23, 24 In this work, we focus on decoding devices but it should be clear that our design incentives promise a generalisation to new schemes of dissipative self-correction. This would amount to a dissipative self-correcting memory. In this setting error correction acts simultaneously with error creation, effectively preventing the creating of long error strings. Indeed, provided that the cellular automaton update rules do not change in time, the decoding process can be interpolated down to continuous time. Our class of models in some sense combines the benefits of active and of passive memories. 15– 22 Prior proposals typically use quantum auxiliary systems instead of classical auxiliary systems for studies of self-correcting memories, where full numerical investigations are prohibitively difficult. Our schemes share some features with other proposals which add an auxiliary system in order to enforce suppression of error creation. Finally, we study a class of long-range fields and their decoding capabilities, and observe that certain fields are too long-range to reliably identify excitation pairs. We also provide detailed analysis of the equilibration and propagation characteristics of the fields generated by the automata, which justifies the choice of tunable parameters in our decoders. Two of our decoders have comparable threshold values to more conventional decoders that require more sophisticated hardware. We provide extensive numerical analysis of three specific decoders, two of which use a 2D automaton and the third using a 3D automaton. Inspired by classical electrostatics and gravitational fields, we identify local update rules that induce such error correcting long-range fields. That way, excitations will tend to collapse together rather than to extend out and create logical errors in the code. The operational principles of our automata are based on the mediation of attractive long-range interactions between excitations. That is, we develop simple cellular automata that efficiently perform active error correction on the toric code. Our design can be implemented via classical hardware composed of small local memories and a small set of local operations only depending on neighbouring memories and neighbouring physical qubits, without constituting universal local processors 13, 14 or demanding an explicit message routing system. In this work, we propose an entirely new approach towards designing topological quantum memories that naturally incorporates parallelization. However, the complexity, time lag, and communications traffic of such a system is not fully understood and has not been simulated. By incorporating a message routing system into a lattice of cores, such long-range communication can be achieved with only nearest-neighbour connections. 8, 12 This improves prospects, but still requires a computer with communication between many spatially separated cores. 10, 11 Some of them offer the possibility of parallelization enabling a runtime logarithmic in the system size. Prior proposals have focused on the development of efficient decoding algorithms, 7– 9 or decoders with high error thresholds. When performing active error correction, it is essential for the decoding process to be much quicker than the decoherence time. 6 Schemes based on sequential measurements of the system’s error syndromes, and subsequent elimination of errors have been suggested to preserve the logical subspace. However, in three or fewer dimensions, excitations propagate at little energy cost under thermal dynamics, rapidly corrupting the encoded information. 1, 2 Topological codes in particular have emerged as the most promising quantum error correcting codes, 3 where the toric code 4, 5 is a paradigmatic example. As quantum coherence is intrinsically fragile, it is clear that increased robustness of the encoded information needs to rely heavily on quantum error correction. Yet, it is the necessary first step in the effort to scale up quantum computing and quantum communication to a commercially viable level. Prolonging the lifetime of quantum information stored in a quantum device is a monumental challenge.
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