Quantum and classical algorithms for nonlinear unitary dynamics
Bibliographic record
Abstract
Quantum algorithms for Hamiltonian simulation and linear differential equations more generally have provided promising exponential speed-ups over classical computers on a set of problems with high real-world interest. However, extending this to a nonlinear problem has proven challenging, with exponential lower bounds having been demonstrated for the time scaling. We provide a quantum algorithm matching these bounds. Specifically, we find that for a non-linear differential equation of the form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mi>u</mml:mi><mml:mo fence="false" stretchy="false">&#x27E9;</mml:mo></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mi>A</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mi>u</mml:mi><mml:mo fence="false" stretchy="false">&#x27E9;</mml:mo><mml:mo>+</mml:mo><mml:mi>B</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mi>u</mml:mi><mml:msup><mml:mo fence="false" stretchy="false">&#x27E9;</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>&#x2297;</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> for evolution of time <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>T</mml:mi></mml:math>, error tolerance <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03F5;</mml:mi></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>c</mml:mi></mml:math> dependent on the strength of the nonlinearity, the number of queries to the differential operators that approaches the scaling of the quantum lower bound of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>e</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>o</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>T</mml:mi><mml:mo fence="false" stretchy="false">&#x2016;</mml:mo><mml:mi>B</mml:mi><mml:mo fence="false" stretchy="false">&#x2016;</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> queries in the limit of strong non-linearity. Finally, we introduce a classical algorithm based on the Euler method allowing comparably scaling to the quantum algorithm in a restricted case, as well as a randomized classical algorithm based on path integration that acts as a true analogue to the quantum algorithm in that it scales comparably to the quantum algorithm in cases where sign problems are absent.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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".