Low-depth Clifford circuits approximately solve MaxCut
Bibliographic record
Abstract
We introduce a quantum-inspired approximation algorithm for MaxCut based on low-depth Clifford circuits. We start by showing that the solution unitaries found by the adaptive quantum approximation optimization algorithm (ADAPT-QAOA) for the MaxCut problem on weighted fully connected graphs are (almost) Clifford circuits. Motivated by this observation, we devise an approximation algorithm for MaxCut, ADAPT-Clifford, that searches through the Clifford manifold by combining a minimal set of generating elements of the Clifford group. Our algorithm finds an approximate solution of MaxCut on an <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>N</a:mi></a:math>-vertex graph by building a depth <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mi>O</b:mi><b:mo>(</b:mo><b:mi>N</b:mi><b:mo>)</b:mo></b:mrow></b:math> Clifford circuit. The algorithm has runtime complexity <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mi>O</c:mi><c:mo>(</c:mo><c:msup><c:mi>N</c:mi><c:mn>2</c:mn></c:msup><c:mo>)</c:mo></c:mrow></c:math> and <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mi>O</d:mi><d:mo>(</d:mo><d:msup><d:mi>N</d:mi><d:mn>3</d:mn></d:msup><d:mo>)</d:mo></d:mrow></d:math> for sparse and dense graphs, respectively, and space complexity <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:mi>O</e:mi><e:mo>(</e:mo><e:msup><e:mi>N</e:mi><e:mn>2</e:mn></e:msup><e:mo>)</e:mo></e:mrow></e:math>, with improved solution quality achieved at the expense of more demanding runtimes. We implement ADAPT-Clifford and characterize its performance on graphs with positive and signed weights. The case of signed weights is illustrated with the paradigmatic Sherrington-Kirkpatrick model, for which our algorithm finds solutions with ground-state mean energy density corresponding to <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:mo>∼</f:mo><f:mn>94</f:mn><f:mo>%</f:mo></f:mrow></f:math> of the Parisi value in the thermodynamic limit. The case of positive weights is investigated by comparing the cut found by ADAPT-Clifford with the cut found with the Goemans-Williamson (GW) algorithm. For both sparse and dense instances we provide copious evidence that, up to hundreds of nodes, ADAPT-Clifford finds cuts of lower energy than GW. Published by the American Physical Society 2024
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".