Note on Logical Gates by Gauge Field Formalism of Quantum Error Correction

TL;DR

This paper develops a gauge field formalism to systematically construct and analyze logical gates for CSS quantum error-correcting codes, linking algebraic topology and quantum field theory.

Contribution

It extends the gauge field formalism to include a broad class of logical gates for CSS codes, providing explicit decompositions and topological insights.

Findings

Logical gates expressed as exponentials of gauge fields.

Logical action depends only on (co)homology classes.

Provides a systematic method for logical gate construction.

Abstract

The gauge field formalism, or operator-valued cochain formalism, has recently emerged as a powerful framework for describing quantum Calderbank-Shor-Steane (CSS) codes. In this work, we extend this framework to construct a broad class of logical gates for general CSS codes, including the S, Hadamard, T, and (multi-)controlled-Z gates, under the condition where fault-tolerance or circuit-depth optimality is not necessarily imposed. We show that these logical gates can be expressed as exponential of polynomial functions of the electric and magnetic gauge fields, which allows us to derive explicit decompositions into physical gates. We further prove that their logical action depends only on the (co)homology classes of the corresponding logical qubits, establishing consistency as logical operations. Our results provide a systematic method for formulating logical gates for general CSS codes, offering new insights into the interplay between quantum error correction, algebraic topology, and quantum field theory.