Targeted Clifford logical gates for hypergraph product codes

TL;DR

This paper develops a framework for designing logical Clifford gates for hypergraph product codes, enabling explicit logical operations for these quantum error-correcting codes, exemplified by the $[[18,2,3]]$ toric code.

Contribution

It introduces a method to construct explicit logical Clifford gates for hypergraph product codes using symplectic matrices, expanding the toolkit for quantum error correction.

Findings

Derived symplectic matrices for Clifford gates

Constructed explicit logical circuits for hypergraph product codes

Demonstrated the approach on the $[[18,2,3]]$ toric code

Abstract

Starting with an explicit framework for designing logical Clifford circuits for CSS codes, we construct logical gates for Hypergraph Product Codes. We first derive symplectic matrices for CNOT, CZ, Phase, and Hadamard operators, which together generate the Clifford group. This enables us to design explicit transformations that result in targeted logical gates for arbitrary codes in this family. As a concrete example, we give logical circuits for the $[[18,2,3]]$ toric code.