Simulation of quantum circuits by low-rank stabilizer decompositions

Bravyi, Sergey, Browne, Dan, Calpin, Padraic, Campbell, Earl, Gosset, David, & Howard, Mark. (2019). Simulation of quantum circuits by low-rank stabilizer decompositions. Quantum, 3(181). doi: https://doi.org/10.22331/q-2019-09-02-181

Abstract

Recent work has explored using the stabilizer formalism to classically simulate quantum circuits containing a few non-Clifford gates. The computational cost of such methods is directly related to the notion of s t a b i l i z e r rank , which for a pure state ψ is defined to be the smallest integer χ such that ψ is a superposition of χ stabilizer states. Here we develop a comprehensive mathematical theory of the stabilizer rank and the related approximate stabilizer rank. We also present a suite of classical simulation algorithms with broader applicability and significantly improved performance over the previous state-of-the-art. A new feature is the capability to simulate circuits composed of Clifford gates and arbitrary diagonal gates, extending the reach of a previous algorithm specialized to the Clifford+T gate set. We implemented the new simulation methods and used them to simulate quantum algorithms with 40-50 qubits and over 60 non-Clifford gates, without resorting to high-performance computers. We report a simulation of the Quantum Approximate Optimization Algorithm in which we process superpositions of χ ∼ 10 6 stabilizer states and sample from the full n -bit output distribution, improving on previous simulations which used ∼ 10 3 stabilizer states and sampled only from single-qubit marginals. We also simulated instances of the Hidden Shift algorithm with circuits including up to 64 T gates or 16 CCZ gates; these simulations showcase the performance gains available by optimizing the decomposition of a circuit's non-Clifford components.

Funder

Engineering and Physical Sciences Research Council
IBM Research Frontiers Institute

Publisher

Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften

Rights

Attribution-NonCommercial-NoDerivs 3.0 Ireland

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Mathematics (Scholarly Articles)