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

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

Description

A \([[7,1,3]]\) self-dual 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 [3,4], and the code''s encoder-respecting form is the graph of the full cube [5].

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 \(PGL(3,2)\) [6]. It is one of sixteen distinct \([[7,1,3]]\) codes [7].

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 ([8], Fig. 10.14).Fault-tolerant logical zero and logical plus state preparation on all-to-all and 2D grid qubit connectivity [9].

Transversal Gates

The single-qubit Clifford group [10,11].

Gates

Fault-tolerant approximations of arbitrary single-qubit gates [12,13].Fault-tolerant logical zero and magic state preparation [14]. Magic-state preparation converts unbiased noise into biased noise [15].Pieceable fault-tolerant CCZ gate [16].

Decoding

Shor error correction fidelity calculation [17,18].

Fault Tolerance

A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[15,1,3]]\) code [19–22].A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[10,1,2]]\) code [23].Fault-tolerant logical zero and magic state preparation [14]. Magic-state preparation converts unbiased noise into biased noise [15].Fault-tolerant logical zero and logical plus state preparation on all-to-all and 2D grid qubit connectivity [9].Pieceable fault-tolerant CCZ gate [16].Syndrome measurement can be done with ancillary flag qubits [24,25] or with no extra qubits [26]. The depth of syndrome extraction circuits can be lowered by using past syndrome values [27].

Realizations

Trapped-ion devices: seven-qubit device in Blatt group [28]. Ten-qubit QCCD device by Quantinuum [29] 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 [30]). Fault-tolerant universal two-qubit gate set using T injection by Monz group [31]. 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 [32]; 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 [33]. Fault-tolerant universal gate set via code switching between the Steane code and the \([[10,1,2]]\) code [23]. 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 [34]. The quantum Fourier transform on three code blocks [35]. Fault-tolerant transversal and lattice-surgery state teleportation protocols as well as Knill error correction [36]. Rains shadow enumerators have been measured [37].Rydberg atom arrays: Lukin group [38]; ten logical qubits, transversal CNOT gate performed, logical ten-qubit GHZ state initialized with break-even fidelity, and fault-tolerant logical two-qubit GHZ state initialized [39].

Notes

The Steane code can be used for entanglement purification [40].

Cousins

Cyclic quantum code— The Steane code is equivalent to a cyclic code via qubit permutations [41; 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 [41; 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 [42].
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 \(\alpha > 1.2\) [43].
Tensor-network code— The Steane code can be built from two \([[4,2,2]]\) codes in the quantum Lego code framework [44].
\(((7,2,3))\) Pollatsek-Ruskai code— The Pollatsek-Ruskai code can be continuously deformed to the Steane code [45].
\([[m 2^m / (m+1), 2^m / (m+1)]]\) Khesin-Lu-Shor code— The encoder-respecting form of both the Steane and Khesin-Lu-Shor codes is the graph of a hypercube [5].
\([[10,1,2]]\) CSS code— A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[10,1,2]]\) code [23].
\([[4,2,2]]\) Four-qubit code— The Steane code can be built from two \([[4,2,2]]\) codes in the quantum Lego code framework [44].
\([[5,1,2]]\) rotated surface code— The \([[5,1,2]]\) rotated surface code can be obtained by morphing the Steane code [46].
Quantum divisible code— A fault-tolerant \(T\) gate on the Steane code can be obtained by concatenating with particular quantum divisible codes [47].
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 [48].
\([[15,1,3]]\) 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 [49]. A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[15,1,3]]\) code [19–22,50]. An \([[105,1,3]]\) alternative concatenation of the \([[15,1,3]]\) and Steane codes allows for a universal gate set consisting of gates that are close to transversal [51,52].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum

Coherent-parity-check (CPC) code

Qubit CSS codeCSS Stabilizer Qubit Hamiltonian-based QECC Quantum

Generalized homological-product qubit CSS codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum

Quantum rainbow code

Quantum pin codeStabilizer Hamiltonian-based Qubit QECC Quantum

Color code

2D color codeGeneralized homological-product Twist-defect color Lattice stabilizer QLDPC CSS Stabilizer Qubit Abelian topological Topological Hamiltonian-based QECC Quantum

Honeycomb (6.6.6) color code

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 codeColor Quantum Reed-Muller Generalized homological-product QLDPC CSS Stabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum

\([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming codeQuantum Reed-Muller Qubit Generalized homological-product QLDPC CSS Stabilizer Hamiltonian-based Small-distance block quantum QECC Quantum

\([[2^{2r-1}-1,1,2^r-1]]\) quantum punctured Reed-Muller codeQuantum Reed-Muller Qubit Generalized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum

Quantum quadratic-residue (QR) codeCSS Stabilizer Hamiltonian-based QECC Quantum

The Steane code is a qubit quantum QR code [53,54].

Quantum data-syndrome (QDS) codeStabilizer Hamiltonian-based Qubit QECC Quantum

There exists a set of stabilizer generators for the Steane code that make it a QDS code [55].

Finite-geometry (FG) QLDPC codeCSS QLDPC Stabilizer Qubit Hamiltonian-based QECC Quantum

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 [56; Def. 4.9].

Group-representation codeQECC Quantum

The Steane code is a group-representation code with \(G\) being the \(2O\) subgroup of \(SU(2)\) [57].

Concatenated Steane codeCSS Stabilizer Qubit Concatenated quantum HQECC Hamiltonian-based QECC Quantum

The concatenated Steane code at level \(m=1\) is the Steane code.

Planar-perfect-tensor codeQECC Quantum

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.