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Fault-Tolerant Logical Clifford Gates from Code Automorphisms

Hasan Sayginel1,2,*,†, Stergios Koutsioumpas1,*, Mark Webster1,*, Abhishek Rajput3, and Dan E. Browne1

  • *These authors contributed equally to this work.
  • †Contact author: hasan.sayginel.17@ucl.ac.uk
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PRX Quantum 6, 030343 – Published 5 September, 2025

DOI: https://doi.org/10.1103/vf7v-cpq9

Abstract

We study the implementation of fault-tolerant logical Clifford gates on stabilizer quantum error-correcting codes based on their symmetries. Our approach is to map the stabilizer code to a binary linear code, compute its automorphism group, and impose constraints based on the Clifford operators permitted. We provide a rigorous formulation of the method for finding automorphisms of stabilizer codes and generalize the ZX dualities previously introduced for Calderbank-Shor-Steane (CSS) codes to non-CSS codes. We provide a python package implementing our algorithms that uses the computational-algebra magma software system for certain subroutines. Our algorithms map automorphism-group generators to physical circuits, calculate Pauli corrections based on the destabilizers of the code, and determine their logical action. We discuss the fault tolerance of the circuits and include examples of gates through automorphisms for the [[4,2,2]] and perfect [[5,1,3]] codes, bivariate bicycle codes, and the best-known distance codes.

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Popular Summary

While quantum computing holds much promise to solve problems that are beyond classical computers, quantum computers are highly susceptible to errors arising from environmental noise and imperfectly implemented operations. Quantum error correction employs stabilizer codes to protect quantum information from such errors. Errors can be detected and actively corrected using such codes, creating a quantum memory that preserves quantum information over time. To implement quantum algorithms, we will also need to perform logical operations on the stored quantum information. These logical operations will need to be fault tolerant—i.e., any errors introduced by applying them can also be detected and corrected.

In this paper, we show how to identify a class of fault-tolerant logical operators for any stabilizer code. We focus on swap-transversal logical operators that involve swapping qubits and then applying certain single-qubit operations called Clifford gates. Our methods work by mapping the stabilizer code to a classical binary code. Permutations of the bits of the binary code that preserve the code are called permutation automorphisms. We show that these permutation automorphisms correspond to swap-transversal logical operators of the stabilizer code. By changing the mapping, we can change which single-qubit Clifford gates are allowed in the logical operators.

We apply our methods to some well-known stabilizer-code families. For instance, we show that the best-known-distance stabilizer codes listed on http://www.codetables.de that encode one logical qubit typically have the widest possible range of swap-transversal logical operators. We also apply our techniques to the bivariate bicycle codes of IBM. We find a much wider range of logical operators for these than was previously known and study codes with up to 756 physical qubits.

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Outline

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PRX Quantum 6, 030343– Published 5 September, 2025
  • Received 22 December 2024
  • Revised 2 May 2025
  • Accepted 10 July 2025
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