Approximate symmetries and quantum error correction

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.

Verifiable quantum secret sharing scheme based on orthogonal product states

“…Furthermore, significant experimental progress has been made in the domain of QSS schemes, underscoring the feasibility of quantum-based approaches to secure information dissemination. [20][21][22][23] The majority of QSS schemes employ either quantum entanglement or single-particle quantum states as their foundation. Although quantum entanglement does occupy a critical position within the realm of QSS, its presence is not imperative for the functioning of QSS protocols.…”