Faster Born probability estimation via gate merging and frame optimisation

Nikolaos Koukoulekidis1, Hyukjoon Kwon1,2, Hyejung H. Jee3, David Jennings1,4, and M. S. Kim1

1Department of Physics, Imperial College London, London SW7 2AZ, UK
2Korea Institute for Advanced Study, Seoul, 02455, Korea
3Department of Computing, Imperial College London, London SW7 2AZ, UK
4School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, UK

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Outcome probability estimation via classical methods is an important task for validating quantum computing devices. Outcome probabilities of any quantum circuit can be estimated using Monte Carlo sampling, where the amount of negativity present in the circuit frame representation quantifies the overhead on the number of samples required to achieve a certain precision. In this paper, we propose two classical sub-routines: circuit gate merging and frame optimisation, which optimise the circuit representation to reduce the sampling overhead. We show that the runtimes of both sub-routines scale polynomially in circuit size and gate depth. Our methods are applicable to general circuits, regardless of generating gate sets, qudit dimensions and the chosen frame representations for the circuit components. We numerically demonstrate that our methods provide improved scaling in the negativity overhead for all tested cases of random circuits with Clifford+$T$ and Haar-random gates, and that the performance of our methods compares favourably with prior quasi-probability simulators as the number of non-Clifford gates increases.

Quantum computers are expected to provide significant speed-ups compared to classical computing for certain tasks. Determining the set of computational tasks that can be accelerated by the use of quantum computers is a notoriously hard question. A natural approach towards carving out this set consists of studying how well classical computers can simulate these tasks. A large family of such classical simulators relies on representing the tasks as stochastic processes that propagate an input state probabilistically. Quantum tasks contain negative probabilities that in principle require exponential runtimes to simulate classically. In this work, we introduce an efficient optimisation to the classical simulation strategy which works by minimising the negative probabilities that arise during the process. Our results are of practical significance as they provide a faster way to simulate quantum computers, while on a fundamental level, they probe the extent to which negative probabilities are an indicator of non-classical physics.

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[3] Denis A. Kulikov, Vsevolod I. Yashin, Aleksey K. Fedorov, and Evgeniy O. Kiktenko, "Minimizing the negativity of quantum circuits in overcomplete quasiprobability representations", Physical Review A 109 1, 012219 (2024).

[4] Tobias Haug and M. S. Kim, "Scalable Measures of Magic Resource for Quantum Computers", PRX Quantum 4 1, 010301 (2023).

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