Matthew P.A. Fisher – UC Santa Barbara
Wednesday, December 18th , 10:45AM EST
Quantum Dynamics of the 1d Repetition Code
Fault tolerant error correction thresholds for quantum codes are traditionally obtained via map pings to classical statistical mechanics models. For example, the 1d repetition code subject to bit-ip noise and faulty measurements, which has a dynamical Ising Z2 symmetry, is mapped to the classical 2d random bond Ising model. Here, we revisit the 1d repetition code, and develop an exact stabilizer expansion" of the full time evolving density matrix which gives a dual representation of the classical 2d random bond Ising model. However, with generic Z2 respecting dynamics the stabilizer expansion breaks down and a full quantum description is required. The resulting steady state of the quantum dynamics has three possible phases which can be characterized by the
spontaneous breaking of strong" and weak" Ising symmetries. Classicality follows if the strong, but not weak, symmetry is spontaneously broken - recovering the 2d classical random bond Ising model. If neither symmetry is broken one has a non-trivial mixed state density matrix that describes a quantum paramagnet". And with both strong and weak symmetry breaking the steady state retains all encoded information of the quantum code.