Continuous Symmetries and Approximate Quantum Error Correction
February 11th, 2019 PHILIPPE FAIST Institute for Quantum Information and Matter, Caltech

Quantum error correction and symmetry arise in many areas of physics, including quantum many-body systems, quantum metrology in the presence of noise, fault-tolerant quantum computation, and holographic quantum gravity. We discuss the compatibility of these two important principles. If a logical quantum system is encoded in n physical subsystems, we say that the code is covariant with respect to a symmetry group G if a G transformation on the logical system can be realized by performing transformations on the individual subsystems. For the case of a G-covariant code where G is a continuous group, we derive a lower bound on the infidelity of error correction following erasure of a subsystem. This bound approaches zero when the number of subsystems n is large or when the dimension d of each subsystem is large. We exhibit examples of codes which achieve approximately the same scaling of infidelity with n or d as the lower bound. We then prove an approximate version of the Eastin-Knill theorem, showing that if a code admits a universal set of transversal gates and corrects against erasure with a fixed accuracy, then for each logical qubit we need a number of physical qubits per subsystem that is is inversely proportional to the error parameter; in a regime where the code can accurately resolve individual logical basis states of a mixed state, the number of qubits per physical subsystem is exponential in the number of logical qubits. Furthermore, we construct codes covariant with respect to the full logical unitary group which achieve good accuracy when the subsystem dimension d is large (using random codes) or when the number of subsystems n is large (using codes based on many-body W-states). We also develop a framework to systematically construct codes that are covariant with respect to general groups, obtaining natural generalizations of standard qubit codes to, for instance, oscillators and rotors. In the context of the AdS/CFT correspondence, our approach provides insight into how time evolution in the bulk corresponds to time evolution on the boundary without violating the Eastin-Knill theorem, and our five-rotor code can be stacked to form a covariant holographic code.

Seminar, February 11, 2019, 15:00. ICFO’s Seminar Room

Hosted by Prof. Antonio Acín