A Quantum Approximate Optimization Algorithm Edward Farhi Jeffrey Goldstone Sam Gutmann

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 A Quantum Approximate Optimization Algorithm 2014 arXiv 1411.4028 farhi2014-qaoa farhi2014-qaoa.txt, the project's staged copy 069238279c4d89a049de70dc270a14f002d11073 arxiv-open quantum-computing

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 0 spans located by searching the stored text for the quote's exact words, allowing any whitespace between them Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. mathematical-proof Quotation withheld because this record is not cleared for redistribution. THE WHOLE ALGORITHM IS ONE INTEGER KNOB: the state is built by alternating two operators p times over a uniform starting superposition, and the paper's claim is that the quality of the approximation improves as p is raised about this quote: 2026-08-16: A WORD IS SPLIT BY A SPURIOUS SPACE INSIDE THIS QUOTE - "approx- imation" for "approximation". The extraction inserts a space at what was a line break in the PDF, and it does this throughout this file. The quote is an exact substring of the staged text and the meaning is unaffected, but A READER SEARCHING FOR THE UNBROKEN WORD WILL NOT FIND IT, which is the whole reason this is recorded. No pattern in our tools detects it: both halves of a broken word are plausible tokens, so it is invisible to every guard and was found by eye. Quotation withheld because this record is not cleared for redistribution. the first operator is made diagonal in the computational basis, which buys two things at once - every term in the product COMMUTES, and each term's locality is no worse than the locality of the clause it came from Quotation withheld because this record is not cleared for redistribution. THE PARAMETER RANGE IS DERIVED FROM A PROPERTY OF THE OBJECTIVE, NOT CHOSEN: because the objective has integer eigenvalues, one angle can be restricted to lie between 0 and 2 pi about this quote: 2026-08-16: A WORD IS SPLIT BY A SPURIOUS SPACE INSIDE THIS QUOTE - "eigen- values" for "eigenvalues". The extraction inserts a space at what was a line break in the PDF, and it does this throughout this file. The quote is an exact substring of the staged text and the meaning is unaffected, but A READER SEARCHING FOR THE UNBROKEN WORD WILL NOT FIND IT, which is the whole reason this is recorded. No pattern in our tools detects it: both halves of a broken word are plausible tokens, so it is invisible to every guard and was found by eye. Quotation withheld because this record is not cleared for redistribution. MONOTONICITY IS STRUCTURAL RATHER THAN EMPIRICAL: the maximisation at p minus one IS a constrained maximisation at p, so the best value at each depth can never fall below the best value at the depth beneath it Quotation withheld because this record is not cleared for redistribution. the algorithm is used by repetition rather than by a single shot: measure in the computational basis, evaluate the objective on the string obtained, and repeat with the SAME angles - enough repetitions produce a string whose value is very near or above the expectation about this quote: 2026-08-16: A WORD IS SPLIT BY A SPURIOUS SPACE INSIDE THIS QUOTE - "a strin g z" for "a string z". The extraction inserts a space at what was a line break in the PDF, and it does this throughout this file. The quote is an exact substring of the staged text and the meaning is unaffected, but A READER SEARCHING FOR THE UNBROKEN WORD WILL NOT FIND IT, which is the whole reason this is recorded. No pattern in our tools detects it: both halves of a broken word are plausible tokens, so it is invisible to every guard and was found by eye. Quotation withheld because this record is not cleared for redistribution. the authors name the difficulty rather than presenting the method as complete - it is not obvious in advance how to pick good angles - and for p that does not grow with n they offer a search over a fine grid on the compact parameter set Quotation withheld because this record is not cleared for redistribution. the grid search is made tractable by a BOUND ON THE PARTIAL DERIVATIVES of the objective's expectation, which is what turns a search over a continuous compact set into an efficient one Quotation withheld because this record is not cleared for redistribution. for fixed p and bounded clause participation the angles are found by an EFFICIENT CLASSICAL CALCULATION rather than by quantum search, and those angles are then used to run the quantum computer and produce the measured string about this quote: 2026-08-16: A WORD IS SPLIT BY A SPURIOUS SPACE INSIDE THIS QUOTE - "an e fficient" for "an efficient". The extraction inserts a space at what was a line break in the PDF, and it does this throughout this file. The quote is an exact substring of the staged text and the meaning is unaffected, but A READER SEARCHING FOR THE UNBROKEN WORD WILL NOT FIND IT, which is the whole reason this is recorded. No pattern in our tools detects it: both halves of a broken word are plausible tokens, so it is invisible to every guard and was found by eye. Quotation withheld because this record is not cleared for redistribution. THE CONSTRAINT IS ENFORCED BY THE BASIS: for the independent-set variant the orthonormal basis consists only of strings that correspond to independent sets, so the space is exponentially large but smaller than the full 2^n, and is NOT a simple tensor product of qubits about this quote: 2026-08-16: A WORD IS SPLIT BY A SPURIOUS SPACE INSIDE THIS QUOTE - "an orthonormal ba sis" for "an orthonormal basis". The extraction inserts a space at what was a line break, throughout this file. Separately, this span was NARROWED: it originally ran through equation (39), whose large brace is stored as Private Use Area codepoints U+F8F1-U+F8F3. An internal tool refused it; the merge gate's dry run had reported it clean. Quotation withheld because this record is not cleared for redistribution. the mixing operator is likewise restricted rather than penalised: it is the adjacency matrix of the hypercube confined to the legal strings, so the walk it generates can only ever move between independent sets Quotation withheld because this record is not cleared for redistribution. about this quote: QUOTE RECOVERED FROM THE PAPER'S OWN CHARACTERS. The recorded quote was LETTERS-ONLY IDENTICAL to this stretch of the staged text but not an exact substring of it - spaces, hyphens, ligature glyphs or mathematical operators differed, so it could never anchor. This is the file's own text between the same two letters: not produced by a rule about the damage, copied out of the file. Round-trip checked (the recovered span is letters-only equal to the original) and length-checked. The text as originally recorded is kept in evidence as extracted. Update, 15 September 2026: CLEARED evidence status='quote-not-in-current-staged-text' - the quote now locates in the current staged text, so the condition the marker recorded no longer holds. Quotation withheld because this record is not cleared for redistribution. what this limits: the p = 1 performance reported here for 3-regular graphs: the authors state it Quotation withheld because this record is not cleared for redistribution. what this limits: any extrapolation from p = 1 to larger p: the authors state the p = 2 case on 3-regular graphs is more complicated and that they show only partial results for it Quotation withheld because this record is not cleared for redistribution. what this limits: the claim that approximation quality improves with p: the depth of the circuit grows linearly with p times, at worst, the number of constraints, so the two move together Quotation withheld because this record is not cleared for redistribution. what this limits: results reported at p = 2 on 3-regular graphs: the authors state the algorithm at that depth cannot detect whether the graph is bipartite, that is, completely satisfiable Quotation withheld because this record is not cleared for redistribution. MaxCut Quotation withheld because this record is not cleared for redistribution. QAOA Quotation withheld because this record is not cleared for redistribution. approximation ratio Quotation withheld because this record is not cleared for redistribution. classical algorithms Quotation withheld because this record is not cleared for redistribution. circuit depth Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. QAOA classical algorithms strength of the relation: 1 Audit 2026-09-28: DOUBTFUL - read against its full span and kept, but endpoint by anaphora: 'This p = 1 result' = the QAOA's result (previous sentence). Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. about this quote: QUOTE RECOVERED FROM THE PAPER'S OWN CHARACTERS. The recorded quote was LETTERS-ONLY IDENTICAL to this stretch of the staged text but not an exact substring of it - spaces, hyphens, ligature glyphs or mathematical operators differed, so it could never anchor. This is the file's own text between the same two letters: not produced by a rule about the damage, copied out of the file. Round-trip checked (the recovered span is letters-only equal to the original) and length-checked. The text as originally recorded is kept in evidence as extracted. Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. approximation ratio about this quote: QUOTE RECOVERED FROM THE PAPER'S OWN CHARACTERS. The recorded quote was LETTERS-ONLY IDENTICAL to this stretch of the staged text but not an exact substring of it - spaces, hyphens, ligature glyphs or mathematical operators differed, so it could never anchor. This is the file's own text between the same two letters: not produced by a rule about the damage, copied out of the file. Round-trip checked (the recovered span is letters-only equal to the original) and length-checked. The text as originally recorded is kept in evidence as extracted. Quotation withheld because this record is not cleared for redistribution. combinatorial optimization problem Quotation withheld because this record is not cleared for redistribution. MaxCut about this quote: QUOTE RECOVERED FROM THE PAPER'S OWN CHARACTERS. The recorded quote was LETTERS-ONLY IDENTICAL to this stretch of the staged text but not an exact substring of it - spaces, hyphens, ligature glyphs or mathematical operators differed, so it could never anchor. This is the file's own text between the same two letters: not produced by a rule about the damage, copied out of the file. Round-trip checked (the recovered span is letters-only equal to the original) and length-checked. The text as originally recorded is kept in evidence as extracted. Quotation withheld because this record is not cleared for redistribution. mean Quotation withheld because this record is not cleared for redistribution. standard-deviation Quotation withheld because this record is not cleared for redistribution.

The full text of this article is not redistributable, so it is not included here. The evidence sentences are withheld with it; the tag layer and its grounding status remain. The article itself is at https://arxiv.org/abs/1411.4028.