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## Efficient decoding of random errors for quantum expander codes

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Omar Fawzi
Antoine Grospellier
Anthony Leverrier

#### Abstract

We show that quantum expander codes, a constant-rate family of quantum LDPC codes, with the quasi-linear time decoding algorithm of Leverrier, Tillich and Z\'emor can correct a constant fraction of random errors with very high probability. This is the first construction of a constant-rate quantum LDPC code with an efficient decoding algorithm that can correct a linear number of random errors with a negligible failure probability. Finding codes with these properties is also motivated by Gottesman's construction of fault tolerant schemes with constant space overhead. In order to obtain this result, we study a notion of $\alpha$-percolation: for a random subset $W$ of vertices of a given graph, we consider the size of the largest connected $\alpha$-subset of $W$, where $X$ is an $\alpha$-subset of $W$ if $|X \cap W| \geq \alpha |X|$.

### Dates and versions

hal-01671348 , version 1 (22-12-2017)

### Identifiers

• HAL Id : hal-01671348 , version 1
• ARXIV :

### Cite

Omar Fawzi, Antoine Grospellier, Anthony Leverrier. Efficient decoding of random errors for quantum expander codes. 2017. ⟨hal-01671348⟩

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