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

\([[7,1,3]]\) 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.

Figure I: Stabilizer generators of the Steane code.

The Steane code can also be thought of as a code on all corners of a cube except one [3].

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 \(PGL(3,2)\) [4]. It is one of sixteen distinct \([[7,1,3]]\) codes [5].

Protection

The Steane code is a distance 3 code. It detects errors on 2 qubits, corrects errors on 1 qubit.

Encoding

Nine CNOT and four Hadamard gates ([6], Fig. 10.14).Fault-tolerant logical zero and logical plus state preparation on all-to-all and 2D grid qubit connectivity [7].

Transversal Gates

All single-qubit Clifford gates, which realize the \(2O\) binary octahedral subgroup of \(SU(2)\) [8,9].

Gates

Fault-tolerant logical zero and magic state preparation [10]. Magic-state preparation converts unbiased noise into biased noise [11].Pieceable fault-tolerant CCZ gate [12].

Fault Tolerance

A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[15,1,3]]\) code [13–16].A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[10,1,2]]\) code [17].Fault-tolerant logical zero and magic state preparation [10]. Magic-state preparation converts unbiased noise into biased noise [11].Fault-tolerant logical zero and logical plus state preparation on all-to-all and 2D grid qubit connectivity [7].Pieceable fault-tolerant CCZ gate [12].Syndrome measurement can be done with ancillary flag qubits [18,19] or with no extra qubits [20]. The depth of syndrome extraction circuits can be lowered by using past syndrome values [21].

Realizations

Trapped-ion devices: seven-qubit device in Blatt group [22]. Ten-qubit QCCD device by Quantinuum [23] realizing repeated syndrome extraction, real-time look-up-table decoding (yielding lower logical SPAM error rate than physical SPAM), and non-fault-tolerant magic-state distillation (see APS Physics Synopsis [24]). Fault-tolerant universal two-qubit gate set using T injection by Monz group [25]. Logical CNOT gate and Bell-pair creation between two logical qubits (yielding a logical fidelity higher than physical), including rounds of correction and fault-tolerant primitives such as flag qubits and pieceable fault tolerance, on a 20-qubit device by Quantinuum [26]; logical fidelity interval of the combined preparation-CNOT-measurement procedure was higher than that of the unencoded physical qubits. Multiple rounds of Steane error correction [27]. Fault-tolerant universal gate set via code switching between the Steane code and the \([[10,1,2]]\) code [17]. Post-selected fault-tolerant logical Bell-state preparation with logical error rates at least 10 times lower than physical rate on a device by Quantinuum [28]. The quantum Fourier transform on three code blocks [29]. Fault-tolerant transversal and lattice-surgery state teleportation protocols as well as Knill error correction [30].Rydberg atom arrays: Lukin group [31]; transversal CNOT gate performed on distance \(3\), \(5\), and \(7\) codes, logical ten-qubit GHZ state initialized with break-even fidelity, fault-tolerant logical two-qubit GHZ state initialized [32].

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 [33,34].
Quantum data-syndrome (QDS) code — There exists a set of stabilizer generators for the Steane code that make it a QDS code [35].
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 [36; Def. 4.9].
Group-representation code — The Steane code is a group-representation code with \(G\) being the \(2O\) subgroup of \(SU(2)\) [37].
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 [38; 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 [38; 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 [39].
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.
Concatenated cat code — Two-component cat codes concatenated with Steane and Golay codes are estimated to be fault tolerant against photon loss noise with rate \(\eta < 5\times 10^{-4}\) provided that \(\alpha > 1.2\) [40].
Quantum Lego code — The Steane code can be built from two \([[4,2,2]]\) codes in the quantum Lego code framework [41].
\([[10,1,2]]\) CSS code — A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[10,1,2]]\) code [17].
\([[4,2,2]]\) Four-qubit code — The Steane code can be built from two \([[4,2,2]]\) codes in the quantum Lego code framework [41].
\([[5,1,2]]\) rotated surface code — The \([[5,1,2]]\) rotated surface code can be obtained by morphing the Steane code [42].
Quantum divisible code — A fault-tolerant \(T\) gate on the Steane code can be obtained by concatenating with particular quantum divisible codes [43].
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 [44].
\([[15,1,3]]\) 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 [45]. A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[15,1,3]]\) code [13–16,46]. The \([[105,1]]\) concatenation of the \([[15,1,3]]\) and Steane codes allows for a universal gate set consisting of gates that are transversal w.r.t. to two different partitions [47,48].

References

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Page edit log

Remmy Zen (2024-07-15) — most recent
Eric Huang (2024-03-18)
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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/steane/steane.yml.