Symmetry-accelerated classical simulation of Clifford-dominated circuits

TL;DR

This paper introduces symmetry-based methods to significantly improve the efficiency of classical simulations of Clifford-dominated quantum circuits, enabling larger and more complex circuit simulations.

Contribution

It develops symmetry reduction techniques that drastically lower computational costs for stabilizer extent calculations, allowing optimal decompositions on larger unitaries.

Findings

Achieved optimal decompositions for unitaries on up to seven qubits using standard laptops.

Demonstrated exponential runtime improvements in simulating quantum Fourier transforms.

Provided new insights into non-stabilizer properties of certain quantum gates and states.

Abstract

Classical simulation of quantum circuits plays a crucial role in validating quantum hardware and delineating the boundaries of quantum advantage. Among the most effective simulation techniques are those based on the stabilizer extent, which quantifies the overhead of representing non-Clifford operations as linear combinations of Clifford unitaries. However, finding optimal decompositions rapidly becomes intractable as it constitutes a superexponentially large optimization problem. In this work, we exploit symmetries in the computation of the stabilizer extent, proving that for real, diagonal, and real-diagonal unitaries, the optimization can be restricted to the corresponding subgroups of the Clifford group without loss of optimality. This ``strong symmetry reduction'' drastically reduces computational cost, enabling optimal decompositions of unitaries on up to seven qubits using a standard laptop--far beyond previous two-qubit limits. Additionally, we employ a ``weak symmetry reduction'' method that leverages additional invariances to shrink the search space further. Applying these results, we demonstrate exponential runtime improvements in classical simulations of quantum Fourier transform circuits and measurement-based quantum computations on the Union Jack lattice, as well as new insights into the non-stabilizer properties of multi-controlled-phase gates and unitaries generating hypergraph states. Our findings establish symmetry exploitation as a powerful route to scale classical simulation techniques and deepen the resource-theoretic understanding of quantum advantage.