Approximate symmetries and quantum error correction

TL;DR

This paper systematically analyzes the tension between continuous symmetries and quantum error correction, establishing quantitative bounds and exploring implications for fault-tolerant quantum computation and physics.

Contribution

It introduces measures of approximate symmetries, derives trade-off bounds with QEC accuracy, and provides explicit code examples nearly saturating these bounds.

Findings

Derived quantitative limitations on transversally implementable logical gates.

Established trade-off bounds between approximate symmetries and QEC accuracy.

Presented explicit quantum codes from Reed–Muller and thermodynamic codes that nearly saturate bounds.

Abstract

Quantum error correction (QEC) is a key concept in quantum computation as well as many areas of physics. There are fundamental tensions between continuous symmetries and QEC. One vital situation is unfolded by the Eastin--Knill theorem, which forbids the existence of QEC codes that admit transversal continuous symmetry actions (transformations). Here, we systematically study the competition between continuous symmetries and QEC in a quantitative manner. We first define a series of meaningful measures of approximate symmetries motivated from different perspectives, and then establish a series of trade-off bounds between them and QEC accuracy utilizing multiple different methods. Remarkably, the results allow us to derive general quantitative limitations of transversally implementable logical gates, an important topic in fault-tolerant quantum computation. As concrete examples, we showcase two explicit types of quantum codes, obtained from quantum Reed--Muller codes and thermodynamic codes, respectively, that nearly saturate our bounds. Finally, we discuss several potential applications of our results in physics.