpha a_\alpha T_\alpha\Omega t = ∑ α a α T α Ω e i q ( t ) e^{i q(t)} e i q ( t ) − ∥ t ∥ 2 / 2 -\lVert t\rVert^2/2 − ∥ t ∥ 2 /2 ∥ t ∥ 2 = ∑ α , λ a α a λ ⟨ { T α , T λ } / 2 ⟩ 0 \lVert t\rVert^2=\sum_{\alpha,\lambda}a_\alpha a_\lambda\langle\{T_\alpha,T_\lambda\}/2\rangle_0 ∥ t ∥ 2 = ∑ α , λ a α a λ ⟨{ T α , T λ } /2 ⟩ 0 (26) is exactly the required one. Twice differentiating K ( x ) ∗ K ( x ) = 1 K(x)^{*}K(x)=1 K ( x ) ∗ K ( x ) = 1
We next prove uniform bounds for the product U D ( a ) = K D ( a / D ) ⊗ D U_D(a)=K_D(a/\sqrt{D})^{\otimes D} U D ( a ) = K D ( a / D ) ⊗ D A = Q Λ Q A=Q\Lambda Q A = Q Λ Q B D ( a ) = K D ( a / D ) ∗ A K D ( a / D ) B_D(a)=K_D(a/\sqrt{D})^{*}AK_D(a/\sqrt{D}) B D ( a ) = K D ( a / D ) ∗ A K D ( a / D )
∣ ⟨ B D ( a ) ⟩ 0 ∣ ≤ D − 1 P ( a ) , \lvert\langle B_D(a)\rangle_0\rvert\le D^{-1}P(a), ∣⟨ B D ( a ) ⟩ 0 ∣ ≤ D − 1 P ( a ) , ∥ Λ m Q B D ( a ) Ω ∥ + ∥ Λ m Q B D ( a ) ∗ Ω ∥ ≤ D − 1 / 2 P m ( a ) , \lVert\Lambda^mQB_D(a)\Omega\rVert+\lVert\Lambda^mQB_D(a)^{*}\Omega\rVert\le D^{-1/2}P_m(a), ∥ Λ m Q B D ( a ) Ω ∥ + ∥ Λ m Q B D ( a ) ∗ Ω ∥ ≤ D − 1/2 P m ( a ) , and the complementary block has polynomial smooth bounds. Here and below P m P_m P m 1 + ∣ a ∣ 1+\lvert a\rvert 1 + ∣ a ∣ m m m (25) , now applied to the unnormalized sum of the B D ( a ) B_D(a) B D ( a )
U D ( a ) ∗ W U D ( a ) = W 0 + ∑ j B D ( a ) ( j ) . U_D(a)^{*}WU_D(a)=W_0+\sum_j B_D(a)^{(j)}. U D ( a ) ∗ W U D ( a ) = W 0 + j ∑ B D ( a ) ( j ) . has polynomial bounds from any sufficiently high input weight to any prescribed output weight, uniformly in D D D W m U D = U D ( U D ∗ W U D ) m W^mU_D=U_D(U_D^{*}WU_D)^m W m U D = U D ( U D ∗ W U D ) m U D U_D U D
For derivatives,
U D ( a ) ∗ ∂ α U D ( a ) = D − 1 / 2 ∑ j ( K D ∗ ∂ α K D ) ( a / D ) ( j ) . U_D(a)^{*}\partial_\alpha U_D(a)=D^{-1/2}\sum_j(K_D^{*}\partial_\alpha K_D)(a/\sqrt{D})^{(j)}. U D ( a ) ∗ ∂ α U D ( a ) = D − 1/2 j ∑ ( K D ∗ ∂ α K D ) ( a / D ) ( j ) . The one-copy vacuum mean is O ( D − 1 / 2 P ( a ) ) O(D^{-1/2}P(a)) O ( D − 1/2 P ( a )) k ≥ 2 k\ge2 k ≥ 2 D 1 − k / 2 ≤ 1 D^{1-k/2}\le1 D 1 − k /2 ≤ 1
Weak convergence on the total set of exponential vectors, together with equality of the limiting norms, proves strong convergence. Higher-weight convergence follows from (21) ; derivative convergence follows from finite differences and the uniform bounds on one further derivative. This also gives local uniformity in a a a U D , U U_D,U U D , U ∥ ( U D ∗ − U ∗ ) ξ ∥ = ∥ ξ − U D U ∗ ξ ∥ \lVert(U_D^{*}-U^{*})\xi\rVert=\lVert\xi-U_DU^{*}\xi\rVert ∥( U D ∗ − U ∗ ) ξ ∥ = ∥ ξ − U D U ∗ ξ ∥
To verify the unscaled-parameter assertion, the only additional estimate concerns a logarithmic derivative A ν ( x ; θ ) = K ( x ; θ ) ∗ ∂ θ ν K ( x ; θ ) A_\nu(x;\theta)=K(x;\theta)^{*}\partial_{\theta_\nu}K(x;\theta) A ν ( x ; θ ) = K ( x ; θ ) ∗ ∂ θ ν K ( x ; θ ) A ν ( 0 ; θ ) = 0 A_\nu(0;\theta)=0 A ν ( 0 ; θ ) = 0
⟨ ∂ x α A ν ( 0 ; θ ) ⟩ 0 = ∂ θ ν ⟨ ∂ x α K ( 0 ; θ ) ⟩ 0 = 0. \langle\partial_{x^\alpha}A_\nu(0;\theta)\rangle_0=\partial_{\theta_\nu}\langle\partial_{x^\alpha}K(0;\theta)\rangle_0=0. ⟨ ∂ x α A ν ( 0 ; θ ) ⟩ 0 = ∂ θ ν ⟨ ∂ x α K ( 0 ; θ ) ⟩ 0 = 0. Its vacuum entry at a / D a/\sqrt{D} a / D O ( D − 1 P ( a ) ) O(D^{-1}P(a)) O ( D − 1 P ( a )) O ( D − 1 / 2 P ( a ) ) O(D^{-1/2}P(a)) O ( D − 1/2 P ( a )) O ( D − 1 / 2 P ( a ) ) O(D^{-1/2}P(a)) O ( D − 1/2 P ( a )) D D D θ \theta θ a a a θ \theta θ
Remark 2.8. The anticommutator in (26) is a quadratic form on smooth vectors; its vacuum expectation is unambiguous because all jets preserve that domain. Essential self-adjointness of the one-copy derivative T α T_\alpha T α T α Ω T_\alpha\Omega T α Ω
Smooth limits for ordinary root maps
We now prove that the ordinary root maps introduced in Section 2.2 have smooth limits, and hence that their limiting insertions have the root-shift energy computed there. At height h h h u h u_h u h v P , h = u h ∗ P u h Ω h v_{P,h}=u_h^* P u_h\Omega_h v P , h = u h ∗ P u h Ω h P P P h h h h − 1 h-1 h − 1
v Z , h − 1 = u h − 1 ∗ Z u h − 1 Ω h − 1 , u h ⟼ e − i t Z q ( v Z , h − 1 ) u h . v_{Z,h-1}=u_{h-1}^* Z u_{h-1}\Omega_{h-1}, \qquad u_h \longmapsto e^{-itZq(v_{Z,h-1})}u_h. v Z , h − 1 = u h − 1 ∗ Z u h − 1 Ω h − 1 , u h ⟼ e − i tZ q ( v Z , h − 1 ) u h . Here q ( v Z , h − 1 ) q(v_{Z,h-1}) q ( v Z , h − 1 ) H h \mathcal{H}_h H h v P = u ∗ P u Ω v_P=u^*Pu\Omega v P = u ∗ P u Ω q q q
Lemma 2.9 (Ordinary root updates). Ordinary finite words have smooth root limits and smooth adjoint limits at every sufficient finite height. Starting at u = 1 u=1 u = 1 β \beta β t t t
u ⟼ e − i β X u , u ⟼ e − i t Z q ( v Z ) u . (27) u \longmapsto e^{-i\beta X}u,\qquad u \longmapsto e^{-itZq(v_Z)}u. \tag*{(27)} u ⟼ e − i β X u , u ⟼ e − i tZ q ( v Z ) u . (27) The cost update leaves v Z v_Z v Z Z Z Z P P P v P v_P v P
u ⟼ e − i t Z s u . u \longmapsto e^{-itZ_s}u. u ⟼ e − i t Z s u . All these assertions are unchanged when the top root has D + 1 D+1 D + 1 D D D D − 1 / 2 D^{-1/2} D − 1/2 W W W v Z v_Z v Z v W = E ( v Z ) v_W=\mathcal{E}(v_Z) v W = E ( v Z ) E \mathcal{E} E
Proof. At finite D D D
exp ( − i t Z D − 1 / 2 ∑ j = 1 D U j ∗ Z j U j ) . \exp\left(-itZD^{-1/2}\sum_{j=1}^{D}U_j^*Z_jU_j\right). exp ( − i tZ D − 1/2 j = 1 ∑ D U j ∗ Z j U j ) . The one-copy observable is centered, because Z Z Z v Z v_Z v Z Z Z Z (27) ; the scalar correction is zero for an exponential of a fixed observable. The smooth estimates follow inductively from the same lemma and Lemma 2.5. Mixers are local, and a seed gate is multiplication by a two-by-two unitary; its input derivatives have polynomial seed bounds. Both therefore preserve the smooth domain. The update commutes with the root Z Z Z v Z v_Z v Z
For the full-root assertion use M = D + 1 M=D+1 M = D + 1 q / D q/\sqrt{D} q / D M / D → 1 M/D\to1 M / D → 1
The finite-degree ordinary insertion vectors meet the hypotheses of Proposition 2.2: parity makes them centered, the construction preserves rooted symmetry, and the smooth weight dominates 1 + N 1+N 1 + N Z i Z j Z_iZ_j Z i Z j v W = E ( v Z ) v_W=\mathcal{E}(v_Z) v W = E ( v Z )
Substitution of commuting controls
The ordinary cost update uses a scalar time multiplying one field. A controlled operation instead inserts a commuting list of signal operators into the parameters of a local matrix. The following lemma makes that substitution legitimate, including its derivatives and adjoints; no choice of independent Gaussian coordinates is needed.
Lemma 2.10 (Smooth commuting substitution). Let S D = ( S D , 1 , … , S D , b ) S_D=(S_{D,1},\ldots,S_{D,b}) S D = ( S D , 1 , … , S D , b ) m m m
∥ W D m e i t ⋅ S D ξ ∥ ≤ C m ( 1 + ∣ t ∣ ) k m ∥ W D r m ξ ∥ . (28) \lVert W_D^m e^{it\cdot S_D}\xi\rVert\le C_m(1+|t|)^{k_m}\lVert W_D^{r_m}\xi\rVert. \tag*{(28)} ∥ W D m e i t ⋅ S D ξ ∥ ≤ C m ( 1 + ∣ t ∣ ) k m ∥ W D r m ξ ∥ . (28) uniformly in D , t D,t D , t F D ( x ) F_D(x) F D ( x ) F D ( x ) ∗ F_D(x)^* F D ( x ) ∗ x x x F D ( x ) F_D(x) F D ( x ) S D , j S_{D,j} S D , j F D ( S D ) F_D(S_D) F D ( S D ) F D ( x ) F_D(x) F D ( x ) x x x F D ( S D ) F_D(S_D) F D ( S D ) S S S
For a real smooth function f ( s , x ) f(s,x) f ( s , x ) x x x s s s e i t f ( s , S D ) e^{itf(s,S_D)} e i t f ( s , S D ) t t t s s s S D , j S_{D,j} S D , j f ( s , S D ) f(s,S_D) f ( s , S D ) S D , j S_{D,j} S D , j
Proof. First identify the limiting signal operators. Write U ( t ) = lim D e i t ⋅ S D U(t)=\lim_D e^{it\cdot S_D} U ( t ) = lim D e i t ⋅ S D t t t − t -t − t U ( t ) U(t) U ( t ) ξ \xi ξ
∥ ( e i t ⋅ S D − 1 ) ξ ∥ ≤ ∑ α = 1 b ∣ t α ∣ ∥ S D , α ξ ∥ . \lVert(e^{it\cdot S_D}-1)\xi\rVert\le\sum_{\alpha=1}^{b}|t_\alpha|\lVert S_{D,\alpha}\xi\rVert. ∥( e i t ⋅ S D − 1 ) ξ ∥ ≤ α = 1 ∑ b ∣ t α ∣ ∥ S D , α ξ ∥ . The uniform polynomial bounds give strong continuity on this dense domain, hence on the full space. The strongly continuous joint unitary group therefore determines strongly commuting self-adjoint generators S 1 , … , S b S_1,\ldots,S_b S 1 , … , S b F ( S ) F(S) F ( S )
We give the estimates, which also specify the substitution on the smooth domain when F D F_D F D e i t ⋅ S D ξ e^{it\cdot S_D}\xi e i t ⋅ S D ξ D D D m m m r m , k m r_m,k_m r m , k m (28) . Choose an integer q > k m + b q>k_m+b q > k m + b N N N
G ( x ) = ( 1 + ∣ x ∣ 2 ) − N F ( x ) G(x)=(1+|x|^2)^{-N}F(x) G ( x ) = ( 1 + ∣ x ∣ 2 ) − N F ( x ) and its first q q q x x x r m r_m r m N N N G ^ ( t ) = ( 2 π ) − b ∫ e − i t ⋅ x G ( x ) d x \widehat{G}(t)=(2\pi)^{-b}\int e^{-it\cdot x}G(x)\,dx G ( t ) = ( 2 π ) − b ∫ e − i t ⋅ x G ( x ) d x
∥ W r m G ^ ( t ) η ∥ ≤ C ( 1 + ∣ t ∣ ) − q ∥ W r η ∥ . \lVert W^{r_m}\widehat{G}(t)\eta\rVert\le C(1+|t|)^{-q}\lVert W^r\eta\rVert. ∥ W r m G ( t ) η ∥ ≤ C ( 1 + ∣ t ∣ ) − q ∥ W r η ∥ . Therefore
F ( S ) ξ = ∫ R b e i t ⋅ S G ^ ( t ) ( 1 + ∣ S ∣ 2 ) N ξ d t (29) F(S)\xi=\int_{\mathbb{R}^b}e^{it\cdot S}\widehat{G}(t)(1+|S|^2)^N\xi\,dt \tag*{(29)} F ( S ) ξ = ∫ R b e i t ⋅ S G ( t ) ( 1 + ∣ S ∣ 2 ) N ξ d t (29) converges absolutely in the m m m (20) . The polynomial on the right acts first; its bound can be absorbed by increasing the input weight last. This order of choices avoids any circular dependence of exponents.
Integration by parts in t t t N N N 1 + ∣ x ∣ 2 1+|x|^2 1 + ∣ x ∣ 2 1 − Δ t 1-\Delta_t 1 − Δ t 1 + ∣ S ∣ 2 1+|S|^2 1 + ∣ S ∣ 2
F ( S ) G ( S ) = ( F G ) ( S ) . F(S)G(S)=(FG)(S). F ( S ) G ( S ) = ( F G ) ( S ) . In this identity the order of F F F G G G S S S F ( S ) ∗ = F ∗ ( S ) F(S)^*=F^*(S) F ( S ) ∗ = F ∗ ( S ) 1 1 1
For convergence, first restrict x x x ( 1 + ∣ x ∣ 2 ) − N (1+|x|^2)^{-N} ( 1 + ∣ x ∣ 2 ) − N (21) . For coefficients merely measurable in s s s s s s
Finally, every x x x e i t f ( s , x ) e^{itf(s,x)} e i t f ( s , x ) ∣ t ∣ |t| ∣ t ∣ ∣ x ∣ |x| ∣ x ∣ ∣ s ∣ |s| ∣ s ∣
An important supply of lists satisfying the hypotheses is the following. If M 1 , … , M b M_1,\ldots,M_b M 1 , … , M b K ( x ) = exp ( i ∑ α x α M α ) K(x)=\exp(i\sum_\alpha x_\alpha M_\alpha) K ( x ) = exp ( i ∑ α x α M α )
( D − 1 / 2 ∑ j M α ( j ) ) α = 1 b ⟶ ( q ( M α Ω ) ) α = 1 b . (30) \left(D^{-1/2}\sum_j M_\alpha^{(j)}\right)_{\alpha=1}^{b}
\longrightarrow
\left(q(M_\alpha\Omega)\right)_{\alpha=1}^{b}. \tag*{(30)} ( D − 1/2 j ∑ M α ( j ) ) α = 1 b ⟶ ( q ( M α Ω ) ) α = 1 b . (30) The limiting fields commute: the imaginary part of their Gram matrix is the vacuum expectation of a commutator of the original signals, divided by 2 i 2i 2 i
Exclusion calculations and controlled gates
We now prove the stronger calculus needed for synthesis. Its objects are temporary operations, not additional gates in the final QAOA word. The purpose of the calculus is to justify replacing such operations by finite products inside an already chosen surrounding circuit.
An even mask is a measurable subset of S S S ι \iota ι I , J I,J I , J
( A I J f ) i = 1 I ( s i ) D − 1 / 2 ∑ j ∼ i 1 J ( s j ) f j (31) (A_{IJ}f)_i = \mathbf{1}_I(s_i)D^{-1/2}\sum_{j\sim i}\mathbf{1}_J(s_j)f_j \tag*{(31)} ( A I J f ) i = 1 I ( s i ) D − 1/2 j ∼ i ∑ 1 J ( s j ) f j (31) Masks and the coefficients attached to them are part of a fixed finite expression. They are not optimized after sampling a tree or its seeds.
Definition 2.11 (Exclusion calculation and gate). A finite exclusion calculation uses local seeds, one prescribed commuting spin axis at each control site, smooth pointwise functions of finite lists of messages, and normalized sums (2.24). An axis may depend on the site’s mask. A formula is split into finitely many named roles when different occurrences need different masks. The following conditions are imposed.
Every transmitted field is odd under simultaneous seed reflection and spin flip. Every pointwise function is smooth in its aggregate inputs, with polynomial bounds for every derivative. Its local-seed coefficients are measurable and uniformly bounded, unless the seed-moment variant is specified.
In every normalized sum, the receiving mask is disjoint from every role mask occurring below that sum. Thus all ancestor receiving types are excluded from the descendant calculation. In particular a message sent across a cut does not use the site on the other side of that cut.
A gate chooses a target mask and applies a two-by-two unitary V V V V V V
All parameter operators of this one gate belong to the same commuting control algebra. The gate, including its two-by-two matrix, preserves the joint symmetry.
The global gate is the product of these controlled matrices. Its factors commute: their targets are distinct and none is a control for another factor of the same gate. A finite-range local observable is therefore affected by only finitely many of them at finite D D D
The exclusions make the information across a removed edge depend on finitely many boundary inputs. We next show that the resulting finite recursion respects the weighted limits already proved. The replacement assertion is formulated inside an existing circuit, which is the form required by the synthesis argument.
Theorem 2.12 (Smooth gate calculus). Fix a finite sequence of exclusion gates, ordinary mixers and costs, and local seed gates. Its root interaction-picture maps, their adjoints, and their root insertion vectors have smooth limits under the occupation embeddings. They have uniform smooth bounds at every sufficient finite height. The same limit is obtained at a full root with D + 1 D+1 D + 1
An exclusion gate may be replaced inside any such fixed sequence by a sequence of pointwise unitary matrices with the same parameter calculations, provided every fixed number of aggregate-input derivatives converges on compact sets and has uniform polynomial bounds. Convergence almost everywhere in the local seed, or convergence in seed measure followed by a subsequence, is sufficient for bounded measurable local-seed coefficients. The resulting root maps, adjoints, and insertion vectors converge smoothly, and their energies converge.
If local-seed coefficients are uniformly bounded, these bounds hold conditionally on the root seed, with constants independent of its value, using W 0 = 1 W_0=1 W 0 = 1 c c c m m m C m C_m C m
∥ W m 1 J c ∥ ≤ C m μ ( J ) . (32) \lVert W_m 1_J c\rVert\le C_m\sqrt{\mu(J)}. \tag*{(32)} ∥ W m 1 J c ∥ ≤ C m μ ( J ) . (32) The same statements hold for any further scalar boundary-input jets of a fixed exclusion calculation. They also hold, including adjoint jets, for a finite list of external unscaled parameters, locally uniformly in those parameters, when the matrix and signal hypotheses hold jointly for their mixed jets.
Proof. We give both the finite-tree recursion and the analytic induction. This is necessary because a global operator-norm error proportional to the number of vertices would not imply the assertion.
Boundary inputs. Unroll the fixed expression and index its normalized-sum occurrences by their named roles, retaining repeated occurrences separately. These indices form the finite boundary-channel list α = 1 , … , b \alpha=1,\ldots,b α = 1 , … , b A I α J α f α A_{I_\alpha}J_\alpha f_\alpha A I α J α f α y α y_\alpha y α I α I_\alpha I α y = ( y 1 , … , y b ) y=(y_1,\ldots,y_b) y = ( y 1 , … , y b ) m α ( y ) = 1 J α ( s ) f α ( y ) m_\alpha(y)=1_{J_\alpha}(s)f_\alpha(y) m α ( y ) = 1 J α ( s ) f α ( y ) m α ( y ) / D m_\alpha(y)/\sqrt{D} m α ( y ) / D V ( y ) V(y) V ( y ) m ( y ) m(y) m ( y )
We first derive the factorization in the physical coordinates, before conjugating by the preceding circuit. Retain the target operations at distance at most d d d G [ d ] ( y ) G^{[d]}(y) G [ d ] ( y ) V p h y s ( y ) V_{\mathrm{phys}}(y) V phys ( y ) m p h y s ( y ) m_{\mathrm{phys}}(y) m phys ( y ) d = 0 d=0 d = 0 V p h y s ( y ) V_{\mathrm{phys}}(y) V phys ( y ) d ≥ 1 d\ge1 d ≥ 1
G [ d ] ( y ) = V p h y s ( y ) ∏ j c h i l d G j [ d − 1 ] ( D − 1 / 2 m p h y s ( y ) ) . G^{[d]}(y)=V_{\mathrm{phys}}(y)\prod_{j\ \mathrm{child}}G_j^{[d-1]}\left(D^{-1/2}m_{\mathrm{phys}}(y)\right). G [ d ] ( y ) = V phys ( y ) j child ∏ G j [ d − 1 ] ( D − 1/2 m phys ( y ) ) . The same formula list is evaluated in each child with its receiving masks imposed there. The exclusion rule ensures that the input sent to a child does not feed back through that child’s receiving type.
The zero-boundary child operation is the baseline to factor out. Put
B [ d ] = ∏ j G j [ d − 1 ] ( 0 ) , P = ⨂ j U j , U = P u . B^{[d]} = \prod_j G_j^{[d-1]}(0), \qquad P = \bigotimes_j U_j, \qquad U = Pu. B [ d ] = j ∏ G j [ d − 1 ] ( 0 ) , P = j ⨂ U j , U = P u . where U U U U j U_j U j d = 0 d=0 d = 0 B [ 0 ] = 1 B^{[0]}=1 B [ 0 ] = 1
K j [ d − 1 ] ( a ) = U j ∗ G j [ d − 1 ] ( 0 ) ∗ G j [ d − 1 ] ( a ) U j . K_j^{[d-1]}(a) = U_j^*G_j^{[d-1]}(0)^*G_j^{[d-1]}(a)U_j. K j [ d − 1 ] ( a ) = U j ∗ G j [ d − 1 ] ( 0 ) ∗ G j [ d − 1 ] ( a ) U j . At the current branch use the same definition,
K [ d ] ( y ) = U ∗ G [ d ] ( 0 ) ∗ G [ d ] ( y ) U . K^{[d]}(y) = U^*G^{[d]}(0)^*G^{[d]}(y)U. K [ d ] ( y ) = U ∗ G [ d ] ( 0 ) ∗ G [ d ] ( y ) U . Thus K [ d ] ( 0 ) = 1 K^{[d]}(0)=1 K [ d ] ( 0 ) = 1
Conjugate the physical signals by P P P V ( y ) = P ∗ V p h y s ( y ) P V(y)=P^*V_{\mathrm{phys}}(y)P V ( y ) = P ∗ V phys ( y ) P m ( y ) = P ∗ m p h y s ( y ) P m(y)=P^*m_{\mathrm{phys}}(y)P m ( y ) = P ∗ m phys ( y ) P U j U_j U j E [ d ] ( y ) = P ∗ ( B [ d ] ) ∗ G [ d ] ( y ) P E^{[d]}(y)=P^*(B^{[d]})^*G^{[d]}(y)P E [ d ] ( y ) = P ∗ ( B [ d ] ) ∗ G [ d ] ( y ) P
E [ d ] ( y ) = V ( y ) ∏ j c h i l d K j [ d − 1 ] ( D − 1 / 2 m ( y ) ) , (33) E^{[d]}(y) = V(y)\prod_{j\ \mathrm{child}} K_j^{[d-1]}\left(D^{-1/2}m(y)\right), \tag*{(33)} E [ d ] ( y ) = V ( y ) j child ∏ K j [ d − 1 ] ( D − 1/2 m ( y ) ) , (33) K [ d ] ( y ) = u ∗ E [ d ] ( 0 ) ∗ E [ d ] ( y ) u . (34) K^{[d]}(y) = u^*E^{[d]}(0)^*E^{[d]}(y)u. \tag*{(34)} K [ d ] ( y ) = u ∗ E [ d ] ( 0 ) ∗ E [ d ] ( y ) u . (34) For d = 0 d=0 d = 0 E [ 0 ] ( y ) = V ( y ) E^{[0]}(y)=V(y) E [ 0 ] ( y ) = V ( y ) d = 1 d=1 d = 1 m ( 0 ) m(0) m ( 0 ) E [ d ] ( 0 ) E^{[d]}(0) E [ d ] ( 0 ) K [ d ] K^{[d]} K [ d ]
The second line follows by canceling B [ d ] ( B [ d ] ) ∗ B^{[d]}(B^{[d]})^* B [ d ] ( B [ d ] ) ∗
G [ d ] ( 0 ) U = B [ d ] P E [ d ] ( 0 ) u . G^{[d]}(0)U = B^{[d]}PE^{[d]}(0)u. G [ d ] ( 0 ) U = B [ d ] P E [ d ] ( 0 ) u . The new child circuits are G j [ d − 1 ] ( 0 ) U j G_j^{[d-1]}(0)U_j G j [ d − 1 ] ( 0 ) U j
u n e w = E [ d ] ( 0 ) u . (35) u_{\mathrm{new}} = E^{[d]}(0)u. \tag*{(35)} u new = E [ d ] ( 0 ) u . (35) Only targets within the finite influence distance can respond to a cut input. Taking d d d
The factors and their parameter arguments commute before the preceding circuit is applied: targets are disjoint from controls, and each site supplies only its prescribed axis. More explicitly, let A j \mathcal{A}_j A j j j j a a a G j [ d ] ( a ) G_j^{[d]}(a) G j [ d ] ( a ) A j \mathcal{A}_j A j
K j [ d ] ( a ) ∈ ( U j ∗ A j U j ) ′ . K_j^{[d]}(a) \in\left(U_j^*\mathcal{A}_jU_j\right)'. K j [ d ] ( a ) ∈ ( U j ∗ A j U j ) ′ . Before the final root conjugation by u u u m ( y ) m(y) m ( y ) U j ∗ A j U j U_j^* A_j U_j U j ∗ A j U j K [ d ] ( a ) K^{[d]}(a) K [ d ] ( a ) j j j m ( y ) m(y) m ( y )
Zero-boundary signals. Assume by induction that the preceding maps u u u u ∗ u^* u ∗ u u u
Here is the induction that also controls exponentials. Local commuting seed/axis variables plainly have joint exponentials with polynomial smooth bounds. Suppose this is true for the child signal list. Lemma 2.7 gives (30) , including the polynomial frequency bounds on joint exponentials, and Lemma 2.6 gives polynomial convergence. Lemma 2.10 then computes each smooth pointwise function and gives polynomial bounds for the joint exponentials of every new finite signal list. Conjugating by u u u u ∗ u^* u ∗
Propagation of a gate across a cut. Proceed next in d d d (33) –(34) . For d = 0 d=0 d = 0 V ( y ) V(y) V ( y ) K [ d − 1 ] ( a ) K^{[d-1]}(a) K [ d − 1 ] ( a ) a = 0 a=0 a = 0 P K [ d − 1 ] ( a ) P = K [ d − 1 ] ( − a ) P K^{[d-1]}(a) P=K^{[d-1]}(-a) P K [ d − 1 ] ( a ) P = K [ d − 1 ] ( − a ) P Ω = Ω P\Omega=\Omega P Ω = Ω
For scalar a a a ∂ α K [ d − 1 ] ( 0 ) = i T α \partial_\alpha K^{[d-1]}(0)=iT_\alpha ∂ α K [ d − 1 ] ( 0 ) = i T α
exp { i ∑ α a α q ( T α Ω ) + 1 2 ∑ α , λ a α a λ ⟨ K α λ + 1 2 { T α , T λ } ⟩ 0 } (36) \exp\left\{i\sum_\alpha a_\alpha q(T_\alpha\Omega)+\frac{1}{2}\sum_{\alpha,\lambda}a_\alpha a_\lambda\left\langle K_{\alpha\lambda}+\frac{1}{2}\{T_\alpha,T_\lambda\}\right\rangle_0\right\} \tag*{(36)} exp ⎩ ⎨ ⎧ i α ∑ a α q ( T α Ω ) + 2 1 α , λ ∑ a α a λ ⟨ K α λ + 2 1 { T α , T λ } ⟩ 0 ⎭ ⎬ ⎫ (36) The second term is scalar for scalar a a a a a a m ( y ) m(y) m ( y ) a = m ( y ) a=m(y) a = m ( y ) V ( y ) V(y) V ( y ) E [ d ] ( y ) E^{[d]}(y) E [ d ] ( y ) (34) completes it for K [ d ] ( y ) K^{[d]}(y) K [ d ] ( y )
This proves (35) with smooth bounds and limits. It adds one gate to the preceding sequence. Ordinary cost and mixer updates may be inserted between these gates by Lemma 2.9; its proof requires only smooth bounds on the preceding maps and preservation of symmetry. Local seed gates are handled as in that lemma. Induction on the finite sequence proves the first assertion. The same proof with D + 1 D+1 D + 1 D D D ( D + 1 ) / D → 1 (D+1)/D \to1 ( D + 1 ) / D → 1
External parameters can be carried through this induction without being scaled. At the child-product step use the unscaled-parameter assertion of Lemma 2.7; at the substitution step use the mixed-jet version of Lemma 2.10. Each unitary strong limit is unitary, so adjoint convergence follows from the identity in the proof of Lemma 2.7; its jet version follows by finite differences with the adjoint jet bounds. For real commuting signals the adjoint is the same signal, and compositions use the reversed products of the already convergent adjoint families. Thus adjoint convergence is established at each induction step, rather than inferred from bounds alone.
Continuity inside a sequence. Repeat these two inductions with a further approximation index. Compact convergence of a fixed number of matrix derivatives and uniform polynomial bounds are precisely the hypotheses of commuting substitution. The normalized sum and product lemmas allow the child maps themselves to vary with this index. Hence every new root map and its adjoint converge on smooth vectors with uniform smooth bounds. A fixed finite surrounding sequence is handled by induction and the product rule. The assertion does not make a bound uniform in the length of that sequence. For a pointwise product formula on a single target and a common commuting control algebra, the entire product is first regarded as one matrix-valued function; the theorem then applies to its smooth matrix limit. This is how arbitrarily long local commutator or sum formulas can be used without an estimate proportional to graph size. For measurable seed limits, dominated convergence in the root seed and then the same child induction give the stated extension. Norm continuity (2.7) proves energy convergence.
Conditional bounds and masks. Take W 0 = 1 W_0=1 W 0 = 1 J J J
The gate theorem controls a whole unitary. In the helper contractions we also need its first variation with respect to time: this is a Hamiltonian response to boundary inputs. The next corollary derives that response from the same unitary limit, so that the scalar second-order term and lower-level responses are retained.
Corollary 2.13 (Additive cut responses). Fix a preceding finite sequence as in Theorem 2.12. Suppose a temporary Hamiltonian is diagonal in a common collection of site axes, with coefficients given by an exclusion calculation. Require its local Hamiltonian matrices and every aggregate-input derivative to have polynomial bounds as in Definition 2.11. In a cut branch let H D ( a ) H_D(a) H D ( a ) a a a
H D ( − a ) = P H D ( a ) P , P Ω = Ω , (37) H_D(-a)=\mathcal{P}H_D(a)\mathcal{P}, \qquad\mathcal{P}\Omega=\Omega, \tag*{(37)} H D ( − a ) = P H D ( a ) P , P Ω = Ω , (37) *where P \mathcal{P} P H α = lim D ∂ α H D ( 0 ) H_\alpha=\lim_D \partial_\alpha H_D(0) H α = lim D ∂ α H D ( 0 ) H α λ = lim D ∂ α ∂ λ H D ( 0 ) H_{\alpha\lambda}=\lim_D \partial_\alpha\partial_\lambda H_D(0) H α λ = lim D ∂ α ∂ λ H D ( 0 ) H α H_\alpha H α H D ( 0 ) H_D(0) H D ( 0 ) D D D a / D a/\sqrt{D} a / D
∑ j = 1 D [ H D ( a / D ) − H D ( 0 ) ] ( j ) . \sum_{j=1}^{D}\left[H_D(a/\sqrt{D})-H_D(0)\right]^{(j)}. j = 1 ∑ D [ H D ( a / D ) − H D ( 0 ) ] ( j ) . Its smooth limit is
∑ α a α q ( H α Ω ) + 1 2 ∑ α , λ a α a λ ⟨ H α λ ⟩ 0 . (38) \sum_{\alpha}a_{\alpha q}(H_{\alpha}\Omega)+\frac{1}{2}\sum_{\alpha,\lambda}a_{\alpha}a_{\lambda}\langle H_{\alpha\lambda}\rangle_{0}. \tag*{(38)} α ∑ a α q ( H α Ω ) + 2 1 α , λ ∑ a α a λ ⟨ H α λ ⟩ 0 . (38) The correction is realized as the self-adjoint generator of the limiting time group of the child cut ratios; the display specifies its action on the smooth domain. We use the same realization after commuting substitution and for the correction generators below. The same formula applies recursively to lower cut responses and to their input jets. It is continuous with the smooth bounds of Theorem 2.12, for a fixed finite list of channels. Commuting parameter operators may replace a a a
In particular, if a sequence of such correction generators R j R_j R j ∥ R j ξ ∥ ≤ δ j ∥ W r ξ ∥ \lVert R_j\xi\rVert\leq\delta_j\lVert W^r\xi\rVert ∥ R j ξ ∥ ≤ δ j ∥ W r ξ ∥ δ j → 0 \delta_j\to0 δ j → 0
∥ ( e i t R j − 1 ) ξ ∥ ≤ ∣ t ∣ δ j ∥ W r ξ ∥ . (39) \left\lVert\left(e^{itR_j}-1\right)\xi\right\rVert\leq|t|\delta_j\lVert W^r\xi\rVert. \tag*{(39)} ( e i t R j − 1 ) ξ ≤ ∣ t ∣ δ j ∥ W r ξ ∥ . (39) If their exponentials and adjoints also have uniform smooth bounds, these errors tend to zero in every fixed lower graph norm and may be inserted in a fixed surrounding sequence.
Proof. Differentiating (37) at zero gives P H α P = − H α PH_{\alpha}P=-H_{\alpha} P H α P = − H α t t t
K D ( a ; t ) = exp ( i t [ H D ( a ) − H D ( 0 ) ] ) . K_D(a;t)=\exp\left(it[H_D(a)-H_D(0)]\right). K D ( a ; t ) = exp ( i t [ H D ( a ) − H D ( 0 )] ) . All current Hamiltonian terms commute before conjugation, and simultaneous conjugation preserves this property. In particular H D ( a ) H_D(a) H D ( a ) H D ( 0 ) H_D(0) H D ( 0 ) a a a e i t h ( x ) e^{ith(x)} e i t h ( x ) t , x t,x t , x h h h K D ( a ; t ) K_D(a;t) K D ( a ; t )
H D ( a ) − H D ( 0 ) = 1 i ∂ t K D ( a ; 0 ) , ∂ a α H D ( a ) = 1 i a α ∂ a α ∂ t K D ( a ; 0 ) ( ∣ α ∣ ≥ 1 ) . H_D(a)-H_D(0)=\frac{1}{i}\partial_tK_D(a;0),\qquad
\partial_a^\alpha H_D(a)=\frac{1}{i}a^\alpha\partial_a^\alpha\partial_tK_D(a;0)\quad(|\alpha|\geq1). H D ( a ) − H D ( 0 ) = i 1 ∂ t K D ( a ; 0 ) , ∂ a α H D ( a ) = i 1 a α ∂ a α ∂ t K D ( a ; 0 ) ( ∣ α ∣ ≥ 1 ) . These are the required bounds for the response jets; no bound for the possibly extensive H D ( 0 ) H_D(0) H D ( 0 )
For the child-product limit compute the first two boundary jets at zero:
K α ( t ) = i t H α , K α λ ( t ) = i t H α λ − t 2 2 { H α , H λ } . K_\alpha(t)=itH_\alpha,\qquad K_{\alpha\lambda}(t)=itH_{\alpha\lambda}-\frac{t^2}{2}\{H_\alpha,H_\lambda\}. K α ( t ) = i t H α , K α λ ( t ) = i t H α λ − 2 t 2 { H α , H λ } . In Lemma 2.7 one has T α = t H α T_\alpha=tH_\alpha T α = t H α D D D i t it i t (38) . Differentiating at t = 0 t=0 t = 0 t t t (38) ; no separate choice of closure of the displayed expression is used. The exclusions and Lemma 2.10 allow operator inputs and preserve the group law and this generator identification. Repeating this argument at the finitely many affected levels proves the recursive assertion.
For R j R_j R j ∣ e i t x − 1 ∣ ≤ ∣ t x ∣ |e^{itx}-1| \le|tx| ∣ e i t x − 1∣ ≤ ∣ t x ∣ (39) . Uniform smooth bounds and (21) then give higher-norm convergence; applying the same reasoning to the adjoints and using finite composition proves the final statement.
Remark 2.14 (Scope of uniformity). The constants in the calculus are uniform in degree and in the stated pointwise approximation limits, for a fixed finite calculation and fixed surrounding sequence. They are not automatically uniform when the number of roles or channels tends to infinity. In a refinement argument, such as the helper contraction below, that additional uniformity must be proved from estimates on the combined columns or kernels. Once it is proved, the same smooth-domain and continuity arguments apply. Scalar phases depending only on the root seed commute with root observables and do not change insertion vectors or the child conjugations in the cut recursion.
We now construct a bounded classical output whose rooted energy is arbitrarily close to P ∗ P_* P ∗
We use the following results from the complete companion Full support of the zero-temperature Sherrington–Kirkpatrick order parameter [19 ] . We state the normalization and the finite formula explicitly because the rooted realization below depends on their exact form.
Let U \mathcal{U} U γ : [ 0 , 1 ) → [ 0 , ∞ ) \gamma: [0,1) \to[0,\infty) γ : [ 0 , 1 ) → [ 0 , ∞ ) ∫ 0 1 γ ( t ) d t < ∞ \int_0^1 \gamma(t)\,\mathrm{d}t < \infty ∫ 0 1 γ ( t ) d t < ∞
Φ t = − 1 2 ( Φ x x + γ ( t ) Φ x 2 ) , Φ ( 1 , x ) = ∣ x ∣ , \Phi_t = -\frac{1}{2}\left(\Phi_{xx}+\gamma(t)\Phi_x^2\right), \qquad\Phi(1,x)=|x|, Φ t = − 2 1 ( Φ xx + γ ( t ) Φ x 2 ) , Φ ( 1 , x ) = ∣ x ∣ , in its integrable-coefficient control interpretation, and put
P ( γ ) = Φ γ ( 0 , 0 ) − 1 2 ∫ 0 1 t γ ( t ) d t . \mathcal{P}(\gamma)=\Phi_\gamma(0,0)-\frac{1}{2}\int_0^1 t\gamma(t)\,\mathrm{d}t. P ( γ ) = Φ γ ( 0 , 0 ) − 2 1 ∫ 0 1 t γ ( t ) d t . The ground-state formula and attainment of Auffinger and Chen [2 ] identify min γ ∈ U P ( γ ) \min_{\gamma\in\mathcal{U}}\mathcal{P}(\gamma) min γ ∈ U P ( γ ) P ∗ P_* P ∗ (2) . Fix any such minimizer. Throughout this section γ ( t ) \gamma(t) γ ( t )
For clarity, the covariance of the unordered-pair Hamiltonian is E h n , J ( σ 1 ) h n , J ( σ 2 ) = n R 12 2 / 2 − 1 / 2 \mathbb{E}h_{n,J}(\sigma^1)h_{n,J}(\sigma^2)=nR_{12}^2/2-1/2 E h n , J ( σ 1 ) h n , J ( σ 2 ) = n R 12 2 /2 − 1/2 R 12 = n − 1 ∑ i σ i 1 σ i 2 R_{12}=n^{-1}\sum_i\sigma_i^1\sigma_i^2 R 12 = n − 1 ∑ i σ i 1 σ i 2 1 / 2 1/2 1/2 n ξ ( R 12 ) n\xi(R_{12}) n ξ ( R 12 )
Write u = Φ γ , x u = \Phi_{\gamma,x} u = Φ γ , x a = Φ γ , x x a = \Phi_{\gamma,xx} a = Φ γ , xx
d X t = γ ( t ) u ( t , X t ) d t + d B t , X 0 = 0. \mathrm{d}X_t = \gamma(t)u(t,X_t)\,\mathrm{d}t + \mathrm{d}B_t, \qquad X_0 = 0. d X t = γ ( t ) u ( t , X t ) d t + d B t , X 0 = 0. The companion proves full relative Stieltjes support on [ 0 , 1 ) [0,1) [ 0 , 1 ) μ γ ( [ 0 , t ] ) = γ ( t ) \mu_\gamma([0,t])=\gamma(t) μ γ ([ 0 , t ]) = γ ( t )
E u ( t , X t ) 2 = t , E a ( t , X t ) 2 = 1 , P ∗ = ∫ 0 1 E a ( t , X t ) d t . (40) \mathbb{E}u(t,X_t)^2=t,\qquad\mathbb{E}a(t,X_t)^2=1,\qquad P_*=\int_0^1 \mathbb{E}a(t,X_t)\,\mathrm{d}t. \tag*{(40)} E u ( t , X t ) 2 = t , E a ( t , X t ) 2 = 1 , P ∗ = ∫ 0 1 E a ( t , X t ) d t . (40) Here u u u a a a ∣ u ∣ ≤ 1 |u|\leq1 ∣ u ∣ ≤ 1 [19 ] ; their complete analytic proofs belong to that companion, not to the rooted-space argument below. In particular, Cauchy–Schwarz gives
0 ≤ P ∗ − ∫ 0 T E a ( t , X t ) d t ≤ 1 − T , 0 < T < 1. (41) 0\leq P_*-\int_0^T \mathbb{E}a(t,X_t)\,\mathrm{d}t\leq1-T,\qquad0<T<1. \tag*{(41)} 0 ≤ P ∗ − ∫ 0 T E a ( t , X t ) d t ≤ 1 − T , 0 < T < 1. (41) No smooth density of μ γ \mu_\gamma μ γ T ↑ 1 T\uparrow1 T ↑ 1
The use of the Parisi gradient and a normalized Euler martingale follows the classical constructions of Montanari [18 ] , Section 3.1 and El Alaoui, Montanari, and Sellke [8 ] , Proposition 5.3. The precise shifted-coefficient identity needed here is the scalar companion input stated below.
For fixed 0 < T < 1 0<T<1 0 < T < 1 N ≥ 1 N\geq1 N ≥ 1 Δ = T / N \Delta=T/N Δ = T / N t j = j Δ t_j=j\Delta t j = j Δ Z 1 , … , Z N + 1 Z_1,\ldots,Z_{N+1} Z 1 , … , Z N + 1
Y 0 = 0 , Y j = Y j − 1 + Δ Z j + Δ γ ( t j − 1 ) u ( t j − 1 , Y j − 1 ) , (42) Y_0=0,\qquad Y_j=Y_{j-1}+\sqrt{\Delta}Z_j+\Delta\gamma(t_{j-1})u(t_{j-1},Y_{j-1}), \tag*{(42)} Y 0 = 0 , Y j = Y j − 1 + Δ Z j + Δ γ ( t j − 1 ) u ( t j − 1 , Y j − 1 ) , (42) c j = ( E a ( t j − 1 , Y j − 1 ) 2 ) − 1 / 2 , g j = c j a ( t j − 1 , Y j − 1 ) Z j , 1 ≤ j ≤ N . (43) c_j=\left(\mathbb{E}a(t_{j-1},Y_{j-1})^2\right)^{-1/2},\qquad g_j=c_ja(t_{j-1},Y_{j-1})Z_j,\qquad1\leq j\leq N. \tag*{(43)} c j = ( E a ( t j − 1 , Y j − 1 ) 2 ) − 1/2 , g j = c j a ( t j − 1 , Y j − 1 ) Z j , 1 ≤ j ≤ N . (43) For all sufficiently large N N N c j c_j c j Φ G \Phi_{\mathrm{G}} Φ G
H N = u ( T , Y N ) + 2 Φ G ( Z N + 1 ) − 1 , F N = sgn ( H N ) , (44) H_N=u(T,Y_N)+2\Phi_{\mathrm{G}}(Z_{N+1})-1,\qquad F_N=\operatorname{sgn}(H_N), \tag*{(44)} H N = u ( T , Y N ) + 2 Φ G ( Z N + 1 ) − 1 , F N = sgn ( H N ) , (44) A j = E [ F N g j ] ( 1 ≤ j ≤ N ) , B i = E [ F N Z i ] ( 1 ≤ i ≤ N + 1 ) , A 0 = 0. (45) A_j=\mathbb{E}[F_Ng_j]\quad(1\leq j\leq N),\qquad B_i=\mathbb{E}[F_NZ_i]\quad(1\leq i\leq N+1),\qquad A_0=0. \tag*{(45)} A j = E [ F N g j ] ( 1 ≤ j ≤ N ) , B i = E [ F N Z i ] ( 1 ≤ i ≤ N + 1 ) , A 0 = 0. (45) Conditional on Z 1 , … , Z N Z_1,\ldots,Z_N Z 1 , … , Z N 2 Φ G ( Z N + 1 ) − 1 2\Phi_{\mathrm{G}}(Z_{N+1})-1 2 Φ G ( Z N + 1 ) − 1 ( − 1 , 1 ) (-1,1) ( − 1 , 1 ) ∣ u ∣ ≤ 1 |u|\leq1 ∣ u ∣ ≤ 1
E [ F N ∣ Z 1 , … , Z N ] = u ( T , Y N ) . \mathbb{E}[F_N\mid Z_1,\ldots,Z_N]=u(T,Y_N). E [ F N ∣ Z 1 , … , Z N ] = u ( T , Y N ) . Thus sign rounding preserves the terminal gradient as its conditional mean. The same uniformity makes the value of the sign at zero immaterial. Proposition 7.2 of [19 ] states that
lim N → ∞ ∑ j = 0 N A j B j + 1 = ∫ 0 T E a ( t , X t ) d t . (46) \lim_{N\to\infty}\sum_{j=0}^{N}A_jB_{j+1}=\int_0^T\mathbb{E}a(t,X_t)\,\mathrm{d}t. \tag*{(46)} N → ∞ lim j = 0 ∑ N A j B j + 1 = ∫ 0 T E a ( t , X t ) d t . (46) The mesh limit in this assertion is taken at fixed T T T T T T
Exact realization in the rooted Gaussian space
Recall that a real centered child vector f f f G ( f ) = q ( f ) \mathrm{G}(f)=q(f) G ( f ) = q ( f ) E [ G ( f ) G ( g ) ] = ⟨ f , g ⟩ \mathbb{E}[\mathrm{G}(f)\mathrm{G}(g)]=\langle f,g\rangle E [ G ( f ) G ( g )] = ⟨ f , g ⟩
Lemma 3.1 (A finite Gaussian formula and its rooted energy). Fix T T T N N N (42) –(45) , with all c j c_j c j F N F_N F N N + 1 N+1 N + 1
E ( F N ) = ∑ j = 0 N A j B j + 1 . (47) \mathcal{E}(F_N)=\sum_{j=0}^{N} A_jB_{j+1}. \tag*{(47)} E ( F N ) = j = 0 ∑ N A j B j + 1 . (47) The formula is bounded by one, although its transmitted columns need not be bounded.
Proof. Let S S S g 0 = S g_0=S g 0 = S
Z j + 1 = G ( g j ) , 0 ≤ j ≤ N , Z_{j+1}=\mathrm{G}(g_j),\qquad0\leq j\leq N, Z j + 1 = G ( g j ) , 0 ≤ j ≤ N , and use (42) –(43) to form Y j , g j Y_j,g_j Y j , g j Z j Z_j Z j c j c_j c j
At each step identify all earlier vectors by the compatible empty-descendant embeddings of Section 2. These embeddings preserve the earlier Gaussian positions: lifting an embedded child vector gives the embedded copy of its previous lift. Thus the recursion uses the same earlier variables at the next height. Inductively g j g_j g j j j j Z j + 1 Z_{j+1} Z j + 1 j + 1 j+1 j + 1
We show inductively that g 0 , … , g j g_0,\ldots,g_j g 0 , … , g j g 0 g_0 g 0 g j − 1 g_{j-1} g j − 1 Z 1 , … , Z j Z_1,\ldots,Z_j Z 1 , … , Z j j ≥ 1 j\geq1 j ≥ 1 c j a ( t j − 1 , Y j − 1 ) c_ja(t_{j-1},Y_{j-1}) c j a ( t j − 1 , Y j − 1 ) Z 1 , … , Z j − 1 Z_1,\ldots,Z_{j-1} Z 1 , … , Z j − 1 Z j Z_j Z j g j g_j g j g i g_i g i i ≥ 1 i\geq1 i ≥ 1 (43) gives ∥ g j ∥ = 1 \lVert g_j\rVert=1 ∥ g j ∥ = 1 S S S ⟨ g 0 , g j ⟩ = 0 \langle g_0,g_j\rangle=0 ⟨ g 0 , g j ⟩ = 0 Z 1 , … , Z N + 1 Z_1,\ldots,Z_{N+1} Z 1 , … , Z N + 1
The map u ( t , ⋅ ) u(t,\cdot) u ( t , ⋅ ) a ( t , ⋅ ) a(t,\cdot) a ( t , ⋅ ) Y j Y_j Y j g j g_j g j Φ G ( − z ) = 1 − Φ G ( z ) \Phi_{\mathrm{G}}(-z)=1-\Phi_{\mathrm{G}}(z) Φ G ( − z ) = 1 − Φ G ( z ) F N F_N F N F N F_N F N
It remains to compute its energy, not just its distribution. The vector F N F_N F N G ( g 0 ) , … , G ( g N ) \mathrm{G}(g_0),\ldots,\mathrm{G}(g_N) G ( g 0 ) , … , G ( g N ) q ( g ) = a ∗ ( g ) + a ( g ) q(g)=a^*(g)+a(g) q ( g ) = a ∗ ( g ) + a ( g ) ∥ g ∥ 2 \lVert g\rVert^2 ∥ g ∥ 2
L ∗ F N = ∑ j = 0 N E [ F N Z j + 1 ] g j = ∑ j = 0 N B j + 1 g j . L^*F_N=\sum_{j=0}^{N}\mathbb{E}[F_NZ_{j+1}]g_j=\sum_{j=0}^{N}B_{j+1}g_j. L ∗ F N = j = 0 ∑ N E [ F N Z j + 1 ] g j = j = 0 ∑ N B j + 1 g j . There are no other first-chaos coefficients: F N F_N F N F N F_N F N ⟨ F N , g 0 ⟩ = 0 = A 0 \langle F_N,g_0\rangle=0=A_0 ⟨ F N , g 0 ⟩ = 0 = A 0 g j g_j g j A j A_j A j
E ( F N ) = Re ⟨ F N , L F N ⟩ = Re ⟨ L ∗ F N , F N ⟩ = ∑ j = 0 N A j B j + 1 . \mathcal{E}(F_N)=\operatorname{Re}\langle F_N,LF_N\rangle=\operatorname{Re}\langle L^*F_N,F_N\rangle=\sum_{j=0}^{N}A_jB_{j+1}. E ( F N ) = Re ⟨ F N , L F N ⟩ = Re ⟨ L ∗ F N , F N ⟩ = j = 0 ∑ N A j B j + 1 . This proves the claimed identity, with the Gaussian variance and root-shift normalizations used throughout the paper.
Combining (41) , (46) and (47) produces a bounded near-optimal rooted vector by first choosing T T T N N N F N , k = tanh ( k H N ) F_{N,k}=\tanh(kH_N) F N , k = tanh ( k H N ) (45) and dominated convergence give F N , k → F N F_{N,k}\to F_N F N , k → F N L 2 L^2 L 2
∣ E ( f ) − E ( g ) ∣ ≤ ( ∥ f ∥ + ∥ g ∥ ) ∥ f − g ∥ . (48) \left|\mathcal{E}(f)-\mathcal{E}(g)\right|\leq\left(\lVert f\rVert+\lVert g\rVert\right)\lVert f-g\rVert. \tag*{(48)} ∣ E ( f ) − E ( g ) ∣ ≤ ( ∥ f ∥ + ∥ g ∥ ) ∥ f − g ∥ . (48) Both outputs have norm at most one. Thus for every ε > 0 \varepsilon>0 ε > 0 T , N , k T,N,k T , N , k
z = F N , k , ∣ z ∣ ≤ 1 , E ( z ) ≥ P ∗ − ε . (49) z=F_{N,k},\qquad|z|\leq1,\qquad\mathcal{E}(z)\geq P_*-\varepsilon. \tag*{(49)} z = F N , k , ∣ z ∣ ≤ 1 , E ( z ) ≥ P ∗ − ε . (49) For the already fixed T , N , k T,N,k T , N , k u , a u,a u , a k k k
Two further operations are still necessary: bounding every transmitted column and imposing type exclusions. We perform them directly in the limiting Hilbert space. No finite-degree edge-energy convergence is asserted for the discontinuous raw F N F_N F N
Bounded trigonometric approximation
For use with either classical construction, let π ( x ) = max ( − 1 , min ( x , 1 ) ) \pi(x)=\max(-1,\min(x,1)) π ( x ) = max ( − 1 , min ( x , 1 )) π ( z ) = z \pi(z)=z π ( z ) = z (49) .
Lemma 3.2 (Bounded trigonometric approximation). Let m ∗ = π ( z ) m^*=\pi(z) m ∗ = π ( z ) z z z η > 0 \eta>0 η > 0 θ \theta θ
∥ sin ( 2 θ ) − m ∗ ∥ < η . (50) \lVert\sin(2\theta)-m^*\rVert<\eta. \tag*{(50)} ∥ sin ( 2 θ ) − m ∗ ∥ < η . (50) Every sent column in the formula for θ \theta θ
Proof. Replace a Gaussian local seed s s s s R s_R s R ∣ s R ∣ ≤ ∣ s ∣ |s_R|\leq|s| ∣ s R ∣ ≤ ∣ s ∣ s R → s s_R\to s s R → s (17) at each aggregation, Gaussian moment bounds for its finite list of fields, and the polynomial bounds for the derivatives of the pointwise maps.
Fix 0 < κ < 1 0 < \kappa< 1 0 < κ < 1
θ κ = 1 2 arcsin ( ( 1 − κ ) m ∗ ) \theta_\kappa= \frac{1}{2}\arcsin((1-\kappa)m^*) θ κ = 2 1 arcsin (( 1 − κ ) m ∗ ) is bounded and odd, and is a Lipschitz function of the clipped output. It satisfies sin ( 2 θ κ ) = ( 1 − κ ) m ∗ \sin(2\theta_\kappa) = (1-\kappa)m^* sin ( 2 θ κ ) = ( 1 − κ ) m ∗ L p L^p L p p p p F ( x ) ↦ ( F ( x ) − F ( − x ) ) / 2 F(x) \mapsto(F(x)-F(-x))/2 F ( x ) ↦ ( F ( x ) − F ( − x )) /2
Once an outer map has been replaced by a fixed bounded Lipschitz map, perturbations of its inputs propagate continuously in L 2 L^2 L 2 (17) , jointly for the finite list of child columns. We can therefore choose the approximations at the next lower level sufficiently accurately to preserve the already chosen output accuracy. There are finitely many operations, so induction supplies a formula in which every transmitted scalar function is bounded, smooth, and odd. No uniform bound on Lipschitz constants over the sequence of approximation choices is needed; each outer choice is fixed before its inputs are approximated.
Finally a bounded smooth cutoff supported strictly inside a box can be extended periodically to a larger box. Fourier approximation of this smooth periodic function gives finite trigonometric sums converging uniformly, with any prescribed finite number of derivatives, on that box. To control the complement, first fix the box so that its probability is small and choose the Fourier approximation uniformly on the entire period; both the cutoff and its approximant then have a common finite sup norm. The same output-to-leaf continuity argument applies. Odd symmetrization leaves only sine terms sin ( λ ⋅ x ) \sin(\lambda\cdot x) sin ( λ ⋅ x ) η / 4 \eta/4 η /4 κ < η / 2 \kappa< \eta/2 κ < η /2 x ↦ sin ( 2 x ) x \mapsto\sin(2x) x ↦ sin ( 2 x ) (50) .
The synthesis interface
A mask is an event in the local seed space invariant under seed sign reversal. Masks at different vertices are evaluated on their respective seeds. We use a finite partition into masks and write p R p_R p R R R R R , S R,S R , S
( A R S f ) i = 1 R ( i ) D − 1 / 2 ∑ j ∼ i 1 S ( j ) f j . (A_{RS}f)_i = \mathbf{1}_R(i)D^{-1/2}\sum_{j\sim i}\mathbf{1}_S(j)f_j. ( A R S f ) i = 1 R ( i ) D − 1/2 j ∼ i ∑ 1 S ( j ) f j . This notation first denotes a finite-tree operation. Its rooted limit is provided by Theorem 2.12.
The scalar formulas considered here have finite expression trees. Their leaves are bounded measurable functions of the local seed. An internal operation is either a pointwise function of finitely many already constructed quantities or an aggregation A R S A_{RS} A R S
Definition 3.3 (Exclusion formulas). An aggregation from S S S R R R excluded if the formula sent from S S S R R R
The distinction between an expression tree and the underlying regular tree is useful. A formula may reuse a mask in different expression branches. The exclusion rule forbids its reuse along a single active branch. It therefore forbids an immediate return across a cut, without requiring independent values for two formulas that happen to use the same descendant subtree.
Colors, exclusions, and helpers
A descendant formula omits the parent vertex in its finite-tree interpretation. We now impose the stronger type exclusions needed for synthesis. We estimate their cost in the limiting classical space, where (17) makes the cost of deleting small-probability types explicit.
Let s r a w s_{\mathrm{raw}} s raw ( 0 , 1 ) (0,1) ( 0 , 1 ) C C C S S S
s r a w ⟼ ( C ( s r a w ) , S ( s r a w ) ) , s_{\mathrm{raw}} \longmapsto\left(C(s_{\mathrm{raw}}),S(s_{\mathrm{raw}})\right), s raw ⟼ ( C ( s raw ) , S ( s raw ) ) , C ( − s r a w ) = C ( s r a w ) , S ( − s r a w ) = − S ( s r a w ) (51) C(-s_{\mathrm{raw}})=C(s_{\mathrm{raw}}), \qquad S(-s_{\mathrm{raw}})=-S(s_{\mathrm{raw}}) \tag*{(51)} C ( − s raw ) = C ( s raw ) , S ( − s raw ) = − S ( s raw ) (51) whose image has the product law of the chosen color and a standard Gaussian. Every colored formula below is evaluated at this pair at each vertex: its Gaussian input is S ( s r a w ) S(s_{\mathrm{raw}}) S ( s raw )
One may first adjoin an independent color to the color-free seed space and then pull back by (51) . The local pullback f ( C , S ) ↦ f ( C ( s r a w ) , S ( s r a w ) ) f(C,S)\mapsto f(C(s_{\mathrm{raw}}),S(s_{\mathrm{raw}})) f ( C , S ) ↦ f ( C ( s raw ) , S ( s raw )) L L L 1 , … , M 1,\ldots,M 1 , … , M
max 1 ≤ c ≤ M Pr ( C = c ) ≤ α , Pr ( C is a helper ) = p H . \max_{1\leq c\leq M}\Pr(C=c)\leq\alpha,\qquad\Pr(C\text{ is a helper})=p_H. 1 ≤ c ≤ M max Pr ( C = c ) ≤ α , Pr ( C is a helper ) = p H . Every helper type may have a prescribed positive probability, subject to the total being sufficiently small. The number of helper types needed later is finite, at most one per nested propagation depth.
Lemma 3.4 (Type exclusion at small cost). *Fix a descendant formula in which every transmitted column and the output are given by jointly odd bounded smooth functions of the local odd seed and the odd aggregate inputs, with bounded first derivatives. Let its propagation depth be h h h f f f f c o l f^{\mathrm{col}} f col
∥ f c o l − f ∥ 2 ≤ C ( α + p H ) , (52) \lVert f^{\mathrm{col}}-f\rVert_2 \le C(\alpha+p_H), \tag*{(52)} ∥ f col − f ∥ 2 ≤ C ( α + p H ) , (52) where C C C M M M
Proof. Unroll the fixed expression into a finite expression tree. For each occurrence, each possible current nonhelper color c c c A A A A ∪ { c } A \cup\{c\} A ∪ { c } d d d A ∪ { c } A \cup\{c\} A ∪ { c } d d d A ∪ { c , d } A \cup\{c,d\} A ∪ { c , d } h + 1 h+1 h + 1 c c c G G G
These copies also define an orientation-free all-neighbor rule. When a recursive call crosses from a vertex i i i c c c j j j d d d j j j c c c j j j i i i i i i D − 1 / 2 D^{-1/2} D − 1/2 A R S A_{RS} A R S D + 1 D+1 D + 1 D D D
Here is a uniform error estimate. Give the original formula the extra independent colors but ignore them in its computation. At one aggregate, let g g g ∣ g ∣ ≤ K \lvert g\rvert\le K ∣ g ∣ ≤ K g A , d g^{A,d} g A , d d d d (17) . The modified aggregate has child column
g ~ = ∑ d ∉ A ∪ { c } , d n o n h e l p e r 1 { C = d } g A ∪ { c } , d . \widetilde{g}=\sum_{\substack{d\notin A\cup\{c\},\ d\ \mathrm{nonhelper}}} \mathbf{1}_{\{C=d\}}g^{A\cup\{c\},d}. g = d ∈ / A ∪ { c } , d nonhelper ∑ 1 { C = d } g A ∪ { c } , d . The color masks are orthogonal, and g g g
∥ g ~ − g ∥ 2 ≤ K 2 ( p H + ( ∣ A ∣ + 1 ) α ) + max d ∉ A ∪ { c } E [ ∣ g A ∪ { c } , d − g ∣ 2 ∣ C = d ] . \lVert\widetilde{g}-g\rVert^2 \le K^2\bigl(p_H+(\lvert A\rvert+1)\alpha\bigr)
+\max_{d\notin A\cup\{c\}}\mathbb{E}\left[\left.\lvert g^{A\cup\{c\},d}-g\rvert^2\right|C=d\right]. ∥ g − g ∥ 2 ≤ K 2 ( p H + (∣ A ∣ + 1 ) α ) + d ∈ / A ∪ { c } max E [ ∣ g A ∪ { c } , d − g ∣ 2 C = d ] . Every masked column is centered: color masks are even and the conditional formula is odd in the Gaussian seeds. Thus the Gaussian-isometry identity applies without a mean correction. For a list of k k k K ′ K' K ′ ( K ′ ) 2 (K')^2 ( K ′ ) 2 C ( α + p H ) C(\alpha+p_H) C ( α + p H ) ∥ f ∥ ∞ 2 p H \lVert f\rVert_\infty^2p_H ∥ f ∥ ∞ 2 p H
This estimate has the fan-in of the original formula, not the number of color copies: disjoint child masks were combined by squared norms before applying a Lipschitz bound. Each nonlinear pointwise map still receives the original finite list of grouped fields: if it originally had k k k F ( S , Y 1 c o l , … , Y k c o l ) F(S,Y^{\mathrm{col}}_1,\ldots,Y^{\mathrm{col}}_k) F ( S , Y 1 col , … , Y k col )
Y r c o l = ∑ d G ( 1 { C = d } G r , d ) = G ( ∑ d 1 { C = d } G r , d ) . Y^{\mathrm{col}}_r=\sum_d G(\mathbf{1}_{\{C=d\}}\mathcal{G}_{r,d})=G\left(\sum_d\mathbf{1}_{\{C=d\}}\mathcal{G}_{r,d}\right). Y r col = d ∑ G ( 1 { C = d } G r , d ) = G ( d ∑ 1 { C = d } G r , d ) . The conditional derivative bounds in these k k k M M M M M M (52) and the bounds in the original grouped coordinates are asserted to be independent of M M M
Proposition 3.5 (Classical near-optimality in synthesis form). For every ε > 0 \varepsilon>0 ε > 0 and every η H > 0 \eta_H>0 η H > 0 there are a finite color partition with helper probability 0 < p H < η H 0<p_H<\eta_H 0 < p H < η H a finite number of helper types, and finite bounded odd angle formulas θ c \theta_c θ c on the nonhelper colors such that the following hold. Each transmitted column is bounded and odd, every pointwise map may be chosen as a finite trigonometric sum in its odd inputs, and every aggregation excludes its current color, all path-ancestor colors, and all helpers. Conditional on the finite labels, all continuous derivatives of the pointwise maps are bounded. The output
m c o l = ∑ c = 1 M 1 { C = c } sin ( 2 θ c ) (53) m^{\mathrm{col}}=\sum_{c=1}^{M}\mathbf{1}_{\{C=c\}}\sin(2\theta_c) \tag*{(53)} m col = c = 1 ∑ M 1 { C = c } sin ( 2 θ c ) (53) is centered, satisfies ∣ m c o l ∣ ≤ 1 \lvert m^{\mathrm{col}}\rvert\le1 ∣ m col ∣ ≤ 1 vanishes on helper roots, and obeys
E ( m c o l ) ≥ P ∗ − ε . (54) \mathcal{E}(m^{\mathrm{col}})\ge P_*-\varepsilon. \tag*{(54)} E ( m col ) ≥ P ∗ − ε . (54) After the uncolored finite formula has been fixed, the total helper probability may be taken arbitrarily small, with all individual helper probabilities positive. All depths, coefficients, label laws, and functions in the conclusion are fixed independently of degree, system size, and disorder.
Proof. Use (49) with error ε / 3 \varepsilon/3 ε /3 m ∗ = z m^*=z m ∗ = z θ \theta θ m = sin ( 2 θ ) m=\sin(2\theta) m = sin ( 2 θ ) ∥ m − m ∗ ∥ < ε / 12 \lVert m-m^*\rVert<\varepsilon/12 ∥ m − m ∗ ∥ < ε /12 (48) loses less than ε / 6 \varepsilon/6 ε /6
Reserve a distinct helper type for each propagation depth. Choose their positive probabilities with total p H < η H p_H<\eta_H p H < η H α \alpha α α , p H \alpha,p_H α , p H ∥ m c o l − m ∥ < ε / 6 \lVert m^{\mathrm{col}}-m\rVert<\varepsilon/6 ∥ m col − m ∥ < ε /6 (48) loses less than ε / 3 \varepsilon/3 ε /3 ε \varepsilon ε
The preceding section supplies a bounded classical formula whose rooted energy is close to the SK optimum. We now implement any formula in that class by quantum controls with independent Gaussian longitudinal seeds. A seeded circuit is a finite word in the cost, the uniform mixer, and the gates
exp ( − i t ∑ i s i Z i ) , t ∈ R , \exp\left(-\mathrm{i}t\sum_i s_i Z_i\right), \qquad t \in\mathbb{R}, exp ( − i t i ∑ s i Z i ) , t ∈ R , where the same independent standard Gaussian family ( s i ) (s_i) ( s i ) s i = s r a w , i s_i=s_{\mathrm{raw},i} s i = s raw , i C ( s i ) , S ( s i ) C(s_i), S(s_i) C ( s i ) , S ( s i )
We use the masks, aggregations A R S A_{RS} A R S
All approximations in this section take place in the infinite-degree rooted representation, inside a fixed surrounding circuit. We first construct elementary masked controls, then use unused helper spins to implement one aggregation, and finally compile a finite expression tree from its outermost operations inward. The helper contraction is the main step. Its proof controls both the visible aggregation and the changes it induces inside every lower formula.
The synthesis statement
Fix a finite collection of such formulas θ R \theta_R θ R R R R
U c l = ∏ i exp ( i θ i Y i ) , m i = sin ( 2 θ i ) . (55) U_{\mathrm{cl}}=\prod_i \exp(\mathrm{i}\theta_iY_i), \qquad m_i=\sin(2\theta_i). \tag*{(55)} U cl = i ∏ exp ( i θ i Y i ) , m i = sin ( 2 θ i ) . (55) The product is locally defined: an output at a vertex uses only a fixed-radius neighborhood. The factors commute because the angles are seed-only. Reserve disjoint helper masks, absent from all these formulas, and set the output angle to zero on them. One helper mask per aggregation depth is sufficient. Let H H H p H p_H p H
Theorem 4.1 (Classical synthesis). For every finite collection of bounded smooth odd seed-only exclusion formulas of Definition 3.3, with disjoint positive-probability helper masks, one for each positive aggregation depth, absent from all their dependencies and with output angle zero on their union H H H η > 0 \eta>0 η > 0 v v v
∥ 1 H c ( v − v c l ) ∥ < η , (56) \left\lVert\mathbf{1}_{H^c}(v-v_{\mathrm{cl}})\right\rVert<\eta, \tag*{(56)} ∥ 1 H c ( v − v cl ) ∥ < η , (56) where v c l v_{\mathrm{cl}} v cl
∣ E ( v ) − E ( m ) ∣ ≤ 2 η + 2 p H . (57) \left|\mathcal{E}(v)-\mathcal{E}(m)\right|\leq2\eta+2\sqrt{p_H}. \tag*{(57)} ∣ E ( v ) − E ( m ) ∣ ≤ 2 η + 2 p H . (57) The circuit and all its angles are chosen independently of the degree, system size, and disorder. Helper probabilities can be arbitrarily small but are fixed and positive when the construction is performed.
The proof will establish convergence in a fixed surrounding circuit, not only convergence on a freshly initialized input. This is needed because a formula is called repeatedly inside commutator products.
We use the smooth spaces of Section 2. At the root let
W = 1 + d Γ ( Λ ) , ∥ T ∥ a ← b = ∥ W a T W − b ∥ , ∥ d ∥ a = ∥ Λ a d ∥ . W = 1 + \mathrm{d}\Gamma(\Lambda), \qquad\lVert T\rVert_{a\leftarrow b} = \lVert W^{a}TW^{-b}\rVert, \qquad\lVert d\rVert_{a} = \lVert\Lambda^{a}d\rVert. W = 1 + d Γ ( Λ ) , ∥ T ∥ a ← b = ∥ W a T W − b ∥ , ∥ d ∥ a = ∥ Λ a d ∥ . The bounded-seed version of the weights is used until the last compilation step. Thus the weights do not differentiate or multiply the root seed. In particular they commute with root masks on the odd subspace. A family of operators is uniformly smooth if for each a a a b , C b,C b , C ∥ T ∥ a ← b ≤ C \lVert T\rVert_{a\leftarrow b} \le C ∥ T ∥ a ← b ≤ C y y y 1 + ∣ y ∣ 1+\lvert y\rvert 1 + ∣ y ∣ T = O s m ( ε ) T=O_{\mathrm{sm}}(\varepsilon) T = O sm ( ε ) C ε C\varepsilon C ε
The conditional bounds in Theorem 2.12 imply the following useful rule. If T Ω T\Omega T Ω R R R
∥ W a 1 R T Ω ∥ ≤ C a p R . (58) \lVert W^{a}{}_1RT\Omega\rVert\le C_a\sqrt{p_R}. \tag*{(58)} ∥ W a 1 R T Ω ∥ ≤ C a p R . (58) For odd columns, the removal of the vacuum in the definition of Λ \Lambda Λ
Elementary masked controls and matrix products
We first identify the elementary gates from which the helper operation will be built. These gates need only the original Gaussian longitudinal control, parity-preserving local rotations, and the cost.
We first treat the following gates as exact:
local X X X Y Y Y Z Z Z
interactions between two disjoint masks in any pair of odd axes in the Y Z YZ Y Z
We justify at the end of this subsection that they lie in the closure of finite seeded circuits. Working with exact elementary masked gates in the interim keeps unused helpers exactly pinned.
We will repeatedly use pointwise matrix sums and commutators. Here is the topology of that use. If F ( x ) , G ( x ) F(x),G(x) F ( x ) , G ( x ) x x x
V h ( x ) = e i h F ( x ) e i h G ( x ) e − i h F ( x ) e − i h G ( x ) . V_h(x)=e^{\mathrm{i}hF(x)}e^{\mathrm{i}hG(x)}e^{-\mathrm{i}hF(x)}e^{-\mathrm{i}hG(x)}. V h ( x ) = e i h F ( x ) e i h G ( x ) e − i h F ( x ) e − i h G ( x ) . Twice integrating its mixed derivative in the two loop times shows
∥ ∂ x α ( V h − 1 ) ∥ ≤ C α h 2 ( 1 + ∣ x ∣ ) r α , ∣ h ∣ ≤ 1. \lVert\partial_x^\alpha(V_h-1)\rVert\le C_\alpha h^2(1+\lvert x\rvert)^{r_\alpha}, \qquad\lvert h\rvert\le1. ∥ ∂ x α ( V h − 1 )∥ ≤ C α h 2 ( 1 + ∣ x ∣ ) r α , ∣ h ∣ ≤ 1. For a differentiated product of N N N O ( h 2 ) O(h^2) O ( h 2 ) N h 2 Nh^2 N h 2 h = N − 1 / 2 h=N^{-1/2} h = N − 1/2 O ( h 3 ) O(h^3) O ( h 3 ) N N N N h Nh N h N h 2 Nh^2 N h 2
When such a matrix formula is used simultaneously at many vertices, the coefficient algebra must be common and commuting, and exclude the pivot mask. Under that condition the entire finite-tree gate is a product of exactly those two-by-two formulas. Its reinterpretation at a new pivot is permitted only after an exact finite-tree identity has been established. These identities are also valid on every cut tree, so the rooted maps agree.
To obtain the local elementary controls from the raw seeded controls, write s = s r a w s = s_{\mathrm{raw}} s = s raw X X X e i t s Z e^{\mathrm{i}tsZ} e i t s Z s s s cos ( 2 t s ) X \cos(2ts)X cos ( 2 t s ) X sin ( 2 t s ) Y \sin(2ts)Y sin ( 2 t s ) Y X X X Y Y Y Z Z Z
We isolate interactions by sign averaging, a standard refocusing device for Ising couplings; see Jones and Knill [14 ] , Section II. Refine the masks to a finite partition R 1 , … , R M R_1,\ldots,R_M R 1 , … , R M ϵ ∈ { − 1 , 1 } M \epsilon\in\{-1,1\}^M ϵ ∈ { − 1 , 1 } M X X X
C ϵ = D − 1 / 2 ∑ { i , j } ϵ R ( i ) ϵ R ( j ) Z i Z j . C_{\epsilon}=D^{-1/2}\sum_{\{i,j\}}\epsilon_{R(i)}\epsilon_{R(j)}Z_iZ_j. C ϵ = D − 1/2 { i , j } ∑ ϵ R ( i ) ϵ R ( j ) Z i Z j . For distinct a , b a,b a , b
C R a R b = 2 − M ∑ ϵ ϵ a ϵ b C ϵ . C_{R_aR_b}=2^{-M}\sum_{\epsilon}\epsilon_a\epsilon_bC_{\epsilon}. C R a R b = 2 − M ϵ ∑ ϵ a ϵ b C ϵ . All the terms on the right commute, so their exponential implements the desired subcost exactly when the masked flips are treated as exact. Separate masked X X X Y Z YZ Y Z
A contraction through unused spins
The elementary local rotations constructed so far depend only on the seed at the rotated vertex. The next construction extends them to a neighbor sum of a fixed lower formula. It uses a disjoint set of spins kept in their X = 1 X=1 X = 1
Let I , J , K I,J,K I , J , K K K K p K > 0 p_K>0 p K > 0 f J f_J f J J J J I , J , K I,J,K I , J , K f f f J J J
H I , J , f = ∑ i ∈ I P i ( A I , J f ) i , P ∈ { Y , Z } . (59) H_{I,J,f}=\sum_{i\in I}P_i(A_{I,J}f)_i,\qquad P\in\{Y,Z\}. \tag*{(59)} H I , J , f = i ∈ I ∑ P i ( A I , J f ) i , P ∈ { Y , Z } . (59) Partition J J J J l J_l J l p l p_l p l f f f
i ∈ I − j ∈ J l − k ∈ K − j ′ ∈ J l ↓ f j ′ (60) \begin{gathered}
i \in I \quad- \quad j \in J_l \quad- \quad k \in K \quad- \quad j' \in J_l \\
\hspace{7cm}\downarrow\\
\hspace{7cm}f_{j'} \tag*{(60)}
\end{gathered} i ∈ I − j ∈ J l − k ∈ K − j ′ ∈ J l ↓ f j ′ (60) j ′ ≠ j j' \ne j j ′ = j j ′ = j \qquad j'=j j ′ = j i − j − k − j i-j-k-j i − j − k − j
The downward arrow denotes the lower calculation of f f f f f f j ′ = j j'=j j ′ = j f f f
Write A i j = D − 1 / 2 1 i ∼ j A_{ij}=D^{-1/2}\mathbf{1}_{i\sim j} A ij = D − 1/2 1 i ∼ j b j = ∑ i ∈ I A j i P i b_j=\sum_{i\in I}A_{ji}P_i b j = ∑ i ∈ I A j i P i J l J_l J l
H l = ∑ i ∈ I , j ∈ J l , k ∈ K A i j A j k P i X j Z k , (61) H_l=\sum_{i\in I,j\in J_l,k\in K} A_{ij}A_{jk}P_iX_jZ_k, \tag*{(61)} H l = i ∈ I , j ∈ J l , k ∈ K ∑ A ij A j k P i X j Z k , (61) G l = ∑ j ∈ J l , k ∈ K A j k X j f j Y k . (62) G_l=\sum_{j\in J_l,k\in K} A_{jk}X_jf_jY_k. \tag*{(62)} G l = j ∈ J l , k ∈ K ∑ A j k X j f j Y k . (62) For example, ( 2 i ) − 1 [ ∑ A i j P i Y j , ∑ A j k Z j Z k ] = H l (2i)^{-1}[\sum A_{ij}P_iY_j,\sum A_{jk}Z_jZ_k]=H_l ( 2 i ) − 1 [ ∑ A ij P i Y j , ∑ A j k Z j Z k ] = H l ( 2 i ) − 1 [ ∑ f j Y j , ∑ A j k Z j Y k ] = G l (2i)^{-1}[\sum f_jY_j,\sum A_{jk}Z_jY_k]=G_l ( 2 i ) − 1 [ ∑ f j Y j , ∑ A j k Z j Y k ] = G l J l J_l J l
At the next pivot, K K K
B k , l = ∑ j ∈ J l A k j X j b j , F k , l = ∑ j ∈ J l A k j X j f j . B_{k,l}=\sum_{j\in J_l}A_{kj}X_jb_j,\qquad F_{k,l}=\sum_{j\in J_l}A_{kj}X_jf_j. B k , l = j ∈ J l ∑ A k j X j b j , F k , l = j ∈ J l ∑ A k j X j f j . They commute with each other and with all spins on K K K K K K − ( 2 i ) − 1 [ H l , G l ] = ∑ k ∈ K X k B k , l F k , l -(2i)^{-1}[H_l,G_l]=\sum_{k\in K}X_kB_{k,l}F_{k,l} − ( 2 i ) − 1 [ H l , G l ] = ∑ k ∈ K X k B k , l F k , l
H ~ P = 1 p K ∑ l ∑ i ∈ I , j , j ′ ∈ J l , k ∈ K P i A i j X j A j k X k A k j ′ X j ′ f j ′ . (63) \widetilde{H}_P=\frac{1}{p_K}\sum_l\sum_{i\in I,j,j'\in J_l,k\in K}P_iA_{ij}X_jA_{jk}X_kA_{kj'}X_{j'}f_{j'}. \tag*{(63)} H P = p K 1 l ∑ i ∈ I , j , j ′ ∈ J l , k ∈ K ∑ P i A ij X j A j k X k A k j ′ X j ′ f j ′ . (63) All its summands commute. At every fixed partition P \mathcal{P} P f f f
We now specify the space on which pinning is used. On a finite light cone with vertex set V V V
Π K , V = ∏ v ∈ V [ 1 − 1 K ( s v ) 1 − X v 2 ] . (64) \Pi_{K,V}=\prod_{v\in V}\left[1-\mathbf{1}_K(s_v)\frac{1-X_v}{2}\right]. \tag*{(64)} Π K , V = v ∈ V ∏ [ 1 − 1 K ( s v ) 2 1 − X v ] . (64) Its range fixes every helper-labeled spin in ∣ + ⟩ |+\rangle ∣ + ⟩
K K = Ran ( 1 − 1 K ( s ) 1 − X 2 ) , \mathcal{K}_K=\operatorname{Ran}\left(1-\mathbf{1}_K(s)\frac{1-X}{2}\right), K K = Ran ( 1 − 1 K ( s ) 2 1 − X ) , H K , 0 = K K , H K , h + 1 = K K ⊗ Γ s ( H K , h ) . \mathcal{H}_{K,0}=\mathcal{K}_K,\qquad\mathcal{H}_{K,h+1}=\mathcal{K}_K\otimes\Gamma_s(\mathcal{H}_{K,h}). H K , 0 = K K , H K , h + 1 = K K ⊗ Γ s ( H K , h ) . where the centered subspace is taken relative to the same vacuum Ω h \Omega_h Ω h H h \mathcal{H}_h H h compressed space . Helper-labeled descendant spaces remain present; only their helper spins are pinned, and the seed law is unchanged.
The ideal propagations, the exact generators (63) , and surrounding gates avoiding K K K 1 K c ( s o ) Z o \mathbf{1}_{K^c}(s_o)Z_o 1 K c ( s o ) Z o o o o 1 K c \mathbf{1}_{K^c} 1 K c
Proposition 4.2 (Simultaneous helper contraction). Consider a fixed finite sequence in the parity-preserving, root-seed-commuting gate-calculus class, containing finitely many propagations (59) , possibly with different I I I J J J f f f P P P K K K X = 1 X=1 X = 1 (63) , and refine their sender partitions so that their largest part probability tends to zero. Then all insertion vectors off K K K f f f
The contraction must control every root cut, including cuts inside f f f K K K J l J_l J l
Estimates and proof of the contraction
Lemma 4.3 (A fixed number of commuting fields). Let Q j = q ( d j ) Q_j=q(d_j) Q j = q ( d j ) 1 ≤ j ≤ k 1\leq j\leq k 1 ≤ j ≤ k k k k ∣ d j ∣ a ≤ C a \lvert d_j\rvert_a\leq C_a ∣ d j ∣ a ≤ C a a a a F s : R k → R F_s:\mathbb{R}^k\to\mathbb{R} F s : R k → R s s s t t t exp ( i t F s ( Q ) ) \exp(i t F_s(Q)) exp ( i t F s ( Q )) F F F d j d_j d j
Proof. The Weyl estimates in Lemma 2.5, applied to ∑ j ξ j d j \sum_j \xi_jd_j ∑ j ξ j d j a a a
∥ W a e i ξ ⋅ Q ξ ∥ ≤ C a ( 1 + ∣ ξ ∣ ) r a ∥ W b a ξ ∥ . \left\|W^a e^{i\xi\cdot Q}\xi\right\|\leq C_a(1+|\xi|)^{r_a}\left\|W^{b_a}\xi\right\|. W a e i ξ ⋅ Q ξ ≤ C a ( 1 + ∣ ξ ∣ ) r a W b a ξ . Only upper bounds for weighted column norms enter this inequality. For an integer N N N g s , t ( x ) = e i t F s ( x ) ( 1 + ∣ x ∣ 2 ) − N g_{s,t}(x)=e^{itF_s(x)}(1+|x|^2)^{-N} g s , t ( x ) = e i t F s ( x ) ( 1 + ∣ x ∣ 2 ) − N ( 1 + ∣ ξ ∣ ) r a (1+|\xi|)^{r_a} ( 1 + ∣ ξ ∣ ) r a N N N s , t s,t s , t
e i t F s ( Q ) = ∫ R k g ^ s , t ( ξ ) e i ξ ⋅ Q d ξ ( 1 + ∣ Q ∣ 2 ) N , (65) e^{itF_s(Q)}=\int_{\mathbb{R}^k}\widehat{g}_{s,t}(\xi)e^{i\xi\cdot Q}\,\mathrm{d}\xi\,(1+|Q|^2)^N, \tag*{(65)} e i t F s ( Q ) = ∫ R k g s , t ( ξ ) e i ξ ⋅ Q d ξ ( 1 + ∣ Q ∣ 2 ) N , (65) with the chosen Fourier normalization absorbed in g ^ \widehat{g} g v ( y ) = ∑ j ξ j d j ( y ) v(y)=\sum_j \xi_jd_j(y) v ( y ) = ∑ j ξ j d j ( y )
∂ y e i q ( v ( y ) ) = e i q ( v ( y ) ) ( i q ( ∂ y v ) + i Im ⟨ v , ∂ y v ⟩ ) . (66) \partial_y e^{iq(v(y))}=e^{iq(v(y))}\left(iq(\partial_yv)+i\operatorname{Im}\langle v,\partial_yv\rangle\right). \tag*{(66)} ∂ y e i q ( v ( y )) = e i q ( v ( y )) ( i q ( ∂ y v ) + i Im ⟨ v , ∂ y v ⟩ ) . (66) Repeated derivatives are the same Weyl operator times finite ordered products of derivative fields and scalar Gram pairings. Each field insertion has a smooth bound polynomial in ∣ ξ ∣ |\xi| ∣ ξ ∣ ξ \xi ξ ( 1 + ∣ Q ∣ 2 ) N (1+|Q|^2)^N ( 1 + ∣ Q ∣ 2 ) N
Lemma 4.4 (Quadratics with many orthogonal columns). Let R l R_l R l p l = Pr ( R l ) p_l=\operatorname{Pr}(R_l) p l = Pr ( R l ) d l , e l d_l,e_l d l , e l R l R_l R l q ( d l ) , q ( e l ) q(d_l),q(e_l) q ( d l ) , q ( e l ) a a a
∥ d l ∥ a + ∥ e l ∥ a ≤ C a p l . \|d_l\|_a+\|e_l\|_a\le C_a\sqrt{p_l}. ∥ d l ∥ a + ∥ e l ∥ a ≤ C a p l . Set
Q = ∑ l ( q ( d l ) q ( e l ) − ⟨ d l , e l ⟩ ) , ρ = ( ∑ l p l 2 ) 1 / 2 . Q=\sum_l\left(q(d_l)q(e_l)-\langle d_l,e_l\rangle\right),\qquad\rho=\left(\sum_l p_l^2\right)^{1/2}. Q = l ∑ ( q ( d l ) q ( e l ) − ⟨ d l , e l ⟩ ) , ρ = ( l ∑ p l 2 ) 1/2 . Then Q = O s m ( ρ ) Q=O_{\mathrm{sm}}(\rho) Q = O sm ( ρ ) ρ → 0 \rho\to0 ρ → 0
Proof. Normal ordering gives a pair-creation kernel, its adjoint, and a one-particle kernel:
Q = a ∗ ( ∑ l d l ⊗ e l ) + a ( ∑ l e l ⊗ d l ) + d Γ ( ∑ l ∣ d l ⟩ ⟨ e l ∣ + ∣ e l ⟩ ⟨ d l ∣ ) . Q=a^*\left(\sum_l d_l\otimes e_l\right)+a\left(\sum_l e_l\otimes d_l\right)+\mathrm{d}\Gamma\left(\sum_l\lvert d_l\rangle\langle e_l\rvert+\lvert e_l\rangle\langle d_l\rvert\right). Q = a ∗ ( l ∑ d l ⊗ e l ) + a ( l ∑ e l ⊗ d l ) + d Γ ( l ∑ ∣ d l ⟩ ⟨ e l ∣ + ∣ e l ⟩ ⟨ d l ∣ ) . where the first kernel is understood symmetrically. Orthogonality of different masks gives
∥ ∑ l Λ a d l ⊗ Λ b e l ∥ H S + ∥ ∑ l ∣ Λ a d l ⟩ ⟨ Λ b e l ∣ ∥ H S ≤ C a , b ρ . \left\|\sum_l\Lambda^a d_l\otimes\Lambda^b e_l\right\|_{\mathrm{HS}}+\left\|\sum_l\lvert\Lambda^a d_l\rangle\langle\Lambda^b e_l\rvert\right\|_{\mathrm{HS}}\le C_{a,b}\rho. l ∑ Λ a d l ⊗ Λ b e l HS + l ∑ ∣ Λ a d l ⟩ ⟨ Λ b e l ∣ HS ≤ C a , b ρ . The same inequalities hold with d , e d,e d , e ( n + 1 ) ( n + 2 ) \sqrt{(n+1)(n+2)} ( n + 1 ) ( n + 2 ) n n n n + 1 n+1 n + 1 W W W Λ \Lambda Λ
For completeness, this smallness of the generator alone is not used to assert an exponential power bound. Every iterated commutator ad W j ( Q ) \operatorname{ad}_W^j(Q) ad W j ( Q )
∥ ad W j ( Q ) ∥ ≤ C j ∥ W c ∥ . \|\operatorname{ad}_W^j(Q)\|\le C_j\|W^c\|. ∥ ad W j ( Q ) ∥ ≤ C j ∥ W c ∥. Writing [ W a , Q ] [W^a,Q] [ W a , Q ] ∥ [ W a , Q ] ∥ ≤ C a ∥ W a ∥ \|[W^a,Q]\|\le C_a\|W^a\| ∥ [ W a , Q ] ∥ ≤ C a ∥ W a ∥ a a a ∥ W e i t Q ξ ∥ \|W e^{itQ}\xi\| ∥ W e i tQ ξ ∥
Finally, spectral calculus gives
∥ ( e i t Q − 1 ) ζ ∥ ≤ ∣ t ∣ ∥ Q ζ ∥ ≤ C ∣ t ∣ ρ ∥ W b ζ ∥ . \left\lVert(e^{itQ}-1)\zeta\right\rVert\le|t|\lVert Q\zeta\rVert\le C|t|\rho\lVert W^{b}\zeta\rVert. ( e i tQ − 1 ) ζ ≤ ∣ t ∣ ∥ Qζ ∥ ≤ C ∣ t ∣ ρ ∥ W b ζ ∥ . The uniform higher-power bounds and interpolation prove the last assertion. The same conclusions hold for U ∗ Q U U^{*}QU U ∗ Q U U U U U U U U ∗ U^{*} U ∗ e i t U ∗ Q U = U ∗ e i t Q U e^{itU^{*}QU}=U^{*}e^{itQ}U e i t U ∗ Q U = U ∗ e i tQ U
Proof of Proposition 4.2. We prove a single replacement with arbitrary preceding root maps from the gate calculus: they preserve the joint parity, commute with root seed multiplication, and have uniform conditional smooth bounds. These hypotheses are preserved by all the gates used here. For the sender partition put
ρ = ( ∑ l p l 2 ) 1 / 2 , ε = max l p l . \rho=\left(\sum_l p_l^2\right)^{1/2},\qquad\varepsilon=\max_l\sqrt{p_l}. ρ = ( l ∑ p l 2 ) 1/2 , ε = l max p l . We will estimate the Hamiltonian correction at each root cut, prove uniform smooth bounds for its exponential, and then iterate the replacement. At helper roots a scalar phase remains; we will check that it cancels from the child signals before the next replacement. The partition changes only the terminal coefficients of the lower calculation: the formula f f f
On the range of (4.10), and hence in its recursive occupation realization, we may set X k = 1 X_k=1 X k = 1 k ∈ K k\in K k ∈ K
The relative root operation. Fix the partition and introduce an unscaled real time t t t t t t H D H_D H D H ~ D \widetilde{H}_D H D T T T d d d T a T_a T a d − 1 d-1 d − 1 F D F_D F D
L F , D = F D ( T ) − ∑ a F D ( T a ) . L_{F,D}=F_D(T)-\sum_a F_D(T_a). L F , D = F D ( T ) − a ∑ F D ( T a ) . All terms in the two Hamiltonians and their branch baselines commute: they are diagonal in the same physical site axes. Factor the preceding circuit as U D = u D U D U_D=u_DU_D U D = u D U D U D = ⨂ a U a , D U_D=\bigotimes_a U_{a,D} U D = ⨂ a U a , D e i t F D ( T a ) U a , D e^{itF_D(T_a)}U_{a,D} e i t F D ( T a ) U a , D e i t F D ( T ) U D e^{itF_D(T)}U_D e i t F D ( T ) U D
u F , D ( t ) = U D ∗ e i t L F , D u D U D . u_{F,D}(t)=U_D^{*}e^{itL_{F,D}}u_DU_D. u F , D ( t ) = U D ∗ e i t L F , D u D U D . Consequently the relative root maps satisfy
T P , D ( t ) : = u H , D ( t ) ∗ u H ~ D ( t ) = U D ∗ e i t ( L ~ H , D − L H , D ) U D . (67) \mathcal{T}_{P,D}(t):=u_{H,D}(t)^{*}u_{\widetilde{H}_D}(t)=U_D^{*}e^{it(\widetilde{L}_{H,D}-L_{H,D})}U_D. \tag*{(67)} T P , D ( t ) := u H , D ( t ) ∗ u H D ( t ) = U D ∗ e i t ( L H , D − L H , D ) U D . (67) This is a unitary group in t t t
At this fixed partition, Theorem 2.12 applies to both complete operations with t t t K K K p K − 1 ∑ l B k , l F k , l p_K^{-1}\sum_l B_{k,l}F_{k,l} p K − 1 ∑ l B k , l F k , l t t t (67) gives a unitary group T P ( t ) \mathcal{T}_P(t) T P ( t ) A P A_P A P
A P ζ = lim D → ∞ U D ∗ ( L H ~ , D − L H , D ) U D ζ . (68) A_P\zeta=\lim_{D\to\infty}U_D^*(L_{\widetilde{H},D}-L_{H,D})U_D\zeta. \tag*{(68)} A P ζ = D → ∞ lim U D ∗ ( L H , D − L H , D ) U D ζ . (68) We now compute this derivative by separating diagonal paths, simple paths, and lower-formula responses. This is an algebraic decomposition of the finite-degree cut difference, not an application of the gate calculus to each separated Hamiltonian. All contributions to (68) use the preceding full frame. Thus top-path expressions first computed in the child frame receive one final conjugation by u u u
The diagonal paths. The terms j = j ′ j=j' j = j ′ (63) equal
∑ i ∈ I , j ∈ J A i j P i f j # { k ∼ j : k ∈ K } D p K . (69) \sum_{i\in I,j\in J} A_{ij}P_i f_j\frac{\#\{k\sim j:k\in K\}}{D p_K}. \tag*{(69)} i ∈ I , j ∈ J ∑ A ij P i f j D p K # { k ∼ j : k ∈ K } . (69) Away from the incidence lost at a K K K O ( D − 1 / 2 ) O(D^{-1/2}) O ( D − 1/2 ) O ( D − 1 ) O(D^{-1}) O ( D − 1 )
If the removed root lies in K K K ( D p K ) − 1 ∑ j ∼ o 1 J ( j ) b j f j (D p_K)^{-1}\sum_{j\sim o}1_J(j)b_jf_j ( D p K ) − 1 ∑ j ∼ o 1 J ( j ) b j f j K K K K K K
The nonbacktracking paths at top masks. Remove the terms j = j ′ j=j' j = j ′ i , j , k , j ′ i,j,k,j' i , j , k , j ′ j ′ ∈ J l j'\in J_l j ′ ∈ J l
R j ′ , l = 1 p K ∑ k ∼ j ′ k ∈ K A j ′ k ∑ j ∼ k , j ∈ J l j ≠ j ′ A k j X j b j . (70) R_{j',l}=\frac{1}{p_K}\sum_{\substack{k\sim j'\\ k\in K}}A_{j'k}\sum_{\substack{j\sim k,\ j\in J_l\\ j\ne j'}}A_{kj}X_jb_j. \tag*{(70)} R j ′ , l = p K 1 k ∼ j ′ k ∈ K ∑ A j ′ k j ∼ k , j ∈ J l j = j ′ ∑ A k j X j b j . (70) Here all inner fields are computed in descendant trees. Let d l d_l d l 1 J l X b 1_{J_l}Xb 1 J l X b (58) gives ∥ d l ∥ a ≤ C a v p l \lVert d_l\rVert_a\le C_av\sqrt{p_l} ∥ d l ∥ a ≤ C a v p l q ( d l ) q(d_l) q ( d l ) K K K
R j ′ , l = O s m ( p l ) conditionally at j ′ ∈ J l . (71) R_{j',l}=O_{\mathrm{sm}}(\sqrt{p_l})\quad\text{conditionally at }j'\in J_l. \tag*{(71)} R j ′ , l = O sm ( p l ) conditionally at j ′ ∈ J l . (71) The constants include p K − 1 p_K^{-1} p K − 1 f f f b b b
At a root in I I I J l J_l J l C ∑ l p l 2 C\sum_l p_l^2 C ∑ l p l 2 J l J_l J l p l p_l p l (71) contributes another. The extra root Hamiltonian is its normalized lift times the root axis P P P O s m ( ρ ) O_{\mathrm{sm}}(\rho) O sm ( ρ )
At a root in J l J_l J l j j j j ′ j' j ′ f f f b b b (71) . The two factors use distinct first-step masks. Thus there is no further same-neighbor term to subtract. The root correction is O s m ( p l ) O_{\mathrm{sm}}(\sqrt{p_l}) O sm ( p l )
At a root in K K K d l , e l d_l,e_l d l , e l 1 J l X b \mathbf{1}_{J_l}X_b 1 J l X b 1 J l X f \mathbf{1}_{J_l}X_f 1 J l X f p K − 1 ∑ l q ( d l ) q ( e l ) p_K^{-1}\sum_l q(d_l)q(e_l) p K − 1 ∑ l q ( d l ) q ( e l )
1 D p K ∑ j ∼ o 1 J ( j ) b j f j ⟶ κ = 1 p K ∑ l ⟨ d l , e l ⟩ . \frac{1}{D_{p_K}}\sum_{j\sim o}\mathbf{1}_{J(j)}b_jf_j \longrightarrow\kappa=\frac{1}{p_K}\sum_l\langle d_l,e_l\rangle. D p K 1 j ∼ o ∑ 1 J ( j ) b j f j ⟶ κ = p K 1 l ∑ ⟨ d l , e l ⟩ . In particular, this is an average on one branch, not the product of two independent lifts. After subtracting it, the correction is
Q K = 1 p K ∑ l ( q ( d l ) q ( e l ) − ⟨ d l , e l ⟩ ) . (72) Q_K=\frac{1}{p_K}\sum_l\bigl(q(d_l)q(e_l)-\langle d_l,e_l\rangle\bigr). \tag*{(72)} Q K = p K 1 l ∑ ( q ( d l ) q ( e l ) − ⟨ d l , e l ⟩ ) . (72) The weighted columns belonging to different parts are orthogonal. Lemma 4.4 gives Q K = O s m ( ρ ) Q_K=O_{\mathrm{sm}}(\rho) Q K = O sm ( ρ ) κ \kappa κ
This list accounts for all top-path cuts. If the root belongs to I , J , K I,J,K I , J , K f f f J J J f f f f f f
Responses through the lower formula. We have accounted for cuts through the four-vertex helper paths. A cut can also change the value of f f f
First work in the physical frame. Write H o f f ( T ) = ∑ l , j ′ ∈ J l ω j ′ , l f j ′ T H_{\mathrm{off}}(T)=\sum_{l,j'\in J_l}\omega_{j',l}f_{j'}^T H off ( T ) = ∑ l , j ′ ∈ J l ω j ′ , l f j ′ T ω j ′ , l = X j ′ R j ′ , l \omega_{j',l}=X_{j'}R_{j',l} ω j ′ , l = X j ′ R j ′ , l v v v T a T_a T a
H o f f ( T ) − ∑ a H o f f ( T a ) = ∑ a , l , j ′ ∈ T a ∩ J l ω j ′ , l ( f j ′ T − f j ′ T a ) . (73) H_{\mathrm{off}}(T)-\sum_a H_{\mathrm{off}}(T_a)=\sum_{a,l,j'\in T_a\cap J_l}\omega_{j',l}\bigl(f_{j'}^T-f_{j'}^{T_a}\bigr). \tag*{(73)} H off ( T ) − a ∑ H off ( T a ) = a , l , j ′ ∈ T a ∩ J l ∑ ω j ′ , l ( f j ′ T − f j ′ T a ) . (73) Only terminals within the fixed influence radius contribute. At an active terminal j ′ ∈ J l j'\in J_l j ′ ∈ J l
X j ′ R j ′ , l ( f j ′ ( y ) − f j ′ ( 0 ) ) . (74) X_{j'}R_{j',l}\bigl(f_{j'}(y)-f_{j'}(0)\bigr). \tag*{(74)} X j ′ R j ′ , l ( f j ′ ( y ) − f j ′ ( 0 ) ) . (74) Whenever this difference is active, its parent lies in a lower mask, outside I , J , K I,J,K I , J , K R j ′ , l R_{j',l} R j ′ , l f f f (74) is zero. This observation ensures that using the downward expression (70) loses no term.
Here is the precise difference recursion. At a root v v v m ( y ) m(y) m ( y ) H v [ d ] ( y ) H_v^{[d]}(y) H v [ d ] ( y ) d d d y y y Δ v [ d ] ( y ) = H v [ d ] ( y ) − H v [ d ] ( 0 ) \Delta_v^{[d]}(y)=\mathcal{H}_v^{[d]}(y)-\mathcal{H}_v^{[d]}(0) Δ v [ d ] ( y ) = H v [ d ] ( y ) − H v [ d ] ( 0 ) (74) , when the root is terminal, plus
∑ α c h i l d [ Δ α [ d − 1 ] ( D − 1 / 2 m ( y ) ) − Δ α [ d − 1 ] ( D − 1 / 2 m ( 0 ) ) ] . (75) \sum_{\alpha\ \mathrm{child}}\left[\Delta_{\alpha}^{[d-1]}\left(D^{-1/2}m(y)\right)-\Delta_{\alpha}^{[d-1]}\left(D^{-1/2}m(0)\right)\right]. \tag*{(75)} α child ∑ [ Δ α [ d − 1 ] ( D − 1/2 m ( y ) ) − Δ α [ d − 1 ] ( D − 1/2 m ( 0 ) ) ] . (75) Only channels of the fixed f f f D − 1 / 2 m ( 0 ) D^{-1/2}m(0) D − 1/2 m ( 0 ) d d d H a \mathcal{H}_a H a H a ( z ) = Δ a ( z ) + H a ( 0 ) \mathcal{H}_a(z)=\Delta_a(z)+\mathcal{H}_a(0) H a ( z ) = Δ a ( z ) + H a ( 0 ) m ( 0 ) m(0) m ( 0 )
Now factor the preceding circuit as U v = ( ⨂ a U a ) u v U_v=\left(\bigotimes_a U_a\right)u_v U v = ( ⨂ a U a ) u v
Δ ^ v ( y ) = u v ∗ [ T v ( y ) + ∑ a { Δ ^ a ( D − 1 / 2 m ( y ) ) − Δ ^ a ( D − 1 / 2 m ( 0 ) ) } ] u v , (76) \widehat{\Delta}_v(y)=u_v^*\left[T_v(y)+\sum_a\left\{\widehat{\Delta}_a\left(D^{-1/2}m(y)\right)-\widehat{\Delta}_a\left(D^{-1/2}m(0)\right)\right\}\right]u_v, \tag*{(76)} Δ v ( y ) = u v ∗ [ T v ( y ) + a ∑ { Δ a ( D − 1/2 m ( y ) ) − Δ a ( D − 1/2 m ( 0 ) ) } ] u v , (76) where Δ ^ a = U a ∗ Δ a U a \widehat{\Delta}_a=U_a^*\Delta_aU_a Δ a = U a ∗ Δ a U a T v ( y ) T_v(y) T v ( y ) (74) , zero at a nonterminal root. The direct term and messages are interpreted in the same child frames. There is no derivative of U a U_a U a u v u_v u v
The required parity covariance is explicit: if Π \Pi Π Π ω Π = − ω \Pi\omega\Pi=-\omega Π ω Π = − ω Π f ( y ) Π = − f ( − y ) \Pi f(y)\Pi=-f(-y) Π f ( y ) Π = − f ( − y ) Π { ω [ f ( y ) − f ( 0 ) ] } Π = ω [ f ( − y ) − f ( 0 ) ] \Pi\{\omega[f(y)-f(0)]\}\Pi=\omega[f(-y)-f(0)] Π { ω [ f ( y ) − f ( 0 )]} Π = ω [ f ( − y ) − f ( 0 )] (75) to the limit by the centered normalized-sum calculation used in the proof of Corollary 2.13. At a fixed partition, the terminal coefficients and their jets are finite products and smooth compositions of centered odd normalized sums, together with the separately estimated bounded-mask averages normalized by D − 1 D^{-1} D − 1 D ⋅ D − 3 / 2 D\cdot D^{-3/2} D ⋅ D − 3/2 D α , D α β D_\alpha,D_{\alpha\beta} D α , D α β
r α = D α Ω , c α β = ⟨ Ω ∣ D α β ∣ Ω ⟩ . r_\alpha=D_\alpha\Omega,\qquad c_{\alpha\beta}=\langle\Omega\mid D_{\alpha\beta}\mid\Omega\rangle. r α = D α Ω , c α β = ⟨ Ω ∣ D α β ∣ Ω ⟩ . Its contribution is
∑ α ( m α ( y ) − m α ( 0 ) ) q ( r α ) + 1 2 ∑ α , β ( m α ( y ) m β ( y ) − m α ( 0 ) m β ( 0 ) ) c α β . (77) \sum_\alpha\left(m_\alpha(y)-m_\alpha(0)\right)q(r_\alpha)
+\frac{1}{2}\sum_{\alpha,\beta}\left(m_\alpha(y)m_\beta(y)-m_\alpha(0)m_\beta(0)\right)c_{\alpha\beta}. \tag*{(77)} α ∑ ( m α ( y ) − m α ( 0 ) ) q ( r α ) + 2 1 α , β ∑ ( m α ( y ) m β ( y ) − m α ( 0 ) m β ( 0 ) ) c α β . (77) followed by the preceding root conjugation. At a nonterminal lower root with no parent input, the Hamiltonian correction instead uses the unsubtracted child contribution
u v ∗ [ ∑ α m α ( 0 ) q ( r α ) + 1 2 ∑ α , β m α ( 0 ) m β ( 0 ) c α β ] u v . u_v^* \left[\sum_{\alpha} m_{\alpha}(0)q(r_{\alpha})+\frac{1}{2}\sum_{\alpha,\beta}m_{\alpha}(0)m_{\beta}(0)c_{\alpha\beta}\right]u_v. u v ∗ α ∑ m α ( 0 ) q ( r α ) + 2 1 α , β ∑ m α ( 0 ) m β ( 0 ) c α β u v . This is the root correction specified after (75) , rather than the value at y = 0 y=0 y = 0
At depth zero, (71) and the derivative bounds of f f f O s m ( ε ) O_{\mathrm{sm}}(\varepsilon) O sm ( ε ) l l l J l J_l J l C p l C\sqrt{p_l} C p l d − 1 d-1 d − 1 r α r_\alpha r α ∣ c α β ∣ \lvert c_{\alpha\beta}\rvert ∣ c α β ∣ C ε C\varepsilon C ε m m m (77) therefore give the same smallness at depth d d d J l J_l J l k k k s s s m m m s + k s+k s + k r α r_\alpha r α
The diagonal degree-factor errors have the same response recursion with terminal coefficient
b j ′ ( # { k ∼ j ′ : k ∈ K } D p K − 1 ) b_{j'}\left(\frac{\#\{k\sim j':k\in K\}}{D_{p_K}}-1\right) b j ′ ( D p K # { k ∼ j ′ : k ∈ K } − 1 ) in place of X j ′ R j ′ , l X_{j'}R_{j',l} X j ′ R j ′ , l K K K I I I
Uniform topology and iteration. After the degree limit, the diagonal degree-factor errors vanish. The preceding computations identify the total action in (68) . At a helper root its scalar part is σ P ( s ) = κ 1 K ( s ) \sigma_P(s)=\kappa1_K(s) σ P ( s ) = κ 1 K ( s )
∣ κ ∣ ≤ p K − 1 ∑ l ∥ d l ∥ ∥ e l ∥ ≤ C p J / p K . \lvert\kappa\rvert\le p_K^{-1}\sum_l\lVert d_l\rVert\lVert e_l\rVert\le Cp_J/p_K. ∣ κ ∣ ≤ p K − 1 l ∑ ∥ d l ∥ ∥ e l ∥ ≤ C p J / p K . This bounded root-seed multiplication commutes with every root map. Hence
S P ( t ) = e − i t σ P T P ( t ) S_P(t)=e^{-it\sigma_P}\mathcal{T}_P(t) S P ( t ) = e − i t σ P T P ( t ) is a strongly continuous unitary group. Its self-adjoint generator R P R_P R P
Table 1.
Since the parts partition the fixed sender mask,
ρ 2 = ∑ l p l 2 ≤ p J max l p l , ε 2 = max l p l . \rho^2 = \sum_l p_l^2 \le pJ \max_l p_l, \qquad\varepsilon^2 = \max_l p_l. ρ 2 = l ∑ p l 2 ≤ p J l max p l , ε 2 = l max p l . Both quantities tend to zero under the refinement hypothesis, with f f f p K > 0 p_K > 0 p K > 0
To obtain uniform smooth bounds for this actual group, consider the full field expression for the remaining correction at each root. At an I I I J l J_l J l K K K Q K Q_K Q K V P ( t ) V_P(t) V P ( t ) u u u K K K
We verify that V P V_P V P S P S_P S P ∂ t V P ( t ) ζ = i R P V P ( t ) ζ \partial_t V_P(t)\zeta= iR_PV_P(t)\zeta ∂ t V P ( t ) ζ = i R P V P ( t ) ζ (68) . For a smooth vector ζ \zeta ζ V P ( s ) ζ V_P(s)\zeta V P ( s ) ζ R P R_P R P
d d s [ S P ( t − s ) V P ( s ) ζ ] = 0. \frac{d}{ds}\left[S_P(t-s)V_P(s)\zeta\right] = 0. d s d [ S P ( t − s ) V P ( s ) ζ ] = 0. Evaluation at s = 0 s=0 s = 0 t t t S P ( t ) ζ = V P ( t ) ζ S_P(t)\zeta= V_P(t)\zeta S P ( t ) ζ = V P ( t ) ζ
Spectral calculus for its generator now yields
∥ [ S P ( t ) − 1 ] ζ ∥ ≤ ∣ t ∣ ∥ R P ζ ∥ ⟶ 0. \left\|[S_P(t)-1]\zeta\right\| \le|t|\left\|R_P\zeta\right\| \longrightarrow0. ∥ [ S P ( t ) − 1 ] ζ ∥ ≤ ∣ t ∣ ∥ R P ζ ∥ ⟶ 0. Uniform higher smooth bounds upgrade this to convergence in every fixed lower graph norm; the same holds for adjoints. The relative factor multiplies the desired root map on the right, so
u H ~ , P ( t ) = u H ( t ) T P ( t ) = e i t σ P u H ( t ) S P ( t ) . u_{\widetilde{H},P}(t) = u_H(t)T_P(t) = e^{it\sigma_P}u_H(t)S_P(t). u H , P ( t ) = u H ( t ) T P ( t ) = e i t σ P u H ( t ) S P ( t ) . This proves convergence of the actual root maps up to their root-seed phase, with uniform conditional smooth bounds. The same argument applies on cut subtrees, and the fixed lower response jets retain the estimates proved above.
A root-seed phase cancels in each descendant Heisenberg conjugation before its vacuum column is lifted; helper-labeled descendant spaces need not be removed. Starting from the fixed initial preceding maps, apply this single-replacement argument successively through the fixed finite sequence. The continuity part of Theorem 2.12 applies at each step with the conditional smooth bounds just established. It gives insertion convergence off K K K
Pointwise functions and the order of compilation
Proposition 4.2 implements an aggregation whenever its lower formula is still available as an ideal call. To remove all ideal calls without losing unused helper spins, we combine it with pointwise synthesis and process the formula from its outermost aggregations inward.
Propagation and elementary local controls generate the required pointwise nonlinearities. Suppose rotations by odd fields h 1 , … , h k h_1,\ldots,h_k h 1 , … , h k X X X a ( s ) a(s) a ( s ) a ( s ) X a(s)X a ( s ) X h h h a ( s ) sin ( 2 h ) Y a(s)\sin(2h)Y a ( s ) sin ( 2 h ) Y c ( s ) c(s) c ( s ) c ( s ) Y c(s)Y c ( s ) Y Z Z Z c ( s ) cos ( 2 h ) Y c(s)\cos(2h)Y c ( s ) cos ( 2 h ) Y
∑ ν a ν ( s ) sin ( ξ ν ⋅ h ) + c ν ( s ) cos ( ξ ν ⋅ h ) , a ν even , c ν odd , (78) \sum_{\nu} a_{\nu}(s)\sin(\xi_{\nu}\cdot h)+c_{\nu}(s)\cos(\xi_{\nu}\cdot h),\qquad a_{\nu}\ \text{even},\quad c_{\nu}\ \text{odd}, \tag*{(78)} ν ∑ a ν ( s ) sin ( ξ ν ⋅ h ) + c ν ( s ) cos ( ξ ν ⋅ h ) , a ν even , c ν odd , (78) can be implemented as an odd-axis Hamiltonian by pointwise matrix sums. This covers joint parity in the local seed and aggregate arguments; it does not require multiplying arbitrary even fields.
The same construction applies to bounded smooth odd functions of a finite list of odd aggregate inputs. To see the required closure precisely, cut off on larger symmetric boxes, periodically extend, and approximate the smooth extension by its Fourier polynomials. The cutoff can be chosen even, and parity projection gives exactly the coefficient parities in (78) . Uniform bounds for derivatives in the aggregate variables are preserved on each fixed compact set. Choose successive Fourier truncations accurate for increasing derivative orders. The cutoffs and periodic extensions can be chosen with polynomial derivative bounds independent of the box: inside the box these follow from the original bounds, and outside its fundamental cell the absolute value of the reduced coordinate is no larger than that of the original coordinate. Fourier errors in the finitely many required derivative norms may be made at most one. A diagonal choice gives the polynomial domination and local all-jet convergence used by Theorem 2.12. Alternatively, if only the terminal insertion is needed, first truncate the finitely many input laws and use their smooth moment bounds. Local seed coefficients remain bounded measurable functions throughout.
We finish by specifying the nesting of all limits. This is also where helper pinning is used. Define expression rank by
rank ( g ( s ) ) = 0 , rank ( F ( h 1 , … , h k ) ) = max j rank ( h j ) , rank ( A R S f ) = 1 + rank ( f ) . \operatorname{rank}(g(s))=0,\qquad\operatorname{rank}(F(h_1,\ldots,h_k))=\max_j\operatorname{rank}(h_j),\qquad\operatorname{rank}(A_{RS}f)=1+\operatorname{rank}(f). rank ( g ( s )) = 0 , rank ( F ( h 1 , … , h k )) = j max rank ( h j ) , rank ( A R S f ) = 1 + rank ( f ) . Local masking does not change rank. Reserve a distinct helper K r K_r K r r r r K q K_q K q q > r q>r q > r K r K_r K r
At the largest remaining rank, replace each pointwise nonlinearity by the pointwise synthesis just described. Take these limits successively in the fixed surrounding circuit. After fixing their finite products to any desired accuracy, the remaining calls at rank r r r A I J f A_{IJ}f A I J f rank ( f ) ≤ r − 1 \operatorname{rank}(f)\le r-1 rank ( f ) ≤ r − 1
Apply Proposition 4.2 to all those propagations simultaneously, using the reserved helper K r K_r K r
With that subdivision fixed, implement all the commutators and sums defining (63) by finite pointwise products. Their convergence holds on the full space, so these finite products may temporarily rotate the helper away from X = 1 X=1 X = 1
The resulting finite word contains only lower-rank formula calls and elementary masked gates. Repeat the same procedure at the next rank, using its new helper. The previously produced finite word may act on higher helpers, but it avoids this new one exactly. Thus the pinning hypothesis holds anew. Source formulas retain all ancestor exclusions, including those introduced by their particular calling masks. A sender subdivision only restricts the local calling mask: exclusion of the original sender mask implies exclusion of each submask. The commutators create further occurrences of the same lower formula, without adding a cross-edge dependency or introducing a control on any still-unused helper.
This induction terminates because the formula rank is finite. At each stage there are only finitely many occurrences in the already fixed surrounding word. Accuracies can therefore be chosen successively from the outside inward. No uniform estimate in the number of commutator factors, the number of formula occurrences created by an earlier stage, or inverse helper density is asserted or needed. The only uniformity in a growing sender subdivision is the one explicitly proved in Proposition 4.2.
At termination, replace the elementary masked gates by their finite seeded-circuit approximations from Section 4.2. All mask-controlled operations are compiled at this final stage. Theorem 2.12 permits the successive replacements inside the fixed finite remaining word, including the raw Gaussian seed control with its moment weight. Exact helper pinning is no longer required after the contraction limits have been fixed. This constructs a finite seeded word with arbitrarily small insertion error off the helpers.
Proof of Theorem 4.1. The finite nested construction proves (56) . To identify the target value, the ideal seed-only rotations give
v c l = m ∣ + ⟩ + cos ( 2 θ ) ∣ − ⟩ , v_{\mathrm{cl}} = m|+\rangle+ \cos(2\theta)|-\rangle, v cl = m ∣ + ⟩ + cos ( 2 θ ) ∣ − ⟩ , where both coefficients are classical functions of seeds; all descendant spins remain in their pointed spin vectors. In ⟨ v c l , L v c l ⟩ \langle v_{\mathrm{cl}, L v_{\mathrm{cl}}\rangle} ⟨ v cl , L v cl ⟩ L L L E ( v c l ) = E ( m ) \mathcal{E}(v_{\mathrm{cl}})=\mathcal{E}(m) E ( v cl ) = E ( m ) m = 0 m=0 m = 0
For any circuit preserving root seed labels, ∥ 1 H v ∥ = p H \lVert1_Hv\rVert=\sqrt{p_H} ∥ 1 H v ∥ = p H Z Z Z Z Z Z L L L
∣ E ( v ) − E ( 1 H c v ) ∣ ≤ 2 p H . |\mathcal{E}(v)-\mathcal{E}(1_{H^c}v)| \le2\sqrt{p_H}. ∣ E ( v ) − E ( 1 H c v ) ∣ ≤ 2 p H . Both projected insertion vectors have norm at most one, so (56) changes their energy by at most 2 η 2\eta 2 η (65) . All approximation limits above compare already-defined fixed-word tree values. After choosing finite approximants, the resulting seeded word and all its parameters are independent of the degree, system size, and disorder.
Removing the longitudinal seed controls
Theorem 4.1 produces a finite word that uses a Gaussian longitudinal field in addition to the cost and mixer. We now remove that extra control while preserving its limiting tree energy. This is the last compilation step: the fixed-word transfer then applies to the resulting ordinary QAOA word.
A seeded word is a finite word in the cost, the mixer, and exp ( − i ∑ i Z i s i ) \exp(-i\sum_i Z_i s_i) exp ( − i ∑ i Z i s i ) s i s_i s i σ \sigma σ E ( v Z ) \mathcal{E}(v_Z) E ( v Z ) v Z v_Z v Z E ( v Z ) \mathcal{E}(v_Z) E ( v Z )
Theorem 5.1 (Removal of seed controls). For every finite seeded word and every ϵ > 0 \epsilon> 0 ϵ > 0 ϵ \epsilon ϵ
The proof uses two parameters, chosen in order. We first choose a small fixed b > 0 b > 0 b > 0 b b b r ↓ 0 r \downarrow0 r ↓ 0 r r r r − 1 r^{-1} r − 1 r − 1 r^{-1} r − 1
We first establish estimates for displaced Fock vectors and prepare an exact two-component state. Finite selective spin rotations then let us propagate its small displaced component through the requested word. The echo calculation produces the seed field; a separate root-shift calculation identifies its energy. Finally, a finite generic perturbation supplies all the nonzero projections required to choose the selective rotations.
The large displacements will not have bounded occupation moments. We therefore keep each displacement separate from a tail vector whose moments are controlled. Write N N N
S = ⋂ k ≥ 0 Dom ( 1 + N ) k , ∥ η ∥ k = ∥ ( 1 + N ) k η ∥ . \mathcal{S}=\bigcap_{k\geq0}\operatorname{Dom}(1+N)^k,\qquad\|\eta\|_k=\|(1+N)^k\eta\|. S = k ≥ 0 ⋂ Dom ( 1 + N ) k , ∥ η ∥ k = ∥ ( 1 + N ) k η ∥. The spin factor is included in this notation. No weight is imposed inside a child-mode direction. A bounded family of inputs means a family η r ∈ S \eta_r\in\mathcal{S} η r ∈ S sup r ∥ η r ∥ k < ∞ \sup_r\|\eta_r\|_k<\infty sup r ∥ η r ∥ k < ∞ k k k b b b r r r
Let
W ( d ) = e i q ( d ) , W P ( d ) = e i P q ( d ) , P 2 = 1 , W(d)=e^{iq(d)},\qquad W_P(d)=e^{iPq(d)},\qquad P^2=1, W ( d ) = e i q ( d ) , W P ( d ) = e i P q ( d ) , P 2 = 1 , where P P P
W ( d ) W ( e ) = e − i Im ⟨ d , e ⟩ W ( d + e ) , W ( d / r ) ∗ q ( v ) W ( d / r ) = q ( v ) − 2 r Im ⟨ v , d ⟩ . (79) W(d)W(e)=e^{-i\operatorname{Im}\langle d,e\rangle}W(d+e),\qquad W(d/r)^*q(v)W(d/r)=q(v)-\frac{2}{r}\operatorname{Im}\langle v,d\rangle. \tag*{(79)} W ( d ) W ( e ) = e − i Im ⟨ d , e ⟩ W ( d + e ) , W ( d / r ) ∗ q ( v ) W ( d / r ) = q ( v ) − r 2 Im ⟨ v , d ⟩ . (79) A packet with scaled center d r d_r d r W ( d r / r ) η r W(d_r/r)\eta_r W ( d r / r ) η r η r \eta_r η r r r r
Lemma 5.2 (Packet estimates). The following estimates are uniform for bounded families of tails.
Displacements with bounded directions preserve all the seminorms ∥ ⋅ ∥ k \|\cdot\|_k ∥ ⋅ ∥ k with bounds depending only on the direction norm and k k k For ∥ d ∥ ≤ C \|d\| \le C ∥ d ∥ ≤ C
W ( r d ) η = η + i r q ( d ) η + O ( r 2 ) ∥ η ∥ 1 in norm. W(rd)\eta=\eta+ir q(d)\eta+O(r^2)\|\eta\|_1 \quad\text{in norm.} W ( r d ) η = η + i r q ( d ) η + O ( r 2 ) ∥ η ∥ 1 in norm. If d r − e r → 0 d_r-e_r\to0 d r − e r → 0 in norm and both directions are bounded, then ( W ( d r ) − W ( e r ) ) η r → 0 (W(d_r)-W(e_r))\eta_r\to0 ( W ( d r ) − W ( e r )) η r → 0 in norm. These assertions also hold for W P W_{P} W P
If ∥ d r − e r ∥ ≥ κ > 0 \|d_r-e_r\|\ge\kappa>0 ∥ d r − e r ∥ ≥ κ > 0 then for every integer M M M
∣ ⟨ W ( d r / r ) η r , W ( e r / r ) ξ r ⟩ ∣ = O ( r M ) . (80) \left|\left\langle W(d_r/r)\eta_r,W(e_r/r)\xi_r\right\rangle\right|=O(r^M). \tag*{(80)} ∣ ⟨ W ( d r / r ) η r , W ( e r / r ) ξ r ⟩ ∣ = O ( r M ) . (80) The conclusion is unchanged by inserting a bounded root spin matrix.
For bounded d r d_r d r v r v_r v r and fixed τ \tau τ conjugating the short probe e − i r τ P q ( v r ) e^{-ir\tau Pq(v_r)} e − i r τ P q ( v r ) through a packet gives
W ( d r / r ) ∗ e − i r τ P q ( v r ) W ( d r / r ) = e − i τ a r P e − i r τ P q ( v r ) , a r = − 2 Im ⟨ v r , d r ⟩ . (81) W(d_r/r)^*e^{-ir\tau Pq(v_r)}W(d_r/r)=e^{-i\tau a_rP}e^{-ir\tau Pq(v_r)},\qquad a_r=-2\operatorname{Im}\langle v_r,d_r\rangle. \tag*{(81)} W ( d r / r ) ∗ e − i r τ P q ( v r ) W ( d r / r ) = e − i τ a r P e − i r τ P q ( v r ) , a r = − 2 Im ⟨ v r , d r ⟩ . (81) Consequently a fixed finite list of short probes preserves its scaled centers and bounded centered-tail moments, without any smallness assumption on a r a_r a r
Proof. The first assertion follows from the creation–annihilation estimates in Lemma 2.5, or by differentiating a Weyl displacement on S S S ∥ q ( d ) 2 η ∥ ≤ C ∥ d ∥ 2 ∥ η ∥ 1 \|q(d)^2\eta\|\le C\|d\|^2\|\eta\|_1 ∥ q ( d ) 2 η ∥ ≤ C ∥ d ∥ 2 ∥ η ∥ 1 P P P
For the second assertion, the Weyl product reduces the question to a matrix coefficient of W ( f r / r ) W(f_r/r) W ( f r / r ) ∥ f r ∥ ≥ κ \|f_r\|\ge\kappa ∥ f r ∥ ≥ κ Q r = q ( i f r / ∥ f r ∥ ) Q_r=q(if_r/\|f_r\|) Q r = q ( i f r /∥ f r ∥ ) (79) translates Q r Q_r Q r 2 ∥ f r ∥ / r 2\|f_r\|/r 2∥ f r ∥/ r Q r Q_r Q r [ − ∥ f r ∥ / ( 2 r ) , ∥ f r ∥ / ( 2 r ) ] [-\|f_r\|/(2r),\|f_r\|/(2r)] [ − ∥ f r ∥/ ( 2 r ) , ∥ f r ∥/ ( 2 r )] C M r M ∥ η ∥ k M C_Mr^M\|\eta\|_{k_M} C M r M ∥ η ∥ k M ∥ Q r M η ∥ ≤ C M ∥ η ∥ k M \|Q_r^M\eta\|\le C_M\|\eta\|_{k_M} ∥ Q r M η ∥ ≤ C M ∥ η ∥ k M (80) . Root spin matrices commute with these projections. Finally, (81) is exactly the second Weyl identity; its right-hand side is a spin rotation followed by a displacement of size O ( r ) O(r) O ( r )
We use exact local unitaries and exact insertion directions throughout. At a given stage, if the local unitary is u r u_r u r
v P , r = u r ∗ P u r Ω . v_{P,r}=u_r^*Pu_r\Omega. v P , r = u r ∗ P u r Ω. All the v P , r v_{P,r} v P , r P P P Z Y ZY Z Y e − i r τ P q ( v P , r ) e^{-ir\tau Pq(v_{P,r})} e − i r τ P q ( v P , r )
Initialization and the exact two-component decomposition
We next prepare the two components that the selective probes will distinguish. Their decomposition is exact, even though they need not be orthogonal. This exactness will keep the special component’s amplitude equal to r r r
Fix once and for all β ⋆ = π / 4 \beta_\star= \pi/4 β ⋆ = π /4 R ⋆ = e − i β ⋆ X R_\star= e^{-i\beta_\star X} R ⋆ = e − i β ⋆ X 0 < r < 1 0<r<1 0 < r < 1
δ r = arcsin r , γ r = b / r , e = Z Ω , ℓ r = Y e − 2 i γ r Z q ( e ) Ω . \delta_r = \arcsin r,\qquad\gamma_r = b/r,\qquad e = Z\Omega,\qquad\ell_r = Y e^{-2i\gamma_r Zq(e)}\Omega. δ r = arcsin r , γ r = b / r , e = Z Ω , ℓ r = Y e − 2 i γ r Z q ( e ) Ω. Apply, in order, cost γ r \gamma_r γ r δ r \delta_r δ r − γ r -\gamma_r − γ r β ⋆ \beta_\star β ⋆
v 2 , r = cos ( 2 δ r ) e + sin ( 2 δ r ) ℓ r . v_{2,r} = \cos(2\delta_r)e + \sin(2\delta_r)\ell_r. v 2 , r = cos ( 2 δ r ) e + sin ( 2 δ r ) ℓ r . Define d 0 , r = γ r ( v 2 , r − e ) d_{0,r} = \gamma_r(v_{2,r}-e) d 0 , r = γ r ( v 2 , r − e ) exact decomposition
u r = a 0 , r U O , r + a 1 , r U S , r , U O , r = R ⋆ W Z ( d 0 , r ) , U S , r = R ⋆ X W Z ( − 2 γ r e − d 0 , r ) , u_r = a_{0,r}U_{O,r} + a_{1,r}U_{S,r},\qquad U_{O,r} = R_\star W_Z(d_{0,r}),\qquad U_{S,r} = R_\star XW_Z(-2\gamma_r e-d_{0,r}), u r = a 0 , r U O , r + a 1 , r U S , r , U O , r = R ⋆ W Z ( d 0 , r ) , U S , r = R ⋆ X W Z ( − 2 γ r e − d 0 , r ) , where
a 0 , r = cos δ r e i ϑ r , a 1 , r = − i r e − i ϑ r , ϑ r = γ r 2 Im ⟨ v 2 , r , e ⟩ . a_{0,r} = \cos\delta_r e^{i\vartheta_r},\qquad a_{1,r} = -ir e^{-i\vartheta_r},\qquad\vartheta_r = \gamma_r^2\operatorname{Im}\langle v_{2,r},e\rangle. a 0 , r = cos δ r e i ϑ r , a 1 , r = − i r e − i ϑ r , ϑ r = γ r 2 Im ⟨ v 2 , r , e ⟩ . Indeed, this is obtained by expanding e − i δ r X e^{-i\delta_r X} e − i δ r X
⟨ e , ℓ r ⟩ = − i e − 2 γ r 2 , d 0 , r = − 2 b r e + 2 b 1 − r 2 ℓ r , \langle e,\ell_r\rangle= -ie^{-2\gamma_r^2},\qquad d_{0,r} = -2br e + 2b\sqrt{1-r^2}\,\ell_r, ⟨ e , ℓ r ⟩ = − i e − 2 γ r 2 , d 0 , r = − 2 b r e + 2 b 1 − r 2 ℓ r , we have ϑ r → 0 \vartheta_r\to0 ϑ r → 0 r r r ∥ d 0 , r − 2 b ℓ r ∥ → 0 \lVert d_{0,r}-2b\ell_r\rVert\to0 ∥ d 0 , r − 2 b ℓ r ∥ → 0
At every later stage retain the exact decomposition obtained by applying all subsequent local updates to the two operators in (5.4) . Denote its actions on an input η \eta η Ψ O , r ( η ) \Psi_{O,r}(\eta) Ψ O , r ( η ) Ψ S , r ( η ) \Psi_{S,r}(\eta) Ψ S , r ( η )
u r η = Ψ O , r ( η ) + Ψ S , r ( η ) , ∥ Ψ S , r ( η ) ∥ = r ∥ η ∥ . (82) u_r\eta= \Psi_{O,r}(\eta) + \Psi_{S,r}(\eta),\qquad\lVert\Psi_{S,r}(\eta)\rVert= r\lVert\eta\rVert. \tag*{(82)} u r η = Ψ O , r ( η ) + Ψ S , r ( η ) , ∥ Ψ S , r ( η )∥ = r ∥ η ∥ . (82) No orthogonality is asserted or needed in this exact split. The echo will depend on the change in the root Z Z Z r r r ψ S , r = Ψ S , r ( Ω ) \psi_{S,r}=\Psi_{S,r}(\Omega) ψ S , r = Ψ S , r ( Ω )
w r = u r ∗ Z ψ S , r / r . w_r = u_r^*Z\psi_{S,r}/r. w r = u r ∗ Z ψ S , r / r . Thus ∥ w r ∥ = 1 \lVert w_r\rVert=1 ∥ w r ∥ = 1 ℓ r \ell_r ℓ r
Lemma 5.3 (The initial high directions). The vectors ℓ r , w r \ell_r,w_r ℓ r , w r
⟨ ℓ r , w r ⟩ ⟶ 0 , ∥ 1 N ≤ M ℓ r ∥ + ∥ 1 N ≤ M w r ∥ ⟶ 0 for every fixed M . \langle\ell_r,w_r\rangle\longrightarrow0,\qquad\lVert\mathbf{1}_{N\le M}\ell_r\rVert+\lVert\mathbf{1}_{N\le M}w_r\rVert\longrightarrow0\quad\text{for every fixed }M. ⟨ ℓ r , w r ⟩ ⟶ 0 , ∥ 1 N ≤ M ℓ r ∥ + ∥ 1 N ≤ M w r ∥ ⟶ 0 for every fixed M . They consequently have vanishing inner products with every bounded family of top-number-smooth vectors.
Proof. Centering follows from the global spin-flip symmetry; every initialization gate is even and the inserted Y Y Y Z Z Z R ⋆ ∗ Z R ⋆ = Y R_\star^*ZR_\star=Y R ⋆ ∗ Z R ⋆ = Y Y X = − i Z YX=-iZ Y X = − i Z (5.6) gives
w r = − Z W Z ( − 2 γ r e − 2 d 0 , r ) Ω + o ( 1 ) , w_r=-ZW_Z(-2\gamma_r e-2d_{0,r})\Omega+o(1), w r = − Z W Z ( − 2 γ r e − 2 d 0 , r ) Ω + o ( 1 ) , where the omitted special–special term has norm r r r d 0 , r d_{0,r} d 0 , r Z Z Z ℓ r \ell_r ℓ r w r w_r w r 4 γ r e + O ( 1 ) 4\gamma_r e+O(1) 4 γ r e + O ( 1 ) N ≤ M N\le M N ≤ M O ( M − k ) O(M^{-k}) O ( M − k ) r ↓ 0 r\downarrow0 r ↓ 0 M → ∞ . □ M\to\infty. \square M → ∞. □
The special part of (5.4) , on any bounded input family, has four packets. Index them by the spin z = ± 1 z=\pm1 z = ± 1 X X X y = ± 1 y=\pm1 y = ± 1 R ⋆ R_{\star} R ⋆ − 2 b z e -2bz e − 2 b z e y y y − z d 0 , r -z d_{0,r} − z d 0 , r
A finite ensemble of spin rotations
On the vacuum, the exact split has an ordinary component with bounded top-number moments and a special component of norm r r r
The selective-rotation argument uses the Lie-bracket and polynomial-approximation approach to ensemble control of Li and Khaneja [15 ] , Section II, Examples 1–2. The finite-family and exact zero-clock requirements needed here are proved below.
Lemma 5.4 (Selective rotations with zero clocks). Let h 1 , … , h m h_1,\ldots,h_m h 1 , … , h m R 2 \mathbb{R}^2 R 2 P 1 , P 2 P_1,P_2 P 1 , P 2 Z Y ZY Z Y c j , a = h j ⋅ P a c_{j,a}=h_j\cdot P_a c j , a = h j ⋅ P a ( Z , Y ) (Z,Y) ( Z , Y ) θ ∈ R \theta\in\mathbb{R} θ ∈ R ( a k , τ k ) (a_k,\tau_k) ( a k , τ k ) a k ∈ { 1 , 2 } a_k\in\{1,2\} a k ∈ { 1 , 2 }
∑ k : a k = a τ k = 0 ( a = 1 , 2 ) , ∏ k e − i τ k c j , a k P a k = e − i θ X ( 1 ≤ j ≤ m ) . (83) \sum_{k:a_k=a}\tau_k=0\quad(a=1,2),\qquad\prod_k e^{-i\tau_k c_{j,a_k}P_{a_k}}=e^{-i\theta X}\quad(1\leq j\leq m). \tag*{(83)} k : a k = a ∑ τ k = 0 ( a = 1 , 2 ) , k ∏ e − i τ k c j , a k P a k = e − i θ X ( 1 ≤ j ≤ m ) . (83) The product is in chronological order, with later factors on the left. The assertion is exact, not merely an assertion about closure.
Proof. Choose the axes so that every c j , a ≠ 0 c_{j,a}\ne0 c j , a = 0 ( c j , 1 2 , c j , 2 2 ) (c_{j,1}^2,c_{j,2}^2) ( c j , 1 2 , c j , 2 2 ) c j , 1 c j , 2 c_{j,1}c_{j,2} c j , 1 c j , 2
In G = S U ( 2 ) m × R 2 G=SU(2)^m\times\mathbb{R}^2 G = S U ( 2 ) m × R 2 − i c j , a P a -ic_{j,a}P_a − i c j , a P a a a a i X iX i X c j , 1 c j , 2 c_{j,1}c_{j,2} c j , 1 c j , 2 c j , 1 2 c_{j,1}^2 c j , 1 2 c j , 2 2 c_{j,2}^2 c j , 2 2
c j , 1 c j , 2 F ( c j , 1 2 , c j , 2 2 ) i X c_{j,1}c_{j,2}F(c_{j,1}^2,c_{j,2}^2)iX c j , 1 c j , 2 F ( c j , 1 2 , c j , 2 2 ) i X for every real polynomial F F F
For completeness, exponentials of this Lie-algebra vector are reachable by finite products of the two original one-parameter groups. Let R \mathcal{R} R g 0 \mathfrak{g}_0 g 0 R \mathcal{R} R g 0 \mathfrak{g}_0 g 0 g 0 \mathfrak{g}_0 g 0 R \mathcal{R} R X X X (83) . □ \square □
The preceding lemma treats fixed real coefficients. In the circuit, the coefficients come from actual insertion vectors and change during the probes. The next estimate controls that change without replacing the insertion vectors by approximants before a singular rescaling.
Lemma 5.5 (Accuracy of selective probes). Consider a stage whose vacuum state u r Ω u_r\Omega u r Ω o ( r ) o(r) o ( r ) O ( r ) O(r) O ( r )
At a standard stage all ordinary centers are zero.
At a huge-cost stage the ordinary part is ψ + + ψ − \psi_+ + \psi_- ψ + + ψ − Z ψ y = y ψ y Z\psi_y = y\psi_y Z ψ y = y ψ y − t y v r -tyv_r − t y v r t ≠ 0 t \ne0 t = 0 v r = u r ∗ Z u r Ω v_r = u_r^* Z u_r\Omega v r = u r ∗ Z u r Ω d j , r d_{j,r} d j , r lim Im ⟨ v r , d j , r ⟩ ≠ 0 \lim\operatorname{Im}\langle v_r,d_{j,r}\rangle\ne0 lim Im ⟨ v r , d j , r ⟩ = 0
Assume the limiting special projection pairs
h j = lim r ↓ 0 − 2 ( Im ⟨ v Z , r , d j , r ⟩ , Im ⟨ v Y , r , d j , r ⟩ ) h_j = \lim_{r\downarrow0} -2\bigl(\operatorname{Im}\langle v_{Z,r},d_{j,r}\rangle,\operatorname{Im}\langle v_{Y,r},d_{j,r}\rangle\bigr) h j = r ↓ 0 lim − 2 ( Im ⟨ v Z , r , d j , r ⟩ , Im ⟨ v Y , r , d j , r ⟩ ) exist and are nonzero. Use the axes and finite pulse list of Lemma 5.4 for these pairs, replacing its spin generators by actual toggled probes of angles r τ k r\tau_k r τ k o ( r ) o(r) o ( r ) X X X o ( 1 ) o(1) o ( 1 )
Proof. Let V j V_j V j v P , r ( j ) = ( V j u r ) ∗ P ( V j u r ) Ω v_{P,r}^{(j)}=(V_j u_r)^*P(V_j u_r)\Omega v P , r ( j ) = ( V j u r ) ∗ P ( V j u r ) Ω
At a huge-cost stage there is the exact identity
⟨ v P , r ( j ) , v r ⟩ = ⟨ P V j ψ , V j Z ψ ⟩ , ψ = u r Ω . (84) \langle v_{P,r}^{(j)},v_r\rangle=\langle P V_j\psi,V_j Z\psi\rangle,\qquad\psi=u_r\Omega. \tag*{(84)} ⟨ v P , r ( j ) , v r ⟩ = ⟨ P V j ψ , V j Z ψ ⟩ , ψ = u r Ω. (84) Group the ordinary packets by their original sign y y y y y y y ⟨ P V j ψ y , V j ψ y ⟩ y\langle P V_j\psi_y,V_j\psi_y\rangle y ⟨ P V j ψ y , V j ψ y ⟩ 2 ∣ t ∣ 2|t| 2∣ t ∣ ∥ v r ∥ = 1 \lVert v_r\rVert=1 ∥ v r ∥ = 1 v r v_r v r r r r O ( r 2 ) O(r^2) O ( r 2 ) o ( r ) o(r) o ( r )
Im ⟨ v P , r ( j ) , v r ⟩ = o ( r ) . (85) \operatorname{Im}\langle v_{P,r}^{(j)},v_r\rangle=o(r). \tag*{(85)} Im ⟨ v P , r ( j ) , v r ⟩ = o ( r ) . (85) More quantitatively, let ε r = o ( r ) \varepsilon_r=o(r) ε r = o ( r ) M M M
max j ∣ Im ⟨ v P , r ( j ) , v r ⟩ ∣ ≤ C M ( r 2 + ε r + r M ) . (86) \max_j\left|\operatorname{Im}\langle v_{P,r}^{(j)},v_r\rangle\right|\le C_M\left(r^2+\varepsilon_r+r^M\right). \tag*{(86)} j max Im ⟨ v P , r ( j ) , v r ⟩ ≤ C M ( r 2 + ε r + r M ) . (86) The constant depends on the finite pulse list, a positive lower bound on center separations, and finitely many designated vacuum-tail seminorms. The exact probe prefixes preserve the norm of the discarded remainder; no number-operator estimate is imposed on it. This derivation used only center preservation, so there is no circular small-phase assumption.
By (81) , an ordinary center − t y v r -t y v_r − t y v r o ( r ) o(r) o ( r ) O ( r ) O(r) O ( r ) O ( r ) O(r) O ( r )
v P , r ( j + 1 ) − v P , r ( j ) = ( V j u r ) ∗ ( V ∗ P V − P ) ( V j u r ) Ω v_{P,r}^{(j+1)}-v_{P,r}^{(j)}=(V_j u_r)^*(V^*PV-P)(V_j u_r)\Omega v P , r ( j + 1 ) − v P , r ( j ) = ( V j u r ) ∗ ( V ∗ P V − P ) ( V j u r ) Ω for the next probe V V V ( P V − V P ) ( V j u r ) Ω (PV-VP)(V_j u_r)\Omega ( P V − V P ) ( V j u r ) Ω
∥ v P , r ( j ) − v P , r ( 0 ) ∥ = O ( r ) (87) \lVert v_{P,r}^{(j)}-v_{P,r}^{(0)}\rVert=O(r) \tag*{(87)} ∥ v P , r ( j ) − v P , r ( 0 ) ∥ = O ( r ) (87) for the fixed finite list.
After removing an ordinary center, expand the displacement in each probe to first order. Equation (85) supplies only an o ( r ) o(r) o ( r ) v P , r ( j ) v_{P,r}^{(j)} v P , r ( j ) O ( r 2 ) O(r^2) O ( r 2 ) (87) . The first-order sum is therefore
− i r ∑ a = 1 2 ( ∑ k : a k = a τ k ) P a q ( v P a , r ( 0 ) ) = 0. -ir\sum_{a=1}^{2}\left(\sum_{k:a_k=a}\tau_k\right)P_aq\bigl(v_{P_a,r}^{(0)}\bigr)=0. − i r a = 1 ∑ 2 ( k : a k = a ∑ τ k ) P a q ( v P a , r ( 0 ) ) = 0. Here is a uniform remainder bound for this cancellation. For a fixed ordinary center d r d_r d r B r ( d r ) \mathcal{B}_r(d_r) B r ( d r ) W ( d r / r ) W(d_r/r) W ( d r / r )
Δ r = max j ∥ v P j , r ( j ) − v P j , r ( 0 ) ∥ , α r ( d r ) = max j 2 ∣ τ j Im ⟨ v P j , r ( j ) , d r ⟩ ∣ . \Delta_r=\max_j\lVert v_{P_j,r}^{(j)}-v_{P_j,r}^{(0)}\rVert,\qquad
\alpha_r(d_r)=\max_j 2\lvert\tau_j\operatorname{Im}\langle v_{P_j,r}^{(j)},d_r\rangle\rvert. Δ r = j max ∥ v P j , r ( j ) − v P j , r ( 0 ) ∥ , α r ( d r ) = j max 2 ∣ τ j Im ⟨ v P j , r ( j ) , d r ⟩∣ . Then for every designated centered tail η \eta η
∥ ( B r ( d r ) − 1 ) η ∥ ≤ C K [ ( r 2 + r Δ r ) ∥ η ∥ 1 + α r ( d r ) ∥ η ∥ ] . (88) \lVert(\mathcal{B}_r(d_r)-1)\eta\rVert\le C_K\left[(r^2+r\Delta_r)\lVert\eta\rVert_1+\alpha_r(d_r)\lVert\eta\rVert\right]. \tag*{(88)} ∥( B r ( d r ) − 1 ) η ∥ ≤ C K [ ( r 2 + r Δ r ) ∥ η ∥ 1 + α r ( d r ) ∥ η ∥ ] . (88) Indeed, remove each spin phase by its operator-norm bound and replace each displacement direction by its starting value. The latter replacement costs at most C r Δ r ∥ η ∥ 1 Cr\Delta_r\lVert\eta\rVert_1 C r Δ r ∥ η ∥ 1 r r r λ \lambda λ λ = 0 \lambda=0 λ = 0 ∥ q ( v ) q ( w ) η ∥ ≤ C ∥ v ∥ ∥ w ∥ ∥ η ∥ 1 \lVert q(v)q(w)\eta\rVert\le C\lVert v\rVert\lVert w\rVert\lVert\eta\rVert_1 ∥ q ( v ) q ( w ) η ∥ ≤ C ∥ v ∥ ∥ w ∥ ∥ η ∥ 1 λ = r \lambda=r λ = r C K r 2 ∥ η ∥ 1 C_Kr^2\lVert\eta\rVert_1 C K r 2 ∥ η ∥ 1 r r r C K C_K C K α r = 0 \alpha_r=0 α r = 0 (86) gives α r ≤ C ( r 2 + ε r + r M ) = o ( r ) \alpha_r\le C(r^2+\varepsilon_r+r^M)=o(r) α r ≤ C ( r 2 + ε r + r M ) = o ( r ) Δ r = O ( r ) \Delta_r=O(r) Δ r = O ( r ) o ( r ) o(r) o ( r ) ∥ η ∥ 1 \lVert\eta\rVert_1 ∥ η ∥ 1 ρ r \rho_r ρ r 2 ∥ ρ r ∥ 2\lVert\rho_r\rVert 2 ∥ ρ r ∥
On a special packet the change of its phase coefficient during the sequence is O ( r ) O(r) O ( r ) (87) and the boundedness of its scaled center. The small displacements contribute O ( r ) O(r) O ( r ) c j , a c_{j,a} c j , a (83) . Multiplication by the packet’s amplitude O ( r ) O(r) O ( r ) o ( r ) o(r) o ( r )
The packet invariant and the singular echo
We now apply the selective estimates to the exact initialized decomposition and propagate it through costs, mixers, and simulated seed gates. The required statements include moving input vectors: later insertion estimates must be allowed to use the approximating circuit itself as an input.
We call the times between simulated gates standard stages . A second kind of stage occurs immediately after the forward cost in a seed echo. The induction will maintain the following assertions, with the same center list for every bounded input family.
At a standard stage there is a unitary A r A_r A r ζ r \zeta_r ζ r
Ψ O , r ( η r ) = ζ r A r η r + o ( r ) . \Psi_{O,r}(\eta_r)=\zeta_r A_r\eta_r+o(r). Ψ O , r ( η r ) = ζ r A r η r + o ( r ) . The operators A r , A r ∗ A_r,A_r^* A r , A r ∗
The special part has a finite packet expansion up to o ( r ) o(r) o ( r ) r r r
d z y , r = − 2 b z e + y D r , z , y ∈ { − 1 , 1 } , d_{zy,r}=-2bze+yD_r,\qquad z,y\in\{-1,1\}, d z y , r = − 2 b z e + y D r , z , y ∈ { − 1 , 1 } , and output spin y y y D r D_r D r O ( r ) O(r) O ( r ) S \mathcal{S} S
At a standard stage the special insertion is transported:
u r ∗ Z ψ S , r / r = w r + o ( 1 ) , u_r^*Z\psi_{S,r}/r=w_r+o(1), u r ∗ Z ψ S , r / r = w r + o ( 1 ) , where w r w_r w r (5.6) . Between a special flip and its undoing the right-hand side instead has a minus sign.
Here and below the norm remainders need not have bounded number moments. They are kept as separate remainder vectors and carried through later unitaries by their norm bounds. Only the designated tails require moment estimates.
The initialization satisfies this invariant with A r = U O , r A_r=U_{O,r} A r = U O , r D r = 0 D_r=0 D r = 0 ζ r = e i ϑ r \zeta_r=e^{i\vartheta_r} ζ r = e i ϑ r O ( r 2 ) O(r^2) O ( r 2 ) (5.15) is initially exact. The following observation will be used repeatedly.
Lemma 5.6 (Insertion approximation from forward action). If (82) and (5.13) *hold uniformly on bounded input families, then
∥ u r ∗ P u r Ω − A r ∗ P A r Ω ∥ = O ( r ) ( P = Y , Z ) . (89) \left\|u_r^*P u_r\Omega-A_r^*P A_r\Omega\right\|=O(r)\qquad(P=Y,Z). \tag*{(89)} ∥ u r ∗ P u r Ω − A r ∗ P A r Ω ∥ = O ( r ) ( P = Y , Z ) . (89) In particular the actual insertion directions have bounded smooth approximants. No approximation to u r ∗ u_r^* u r ∗
Proof. Put v ^ P , r = A r ∗ P A r Ω \widehat v_{P,r}=A_r^*P A_r\Omega v P , r = A r ∗ P A r Ω A r v ^ P , r = P A r Ω A_r\widehat v_{P,r}=P A_r\Omega A r v P , r = P A r Ω
∥ u r ∗ P u r Ω − v ^ P , r ∥ = ∥ P u r Ω − u r v ^ P , r ∥ ≤ ∥ P ( u r − ζ r A r ) Ω ∥ + ∥ ( u r − ζ r A r ) v ^ P , r ∥ . \left\|u_r^*P u_r\Omega-\widehat v_{P,r}\right\|=\left\|P u_r\Omega-u_r\widehat v_{P,r}\right\|\leq\left\|P(u_r-\zeta_r A_r)\Omega\right\|+\left\|(u_r-\zeta_r A_r)\widehat v_{P,r}\right\|. ∥ u r ∗ P u r Ω − v P , r ∥ = ∥ P u r Ω − u r v P , r ∥ ≤ ∥ P ( u r − ζ r A r ) Ω ∥ + ∥ ( u r − ζ r A r ) v P , r ∥ . Each term is O ( r ) O(r) O ( r )
For this induction, assume that the nonzero projection hypotheses of Lemma 5.5 hold at every required probe start. We call these the visibility conditions . The final subsection proves that one perturbation of the finite target word ensures all of them and then constructs the probes in order.
An unscaled cost uses the actual insertion v Z , r v_{Z,r} v Z , r A r A_r A r e − i c Z q ( v Z , r ) e^{-ic_Zq(v_{Z,r})} e − i c Z q ( v Z , r ) y y y Z Z Z
To implement a bulk mixer R θ = e − i θ X R_\theta=e^{-i\theta X} R θ = e − i θ X R θ − 1 R_\theta^{-1} R θ − 1 R θ R_\theta R θ o ( r ) o(r) o ( r ) R θ R_\theta R θ o ( 1 ) o(1) o ( 1 ) Z Z Z o ( 1 ) o(1) o ( 1 ) r r r
For a longitudinal seed gate use a fixed t ≠ 0 t\ne0 t = 0
cost t / r , selective ( − i X ) , cost − t / r , selective ( i X ) . (90) \text{cost }t/r,\qquad\text{selective }(-iX),\qquad\text{cost }-t/r,\qquad\text{selective }(iX). \tag*{(90)} cost t / r , selective ( − i X ) , cost − t / r , selective ( i X ) . (90) The next lemma verifies both the singular cancellation and its error scale.
Lemma 5.7 (One echo). Under the invariant and the visibility conditions, the echo (90) preserves the invariant. Up to a scalar of modulus one its low update is
A r ⟼ W Z ( g r ) A r , g r = t r ( v r ′ − v r ) = − 2 t w r + o ( 1 ) , A_r\longmapsto W_Z(g_r)A_r,\qquad g_r=\frac{t}{r}(v'_r-v_r)=-2tw_r+o(1), A r ⟼ W Z ( g r ) A r , g r = r t ( v r ′ − v r ) = − 2 t w r + o ( 1 ) , where v r v_r v r Z Z Z v r ′ v'_r v r ′
D r ⟼ D r − t ( v r + v r ′ ) . D_r\longmapsto D_r-t(v_r+v'_r). D r ⟼ D r − t ( v r + v r ′ ) . All ordinary errors are o ( r ) o(r) o ( r )
Proof. Write u i n , r u_{\mathrm{in},r} u in , r V r V_r V r
u c o s t , r = e − i t Z q ( v r ) / r u i n , r , u f l i p , r = V r u c o s t , r . u_{\mathrm{cost},r}=e^{-itZq(v_r)/r}u_{\mathrm{in},r},\qquad u_{\mathrm{flip},r}=V_ru_{\mathrm{cost},r}. u cost , r = e − i tZ q ( v r ) / r u in , r , u flip , r = V r u cost , r . Their actual Z Z Z
u i n , r ∗ Z u i n , r Ω = u c o s t , r ∗ Z u c o s t , r Ω = v r , u_{\mathrm{in},r}^{*}Zu_{\mathrm{in},r}\Omega=u_{\mathrm{cost},r}^{*}Zu_{\mathrm{cost},r}\Omega=v_r, u in , r ∗ Z u in , r Ω = u cost , r ∗ Z u cost , r Ω = v r , u f l i p , r ∗ Z u f l i p , r Ω = v r ′ . u_{\mathrm{flip},r}^{*}Zu_{\mathrm{flip},r}\Omega=v'_r. u flip , r ∗ Z u flip , r Ω = v r ′ . The first equality holds because the forward cost commutes with the root Z Z Z y y y − t y v r -tyv_r − t y v r d Z y , r − t y v r d_{Zy,r}-tyv_r d Z y , r − t y v r v r v_r v r − t y v r -tyv_r − t y v r
Denote the exact ordinary and special components after that forward cost by Ψ O , r c o s t \Psi^{\mathrm{cost}}_{O,r} Ψ O , r cost ψ S , r c o s t \psi^{\mathrm{cost}}_{S,r} ψ S , r cost o ( r ) o(r) o ( r ) Z Z Z − i X -iX − i X o ( r ) o(r) o ( r ) Z Z Z X Z X = − Z XZX=-Z X Z X = − Z
v r ′ − v r = u c o s t , r ∗ ( V r ∗ Z V r − Z ) ( Ψ O , r c o s t Ω + ψ S , r c o s t ) = − 2 u c o s t , r ∗ Z ψ S , r c o s t + o ( r ) = − 2 r w r + o ( r ) . (91) \begin{aligned}
v'_r-v_r
&=u_{\mathrm{cost},r}^{*}(V_r^{*}ZV_r-Z)(\Psi^{\mathrm{cost}}_{O,r}\Omega+\psi^{\mathrm{cost}}_{S,r})\\
&=-2u_{\mathrm{cost},r}^{*}Z\psi^{\mathrm{cost}}_{S,r}+o(r)
=-2rw_r+o(r). \tag*{(91)}
\end{aligned} v r ′ − v r = u cost , r ∗ ( V r ∗ Z V r − Z ) ( Ψ O , r cost Ω + ψ S , r cost ) = − 2 u cost , r ∗ Z ψ S , r cost + o ( r ) = − 2 r w r + o ( r ) . (91) The last equality uses transport, which the forward cost preserves. The same calculation shows that transport changes sign at this flip.
On an ordinary input the reverse cost combines with the forward cost as
e i t Z q ( v r ′ ) / r e − i t Z q ( v r ) / r = ξ r W Z ( t r ( v r ′ − v r ) ) , ∣ ξ r ∣ = 1. (92) e^{itZq(v'_r)/r}e^{-itZq(v_r)/r}=\xi_r W_Z\left(\frac{t}{r}(v'_r-v_r)\right),\qquad|\xi_r|=1. \tag*{(92)} e i tZ q ( v r ′ ) / r e − i tZ q ( v r ) / r = ξ r W Z ( r t ( v r ′ − v r ) ) , ∣ ξ r ∣ = 1. (92) The Weyl phase ξ r \xi_r ξ r Z 2 = 1 Z^2=1 Z 2 = 1 (91) makes the residual direction bounded and proves (5.18) . The ordinary o ( r ) o(r) o ( r ) r r r (92) used the two actual directions, and only then was (91) invoked.
A special packet has spin − y -y − y − t y v r − t y v r ′ -tyv_r-tyv'_r − t y v r − t y v r ′ (5.19) . Here y y y (92) has already removed the huge ordinary centers. It acts trivially to o ( r ) o(r) o ( r ) y y y r r r W Z ( g r ) A r W_Z(g_r)A_r W Z ( g r ) A r O ( r ) O(r) O ( r )
The special insertion remains transported because each cost commutes with Z Z Z Z Z Z o ( r ) o(r) o ( r ) A r , A r ∗ A_r,A_r^* A r , A r ∗ v r ′ − v r = O ( r ) v'_r-v_r=O(r) v r ′ − v r = O ( r ) D r D_r D r
This induction also proves its claimed uniformity on input families. The exact decomposition (82) is linear in the input. At initialization every input has the same four special centers. Unscaled costs and compensated mixers preserve that center list, and (5.19) updates it independently of the tail. The probe parameters are determined on the vacuum, but Lemma 5.5 acts uniformly on the whole family of tails having these centers. Every discarded vector has norm o ( r ) o(r) o ( r ) r ↓ 0 r\downarrow0 r ↓ 0 o ( r ) o(r) o ( r )
The external-mode limit and the energy pairing
The echo has now produced the bounded direction − 2 t w r + o ( 1 ) -2t w_r+o(1) − 2 t w r + o ( 1 )
We now identify the limiting low circuit and at the same time establish the limits of the projections used to select the probes. Introduce at each vertex a two-dimensional real external mode space with orthonormal vectors ℓ , w \ell,w ℓ , w E e x t = span C { ℓ , w } E_{\mathrm{ext}}=\operatorname{span}_{\mathbb{C}}\{\ell,w\} E ext = span C { ℓ , w }
H s e e d , 0 = C 2 ⊗ Γ s ( E e x t ) , H s e e d , h + 1 = C 2 ⊗ Γ s ( H s e e d , h ∘ ⊕ E e x t ) . \mathcal{H}_{\mathrm{seed},0}=\mathbb{C}^{2}\otimes\Gamma_s(E_{\mathrm{ext}}),\qquad
\mathcal{H}_{\mathrm{seed},h+1}=\mathbb{C}^{2}\otimes\Gamma_s\left(\mathcal{H}_{\mathrm{seed},h}^{\circ}\oplus E_{\mathrm{ext}}\right). H seed , 0 = C 2 ⊗ Γ s ( E ext ) , H seed , h + 1 = C 2 ⊗ Γ s ( H seed , h ∘ ⊕ E ext ) . The vacuum is ∣ + ⟩ ⊗ Ω |+\rangle\otimes\Omega ∣ + ⟩ ⊗ Ω h + 1 h+1 h + 1
The formal low circuit starts with
A f o r m = R ⋆ e 2 i b Z q ( ℓ ) . (93) A^{\mathrm{form}} = R_{\star}e^{2ibZq(\ell)}. \tag*{(93)} A form = R ⋆ e 2 ib Z q ( ℓ ) . (93) A bulk mixer and an unscaled cost have their usual seeded rules. An echo with parameter t t t e − 2 i t Z q ( w ) e^{-2itZq(w)} e − 2 i tZ q ( w ) q ( w ) q(w) q ( w ) q ( ℓ ) q(\ell) q ( ℓ ) b b b t = σ / 2 t=\sigma/2 t = σ /2 σ \sigma σ
At a standard stage write v f o r m = ( A f o r m ) ∗ P A f o r m Ω v^{\mathrm{form}}=(A^{\mathrm{form}})^*P A^{\mathrm{form}}\Omega v form = ( A form ) ∗ P A form Ω A f o r m A^{\mathrm{form}} A form e = Z Ω e=Z\Omega e = Z Ω ℓ , w \ell,w ℓ , w
Lemma 5.8 (Finite Gram induction). Suppose the required selective probe lists have been constructed through a given standard stage. The joint Gram matrix of e e e ℓ r , w r \ell_r,w_r ℓ r , w r
Proof. Every A r A_r A r d 0 , r d_{0,r} d 0 , r g r g_r g r
For such sums all vacuum inner products reduce to
⟨ W ( f ) Ω , W ( g ) Ω ⟩ = exp ( − 1 2 ∥ g − f ∥ 2 + i Im ⟨ f , g ⟩ ) . (94) \langle W(f)\Omega,W(g)\Omega\rangle
= \exp\left(-\frac{1}{2}\lVert g-f\rVert^2+i\operatorname{Im}\langle f,g\rangle\right). \tag*{(94)} ⟨ W ( f ) Ω , W ( g ) Ω ⟩ = exp ( − 2 1 ∥ g − f ∥ 2 + i Im ⟨ f , g ⟩ ) . (94) Thus only finitely many direction inner products are needed at a step. The initial directions satisfy d 0 , r = 2 b ℓ r + o ( 1 ) d_{0,r}=2b\ell_r+o(1) d 0 , r = 2 b ℓ r + o ( 1 ) g r = − 2 t w r + o ( 1 ) g_r=-2tw_r+o(1) g r = − 2 t w r + o ( 1 ) O ( r ) O(r) O ( r ) O ( r ) O(r) O ( r ) ℓ r , w r \ell_r,w_r ℓ r , w r ⟨ ℓ r , w r ⟩ → 0 \langle\ell_r,w_r\rangle\to0 ⟨ ℓ r , w r ⟩ → 0 (94) now proves the Gram assertion inductively.
Here is the additional calculation for the root-shift energy, so that ordinary Gram convergence is not silently substituted for convergence of a different operator. A low Y Y Y Z Z Z
h = ∑ j = 1 J s j ⊗ W ( f j ) Ω , s j ∈ C 2 . h=\sum_{j=1}^{J}s_j\otimes W(f_j)\Omega,\qquad s_j\in\mathbb{C}^2. h = j = 1 ∑ J s j ⊗ W ( f j ) Ω , s j ∈ C 2 . In the unseeded representation, L h = ∣ + ⟩ ⊗ a ∗ ( h ) Ω Lh=|+\rangle\otimes a^{*}(h)\Omega L h = ∣ + ⟩ ⊗ a ∗ ( h ) Ω W ( f ) Ω W(f)\Omega W ( f ) Ω i e − ∥ f ∥ 2 / 2 f ie^{-\lVert f\rVert^{2}/2}f i e − ∥ f ∥ 2 /2 f
⟨ h , L h ⟩ = − i ∑ j = 1 J ⟨ s j , + ⟩ e − ∥ f j ∥ 2 / 2 ⟨ f j , h ⟩ . (95) \langle h,Lh\rangle=-i\sum_{j=1}^{J}\langle s_j,+\rangle e^{-\lVert f_j\rVert^{2}/2}\langle f_j,h\rangle. \tag*{(95)} ⟨ h , L h ⟩ = − i j = 1 ∑ J ⟨ s j , + ⟩ e − ∥ f j ∥ 2 /2 ⟨ f j , h ⟩ . (95) Each pairing ⟨ f j , h ⟩ \langle f_j,h\rangle ⟨ f j , h ⟩ h h h f j f_j f j h h h h h h O ( r ) O(r) O ( r ) L L L
∣ E ( v ) − E ( h ) ∣ ≤ ( ∥ v ∥ + ∥ h ∥ ) ∥ v − h ∥ , |\mathcal{E}(v)-\mathcal{E}(h)|\leq(\lVert v\rVert+\lVert h\rVert)\lVert v-h\rVert, ∣ E ( v ) − E ( h ) ∣ ≤ (∥ v ∥ + ∥ h ∥) ∥ v − h ∥ , which proves energy convergence. A finite scalar bulk phase cancels from every insertion. The same argument applies just before an echo’s second flip, because its ordinary centers have already canceled and its low update has already been made. □
The standard-stage Gram limits determine the projections needed for ordinary probe starts. For a probe immediately after a cost of size t / r t/r t / r
Lemma 5.9 (The huge-stage transverse projection). Immediately after a forward cost with fixed t ≠ 0 t\ne0 t = 0 Y Y Y S S S
Proof. For a bounded input η r \eta_r η r Ω \Omega Ω η r \eta_r η r − t y v r -tyv_r − t y v r y y y ⟨ u r ∗ Y u r Ω , η r ⟩ \langle u_r^{*}Y u_r\Omega,\eta_r\rangle ⟨ u r ∗ Y u r Ω , η r ⟩ Y Y Y 2 ∣ t ∣ 2|t| 2∣ t ∣ (80) . Every contribution with a special component is O ( r ) O(r) O ( r ) o ( r ) o(r) o ( r ) O ( r ) O(r) O ( r ) v r ′ v'_r v r ′ (5.19) differ by O ( r ) O(r) O ( r ) (91) . Adding the forward displacement − t y v r -tyv_r − t y v r
Visibility and the order of construction
The preceding arguments are conditional only on the visibility conditions. We now choose the target angles and all finite probe lists so that those conditions hold. The order is important: first use the formal low circuit to perturb the target angles, then construct the probe lists successively from limits established at earlier stages, and only afterwards choose r r r λ \lambda λ σ \sigma σ t = σ / 2 t=\sigma/2 t = σ /2 y y y − y -y − y
D f o r m = − 2 ∑ j : reverse cost already applied t j ϑ Z , j f o r m , D^{\mathrm{form}}=-2\sum_{j:\ \text{reverse cost already applied}}t_j\vartheta^{\mathrm{form}}_{Z,j}, D form = − 2 j : reverse cost already applied ∑ t j ϑ Z , j form , where the insertion is taken immediately before that echo. At each listed stage and for each z , y z,y z , y
a z y f o r m ( λ ) = − 2 Im ⟨ v Z f o r m , − 2 b z e + y D f o r m ⟩ ≠ 0. (96) a^{\mathrm{form}}_{zy}(\lambda)=-2\operatorname{Im}\langle v^{\mathrm{form}}_Z,-2bze+yD^{\mathrm{form}}\rangle\ne0. \tag*{(96)} a z y form ( λ ) = − 2 Im ⟨ v Z form , − 2 b z e + y D form ⟩ = 0. (96) Each function in (96) is real analytic in the finitely many angles. One may verify this without an unbounded-operator analyticity theorem: expand the finite word into Weyl sums and use (94) recursively. These expressions use finitely many sums, products, and exponentials of previous finite Gram expressions.
None of these analytic functions is identically zero. To see this, set all angles of the target word to zero, while keeping the initialization fixed. Every listed formal standard stage is then (93) , and D f o r m = 0 D^{\mathrm{form}}=0 D form = 0
⟨ v Z f o r m , e ⟩ = i e − 8 b 2 , a z y f o r m = 4 b z e − 8 b 2 ≠ 0. \langle v^{\mathrm{form}}_Z,e\rangle=ie^{-8b^2},\qquad a^{\mathrm{form}}_{zy}=4bze^{-8b^2}\ne0. ⟨ v Z form , e ⟩ = i e − 8 b 2 , a z y form = 4 b z e − 8 b 2 = 0. The temporary minus sign in special transport before a second flip does not change this formal ordinary insertion or its center formula. Thus the same test point works for every listed stage.
The complement of the finite union of these analytic zero sets, and of the hyperplanes t j = 0 t_j=0 t j = 0
We can now construct the actual pulses successively. At a standard probe start, Lemma 5.8 supplies the limits of both special projection coordinates, and (96) makes their first coordinate nonzero. Choose the finite ensemble-control list from these limiting pairs. Lemma 5.5 implements it with the required errors. At a huge-stage first flip, the first coordinate is unchanged by the forward displacement, while the second coordinate tends to zero by Lemma 5.9. Its limiting pairs are therefore ( a z y f o r m , 0 ) (a^{\mathrm{form}}_{zy},0) ( a z y form , 0 )
Proof of Theorem 5.1. Start with the requested finite seeded word and a tolerance. For a small fixed b > 0 b>0 b > 0 R ⋆ e 2 i b Z q ( ℓ ) R_\star e^{2ibZ_q(\ell)} R ⋆ e 2 ib Z q ( ℓ ) ℓ \ell ℓ b ↓ 0 b\downarrow0 b ↓ 0 b = 0 b=0 b = 0 ∣ + ⟩ \lvert+\rangle ∣ + ⟩
Choose b b b r > 0 r>0 r > 0
Every gate used is a cost or mixer pulse, possibly with negative angle. Toggled probes and all compensations are finite such words; consecutive gates of the same kind can be combined, or zero-angle gates inserted to obtain the stipulated alternating QAOA format. After b b b r r r
The ordinary word and the SK limit
The preceding constructions are now fixed in their order of use. The following error allocation leaves every circuit finite before applying the tree-to-SK identity.
Proof of Theorem 1.1. Fix ε > 0 \varepsilon> 0 ε > 0 ε / 4 \varepsilon/4 ε /4 η H = ( ε / 8 ) 2 \eta_H = (\varepsilon/8)^2 η H = ( ε /8 ) 2 m m m p H p_H p H
E ( m ) ≥ P ∗ − ε / 4 , 2 p H < ε / 4. \mathcal{E}(m) \ge P_* - \varepsilon/4,\qquad2\sqrt{p_H} < \varepsilon/4. E ( m ) ≥ P ∗ − ε /4 , 2 p H < ε /4. Apply Theorem 4.1 with η = ε / 8 \eta= \varepsilon/8 η = ε /8 v v v
E ( v ) ≥ E ( m ) − 2 η − 2 p H > P ∗ − 3 ε / 4. \mathcal{E}(v) \ge\mathcal{E}(m) - 2\eta- 2\sqrt{p_H} > P_* - 3\varepsilon/4. E ( v ) ≥ E ( m ) − 2 η − 2 p H > P ∗ − 3 ε /4. By Theorem 5.1, a finite ordinary word approximates this seeded value to error less than ε / 4 \varepsilon/4 ε /4 P ∗ − ε P_* - \varepsilon P ∗ − ε
All choices just made concern limiting tree calculations and deterministic approximation parameters. After these choices there is a single finite word with real angles independent of D D D n n n J J J v p ( γ , β ) ≥ P ∗ − ε v_p(\gamma,\beta) \ge P_* - \varepsilon v p ( γ , β ) ≥ P ∗ − ε p p p
Remark 6.1 (Order of choices). For the primary route the logical order is: accuracy; a fixed zero-temperature minimizer and stopping time T < 1 T < 1 T < 1 T T T b b b r r r β 0 \beta_0 β 0 T < q β 0 T < q\beta_0 T < q β 0
Proof of the regular MaxCut consequence
We now prove Corollary 1.2. Theorem 1.1 provides one fixed word with SK value close to P ∗ P_* P ∗
Proof of Corollary 1.2. The normalized negative cost used by Basso et al. [4 ] is
C D = − 1 D ∑ { u , v } ∈ E z u z v = 2 D H M C ( G ) − ∣ E ∣ D I . C_D = -\frac{1}{\sqrt{D}}\sum_{\{u,v\}\in E} z_u z_v = \frac{2}{\sqrt{D}}H_{\mathrm{MC}}(G)-\frac{|E|}{\sqrt{D}}I. C D = − D 1 { u , v } ∈ E ∑ z u z v = D 2 H MC ( G ) − D ∣ E ∣ I . Thus its cost gate with angle γ j \gamma_j γ j 2 γ j / D 2\gamma_j/\sqrt{D} 2 γ j / D r p + ( D , γ , β ) r_p^+(D,\gamma,\beta) r p + ( D , γ , β )
ν p ( D , γ , β ) = D 2 r p + ( D , γ , β ) . \nu_p(D,\gamma,\beta)=\frac{\sqrt{D}}{2}r_p^+(D,\gamma,\beta). ν p ( D , γ , β ) = 2 D r p + ( D , γ , β ) . That proposition gives ν p ( D , γ , β ) → v p ( γ , β ) \nu_p(D,\gamma,\beta)\to v_p(\gamma,\beta) ν p ( D , γ , β ) → v p ( γ , β )
For the observable at an edge, backward cancellation of commuting cost gates leaves only cost edges having an endpoint at distance at most p − 1 p-1 p − 1 2 p + 1 2p+1 2 p + 1 only this cone by X X X Z u Z v Z_uZ_v Z u Z v G G G − r p + ( D , γ , β ) -r_p^+(D,\gamma,\beta) − r p + ( D , γ , β )
q p , G ( 2 γ / D , β ) = 1 2 + v p ( D , γ , β ) D . (97) q_{p,G}(2\gamma/\sqrt{D},\beta)=\frac{1}{2}+\frac{v_p(D,\gamma,\beta)}{\sqrt{D}}. \tag*{(97)} q p , G ( 2 γ / D , β ) = 2 1 + D v p ( D , γ , β ) . (97) This is the finite-degree edge formula in the full version of [4 ] , Equations (2.3)–(2.6) and (3.1), in the convention of Proposition 2.1. Choose the word by Theorem 1.1 with v p ( γ , β ) ≥ P ∗ − η / 4 v_p(\gamma,\beta)\ge P_*-\eta/4 v p ( γ , β ) ≥ P ∗ − η /4 D D D v p ( D , γ , β ) ≥ P ∗ − η / 2 v_p(D,\gamma,\beta)\ge P_*-\eta/2 v p ( D , γ , β ) ≥ P ∗ − η /2
For fixed D , p D,p D , p b p , D ( G ) b_{p,D}(G) b p , D ( G ) ( p + 1 ) (p+1) ( p + 1 ) D , p D,p D , p 2 p + 3 2p+3 2 p + 3 [17 ] . Consequently b p , D ( G n , D + 1 ) → 0 b_{p,D}(G_{n,D+1})\to0 b p , D ( G n , D + 1 ) → 0 n → ∞ n\to\infty n → ∞ [ 0 , 1 ] [0,1] [ 0 , 1 ]
∣ q p , G ( 2 γ / D , β ) − 1 2 − v p ( D , γ , β ) D ∣ ≤ b p , D ( G ) . \left|q_{p,G}(2\gamma/\sqrt{D},\beta)-\frac{1}{2}-\frac{v_p(D,\gamma,\beta)}{\sqrt{D}}\right|\le b_{p,D}(G). q p , G ( 2 γ / D , β ) − 2 1 − D v p ( D , γ , β ) ≤ b p , D ( G ) . This proves convergence of the conditional quantum expectation to the tree value at fixed D , p D,p D , p
Dembo, Montanari, and Sen [7 ] , Theorem 1.6(b), with the convention after Equation (1.7), identify the optimal leading correction for the same simple regular model. After dividing their cut size by n d / 2 nd/2 n d /2 d = D + 1 d=D+1 d = D + 1 D ( OPT ( G n , D + 1 ) − 1 / 2 ) \sqrt{D}(\operatorname{OPT}(G_{n,D+1})-1/2) D ( OPT ( G n , D + 1 ) − 1/2 ) P ∗ P_* P ∗ D / ( D + 1 ) → 1 \sqrt{D/(D+1)}\to1 D / ( D + 1 ) → 1 P ∗ P_* P ∗ D D D P ∗ − v p ( D , γ , β ) P_*-v_p(D,\gamma,\beta) P ∗ − v p ( D , γ , β ) η / 2 \eta/2 η /2 D Δ D > η \sqrt{D}\Delta_D>\eta D Δ D > η D \sqrt{D} D η / 4 \eta/4 η /4 n n n
The independent positive-temperature construction
This appendix constructs near-optimal bounded classical formulas without using the zero-temperature scalar companion. Its input is the positive-temperature Parisi measure at a fixed finite inverse temperature. The Parisi diffusion first gives an almost optimal martingale output. A finite-step tree calculation and a strong symmetric-power limit then place that output in the rooted Hilbert space. Finally, the bounded approximation and color-exclusion lemmas of Section 3 put it in the form required by Theorem 4.1.
Besides providing an independent route to the same synthesis input, the appendix retains the finite-degree message calculation and the quantitative fixed-temperature mesh bounds. Temperature is chosen first and held fixed in those estimates. No passage from the log-cosh terminal datum here to the terminal datum ∣ x ∣ |x| ∣ x ∣
The use of a Parisi diffusion and its gradient martingale to choose incremental message-passing coefficients follows the classical construction of Montanari [18 ] , Section 3.1, Lemmas 3.2–3.3, with the more general framework of El Alaoui, Montanari, and Sellke [8 ] , Assumption 1 and Theorem 2. We use the finite-tree calculation of El Alaoui, Montanari, and Sellke [9 ] and convert its output to the synthesis formulas below.
We record explicitly the spin-glass results that enter the argument. For an inverse temperature β 0 > 0 \beta_0 > 0 β 0 > 0 μ β 0 \mu_{\beta_0} μ β 0
γ ( t ) = β 0 μ β 0 ( [ 0 , t ] ) . \gamma(t) = \beta_0\mu_{\beta_0}([0,t]). γ ( t ) = β 0 μ β 0 ([ 0 , t ]) . In field coordinates appropriate to the normalization of this paper, the Parisi PDE and free energy are
ϕ t ( t , x ) = − 1 2 ( ϕ x x ( t , x ) + γ ( t ) ϕ x ( t , x ) 2 ) , ϕ ( 1 , x ) = 1 β 0 log ( 2 cosh ( β 0 x ) ) . (98) \begin{aligned}
\phi_t(t,x) &= -\frac{1}{2}\left(\phi_{xx}(t,x)+\gamma(t)\phi_x(t,x)^2\right),\\
\phi(1,x) &= \frac{1}{\beta_0}\log\left(2\cosh(\beta_0 x)\right). \tag*{(98)}
\end{aligned} ϕ t ( t , x ) ϕ ( 1 , x ) = − 2 1 ( ϕ xx ( t , x ) + γ ( t ) ϕ x ( t , x ) 2 ) , = β 0 1 log ( 2 cosh ( β 0 x ) ) . (98) F β 0 = ϕ ( 0 , 0 ) − 1 2 ∫ 0 1 t γ ( t ) d t = lim n → ∞ 1 n β 0 E log ∑ σ exp ( β 0 h n , J ( σ ) ) . (99) F_{\beta_0} = \phi(0,0)-\frac{1}{2}\int_0^1 t\gamma(t)\,dt
= \lim_{n\to\infty}\frac{1}{n\beta_0}\mathbb{E}\log\sum_{\sigma}\exp\left(\beta_0 h_{n,J}(\sigma)\right). \tag*{(99)} F β 0 = ϕ ( 0 , 0 ) − 2 1 ∫ 0 1 t γ ( t ) d t = n → ∞ lim n β 0 1 E log σ ∑ exp ( β 0 h n , J ( σ ) ) . (99) Equation (99) is the positive-temperature Parisi formula [23 ] , Theorem 1.1, written in the field coordinates of (98) . The covariance of h n , J h_{n,J} h n , J n n n ( R 12 2 − 1 / n ) / 2 (R_{12}^2-1/n)/2 ( R 12 2 − 1/ n ) /2 R 12 = n − 1 ∑ i σ i 1 σ i 2 R_{12}=n^{-1}\sum_i\sigma_i^1\sigma_i^2 R 12 = n − 1 ∑ i σ i 1 σ i 2 O ( 1 / n ) O(1/n) O ( 1/ n ) t 2 / 2 t^2/2 t 2 /2
This route does not need the zero-temperature variational formula, even for existence of the limit defining P ∗ P_* P ∗ A n = n − 1 E max σ h n , J ( σ ) A_n=n^{-1}\mathbb{E}\max_\sigma h_{n,J}(\sigma) A n = n − 1 E max σ h n , J ( σ ) F n , β 0 F_{n,\beta_0} F n , β 0 n n n (99) . The elementary log-sum bounds give
F n , β 0 − log 2 β 0 ≤ A n ≤ F n , β 0 . F_{n,\beta_0}-\frac{\log2}{\beta_0}\le A_n\le F_{n,\beta_0}. F n , β 0 − β 0 log 2 ≤ A n ≤ F n , β 0 . For every fixed finite β 0 \beta_0 β 0 F n , β 0 F_{n,\beta_0} F n , β 0 lim sup n A n − lim inf n A n ≤ ( log 2 ) / β 0 \limsup_n A_n-\liminf_n A_n\le(\log2)/\beta_0 lim sup n A n − lim inf n A n ≤ ( log 2 ) / β 0 β 0 \beta_0 β 0 A n A_n A n
P ∗ ≤ F β 0 ≤ P ∗ + log 2 β 0 . P_*\le F_{\beta_0}\le P_*+\frac{\log2}{\beta_0}. P ∗ ≤ F β 0 ≤ P ∗ + β 0 log 2 . A direct zero-temperature variational formula is also known [2 ] , Theorem 1; the argument here needs only the positive-temperature formula and the preceding squeeze. We use the following precise finite-temperature facts. In the conventions of [16 ] , they are Theorem 1.1, Lemma 2.2, Proposition 2.4, and Lemma 2.5, respectively:
If β 0 > 1 \beta_0 > 1 β 0 > 1 q = q β 0 ∈ ( 0 , 1 ) q = q_{\beta_0} \in(0,1) q = q β 0 ∈ ( 0 , 1 ) supp μ β 0 = [ 0 , q ] \operatorname{supp}\mu_{\beta_0} = [0,q] supp μ β 0 = [ 0 , q ] q q q C ∞ ( [ 0 , q ) ) C^\infty([0,q)) C ∞ ([ 0 , q )) q q q
If X X X W W W
d X t = γ ( t ) ϕ x ( t , X t ) d t + d W t , X 0 = 0 , (100) \mathrm{d}X_t = \gamma(t)\phi_x(t,X_t)\,\mathrm{d}t + \mathrm{d}W_t,\qquad X_0 = 0, \tag*{(100)} d X t = γ ( t ) ϕ x ( t , X t ) d t + d W t , X 0 = 0 , (100) then E ϕ x ( t , X t ) 2 = t \mathbb{E}\phi_x(t,X_t)^2 = t E ϕ x ( t , X t ) 2 = t t ∈ supp μ β 0 t \in\operatorname{supp}\mu_{\beta_0} t ∈ supp μ β 0
All positive-order spatial derivatives of ϕ \phi ϕ [ 0 , 1 ] × R [0,1]\times\mathbb{R} [ 0 , 1 ] × R ϕ \phi ϕ x x x ∣ ϕ x ∣ ≤ 1 |\phi_x| \le1 ∣ ϕ x ∣ ≤ 1 ϕ x x > 0 \phi_{xx} > 0 ϕ xx > 0 [1 ] after the same change of variables.
Differentiating the PDE along (A.4) gives d ϕ x ( t , X t ) = ϕ x x ( t , X t ) d W t \mathrm{d}\phi_x(t,X_t) = \phi_{xx}(t,X_t)\,\mathrm{d}W_t d ϕ x ( t , X t ) = ϕ xx ( t , X t ) d W t
The self-consistency identity E ϕ x ( t , X t ) 2 = t \mathbb{E}\phi_x(t,X_t)^2 = t E ϕ x ( t , X t ) 2 = t t ∈ supp μ β 0 t \in\operatorname{supp}\mu_{\beta_0} t ∈ supp μ β 0 t = 0 t = 0 t = 0 [13 ] . To check the conventions, if u u u log cosh x \log\cosh x log cosh x β 0 2 \beta_0^2 β 0 2 [16 ] , then ϕ ( t , x ) = β 0 − 1 u ( t , β 0 x ) + log 2 \phi(t,x) = \beta_0^{-1}u(t,\beta_0x) + \log2 ϕ ( t , x ) = β 0 − 1 u ( t , β 0 x ) + log 2 [16 ] used here is version 3: its Hamiltonian is written with independent Gaussian terms over all ordered pairs. Combining the two terms for each unordered pair gives the normalization (1.1); the diagonal terms form a centered, configuration-independent Gaussian. They have zero contribution to expected pressure and do not change the Parisi minimizer.
Write
m ( t , x ) = ϕ x ( t , x ) , a ( t , x ) = ϕ x x ( t , x ) , b ( t , x ) = γ ( t ) m ( t , x ) . m(t,x) = \phi_x(t,x),\qquad a(t,x) = \phi_{xx}(t,x),\qquad b(t,x) = \gamma(t)m(t,x). m ( t , x ) = ϕ x ( t , x ) , a ( t , x ) = ϕ xx ( t , x ) , b ( t , x ) = γ ( t ) m ( t , x ) . On every [ 0 , T ] [0,T] [ 0 , T ] T < q T < q T < q γ \gamma γ [ 0 , T ] [0,T] [ 0 , T ] m m m a a a a a a b b b x x x [ 0 , T ] [0,T] [ 0 , T ] a a a x x x b b b
Lemma A.1 (A positive-temperature value bound). For every β 0 > 1 \beta_0 > 1 β 0 > 1 and 0 < T < q 0 < T < q 0 < T < q
∫ 0 T E a ( t , X t ) d t ≥ P ∗ − 1 / 4 + log 2 β 0 − ( q − T ) . (101) \int_0^T \mathbb{E}a(t,X_t)\,\mathrm{d}t \ge P_* - \frac{1/4+\log2}{\beta_0} - (q-T). \tag*{(101)} ∫ 0 T E a ( t , X t ) d t ≥ P ∗ − β 0 1/4 + log 2 − ( q − T ) . (101) Moreover,
E a ( t , X t ) 2 = 1 ( 0 ≤ t ≤ q ) , 1 − q ≤ β 0 − 1 . (102) \mathbb{E}a(t,X_t)^2 = 1\qquad(0 \le t \le q),\qquad1-q \le\beta_0^{-1}. \tag*{(102)} E a ( t , X t ) 2 = 1 ( 0 ≤ t ≤ q ) , 1 − q ≤ β 0 − 1 . (102) Proof. Symmetry gives m ( 0 , 0 ) = 0 m(0,0) = 0 m ( 0 , 0 ) = 0
E m ( t , X t ) 2 = ∫ 0 t E a ( s , X s ) 2 d s . \mathbb{E}m(t,X_t)^2 = \int_0^t \mathbb{E}a(s,X_s)^2\,\mathrm{d}s. E m ( t , X t ) 2 = ∫ 0 t E a ( s , X s ) 2 d s . The left-hand side equals t t t [ 0 , q ] [0,q] [ 0 , q ] a a a X X X γ = β 0 \gamma= \beta_0 γ = β 0 [ q , 1 ] [q,1] [ q , 1 ]
ϕ ( q , x ) = 1 β 0 log ( 2 cosh ( β 0 x ) ) + β 0 ( 1 − q ) 2 , m ( q , x ) = tanh ( β 0 x ) , a ( q , x ) = β 0 sech 2 ( β 0 x ) . (103) \phi(q,x) = \frac{1}{\beta_0}\log\bigl(2\cosh(\beta_0x)\bigr) + \frac{\beta_0(1-q)}{2},\qquad m(q,x) = \tanh(\beta_0x),\qquad a(q,x) = \beta_0\operatorname{sech}^2(\beta_0x). \tag*{(103)} ϕ ( q , x ) = β 0 1 log ( 2 cosh ( β 0 x ) ) + 2 β 0 ( 1 − q ) , m ( q , x ) = tanh ( β 0 x ) , a ( q , x ) = β 0 sech 2 ( β 0 x ) . (103) Thus, by Cauchy–Schwarz and the identity just proved,
1 − q = E sech 2 ( β 0 X q ) ≤ ( E sech 4 ( β 0 X q ) ) 1 / 2 = β 0 − 1 . 1-q=\mathbb{E}\operatorname{sech}^{2}(\beta_0X_q)\leq\left(\mathbb{E}\operatorname{sech}^{4}(\beta_0X_q)\right)^{1/2}=\beta_0^{-1}. 1 − q = E sech 2 ( β 0 X q ) ≤ ( E sech 4 ( β 0 X q ) ) 1/2 = β 0 − 1 . Itô’s formula applied to X t m ( t , X t ) X_t m(t,X_t) X t m ( t , X t ) ϕ ( t , X t ) \phi(t,X_t) ϕ ( t , X t )
∫ 0 q E a ( t , X t ) d t = E X q m ( q , X q ) − ∫ 0 q t γ ( t ) d t , \int_0^q \mathbb{E}a(t,X_t)\,\mathrm{d}t=\mathbb{E}X_qm(q,X_q)-\int_0^q t\gamma(t)\,\mathrm{d}t, ∫ 0 q E a ( t , X t ) d t = E X q m ( q , X q ) − ∫ 0 q t γ ( t ) d t , E ϕ ( q , X q ) = ϕ ( 0 , 0 ) + 1 2 ∫ 0 q t γ ( t ) d t . \mathbb{E}\phi(q,X_q)=\phi(0,0)+\frac{1}{2}\int_0^q t\gamma(t)\,\mathrm{d}t. E ϕ ( q , X q ) = ϕ ( 0 , 0 ) + 2 1 ∫ 0 q t γ ( t ) d t . The stochastic integrals have mean zero because a a a m m m b b b X X X
x tanh ( β 0 x ) ≥ 1 β 0 log ( 2 cosh ( β 0 x ) ) − log 2 β 0 x\tanh(\beta_0x)\geq\frac{1}{\beta_0}\log\left(2\cosh(\beta_0x)\right)-\frac{\log2}{\beta_0} x tanh ( β 0 x ) ≥ β 0 1 log ( 2 cosh ( β 0 x ) ) − β 0 log 2 follows, for example, by differentiating the difference for x ≥ 0 x\geq0 x ≥ 0
∫ 0 q E a ( t , X t ) d t ≥ ϕ ( 0 , 0 ) − 1 2 ∫ 0 q t γ ( t ) d t − β 0 ( 1 − q ) 2 − log 2 β 0 = F β 0 − β 0 ( 1 − q ) 2 4 − log 2 β 0 ≥ P ∗ − 1 / 4 + log 2 β 0 . \begin{aligned}
\int_0^q \mathbb{E}a(t,X_t)\,\mathrm{d}t\geq\phi(0,0)-\frac{1}{2}\int_0^q t\gamma(t)\,\mathrm{d}t-\frac{\beta_0(1-q)}{2}-\frac{\log2}{\beta_0} \\
&=F_{\beta_0}-\frac{\beta_0(1-q)^2}{4}-\frac{\log2}{\beta_0} \\
&\geq P_*-\frac{1/4+\log2}{\beta_0}.
\end{aligned} ∫ 0 q E a ( t , X t ) d t ≥ ϕ ( 0 , 0 ) − 2 1 ∫ 0 q t γ ( t ) d t − 2 β 0 ( 1 − q ) − β 0 log 2 = F β 0 − 4 β 0 ( 1 − q ) 2 − β 0 log 2 ≥ P ∗ − β 0 1/4 + log 2 . Finally 0 ≤ E a ( t , X t ) ≤ ( E a ( t , X t ) 2 ) 1 / 2 = 1 0\leq\mathbb{E}a(t,X_t)\leq(\mathbb{E}a(t,X_t)^2)^{1/2}=1 0 ≤ E a ( t , X t ) ≤ ( E a ( t , X t ) 2 ) 1/2 = 1 [ 0 , q ] [0,q] [ 0 , q ] [ T , q ] [T,q] [ T , q ] q − T q-T q − T
The finite-iteration tree calculation
Here we state the finite-step input from El Alaoui, Montanari, and Sellke with the indices and normalizations needed below. It is the general calculation in [9 ] , Sections 2–4.1, not that paper’s zero-temperature near-optimality theorem.
Fix β 0 > 1 \beta_0>1 β 0 > 1 0 < T < q 0<T<q 0 < T < q δ > 0 \delta>0 δ > 0 L ≥ 1 L\geq1 L ≥ 1 L δ ≤ T L\delta\leq T L δ ≤ T t r = r δ t_r=r\delta t r = r δ ( D + 1 ) (D+1) ( D + 1 ) g i g_i g i u i → j 0 = u i 0 = g i u_{i\to j}^0=u_i^0=g_i u i → j 0 = u i 0 = g i A i → j − 1 = 1 A_{i\to j}^{-1}=1 A i → j − 1 = 1 0 ≤ r < L 0\leq r<L 0 ≤ r < L
u i → j r + 1 = 1 D ∑ v ∈ ∂ i ∖ { j } A v → i r − 1 u v → i r , u i r + 1 = 1 D + 1 ∑ v ∈ ∂ i A v → i r − 1 u v → i r . (104) u_{i\to j}^{r+1}=\frac{1}{\sqrt{D}}\sum_{v\in\partial i\setminus\{j\}}A_{v\to i}^{r-1}u_{v\to i}^{r},\qquad
u_i^{r+1}=\frac{1}{\sqrt{D+1}}\sum_{v\in\partial i}A_{v\to i}^{r-1}u_{v\to i}^{r}. \tag*{(104)} u i → j r + 1 = D 1 v ∈ ∂ i ∖ { j } ∑ A v → i r − 1 u v → i r , u i r + 1 = D + 1 1 v ∈ ∂ i ∑ A v → i r − 1 u v → i r . (104) x i → j r + 1 = x i → j r + δ b ( t r , x i → j r ) + δ u i → j r + 1 , x i → j 0 = δ g i ; (105) x_{i\to j}^{r+1}=x_{i\to j}^{r}+\delta b(t_r,x_{i\to j}^{r})+\sqrt{\delta}\,u_{i\to j}^{r+1},\qquad
x_{i\to j}^{0}=\sqrt{\delta}\,g_i; \tag*{(105)} x i → j r + 1 = x i → j r + δ b ( t r , x i → j r ) + δ u i → j r + 1 , x i → j 0 = δ g i ; (105) x i r + 1 = x i r + δ b ( t r , x i r ) + δ u i r + 1 , x i 0 = δ g i . (106) x_i^{r+1}=x_i^{r}+\delta b(t_r,x_i^{r})+\sqrt{\delta}\,u_i^{r+1},\qquad
x_i^{0}=\sqrt{\delta}\,g_i. \tag*{(106)} x i r + 1 = x i r + δ b ( t r , x i r ) + δ u i r + 1 , x i 0 = δ g i . (106) The coefficients, for 0 ≤ r ≤ L 0\leq r\leq L 0 ≤ r ≤ L
c D , r = ( E a ( t r , x i → j r ) 2 ) − 1 / 2 , A i → j r = c D , r a ( t r , x i → j r ) , B i r = c D , r a ( t r , x i r ) , (107) c_{D,r}=\left(\mathbb{E}a(t_r,x_{i\to j}^{r})^2\right)^{-1/2},\qquad
A_{i\to j}^{r}=c_{D,r}a(t_r,x_{i\to j}^{r}),\qquad
B_i^{r}=c_{D,r}a(t_r,x_i^{r}), \tag*{(107)} c D , r = ( E a ( t r , x i → j r ) 2 ) − 1/2 , A i → j r = c D , r a ( t r , x i → j r ) , B i r = c D , r a ( t r , x i r ) , (107) z i L = δ ∑ r = 1 L B i r − 1 u i r . (108) z_i^L=\sqrt{\delta}\sum_{r=1}^{L}B_i^{r-1}u_i^r. \tag*{(108)} z i L = δ r = 1 ∑ L B i r − 1 u i r . (108) The same constant, computed from the directed-edge law, occurs in A r A^r A r B r B^r B r x i → j r x^r_{i\to j} x i → j r c D , r c_{D,r} c D , r a a a
The one-step lag in A r − 1 u r A^{r-1}u^r A r − 1 u r G i → j r \mathcal{G}^r_{i\to j} G i → j r r r r i i i { i , j } \{i,j\} { i , j } i i i x i → j r − 1 x^{r-1}_{i\to j} x i → j r − 1 A i → j r − 1 A^{r-1}_{i\to j} A i → j r − 1 G i → j r − 1 \mathcal{G}^{r-1}_{i\to j} G i → j r − 1 [9 ] gives, for r ≥ 1 r\ge1 r ≥ 1
u i → j r ∣ G i → j r − 1 ∼ N ( 0 , ( τ i → j r ) 2 ) , u^r_{i\to j}\mid\mathcal{G}^{r-1}_{i\to j}\sim N\left(0,(\tau^r_{i\to j})^2\right), u i → j r ∣ G i → j r − 1 ∼ N ( 0 , ( τ i → j r ) 2 ) , ( τ i → j r ) 2 = 1 D ∑ v ∈ ∂ i ∖ { j } ( A v → i r − 1 ) 2 ( τ v → i r ) 2 , τ 0 = 1. (109) (\tau^r_{i\to j})^2=\frac{1}{D}\sum_{v\in\partial i\setminus\{j\}}(A^{r-1}_{v\to i})^2(\tau^r_{v\to i})^2,\qquad\tau^0=1. \tag*{(109)} ( τ i → j r ) 2 = D 1 v ∈ ∂ i ∖ { j } ∑ ( A v → i r − 1 ) 2 ( τ v → i r ) 2 , τ 0 = 1. (109) The variance recursion also holds at r = 0 r=0 r = 0 A r − 1 u r A^{r-1}u^r A r − 1 u r r = 0 r=0 r = 0 g i g_i g i
Let U 0 , U 1 , … U_0,U_1,\ldots U 0 , U 1 , …
X 0 δ = δ U 0 , X r + 1 δ = X r δ + δ b ( t r , X r δ ) + δ U r + 1 , (110) X^\delta_0=\sqrt{\delta}\,U_0,\qquad X^\delta_{r+1}=X^\delta_r+\delta b(t_r,X^\delta_r)+\sqrt{\delta}\,U_{r+1}, \tag*{(110)} X 0 δ = δ U 0 , X r + 1 δ = X r δ + δ b ( t r , X r δ ) + δ U r + 1 , (110) c δ , r = ( E a ( t r , X r δ ) 2 ) − 1 / 2 , Z L δ = δ ∑ r = 1 L c δ , r − 1 a ( t r − 1 , X r − 1 δ ) U r . (111) c_{\delta,r}=\left(\mathbb{E}a(t_r,X^\delta_r)^2\right)^{-1/2},\qquad Z^\delta_L=\sqrt{\delta}\sum_{r=1}^{L}c_{\delta,r-1}a(t_{r-1},X^\delta_{r-1})U_r. \tag*{(111)} c δ , r = ( E a ( t r , X r δ ) 2 ) − 1/2 , Z L δ = δ r = 1 ∑ L c δ , r − 1 a ( t r − 1 , X r − 1 δ ) U r . (111) At fixed β 0 , T , δ , L \beta_0,T,\delta,L β 0 , T , δ , L c δ , r c_{\delta,r} c δ , r X r δ X^\delta_r X r δ Z L δ Z^\delta_L Z L δ
Lemma A.2 (Finite-step limit). For the preceding construction, as D → ∞ D\to\infty D → ∞ δ , L \delta,L δ , L
c D , r ⟶ c δ , r , 0 ≤ r ≤ L , (112) c_{D,r}\longrightarrow c_{\delta,r},\qquad0\le r\le L, \tag*{(112)} c D , r ⟶ c δ , r , 0 ≤ r ≤ L , (112) ( u i 0 , … , u i L ) ⟹ ( U 0 , … , U L ) , z i L ⟹ Z L δ , (113) (u^0_i,\ldots,u^L_i)\Longrightarrow(U_0,\ldots,U_L),\qquad z^L_i\Longrightarrow Z^\delta_L, \tag*{(113)} ( u i 0 , … , u i L ) ⟹ ( U 0 , … , U L ) , z i L ⟹ Z L δ , (113) lim D → ∞ D + 1 2 D E Z z v L z v L = δ ∑ r = 1 L − 1 c δ , r E a ( t r , X r δ ) , o ∼ v . (114) \lim_{D\to\infty}\frac{D+1}{2\sqrt{D}}\mathbb{E}_\mathbb{Z}z^L_vz^L_v
=\delta\sum_{r=1}^{L-1}c_{\delta,r}\mathbb{E}a(t_r,X^\delta_r),\qquad o\sim v. \tag*{(114)} D → ∞ lim 2 D D + 1 E Z z v L z v L = δ r = 1 ∑ L − 1 c δ , r E a ( t r , X r δ ) , o ∼ v . (114) The sum in (114) is empty when L = 1 L=1 L = 1 The distributional statements include convergence of every fixed moment. The analogous joint Gaussian limit holds for a directed edge. Moreover E ( Z L δ ) 2 = L δ \mathbb{E}(Z^\delta_L)^2=L\delta E ( Z L δ ) 2 = L δ
Proof. We use the bounded-coefficient CLT and edge calculation in [9 ] (Assumptions 2–3, Theorem 3.1, Proposition 2.1, and Lemmas 4.2–4.3). The verification below has two parts: first the degree-dependent normalizers and moment bounds, then the edge value with its finite-step index. The functions a ( t r , ⋅ ) a(t_r,\cdot) a ( t r , ⋅ ) r r r c D , 0 = c δ , 0 c_{D,0}=c_{\delta,0} c D , 0 = c δ , 0 x r x^r x r r − 2 r-2 r − 2 a 2 a^2 a 2
E a ( t r , x i → j r ) 2 ⟶ E a ( t r , X r δ ) 2 > 0. \mathbb{E}a(t_r,x^r_{i\to j})^2\longrightarrow\mathbb{E}a(t_r,X^\delta_r)^2>0. E a ( t r , x i → j r ) 2 ⟶ E a ( t r , X r δ ) 2 > 0. Thus c D , r c_{D,r} c D , r D D D D D D A v → u r − 1 u v → i r A_{v\to u}^{r-1}u_{v\to i}^{r} A v → u r − 1 u v → i r a a a b b b k ≥ 2 k \ge2 k ≥ 2
E ∣ D − 1 / 2 ∑ j = 1 D Y j ∣ k ≤ C k ( ( E Y 1 2 ) k / 2 + D 1 − k / 2 E ∣ Y 1 ∣ k ) . \mathbb{E}\left|D^{-1/2}\sum_{j=1}^{D}Y_j\right|^k \le C_k\left((\mathbb{E}Y_1^2)^{k/2}+D^{1-k/2}\mathbb{E}|Y_1|^k\right). E D − 1/2 j = 1 ∑ D Y j k ≤ C k ( ( E Y 1 2 ) k /2 + D 1 − k /2 E ∣ Y 1 ∣ k ) . Induction over the finite number of updates proves uniform bounds of every order. They imply uniform integrability and hence the asserted moment convergence. The x x x z z z
For the energy, put
R D , ℓ = E [ B o ℓ − 1 B v ℓ − 1 A o → v ℓ − 2 A v → o ℓ − 2 u o → v ℓ − 1 u v → o ℓ − 1 ] . R_{D,\ell}=\mathbb{E}\left[B_o^{\ell-1}B_v^{\ell-1}A_{o\to v}^{\ell-2}A_{v\to o}^{\ell-2}u_{o\to v}^{\ell-1}u_{v\to o}^{\ell-1}\right]. R D , ℓ = E [ B o ℓ − 1 B v ℓ − 1 A o → v ℓ − 2 A v → o ℓ − 2 u o → v ℓ − 1 u v → o ℓ − 1 ] . Proposition 2.1 of [9 ] , with its degree k = D + 1 k=D+1 k = D + 1
D + 1 2 D E z o L z v L = δ ∑ ℓ = 2 L E [ A v → o ℓ − 2 ( u v → o ℓ − 1 ) 2 B o ℓ − 1 B v ℓ − 2 ] + δ 2 D ∑ ℓ = 1 L R D , ℓ . (115) \frac{D+1}{2\sqrt{D}}\mathbb{E}z_o^{L}z_v^{L}
=
\delta\sum_{\ell=2}^{L}\mathbb{E}\left[A_{v\to o}^{\ell-2}(u_{v\to o}^{\ell-1})^2B_o^{\ell-1}B_v^{\ell-2}\right]
+\frac{\delta}{2\sqrt{D}}\sum_{\ell=1}^{L}R_{D,\ell}. \tag*{(115)} 2 D D + 1 E z o L z v L = δ ℓ = 2 ∑ L E [ A v → o ℓ − 2 ( u v → o ℓ − 1 ) 2 B o ℓ − 1 B v ℓ − 2 ] + 2 D δ ℓ = 1 ∑ L R D , ℓ . (115) The second line is O ( D − 1 / 2 ) O(D^{-1/2}) O ( D − 1/2 )
u i r = D D + 1 u i → j r + A j → i r − 2 u j → i r − 1 D + 1 ( 1 ≤ r ≤ L ) . u_i^r=\sqrt{\frac{D}{D+1}}u_{i\to j}^r+\frac{A_{j\to i}^{r-2}u_{j\to i}^{r-1}}{\sqrt{D+1}}
\qquad(1\le r\le L). u i r = D + 1 D u i → j r + D + 1 A j → i r − 2 u j → i r − 1 ( 1 ≤ r ≤ L ) . Its squared difference from u i → j r u_{i\to j}^r u i → j r O ( D − 1 ) O(D^{-1}) O ( D − 1 )
E ∣ x i r − x i → j r ∣ 2 + E ∣ B i r − A i → j r ∣ 2 ≤ C D . (116) \mathbb{E}|x_i^r-x_{i\to j}^r|^2+\mathbb{E}|B_i^r-A_{i\to j}^r|^2\le\frac{C}{D}. \tag*{(116)} E ∣ x i r − x i → j r ∣ 2 + E ∣ B i r − A i → j r ∣ 2 ≤ D C . (116) These are the comparisons in Lemma 4.2 of [9 ] ; in particular, the use of the spatial Lipschitz bound for a a a
Here is the reduction in Lemma 4.3 of that paper in our indices. Set s = ℓ − 1 s=\ell-1 s = ℓ − 1 H = A v → o s − 1 B v s − 1 B o s H=A_{v\to o}^{s-1}B_v^{s-1}B_o^s H = A v → o s − 1 B v s − 1 B o s ( s − 1 ) (s-1) ( s − 1 ) v v v s s s o o o
E [ H ( ( u v → o s ) 2 − 1 ) ] = E [ H ( ( τ v → o s ) 2 − 1 ) ] . \mathbb{E}\left[H\left((u_{v\to o}^{s})^2-1\right)\right]
=
\mathbb{E}\left[H\left((\tau_{v\to o}^{s})^2-1\right)\right]. E [ H ( ( u v → o s ) 2 − 1 ) ] = E [ H ( ( τ v → o s ) 2 − 1 ) ] . The variance estimate in [9 ] , Lemmas 3.2–3.4 is uniform for the present normalizers. Indeed, if K ≥ 1 K\ge1 K ≥ 1 Δ r = E [ ( ( τ i → j r ) 2 − 1 ) 2 ] \Delta_r=\mathbb{E}[((\tau_{i\to j}^{r})^2-1)^2] Δ r = E [(( τ i → j r ) 2 − 1 ) 2 ] E ( A i → j r ) 2 = 1 \mathbb{E}(A_{i\to j}^{r})^2=1 E ( A i → j r ) 2 = 1
Δ r + 1 ≤ 2 K 4 Δ r + 2 ( K 4 + 1 ) D , Δ 0 = 0. \Delta_{r+1}\le2K^4\Delta_r+\frac{2(K^4+1)}{D},
\qquad\Delta_0=0. Δ r + 1 ≤ 2 K 4 Δ r + D 2 ( K 4 + 1 ) , Δ 0 = 0. Thus Δ s = O ( D − 1 ) \Delta_s = O(D^{-1}) Δ s = O ( D − 1 ) O ( D − 1 / 2 ) O(D^{-1/2}) O ( D − 1/2 )
E [ H ( u v → o s ) 2 ] = E [ ( A o → v s ) 2 ( A v → o s − 1 ) 2 ] + O ( D − 1 / 2 ) = E A o → v s E [ ( A v → o s − 1 ) 2 ] + O ( D − 1 / 2 ) = E B o s + O ( D − 1 / 2 ) . \mathbb{E}\left[H\left(u_{v\to o}^{s}\right)^2\right] = \mathbb{E}\left[\left(A_{o\to v}^{s}\right)^2\left(A_{v\to o}^{s-1}\right)^2\right] + O(D^{-1/2})
= \mathbb{E}A_{o\to v}^{s}\mathbb{E}\left[\left(A_{v\to o}^{s-1}\right)^2\right] + O(D^{-1/2})
= \mathbb{E}B_o^{s} + O(D^{-1/2}). E [ H ( u v → o s ) 2 ] = E [ ( A o → v s ) 2 ( A v → o s − 1 ) 2 ] + O ( D − 1/2 ) = E A o → v s E [ ( A v → o s − 1 ) 2 ] + O ( D − 1/2 ) = E B o s + O ( D − 1/2 ) . The middle equality uses the exact independence of the two cavity trees, and the last uses their coefficient normalization and one more node-to-cavity comparison. This proves the quantitative replacement without a convergence rate for c D , r c_{D,r} c D , r δ \delta δ L L L β 0 \beta_0 β 0 T T T r = ℓ − 1 r = \ell- 1 r = ℓ − 1 1 U r U_r U r U 0 , … , U r − 1 U_0,\ldots,U_{r-1} U 0 , … , U r − 1 c δ , r − 1 c_{\delta,r-1} c δ , r − 1 E ( Z L δ ) 2 = δ ∑ r = 1 L 1 = L δ \mathbb{E}(Z_L^\delta)^2 = \delta\sum_{r=1}^{L}1 = L\delta E ( Z L δ ) 2 = δ ∑ r = 1 L 1 = L δ
The degree limit has identified the scalar law Z L δ Z_L^\delta Z L δ [8 ] , Proposition 5.3. We give the estimates for the present fixed-temperature tree construction.
Lemma A.3 (The continuum limit at fixed temperature). Fix β 0 > 1 \beta_0 > 1 β 0 > 1 0 < T < q 0 < T < q 0 < T < q L = ⌊ T / δ ⌋ L = \lfloor T/\delta\rfloor L = ⌊ T / δ ⌋ C = C ( β 0 , T ) C = C(\beta_0,T) C = C ( β 0 , T ) δ \delta δ
max r δ ≤ T E ∣ X r δ − X r δ ∣ 2 ≤ C δ , max r δ ≤ T ∣ c δ , r − 1 ∣ ≤ C δ , (117) \max_{r\delta\leq T}\mathbb{E}\left|X_r^\delta- X_{r\delta}\right|^2 \leq C\delta,\qquad
\max_{r\delta\leq T}\left|c_{\delta,r}-1\right| \leq C\sqrt{\delta}, \tag*{(117)} r δ ≤ T max E X r δ − X r δ 2 ≤ C δ , r δ ≤ T max ∣ c δ , r − 1 ∣ ≤ C δ , (117) E ∣ Z L δ − m ( T , X T ) ∣ 2 ≤ C δ , (118) \mathbb{E}\left|Z_L^\delta- m(T,X_T)\right|^2 \leq C\delta, \tag*{(118)} E Z L δ − m ( T , X T ) 2 ≤ C δ , (118) ∣ δ ∑ r = 1 L − 1 c δ , r E a ( τ r , X r δ ) − ∫ 0 T E a ( t , X t ) d t ∣ ≤ C δ . (119) \left|\delta\sum_{r=1}^{L-1}c_{\delta,r}\mathbb{E}a(\tau_r,X_r^\delta)-\int_0^T\mathbb{E}a(t,X_t)\,\mathrm{d}t\right| \leq C\sqrt{\delta}. \tag*{(119)} δ r = 1 ∑ L − 1 c δ , r E a ( τ r , X r δ ) − ∫ 0 T E a ( t , X t ) d t ≤ C δ . (119) Proof. Take U 0 U_0 U 0 W W W δ U r = W r δ − W ( r − 1 ) δ \sqrt{\delta}U_r = W_{r\delta}-W_{(r-1)\delta} δ U r = W r δ − W ( r − 1 ) δ r ≥ 1 r\geq1 r ≥ 1 τ δ ( t ) = δ ⌊ t / δ ⌋ \tau_\delta(t)=\delta\lfloor t/\delta\rfloor τ δ ( t ) = δ ⌊ t / δ ⌋
X ‾ t δ = δ U 0 + ∫ 0 t b ( τ δ ( s ) , X ⌊ s / δ ⌋ δ ) d s + W t . \overline{X}_t^\delta= \sqrt{\delta}U_0+\int_0^t b\left(\tau_\delta(s),X_{\lfloor s/\delta\rfloor}^\delta\right)\,\mathrm{d}s+W_t. X t δ = δ U 0 + ∫ 0 t b ( τ δ ( s ) , X ⌊ s / δ ⌋ δ ) d s + W t . The drift is bounded and Lipschitz in space and time on [ 0 , T ] [0,T] [ 0 , T ] E ∣ X ‾ s δ − X ⌊ s / δ ⌋ δ ∣ 2 ≤ C δ \mathbb{E}\left|\overline{X}_s^\delta-X_{\lfloor s/\delta\rfloor}^\delta\right|^2\leq C\delta E X s δ − X ⌊ s / δ ⌋ δ 2 ≤ C δ X X X X ‾ δ \overline{X}^\delta X δ
E sup s ≤ T ∣ X ‾ s δ − X s ∣ 2 ≤ C δ . \mathbb{E}\sup_{s\leq T}\left|\overline{X}_s^\delta-X_s\right|^2\leq C\delta. E s ≤ T sup X s δ − X s 2 ≤ C δ . The initial contribution is E ∣ δ U 0 ∣ 2 = δ \mathbb{E}\left|\sqrt{\delta}U_0\right|^2=\delta E δ U 0 2 = δ a a a E a ( t r , X t r ) 2 = 1 \mathbb{E}a(t_r,X_{t_r})^2=1 E a ( t r , X t r ) 2 = 1
max r δ ≤ T ∣ E a ( t r , X r δ ) 2 − 1 ∣ ≤ C δ . \max_{r\delta\leq T}\left|\mathbb{E}a(t_r,X_r^\delta)^2-1\right|\leq C\sqrt{\delta}. r δ ≤ T max E a ( t r , X r δ ) 2 − 1 ≤ C δ . For small δ \delta δ [ 1 / 2 , 3 / 2 ] [1/2,3/2] [ 1/2 , 3/2 ] x ↦ x − 1 / 2 x\mapsto x^{-1/2} x ↦ x − 1/2 (117) .
Equation (111) is the stochastic integral
Z L δ = ∫ 0 L δ c δ , ⌊ t / δ ⌋ a ( τ δ ( t ) , X ⌊ t / δ ⌋ δ ) d W t . Z_L^\delta=\int_0^{L\delta}c_{\delta,\lfloor t/\delta\rfloor}a(\tau_\delta(t),X_{\lfloor t/\delta\rfloor}^\delta)\,\mathrm{d}W_t. Z L δ = ∫ 0 L δ c δ , ⌊ t / δ ⌋ a ( τ δ ( t ) , X ⌊ t / δ ⌋ δ ) d W t . The integrand differs from a ( t , X t ) a(t,X_t) a ( t , X t ) C δ C\delta C δ t ≤ L δ t\leq L\delta t ≤ L δ (117) , the spatial and temporal Lipschitz bounds for a a a E ∣ X t − X τ δ ( t ) ∣ 2 ≤ C δ \mathbb{E}|X_t-X_{\tau_\delta(t)}|^2\leq C\delta E ∣ X t − X τ δ ( t ) ∣ 2 ≤ C δ δ \delta δ (118) , since m ( T , X T ) = ∫ 0 T a ( t , X t ) d W t m(T,X_T)=\int_0^T a(t,X_t)\,\mathrm{d}W_t m ( T , X T ) = ∫ 0 T a ( t , X t ) d W t (119) with its Riemann integral. The omitted first and last intervals have total length at most 3 δ 3\delta 3 δ
We next place this classical construction in the Hilbert space used by Proposition 2.2. This step requires norm convergence, not only a CLT in distribution.
We use the Gaussian lift G G G A r − 1 u r A^{r-1}u^r A r − 1 u r u 0 = g u^0=g u 0 = g D + 1 D+1 D + 1 D D D D + 1 \sqrt{D+1} D + 1 D \sqrt{D} D
Lemma A.4 (The symmetric-power realization). Consider a fixed finite descendant formula whose pointwise functions are smooth with all derivatives of polynomial growth. Assume its leaf seeds have moments of every order and every aggregated child column has mean zero under its finite-degree child law for each D D D G G G f D f_D f D f f f
lim D → ∞ D + 1 2 D E f D ( o ) f D ( v ) = E ( f ) , o ∼ v . (120) \lim_{D\to\infty}\frac{D+1}{2\sqrt{D}}\mathbb{E}f_D(o)f_D(v)=\mathcal{E}(f),\qquad o\sim v. \tag*{(120)} D → ∞ lim 2 D D + 1 E f D ( o ) f D ( v ) = E ( f ) , o ∼ v . (120) Proof. The proof first treats characteristic vectors, then arbitrary pointwise maps, and finally the occupation bound needed for energy. We use induction on formula depth. All intermediate moments are uniformly bounded: normalized centered sums satisfy the moment bound used in Lemma A.2, and polynomial-growth functions preserve the existence of sufficiently high uniform moments.
Suppose the real centered child columns f D , 1 , … , f D , k f_{D,1},\ldots,f_{D,k} f D , 1 , … , f D , k f 1 , … , f k f_1,\ldots,f_k f 1 , … , f k t ∈ R k t\in\mathbb{R}^k t ∈ R k F D = ∑ j t j f D , j F_D=\sum_j t_jf_{D,j} F D = ∑ j t j f D , j F = ∑ j t j f j F=\sum_j t_jf_j F = ∑ j t j f j
h D = exp ( i F D / D ) Ω = α D Ω + ζ D , ζ D ⊥ Ω . h_D=\exp(iF_D/\sqrt{D})\Omega=\alpha_D\Omega+\zeta_D,\qquad\zeta_D\perp\Omega. h D = exp ( i F D / D ) Ω = α D Ω + ζ D , ζ D ⊥ Ω. Taylor expansion in L 2 L^2 L 2
α D = 1 − ∥ F D ∥ 2 2 D + O ( D − 3 / 2 ) , D ζ D ⟶ i F . \alpha_D = 1 - \frac{\lVert F_D\rVert^2}{2D} + O(D^{-3/2}), \qquad\sqrt{D}\,\zeta_D \longrightarrow iF. α D = 1 − 2 D ∥ F D ∥ 2 + O ( D − 3/2 ) , D ζ D ⟶ i F . The j j j
( D j ) 1 / 2 α D D − j ζ D ⊗ j ⟶ e − ∥ F ∥ 2 / 2 ( i F ) ⊗ j j ! . \binom{D}{j}^{1/2}\alpha_D^{D-j}\zeta_D^{\otimes j} \longrightarrow e^{-\lVert F\rVert^2/2}\frac{(iF)^{\otimes j}}{\sqrt{j!}}. ( j D ) 1/2 α D D − j ζ D ⊗ j ⟶ e − ∥ F ∥ 2 /2 j ! ( i F ) ⊗ j . These are the components of e i q ( F ) Ω e^{iq(F)}\Omega e i q ( F ) Ω D ∥ ζ D ∥ 2 = O ( 1 ) D\lVert\zeta_D\rVert^2=O(1) D ∥ ζ D ∥ 2 = O ( 1 ) M = D + 1 M=D+1 M = D + 1 M M M M − 1 / 2 M^{-1/2} M − 1/2 D − 1 / 2 D^{-1/2} D − 1/2 M / D → 1 M/D\to1 M / D → 1 M ∥ ζ D ∥ 2 = O ( 1 ) M\lVert\zeta_D\rVert^2=O(1) M ∥ ζ D ∥ 2 = O ( 1 )
Finite linear combinations of these exponentials, multiplied by bounded measurable functions of the independent root seed, are dense in the limiting L 2 L^2 L 2 L 2 L^2 L 2 S S S Y D Y_D Y D Y Y Y S S S D D D Y D Y_D Y D Y Y Y P ( s , y ) = ∑ j g j ( s ) e i t j ⋅ y P(s,y)=\sum_j g_j(s)e^{it_j\cdot y} P ( s , y ) = ∑ j g j ( s ) e i t j ⋅ y g j g_j g j F ( s , y ) F(s,y) F ( s , y ) y y y s s s ∣ F ( s , y ) − P ( s , y ) ∣ 2 \lvert F(s,y)-P(s,y)\rvert^2 ∣ F ( s , y ) − P ( s , y ) ∣ 2 y y y s s s Y D Y_D Y D s s s s s s
E ∣ F ( S , Y D ) − P ( S , Y D ) ∣ 2 ⟶ E ∣ F ( S , Y ) − P ( S , Y ) ∣ 2 . \mathbb{E}\lvert F(S,Y_D)-P(S,Y_D)\rvert^2 \longrightarrow\mathbb{E}\lvert F(S,Y)-P(S,Y)\rvert^2. E ∣ F ( S , Y D ) − P ( S , Y D ) ∣ 2 ⟶ E ∣ F ( S , Y ) − P ( S , Y ) ∣ 2 . Thus measurability of g j g_j g j
For completeness, the occupation bound is not a consequence of the CLT alone. At a parent, let P j P_j P j j j j H D ( s , S D , 1 , … , S D , k ) H_D(s,S_{D,1},\ldots,S_{D,k}) H D ( s , S D , 1 , … , S D , k ) S D , l = D − 1 / 2 ∑ j f D , l ( j ) S_{D,l}=D^{-1/2}\sum_j f_{D,l}^{(j)} S D , l = D − 1/2 ∑ j f D , l ( j ) N D = ∑ j = 1 D ( I − P j ) N_D=\sum_{j=1}^D(I-P_j) N D = ∑ j = 1 D ( I − P j ) j j j
∥ ( I − P j ) H D ∥ 2 ≤ E ∣ H D − H D ( j ) ∣ 2 ≤ C D . \lVert(I-P_j)H_D\rVert^2\leq\mathbb{E}\lvert H_D-H_D^{(j)}\rvert^2\leq\frac{C}{D}. ∥( I − P j ) H D ∥ 2 ≤ E ∣ H D − H D ( j ) ∣ 2 ≤ D C . Here the derivative of H D H_D H D j , D j,D j , D ⟨ H D , N D H D ⟩ ≤ C \langle H_D,N_DH_D\rangle\leq C ⟨ H D , N D H D ⟩ ≤ C N D N_D N D
The energy identity is now Proposition 2.2 applied to these classical vectors. More explicitly, with M = D + 1 M=D+1 M = D + 1 C / M C/\sqrt{M} C / M O ( M − 1 ) O(M^{-1}) O ( M − 1 ) M \sqrt{M} M ⟨ f , L f ⟩ \langle f,Lf\rangle ⟨ f , L f ⟩ M / ( 2 D ) M/(2\sqrt{D}) M / ( 2 D ) (120) thus gives Re ⟨ f , L f ⟩ \operatorname{Re}\langle f,Lf\rangle Re ⟨ f , L f ⟩ M / D → 1 M/D \to1 M / D → 1
Apply this lemma to the EAMS formulas at fixed δ , L \delta,L δ , L a a a b b b (112) and the coefficient-continuity assertion allow us to replace c D , r c_{D,r} c D , r c δ , r c_{\delta,r} c δ , r z δ z^\delta z δ
E ( z δ ) = δ ∑ r = 1 L − 1 c δ , r E a ( t r , X r δ ) , ∥ z δ ∥ 2 = L δ , z δ has law Z L δ . (121) \mathcal{E}(z^\delta)=\delta\sum_{r=1}^{L-1}c_{\delta,r}\mathbb{E}a(t_r,X_r^\delta),\qquad\lVert z^\delta\rVert^2=L\delta,\qquad z^\delta\text{ has law }Z_L^\delta. \tag*{(121)} E ( z δ ) = δ r = 1 ∑ L − 1 c δ , r E a ( t r , X r δ ) , ∥ z δ ∥ 2 = L δ , z δ has law Z L δ . (121) All coefficients in this vector are independent of degree.
Let π ( x ) = max ( − 1 , min ( x , 1 ) ) \pi(x)=\max(-1,\min(x,1)) π ( x ) = max ( − 1 , min ( x , 1 )) ∣ m ( T , X T ) ∣ ≤ 1 |m(T,X_T)|\le1 ∣ m ( T , X T ) ∣ ≤ 1 (121) and the metric projection property give
∥ z δ − π ( z δ ) ∥ 2 = E ∣ Z L δ − π ( Z L δ ) ∣ 2 ≤ E ∣ Z L δ − m ( T , X T ) ∣ 2 . \lVert z^\delta-\pi(z^\delta)\rVert^2=\mathbb{E}|Z_L^\delta-\pi(Z_L^\delta)|^2\le\mathbb{E}|Z_L^\delta-m(T,X_T)|^2. ∥ z δ − π ( z δ ) ∥ 2 = E ∣ Z L δ − π ( Z L δ ) ∣ 2 ≤ E ∣ Z L δ − m ( T , X T ) ∣ 2 . Only the marginal law of Z L δ Z_L^\delta Z L δ z δ z^\delta z δ (121) . Consequently, by Lemma A.3, ∥ z δ − π ( z δ ) ∥ ≤ C δ \lVert z^\delta-\pi(z^\delta)\rVert\le C\sqrt{\delta} ∥ z δ − π ( z δ )∥ ≤ C δ f = z δ f=z^\delta f = z δ g = π ( z δ ) g=\pi(z^\delta) g = π ( z δ ) (121) proves the following: for every ϵ > 0 \epsilon>0 ϵ > 0 β 0 \beta_0 β 0 T < q T<q T < q q q q δ \delta δ
m ∗ = π ( z δ ) , ∣ m ∗ ∣ ≤ 1 , E ( m ∗ ) ≥ P ∗ − ϵ . (122) m^*=\pi(z^\delta),\qquad|m^*|\le1,\qquad\mathcal{E}(m^*)\ge P_*-\epsilon. \tag*{(122)} m ∗ = π ( z δ ) , ∣ m ∗ ∣ ≤ 1 , E ( m ∗ ) ≥ P ∗ − ϵ . (122) At this point the depth L L L
The next proposition completes this route using only the common bounded-approximation and exclusion lemmas. It gives the same conclusion as Proposition 3.5, with an independent proof from the clipped vector just constructed.
Proposition A.5 (The independent positive-temperature construction). For every ϵ > 0 \epsilon>0 ϵ > 0 η H > 0 \eta_H>0 η H > 0 0 < p H < η H 0<p_H<\eta_H 0 < p H < η H θ c \theta_c θ c
m c o l = ∑ c = 1 M 1 { C = c } sin ( 2 θ c ) (123) m^{\mathrm{col}}=\sum_{c=1}^{M}\mathbf{1}_{\{C=c\}}\sin(2\theta_c) \tag*{(123)} m col = c = 1 ∑ M 1 { C = c } sin ( 2 θ c ) (123) is centered, satisfies ∣ m c o l ∣ ≤ 1 |m^{\mathrm{col}}| \le1 ∣ m col ∣ ≤ 1
E ( m c o l ) ≥ P ∗ − ε . (124) \mathcal{E}(m^{\mathrm{col}}) \ge P_* - \varepsilon. \tag*{(124)} E ( m col ) ≥ P ∗ − ε . (124) After the uncolored finite formula has been fixed, the total helper probability may be taken arbitrarily small, with all individual helper probabilities positive. All depths, coefficients, label laws, and functions in the conclusion are fixed independently of degree, system size, and disorder.
Proof. Use (122) with error ε / 3 \varepsilon/3 ε /3 z δ z^\delta z δ ε / 12 \varepsilon/12 ε /12 ε / 6 \varepsilon/6 ε /6 η H \eta_H η H ε / 6 \varepsilon/6 ε /6 ε / 3 \varepsilon/3 ε /3 (48) . Their sum with the initial loss is less than ε \varepsilon ε
References 