Fault-Tolerant Logical Clifford Gates from Code Automorphisms

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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>Z</a:mi><a:mi>X</a:mi></a:math> dualities previously introduced for Calderbank-Shor-Steane (CSS) codes to non-CSS codes. We provide a package implementing our algorithms that uses the computational-algebra 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 <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mo stretchy="false">[</c:mo><c:mo stretchy="false">[</c:mo><c:mn>4</c:mn><c:mo>,</c:mo><c:mn>2</c:mn><c:mo>,</c:mo><c:mn>2</c:mn><c:mo stretchy="false">]</c:mo><c:mo stretchy="false">]</c:mo></c:math> and perfect <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mo stretchy="false">[</i:mo><i:mo stretchy="false">[</i:mo><i:mn>5</i:mn><i:mo>,</i:mo><i:mn>1</i:mn><i:mo>,</i:mo><i:mn>3</i:mn><i:mo stretchy="false">]</i:mo><i:mo stretchy="false">]</i:mo></i:math> codes, bivariate bicycle codes, and the best-known distance codes.