Khesin-Lu-Shor code

Khesin-Lu-Shor code[1]

Description

A family of \([[m 2^m / (m+1), 2^m / (m+1), d(m)]]\) qubit CSS codes derived from the Hamming code, where \(m = 2^r - 1\). Their encoder-respecting form is the graph of a hypercube in \(m\) dimensions, and input nodes in the graph are codewords of the \([2^r-1,2^r-r-1,3]\) Hamming code [1].

Protection

The code distance satisfies \(\lfloor (m-1)/2 \rfloor \leq d(m) \leq m\) and is conjectured to be \(m\) for \(m \geq 7\) [1].

Decoding

A greedy graph decoder on the hypercube representation corrects at least \(\lfloor (m-1)/4 \rfloor - 1\) Pauli errors [1].

Cousins

\([2^r-1,2^r-r-1,3]\) Hamming code— The encoder-respecting form of the \([[m 2^m / (m+1), 2^m / (m+1), d(m)]]\) Khesin-Lu-Shor code is the graph of a hypercube in \(m = 2^r - 1\) dimensions, and input nodes in the graph are codewords of the \([2^r-1,2^r-r-1,3]\) Hamming code [1].
Hypercube code— The encoder-respecting form of the \([[m 2^m / (m+1), 2^m / (m+1), d(m)]]\) Khesin-Lu-Shor code is the graph of a hypercube in \(m = 2^r - 1\) dimensions, and input nodes in the graph are codewords of the \([2^r-1,2^r-r-1,3]\) Hamming code [1].
\([[7,1,3]]\) Steane code— The encoder-respecting form of both the Steane and Khesin-Lu-Shor codes is the graph of a hypercube [1].