Quantum simulation of Lindbladian dynamics via repeated interactions
Bibliographic record
Abstract
Abstract The Lindblad equation generalizes the Schrödinger equation to quantum systems that undergo dissipative dynamics. The quantum simulation of Lindbladian dynamics is therefore non-unitary, preventing a naive application of state-of-the-art quantum algorithms. Here, we make use of an approximate correspondence between Lindbladian dynamics and evolution based on repeated interaction (RI) CPTP maps to write down a Hamiltonian formulation of the Lindblad dynamics and derive a rigorous error bound on the master equation. Specifically, we show that the number of interactions needed to simulate the Liouvillian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi mathvariant="normal">e</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">L</mml:mi> </mml:mrow> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> within error ε scales in most physical scenarios as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>ν</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>t</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo fence="false" stretchy="false">‖</mml:mo> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">L</mml:mi> </mml:mrow> <mml:msubsup> <mml:mo fence="false" stretchy="false">‖</mml:mo> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo stretchy="false">→</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>ϵ</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> . This is significant because the error in the Lindbladian approximation to the dynamics is not explicitly bounded in existing quantum algorithms for open system simulations. We then provide quantum algorithms to simulate RI maps using an iterative qubitization approach and Trotter–Suzuki formulas, and specifically show that for iterative qubitization the number of operations needed to simulate the dynamics (for a fixed value of ν ) scales as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>α</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mi>t</mml:mi> <mml:mo>+</mml:mo> <mml:mi>ν</mml:mi> <mml:mi>log</mml:mi> <mml:mo></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>ϵ</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>log</mml:mi> <mml:mo></mml:mo> <mml:mi>log</mml:mi> <mml:mo></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>ϵ</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> in the limit where α 0 (the coefficient 1-norm for the system and bath Hamiltonians) asymptotically dominates over the corresponding factor for the interaction Hamiltonian, which is often the case in weak coupling. This scaling would appear to be optimal if the complexity of ν is not considered, which underscores the importance of considering the error in the Liouvillian that we reveal in this work.
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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 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 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".