Abstract
Active quantum error correction on topological codes is one of the most promising routes to long-term qubit storage. In view of future applications, the scalability of the used decoding algorithms in physical implementations is crucial. In this work, we focus on the one-dimensional Majorana chain and construct a strictly local decoder based on a self-dual cellular automaton. We study numerically and analytically its performance and exploit these results to contrive a scalable decoder with exponentially growing decoherence times in the presence of noise. Our results pave the way for scalable and modular designs of actively corrected one-dimensional topological quantum memories.
Highlights
Storing quantum information in a noisy, classical environment is essential for scalable quantum computation and communication [1]
We provide a detailed outline of the methods and approaches used to derive these results: In Subsec. 2.1 we start with a description of the quantum code defined by the degenerate ground state space of the Majorana chain, where dephasing is topologically suppressed and depolarizing errors are forbidden by fermionic parity superselection
We present two binary cellular automata (CA) that are known to perform well on density classification, one of which is self-dual; it can be rewritten in a form that complies with the “quantum handicap”: It naturally takes syndromes as input and produces correction operations as output
Summary
Storing quantum information in a noisy, classical environment is essential for scalable quantum computation and communication [1]. 5.2 we follow this idea and stack copies of TLV in the second dimension perpendicular to the quantum chain The depth of this classical circuit quantifies the hardware overhead required for the retention of the logical qubit in the presence of noise; as it directly relates to the decoding time of TLV, it grows sublinearly with the chain length, so that shallow circuits suffice for reasonably low error rates. We revisit the procedure of quantum error correction using syndrome measurements and demonstrate that it reduces to global majority voting in this particular case This decoding scheme features an exponentially growing lifetime of the encoded logical qubit with the chain length L. This sets the stage for the construction and study of a strictly local, inherently scalable replacement for global majority voting
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