A Quadratic Speedup in the Optimization of Noisy Quantum Optical Circuits

Robbe De Prins1, Yuan Yao2, Anuj Apte3,4, and Filippo M. Miatto2,3

1Photonics Research Group, INTEC, Ghent University – imec, Sint-Pietersnieuwstraat 41, 9000 Ghent, Belgium
2Télécom Paris and Institut Polytechnique de Paris, LTCI, 20 Place Marguerite Perey, 91120 Palaiseau, France
3Xanadu, Toronto, ON, M5G 2C8, Canada
4Kadanoff Center for Theoretical Physics & Enrico Fermi Institute, Department of Physics, University of Chicago, Chicago, IL 60637

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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 $M$-mode noisy circuit with detected modes $D$ and undetected modes $U$, the complexity of our algorithm is $O(M^2 \prod_{i \mskip2mu \in \mskip2mu U} C_i^2 \prod_{i \mskip2mu \in \mskip2mu D} C_i)$, rather than $O(M^2 \prod_{\mskip2mu i \mskip2mu \in \mskip2mu D \mskip3mu \cup \mskip3mu U} C_i^2)$, where $C_i$ is the Fock cutoff of mode $i$. 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.

Animated versions of some figures in the manuscript (GIFs) are included in the Supplementary Materials, as well as here: https://github.com/rdprins/GIFs_NoisyCircuits

Quantum photonic circuits with photon-number-resolving detectors play a pivotal role in the advancement of quantum computing. These circuits have been physically realized to showcase that quantum computers have the potential to surpass classical computers. Similar circuits are being designed to generate complex quantum optical states like GKP states which serve as building blocks to make a practical and useful photonic quantum computer.
Scientists can rely on classical computers to simulate and optimize these circuits. However, such numerical simulations are fundamentally challenging, especially as the size of the circuit grows (if quantum circuits could be simulated efficiently, they wouldn’t be able to outperform classical computers in the first place). More precisely, as circuits grow larger, both the time needed for simulations and the required computer memory increase exponentially. There is little one can do to escape this.
This challenge becomes even greater when we move away from ideal circuits and we take into account that part of the light inevitably escapes from the circuit. Incorporating such realistic effects adds a quadratic increase in computational demands on top of the existing exponential growth. In this manuscript, we introduce a new family of algorithms that can take such real-world effects into account without adding the extra quadratic load. This allows us to simulate and optimize realistic circuits with the same effort as ideal ones.

► BibTeX data

► References

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Cited by

[1] Pranav Chandarana, Koushik Paul, Mikel Garcia-de-Andoin, Yue Ban, Mikel Sanz, and Xi Chen, "Photonic counterdiabatic quantum optimization algorithm", arXiv:2307.14853, (2023).

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