Simulation of quantum circuits by low-rank stabilizer decompositions

TL;DR

This paper advances classical simulation of quantum circuits by developing a comprehensive stabilizer rank theory and algorithms that enable simulating larger and more complex circuits, including those with arbitrary diagonal gates.

Contribution

It introduces a new mathematical framework for stabilizer rank, broadens simulation capabilities to include arbitrary diagonal gates, and demonstrates improved performance on large quantum circuits.

Findings

Simulated quantum algorithms with 40-50 qubits and over 60 non-Clifford gates.

Processed superpositions of approximately 10^6 stabilizer states.

Achieved performance improvements by optimizing non-Clifford component decompositions.

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 stabilizer rank, which for a pure state $\psi$ is defined to be the smallest integer $\chi$ such that $\psi$ is a superposition of $\chi$ 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 $\chi\sim10^6$ stabilizer states and sample from the full n-bit output distribution, improving on previous simulations which used $\sim 10^3$ 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.