Transversal Diagonal Logical Operators for Stabiliser Codes

TL;DR

This paper develops algorithms to identify and construct transversal diagonal logical operators in CSS quantum error correction codes, expanding fault-tolerant quantum computation capabilities beyond traditional Pauli operators.

Contribution

It introduces more efficient algorithms for finding all transversal diagonal logical operators and a method to design CSS codes with specific diagonal logical Clifford hierarchy operators.

Findings

Algorithms for identifying transversal diagonal logical operators are more general and computationally efficient.

A construction method for CSS codes with desired diagonal logical operators is proposed.

Diagonal Clifford hierarchy operators can be represented as diagonal XP operators, enabling broader applications.

Abstract

Storing quantum information in a quantum error correction code can protect it from errors, but the ability to transform the stored quantum information in a fault tolerant way is equally important. Logical Pauli group operators can be implemented on Calderbank-Shor-Steane (CSS) codes, a commonly-studied category of codes, by applying a series of physical Pauli X and Z gates. Logical operators of this form are fault-tolerant because each qubit is acted upon by at most one gate, limiting the spread of errors, and are referred to as transversal logical operators. Identifying transversal logical operators outside the Pauli group is less well understood. Pauli operators are the first level of the Clifford hierarchy which is deeply connected to fault-tolerance and universality. In this work, we study transversal logical operators composed of single- and multi-qubit diagonal Clifford hierarchy gates. We demonstrate algorithms for identifying all transversal diagonal logical operators on a CSS code that are more general or have lower computational complexity than previous methods. We also show a method for constructing CSS codes that have a desired diagonal logical Clifford hierarchy operator implemented using single qubit phase gates. Our methods rely on representing operators composed of diagonal Clifford hierarchy gates as diagonal XP operators and this technique may have broader applications.