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Phys-79 · physics · mathematics · Physics

QAOA reaches the SK ground value when thermodynamic size is taken first

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For zero-field Gaussian SK, let $Q_p$ be the optimum limiting QAOA energy at fixed depth p after n$\to$infinity. Does $Q_p$ tend as p$\to$infinity to the Parisi ground-state value? Equivalently, for each $\epsilon >0$ must one finite depth and deterministic angle vectors independent of n and disorder achieve limiting value at least $P_*-\epsilon $?

Short English statement

For zero-field Gaussian SK, let $Q_p$ be the optimum limiting QAOA energy at fixed depth p after n$\to$infinity. Does $Q_p$ tend as p$\to$infinity to the Parisi ground-state value? Equivalently, for each $\epsilon >0$ must one finite depth and deterministic angle vectors independent of n and disorder achieve limiting value at least $P_*-\epsilon $?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-79-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-79

QAOA is claimed to approach the SK ground-energy density when system size goes to infinity before depth, using deterministic disorder-independent angles but no effective depth bound. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

QAOA reaches the SK ground value when thermodynamic size is taken first

OpenAI's claim

OpenAI presents a proof in the selected manuscript. QAOA is claimed to approach the SK ground-energy density when system size goes to infinity before depth, using deterministic disorder-independent angles but no effective depth bound.

Mathematical statement of the claimed result

Result 1.
For independent standard Gaussian $J_{ij}$, set $C_{n,J}=n^{-1/2}\sum_{i<j}J_{ij}Z_iZ_j$ and $B_n=\sum_iX_i$, with Pauli operators. A depth-$p$ QAOA state is \[ \psi_{n,p,J}=e^{-i\beta_pB_n}e^{-i\gamma_pC_{n,J}}\cdots e^{-i\beta_1B_n}e^{-i\gamma_1C_{n,J}}|+\rangle^{\otimes n}. \] For deterministic parameters, put $v_p(\gamma,\beta)=\lim_n n^{-1}\mathbb E\langle\psi,C_{n,J}\psi\rangle$ and $Q_p=\sup_{\gamma,\beta\in\mathbb R^p}v_p$. The claim is \[ \lim_{p\to\infty}Q_p=P_*,\qquad P_*=\lim_n n^{-1}\mathbb E\max_{\sigma\in\{\pm1\}^n} n^{-1/2}\sum_{i<j}J_{ij}\sigma_i\sigma_j. \] For each accuracy $\varepsilon>0$, one finite depth and parameter vector, independent of both instance and size, achieves $v_p\ge P_*-\varepsilon$. The size limit is taken first.

Outline of the argument

Fixed circuit words have an exact limiting tree calculus. A nearly optimal bounded classical formula is compiled with seeded controls, then longitudinal seeds are removed; the ordinary cost-and-mixer word retains the limiting expected energy in the thermodynamic-first order.

Formal verification

For family 281, the selected value identities express the SK ground energy through an admissible Parisi minimizer and diffusion, without asserting QAOA circuit convergence. A distinct full-support statement covers integrable zero-temperature minimizers. The listed Comparator files are SKFullSupport.lean, SKValue.lean. The inventory does not supply a named declaration for these Comparator entries. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

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