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

\([[7,1,3]]\) Steane code[1,2]

Description

A \([[7,1,3]]\) self-dual CSS code that is the smallest qubit CSS code to correct a single-qubit error [3]. 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 [4,5], and the code''s encoder-respecting form is the graph of the full cube [6].

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 \(PGL(3,2)\) [7]. It is one of sixteen distinct \([[7,1,3]]\) codes [8].

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 ([9], Fig. 10.14).Evolution under stabilizer Hamiltonian [10].Fault-tolerant logical zero and logical plus state preparation on all-to-all and 2D grid qubit connectivity [11].Parity-check encoding with flag-bridge qubits on a square lattice connectivity [12].

Transversal Gates

The single-qubit Clifford group [13,14].

Gates

Fault-tolerant approximations of arbitrary single-qubit gates [15,16].Non-fault-tolerant \(T\) gate [17].Fault-tolerant logical zero and magic state preparation [18]. Magic-state preparation converts unbiased noise into biased noise [19].Pieceable fault-tolerant \(CCZ\) gate [20].

Decoding

Shor error correction fidelity calculation [21–23].

Fault Tolerance

A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[15,1,3]]\) code [24–27].A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[10,1,2]]\) code [28].Fault-tolerant logical zero and magic state preparation [18]. Magic-state preparation converts unbiased noise into biased noise [19].Fault-tolerant logical zero and logical plus state preparation on all-to-all and 2D grid qubit connectivity [11].Pieceable fault-tolerant \(CCZ\) gate [20].Syndrome measurement can be done with ancillary flag qubits [29,30] or with no extra qubits [31]. The depth of syndrome extraction circuits can be lowered by using past syndrome values [32].

Realizations

Trapped-ion devices: seven-qubit device in Blatt group [33]. Ten-qubit QCCD device by Quantinuum [34] 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 [35]). Fault-tolerant universal two-qubit gate set using T injection by Monz group [36]. 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 [37]; 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 [38]. Fault-tolerant universal gate set via code switching between the Steane code and the \([[10,1,2]]\) code [28]. 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 [39]. The quantum Fourier transform on three code blocks [40]. Fault-tolerant transversal and lattice-surgery state teleportation protocols as well as Knill error correction [41]. Rains shadow enumerators have been measured [42]. Inter-block CNOT gates have been characterized via cycle reconstruction [43]. Rydberg atom arrays: Lukin group [44]; 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 [45].

Notes

The Steane code can be used for entanglement purification [46].

Cousins

Cyclic quantum code— The Steane code is equivalent to a cyclic code via qubit permutations [47; 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 [47; 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 [48].
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 [3].
\([7,4,3]\) Hamming code— The Steane code is constructed from the \([7,4,3]\) classical Hamming code via the CSS construction.
Dual-rail quantum code— The KLM protocol, one of the first protocols for fault-tolerant quantum computation, utilizes concatenations of the dual-rail code with a stabilizer code such as the Steane code [49–51].
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 \(\alpha > 1.2\) [52].
Tensor-network code— The Steane code can be built from two \([[4,2,2]]\) codes in the quantum Lego code framework [53].
\([[m 2^m / (m+1), 2^m / (m+1), 2\lfloor m/4 \rfloor + 1]]\) Khesin-Lu-Shor code— The encoder-respecting form of both the Steane and Khesin-Lu-Shor codes is the graph of a hypercube [6].
\([[10,1,2]]\) CSS code— A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[10,1,2]]\) code [28].
\([[4,2,2]]\) Four-qubit code— The Steane code can be built from two \([[4,2,2]]\) codes in the quantum Lego code framework [53].
\([[5,1,2]]\) rotated surface code— The \([[5,1,2]]\) rotated surface code can be obtained by morphing the Steane code [54].
\(((7,2,3))\) Pollatsek-Ruskai code— The Pollatsek-Ruskai code can be continuously deformed to the Steane code [55].
Quantum divisible code— A fault-tolerant \(T\) gate on the Steane code can be obtained by concatenating with particular quantum divisible codes [56].
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 [57].
\([[15,1,3]]\) 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 [58]. A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[15,1,3]]\) code [24–27,59]. 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 [60,61].

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 codeTwist-defect color Qubit Generalized homological-product Lattice stabilizer QLDPC CSS Stabilizer 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

Hermitian qubit codeStabilizer Hamiltonian-based Qubit QECC Quantum

The Steane code is Hermitian [62; Table 6].

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

The Steane code is a qubit quantum QR code [63,64].

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 [65].

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 [66; Def. 4.9].

Group-representation codeQECC Quantum

The Steane code is a group-representation code with \(G\) being the \(2O\) subgroup of \(SU(2)\) [67].

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.