IQIM Postdoctoral and Graduate Student Seminar
Abstract: Quantum error correction (QEC) and continuous symmetries, two key notions in quantum information and physics, are incompatible with each other. Here, we systematically study the competition between the QEC capability and degree of symmetry associated with a quantum code in a quantitative manner. Three meaningful measures of approximate symmetries, respectively based on the violation of covariance conditions over the entire symmetry group or at a local point, and the violation of charge conservation, are defined and considered. Each measure induces a corresponding characterization of approximately covariant codes. We explicate a host of different ideas and techniques that enable us to derive various forms of trade-off relations between the QEC inaccuracy and all symmetry violation measures. More specifically, we introduce two frameworks for establishing the trade-offs respectively based on the notions of charge fluctuation and gate implementation error and employ methods including the Knill--Laflamme conditions as well as quantum metrology and quantum resource theory for the derivation. From the perspective of fault-tolerant quantum computing, we demonstrate fundamental limits on the precision of transversally implementable logical gates for general and stabilizer QEC codes derived from our bounds on symmetry violation. To exemplify nontrivial approximately covariant codes and understand the achievability of the above fundamental limits, we analyze the behaviors of two explicit types of codes: a parametrized extension of the thermodynamic code (which gives a construction of a code family that continuously interpolates between exact QEC and exact symmetry), and the quantum Reed-Muller codes. We show that both codes can saturate the scaling of the bounds for group-global covariance and charge conservation asymptotically, indicating the near-optimality of these bounds and codes.