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Exact amplitudes of sparse quantum circuits in subexponential time, assuming the new #SAT algorithms

Primarily AI-generated textHuman understanding: some partsquant-ph — Quantum Physicscs.CC — Computational Complexitycs.DS — Data Structures and Algorithms

Contributed by Vaibhav Tulsyan ↗

Version 1 / Nov 04, 2026 / CC BY 4.0

Abstract

Classical simulation lower bounds for quantum circuits have been derived from the Exponential Time Hypothesis (ETH). However, in October 2026 OpenAI and Anthropic claimed deterministic subexponential algorithms for #k-SAT, refuting ETH, and neither manuscript mentions quantum circuits. We record the corollary. Assuming three internal results of the Anthropic manuscript, one deterministic Turing machine computes the amplitude ⟨y∣C∣x⟩\langle y|C|x\rangle of a Clifford+T circuit with GG two-qubit gates in arbitrary connectivity exactly, as an element of Z[ω]/2 k\mathbb{Z}[\omega]/\sqrt2^{\,k} with ω=eiπ/4\omega=e^{i\pi/4}, in time 2O(G/log⁡G)2^{O(G/\sqrt{\log G})} whenever the input length LL satisfies log⁡L≤G0.9\log L\le G^{0.9}. The proof reads the sum over paths, one bit per wire segment, as an Anthropic-type network, with a Kronecker map for Z[ω]\mathbb{Z}[\omega] and a Galois bound. We machine-check this theorem in Lean 4 (amplitude_TM_rate', amplitude_TM_rate, amplitude_TM; axioms propext, Classical.choice, Quot.sound) on top of Anthropic's Lean development (sources at https://github.com/xennygrimmato/eth-hamiltonicity). On paper only, the method gives exact marginals and exact sampling, depth-dd circuits in 2O(dn/log⁡n)2^{O(dn/\sqrt{\log n})}, and, through OpenAI's tensor-contraction theorem, polynomial space when the number of Hadamard-type gates is O(G)O(G). Conversely, constant-depth circuits with O(n)O(n) gates encode #3-SAT with O(n)O(n) clauses, so the rate matches that of the refutations. We check the fine-grained supremacy conjectures we found against their sources; a counting form of the Calabro-Impagliazzo-Paturi width reduction refutes the gap conjecture of Morimae and Tamaki's depth-scaling paper. Dense statements (cubic phases, permanents, random circuits) survive. Nothing here concerns approximate sampling, the constants are galactic, and everything is conditional on the claimed refutations, which are under review.

Provenance statement

This work was produced by Vaibhav Tulsyan in sessions with Claude Code (Anthropic). The mathematical arguments, the Lean 4 formalisation and the text were drafted by Claude under my direction. I chose the problems, directed the research, and reviewed the statements of the main theorems together with the paper's own account of what is proved, what is assumed, what is checked numerically and what is not formalised. Every theorem the paper claims is machine-checked in Lean 4 on top of the Anthropic manuscript's own Lean development: the axiom audit shows only propext, Classical.choice and Quot.sound, and fresh leanchecker replays of the full dependency closure passed. Computations, written proofs and informal remarks are labelled as such in the paper and are not claimed as machine-checked. The Lean sources are public at https://github.com/xennygrimmato/eth-hamiltonicity. Independent AI referee passes, exact numerical checks and a citation audit were carried out; the definitions a reader must check to trust the formal statements are listed in the paper. No human expert has yet reviewed the work. All results are conditional on the claimed refutations of the Exponential Time Hypothesis, which are under review.

Tools used

Anthropic
Claude Fable
Anthropic
Claude Opus
Lean
Lean

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Version history

  1. v1 · 15 pagesSubmitted by Vaibhav TulsyanInitial depositCurrentNov 04, 2026