Good Quantum LDPC Codes with Linear Time Decoders

• URL: http://arxiv.org/abs/2206.07750v1
• Date: Wed, 15 Jun 2022 18:23:15 GMT
• Title: Good Quantum LDPC Codes with Linear Time Decoders
• Authors: Irit Dinur, Min-Hsiu Hsieh, Ting-Chun Lin, Thomas Vidick
• Abstract summary: We construct a new explicit family of good quantum low-density parity-check codes which additionally have linear time decoders. One of the main ingredients in the analysis is a proof of an essentiallyoptimal property for the tensor product of two random codes.
• Score: 18.16859234929625
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We construct a new explicit family of good quantum low-density parity-check codes which additionally have linear time decoders. Our codes are based on a three-term chain $(\mathbb{F}_2^{m\times m})^V \quad \xrightarrow{\delta^0}\quad (\mathbb{F}_2^{m})^{E} \quad\xrightarrow{\delta^1} \quad \mathbb{F}_2^F$ where $V$ ($X$-checks) are the vertices, $E$ (qubits) are the edges, and $F$ ($Z$-checks) are the squares of a left-right Cayley complex, and where the maps are defined based on a pair of constant-size random codes $C_A,C_B:\mathbb{F}_2^m\to\mathbb{F}_2^\Delta$ where $\Delta$ is the regularity of the underlying Cayley graphs. One of the main ingredients in the analysis is a proof of an essentially-optimal robustness property for the tensor product of two random codes.

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