A rapidly mixing Markov chain from any gapped quantum many-body system
Bibliographic record
Abstract
We consider the computational task of sampling a bit string <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>x</mml:mi></mml:math> from a distribution <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03C0;</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mo fence="false" stretchy="false">&#x27E8;</mml:mo><mml:mi>x</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mi>&#x03C8;</mml:mi><mml:mo fence="false" stretchy="false">&#x27E9;</mml:mo><mml:msup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03C8;</mml:mi></mml:math> is the unique ground state of a local Hamiltonian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>H</mml:mi></mml:math>. Our main result describes a direct link between the inverse spectral gap of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>H</mml:mi></mml:math> and the mixing time of an associated continuous-time Markov Chain with steady state <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03C0;</mml:mi></mml:math>. The Markov Chain can be implemented efficiently whenever ratios of ground state amplitudes <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo fence="false" stretchy="false">&#x27E8;</mml:mo><mml:mi>y</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mi>&#x03C8;</mml:mi><mml:mo fence="false" stretchy="false">&#x27E9;</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mo fence="false" stretchy="false">&#x27E8;</mml:mo><mml:mi>x</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mi>&#x03C8;</mml:mi><mml:mo fence="false" stretchy="false">&#x27E9;</mml:mo></mml:math> are efficiently computable, the spectral gap of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>H</mml:mi></mml:math> is at least inverse polynomial in the system size, and the starting state of the chain satisfies a mild technical condition that can be efficiently checked. This extends a previously known relationship between sign-problem free Hamiltonians and Markov chains. The tool which enables this generalization is the so-called fixed-node Hamiltonian construction, previously used in Quantum Monte Carlo simulations to address the fermionic sign problem. We implement the proposed sampling algorithm numerically and use it to sample from the ground state of Haldane-Shastry Hamiltonian with up to 56 qubits. We observe empirically that our Markov chain based on the fixed-node Hamiltonian mixes more rapidly than the standard Metropolis-Hastings Markov chain.
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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) | 0.002 | 0.002 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.024 |
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; both teacher heads agree on what is shown here.
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".