Time-dependent Hamiltonian simulation with<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:math>-norm scaling
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Bibliographic record
Abstract
The difficulty of simulating quantum dynamics depends on the norm of the Hamiltonian. When the Hamiltonian varies with time, the simulation complexity should only depend on this quantity instantaneously. We develop quantum simulation algorithms that exploit this intuition. For sparse Hamiltonian simulation, the gate complexity scales with the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:math>norm<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>0</mml:mn></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">d</mml:mi></mml:mrow><mml:mi>τ</mml:mi><mml:mo fence="false" stretchy="false">‖</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:msub><mml:mo fence="false" stretchy="false">‖</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo movablelimits="true" form="prefix">max</mml:mo></mml:mrow></mml:msub></mml:math>, whereas the best previous results scale with<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>t</mml:mi><mml:munder><mml:mo movablelimits="true" form="prefix">max</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>τ</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:munder><mml:mo fence="false" stretchy="false">‖</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:msub><mml:mo fence="false" stretchy="false">‖</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo movablelimits="true" form="prefix">max</mml:mo></mml:mrow></mml:msub></mml:math>. We also show analogous results for Hamiltonians that are linear combinations of unitaries. Our approaches thus provide an improvement over previous simulation algorithms that can be substantial when the Hamiltonian varies significantly. We introduce two new techniques: a classical sampler of time-dependent Hamiltonians and a rescaling principle for the Schrödinger equation. The rescaled Dyson-series algorithm is nearly optimal with respect to all parameters of interest, whereas the sampling-based approach is easier to realize for near-term simulation. These algorithms could potentially be applied to semi-classical simulations of scattering processes in quantum chemistry.
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.002 | 0.001 |
| Scholarly communication | 0.002 | 0.001 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.001 | 0.003 |
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 it