Quantum error correction meets continuous symmetries: fundamental trade-offs and case studies

Quantum error correction and symmetries are two key notions in quantum information and physics. The competition between them has fundamental implications in fault-tolerant quantum computing, many-body physics and quantum gravity. We systematically study the competition between quantum error correction and continuous symmetries associated with a quantum code in a quantitative manner. We derive various forms of trade-off relations between the quantum error correction inaccuracy and three types of symmetry violation measures. We introduce two frameworks for understanding and establishing the trade-offs based on the notions of charge fluctuation and gate implementation error. From the perspective of fault-tolerant quantum computing, we demonstrate fundamental limitations on transversal logical gates. We also analyze the behaviors of two near-optimal codes: a parametrized extension of the thermodynamic code, and quantum Reed-Muller codes.--- Sisi Zhou is a postdoc at Caltech, and will be joining Perimeter Institute as a junior faculty in Fall 2023. She graduated from Department of Physics at Yale University in 2021. Her work focuses on quantum error correction, quantum metrology, and the interplay between them.