Cellular-Automaton Decoders for Topological Quantum Memories

Decoders are one of the necessary building blocks for robust local quantum memories, basic primitives that must arguably be part of any functioning topological quantum computer. While surface codes have raised the prospects for scalability and performance of quantum storage devices, the task of decoding resorts to complicated classical algorithms. Such decoding algorithms are typically designed for traditional centralized computing architectures. This questions whether active decoding can be fast enough to outpace the decoherence time in systems with many qubits, giving rise to significant challenges. In this work, we address this question by recasting error recovery as a dynamical process on a field generating cellular automaton. We envisage quantum systems controlled by a classical hardware composed of small local memories, communicating with nearest neighbors only, and repeatedly performing identical simple update rules. This framework for constructing topological quantum memories does not require any global operations or complex decoding algorithms. Furthermore, the local updates rules do not have to be perfect or synchronized, relaxing many of the former requirements on decoding devices. Hence, our work raises prospects for the technical feasibility of scalable and ultra-fast decoding devices. Through comprehensive numerical study (see Fig. 1c) we find that equipping error syndromes (anyons) with an attractive scalar field of the form 1/rα, can provide a working decoder with an error correction threshold and exponential error suppression. Considering the anyons to move according to the gradient of this field, we give an argument why error correction is possible only if α ≥ 1, i.e. the field is not too long ranged. A key insight is that in a discretized approximation of such fields can be generated via simple and local cellular automaton update rules in an efficient and robust manner. The field takes real values on a periodic D-dimensional square lattice ZL , with anyons constrained to a two-dimensional plane. At time t and location x ∈ ZL the field has a value φt(x), and at later times updates according to