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Symmetry-Accelerated Classical Simulation of Clifford-Dominated Circuits

Giulio Camillo1,*, Filipa C.R. Peres1,†, Markus Heinrich2,‡, and Juani Bermejo-Vega1,3

  • *G.C. and F.C.R.P. contributed equally.
  • †Contact author: fcrperes@onsager.ugr.es
  • ‡Contact author: markus.heinrich@uni-koeln.de
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PRX Quantum 7, 020356 – Published 10 June, 2026

DOI: https://doi.org/10.1103/nnzz-481j

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 insights into the nonstabilizer properties of multicontrolled 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.

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Popular Summary

Faster classical simulation of quantum circuits through symmetry.

Classical simulation of quantum computers plays a double role in today’s quantum technology landscape: It validates the behavior of real hardware and sharpens our understanding of what quantum machines can do that classical ones cannot. Among the most powerful techniques are stabilizer-based simulators, which treat quantum circuits as sums of simple, easy-to-simulate Clifford building blocks. The catch is that finding the decomposition resulting in the fastest simulation is a vast optimization problem that grows superexponentially with the number of qubits. Until now, it could only be solved for the very smallest pieces of a circuit, one or two qubits at a time.

In this work, we show that symmetry provides a significant shortcut. Many of the non-Clifford gates that appear in real quantum algorithms, such as multicontrolled phase gates, Toffoli-like gates, and rotations, carry built-in symmetries. We prove that for real, diagonal, or real-diagonal gates, the optimization can be safely restricted to the corresponding symmetric subset of Clifford operations, without losing the optimal answer. Additional symmetries can be leveraged to further reduce the search space. Together, these reductions enable us to optimally decompose gates acting on up to seven qubits on a standard laptop—several orders of magnitude over the previous state of the art.

Applied to concrete circuits, the method yields striking speedups. Simulating a 16-qubit quantum Fourier transform, a core subroutine of Shor’s factoring algorithm, becomes roughly 300 times faster. Any universal measurement-based quantum computation performed on a “Union Jack” lattice gains an exponential improvement in simulation cost. We also obtain results on the nonstabilizer content of multicontrolled phase gates and hypergraph states, clarifying how entanglement and “magic” combine to fuel quantum advantage. Beyond immediate gains, our findings point toward symmetry as a general organizing principle for classical simulation and raise a conceptual question for future work: Which mathematical symmetries admit such strong reductions, and what does this tell us about the resources that truly power quantum computation?

Article Text

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PRX Quantum 7, 020356– Published 10 June, 2026
  • Received 12 December 2025
  • Accepted 23 April 2026
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