Note on Logical Gates by Gauge Field Formalism of Quantum Error Correction
Junichi Haruna
TL;DR
The paper extends the gauge-field formalism for CSS codes to systematically construct and decompose a broad class of logical gates, including S, Hadamard, T, CZ, and multi-CZ, as exponentials of polynomials in electric and magnetic gauge fields $a$ and $b$. It proves that the logical action of these gates depends only on the (co)homology classes of the associated 1-chains, ensuring well-defined logical operations across code representatives. The framework applies to general CSS codes described by chain complexes (including hypergraphs) and connects quantum error correction with algebraic topology and lattice gauge theory, providing compact, explicit physical decompositions. The work also outlines avenues to incorporate fault-tolerance, extend to non-CSS or higher-dimensional codes, and address non-Abelian gauge structures, aiming to bridge QEC with gauge-theoretic formalisms in practical architectures.
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.