# \([[7,1,3]]\) Steane code[1]

## Description

A \([[7,1,3]]\) CSS code that is the smallest qubit CSS code to correct a single-qubit error [2]. The code is constructed using the classical binary \([7,4,3]\) Hamming code for protecting against both \(X\) and \(Z\) errors.

The code's stabilizer generator matrix blocks \(H_{X}\) and \(H_{Z}\) are both the parity-check matrix for the \([7,4,3]\) Hamming code, \begin{align} H_{X} = H_{Z} = \left(\begin{matrix} 0&0&0&1&1&1&1\\ 0&1&1&0&0&1&1\\ 1&0&1&0&1&0&1 \end{matrix}\right). \tag*{(1)}\end{align}

The stabilizer group for the Steane code has six generators, three \(X\)-type and three \(Z\)-type, which can be thought of as lying on the three trapezoids of the following tiling of the triangle. Figure I.

Logical codewords are \begin{align} \begin{split} |\overline{0}\rangle&=\frac{1}{\sqrt{8}}\Big(|0000000\rangle+|1010101\rangle+|0110011\rangle+|1100110\rangle\\&\,\,\,\,\,\,\,\,+|0001111\rangle+|1011010\rangle+|0111100\rangle+|1101001\rangle\Big)\\|\overline{1}\rangle&=\frac{1}{\sqrt{8}}\Big(|1111111\rangle+|0101010\rangle+|1001100\rangle+|0011001\rangle\\&\,\,\,\,\,\,\,\,+|1110000\rangle+|0100101\rangle+|1000011\rangle+|0010110\rangle\Big)~. \end{split} \tag*{(2)}\end{align}

The automorphism group of the code is \(PGL(3,2)\) [3]. It is one of sixteen distinct \([[7,1,3]]\) codes [4].

## Protection

## Encoding

## Transversal Gates

## Gates

## Fault Tolerance

## Realizations

## Parents

- Honeycomb (6.6.6) color code — Steane code is a 2D color code defined on a seven-qubit patch of the 6.6.6 tiling.
- \([[2^r-1,1,3]]\) simplex code
- \([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code
- \([[2^{2r-1}-1,1,2^r-1]]\) quantum punctured Reed-Muller code
- Quantum quadratic-residue (QR) code — The Steane code is a qubit quantum QR code [31,32].
- Quantum data-syndrome (QDS) code — There exists a set of stabilizer generators for the Steane code that make it a QDS code [33].
- Finite-geometry (FG) QLDPC code — The Steane code is the \(m=1\) member of the \([[2^{2m}+2^{m}+1,1,>2^{m}]]\) PG-QLDPC code family that is constructed from codes corresponding to lines and affine charts in \(PG(2,2^m)\) via the CSS construction [34; Def. 4.9].
- Group-representation code — The Steane code is a group-representation code with \(G\) being the \(2O\) subgroup of \(SU(2)\) [35].
- Concatenated Steane code — The concatenated Steane code at level \(m=1\) is the Steane code.
- Planar-perfect-tensor code — The Steane code is the smallest heptagon holographic code. The encoding of more general heptagon holographic codes is a holographic tensor network consisting of the encoding isometry for the Steane code, which is a planar-perfect tensor.

## Cousins

- Cyclic quantum code — The Steane code is equivalent to a cyclic code via qubit permutations [36; Exam. 1].
- Codeword stabilized (CWS) code — The Steane code is equivalent via a single-qubit Clifford unitary to a CWS code for a particular graph and classical code [36; Exam. 4].
- Graph quantum code — Four non-isomorphic graphs yield graph quantum codes that are equivalent to the Steane code under a single-qubit-Clifford circuit [37].
- EA qubit stabilizer code — The Steane code is globally equivalent to a \([[6,1,3;1]]\) code, which is the smallest EA CSS code with that distance [2].
- \([7,4,3]\) Hamming code — The Steane code is constructed from the \([7,4,3]\) classical Hamming code via the CSS construction.
- Quantum Lego code — The Steane code can be built from two \([[4,2,2]]\) codes in the quantum Lego code framework [38].
- \([[10,1,2]]\) CSS code — A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[10,1,2]]\) code [15].
- \([[4,2,2]]\) Four-qubit code — The Steane code can be built from two \([[4,2,2]]\) codes in the quantum Lego code framework [38].
- \([[5,1,2]]\) rotated surface code — The \([[5,1,2]]\) rotated surface code can be obtained by morphing the Steane code [39].
- Quantum divisible code — A fault-tolerant \(T\) gate on the Steane code can be obtained by concatenating with particular quantum divisible codes [40].
- Raussendorf-Bravyi-Harrington (RBH) cluster-state code — Concatenation of the RBH code with small codes such as the \([[2,1,1]]\) repetition code, \([[4,1,1,2]]\) subsystem code, or Steane code can improve thresholds [41].
- \([[15,1,3]]\) quantum Reed-Muller code — The \([[15,1,3]]\) code can be viewed as a (gauge-fixed) doubled color code obtained from the Steane code via the doubling transformation [42]. A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[15,1,3]]\) code [11–14,43].

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## Page edit log

- Eric Huang (2024-03-18) — most recent
- Victor V. Albert (2022-08-04)
- Victor V. Albert (2022-03-14)
- Joseph T. Iosue (2021-12-19)

## Cite as:

“\([[7,1,3]]\) Steane code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/steane