A Quadratic Speedup in the Optimization of Noisy Quantum Optical Circuits
Bibliographic record
Abstract
Linear optical quantum circuits with photon number resolving (PNR) detectors are used for both Gaussian Boson Sampling (GBS) and for the preparation of non-Gaussian states such as Gottesman-Kitaev-Preskill (GKP), cat and NOON states. They are crucial in many schemes of quantum computing and quantum metrology. Classically optimizing circuits with PNR detectors is challenging due to their exponentially large Hilbert space, and quadratically more challenging in the presence of decoherence as state vectors are replaced by density matrices. To tackle this problem, we introduce a family of algorithms that calculate detection probabilities, conditional states (as well as their gradients with respect to circuit parametrizations) with a complexity that is comparable to the noiseless case. As a consequence we can simulate and optimize circuits with twice the number of modes as we could before, using the same resources. More precisely, for an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>M</mml:mi></mml:math>-mode noisy circuit with detected modes <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>D</mml:mi></mml:math> and undetected modes <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>U</mml:mi></mml:math>, the complexity of our algorithm is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>M</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:munder><mml:mo>&#x220F;</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>i</mml:mi><mml:mspace width="0.111em"/><mml:mo>&#x2208;</mml:mo><mml:mspace width="0.111em"/><mml:mi>U</mml:mi></mml:mrow></mml:munder><mml:msubsup><mml:mi>C</mml:mi><mml:mi>i</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:munder><mml:mo>&#x220F;</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>i</mml:mi><mml:mspace width="0.111em"/><mml:mo>&#x2208;</mml:mo><mml:mspace width="0.111em"/><mml:mi>D</mml:mi></mml:mrow></mml:munder><mml:msub><mml:mi>C</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:math>, rather than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>M</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:munder><mml:mo>&#x220F;</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mspace width="0.111em"/><mml:mi>i</mml:mi><mml:mspace width="0.111em"/><mml:mo>&#x2208;</mml:mo><mml:mspace width="0.111em"/><mml:mi>D</mml:mi><mml:mspace width="0.167em"/><mml:mo>&#x222A;</mml:mo><mml:mspace width="0.167em"/><mml:mi>U</mml:mi></mml:mrow></mml:munder><mml:msubsup><mml:mi>C</mml:mi><mml:mi>i</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>C</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math> is the Fock cutoff of mode <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>i</mml:mi></mml:math>. As a particular case, our approach offers a full quadratic speedup for calculating detection probabilities, as in that case all modes are detected. Finally, these algorithms are implemented and ready to use in the open-source photonic optimization library MrMustard.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.003 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| 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".