\([[7,1,3]]\) 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][4; ID 226]. 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 of 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. Fig. 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 [5,6], and the code’s encoder-respecting form is the graph of the full cube [7].

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 \(PGL(3,2)\) [8]. It is one of sixteen distinct \([[7,1,3]]\) codes [9].

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 [10; Fig. 10.14].Evolution under stabilizer Hamiltonian [11].Fault-tolerant logical zero and logical plus state preparation on all-to-all and 2D grid qubit connectivity [12].Parity-check encoding with flag-bridge qubits on a square lattice connectivity [13].

Transversal Gates

The single-qubit Clifford group [14,15].

Gates

Fault-tolerant approximations of arbitrary single-qubit gates [16,17].Non-fault-tolerant \(T\) gate [18].Fault-tolerant logical zero and magic state preparation [19]. Magic-state preparation converts unbiased noise into biased noise [20].Because transversal Hadamard acts logically on the code, the Steane code serves as a normal self-dual inner code for magic-state distillation. One routine uses 14 noisy \(T\) gates and one noisy input magic state to produce one output with cubic error suppression, and it can be pipelined with the \([[17,1,5]]\) code to obtain fifth-order suppression [21; Sec. I.A].Pieceable fault-tolerant \(CCZ\) gate [22].

Decoding

Shor error correction fidelity calculation [23–25].A \([15,3]\) syndrome-measurement code yields a QDS extension that uses the same 15 measurements as five-fold repetition of the three syndrome bits while achieving lower syndrome-decoding error [26; Fig. 1].Fault-tolerant measurement-free error-correction cycle [27].

Fault Tolerance

A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[15,1,3]]\) code [28–32].A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[10,1,2]]\) code [33].Fault-tolerant logical zero and magic state preparation [19]. Magic-state preparation converts unbiased noise into biased noise [20].Fault-tolerant logical zero and logical plus state preparation on all-to-all and 2D grid qubit connectivity [12].Pieceable fault-tolerant \(CCZ\) gate [22].Syndrome measurement can be done with ancillary flag qubits [34] or with no extra qubits [35]. The depth of syndrome extraction circuits can be lowered by using past syndrome values [36].Computation of ground-state energy of the hydrogen molecule [37].Fault-tolerant measurement-free error-correction cycle [27].

Realizations

Trapped-ion devices: seven-qubit device in Blatt group [38]. Ten-qubit QCCD device by Quantinuum [39] 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 [40]). Fault-tolerant universal two-qubit gate set using T injection by Monz group [41]. 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 [42]; 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 [43]. Fault-tolerant universal gate set via code switching between the Steane code and the \([[10,1,2]]\) code [33]. 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 [44]. The quantum Fourier transform on three code blocks [45]. Fault-tolerant transversal and lattice-surgery state teleportation protocols as well as Knill error correction [46]. Rains shadow enumerators have been measured [47]. Inter-block CNOT gates have been characterized via cycle reconstruction [48]. Code switching between the Steane code and the \([[15,1,3]]\) code as well as magic-state preparation and logical Bell measurements on the Steane code realized on the 28-qubit H2-1 device by Quantinuum [49]. End-to-end fault-tolerant execution of QAOA and HHL circuits, including logical non-Clifford operations, with up to 12 logical qubits on Quantinuum systems [50]. Neutral atom arrays: Lukin group. 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 [51]. Deep-circuit protocols with dozens of logical qubits and hundreds of logical teleportations [52].

Notes

Pedagogical explanation of QEC using the Steane code [53].The Steane code can be used for entanglement purification [54].

Cousins

Cluster-state code— The Steane code is equivalent via a single-qubit Clifford unitary to a cluster-state code for a particular graph and classical code [55; Exam. 4]. Four non-isomorphic graphs yield graph quantum codes that are equivalent to the Steane code under a single-qubit-Clifford circuit [56].
EA qubit stabilizer code— The Steane code is globally equivalent to a \([[6,1,3;1]]\) EA CSS code, which the paper identifies as an example of the smallest one-ebit EA CSS code correcting an arbitrary single-qubit error on the sender’s qubits [3].
\([[6,2,2]]\) \(C_6\) code— In Knill’s \(C_4/C_6\) architecture, noisy \(\ket{\pi/8}\) states are injected using \(C_4/C_6\) logical Bell pairs and then purified by encoding them into the Steane code; Knill also proposed using the Steane code as a final concatenation level for the \(C_4/C_6\) scheme [57].
\([7,4,3]\) 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 [58–60].
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 \(\alpha > 1.2\) [61].
Tensor-network code— The Steane code can be built from two \([[4,2,2]]\) codes in the quantum Lego code framework [62].
Majorana stabilizer code— Applying the CSS-to-Majorana map of [63; Lemma 2] to the \([[7,1,3]]\) Steane code yields a seven-Majorana code encoding half a qubit; pairing two such odd-length copies gives a physical Majorana stabilizer code with odd logical operators [63; Sec. 8].
Khesin-Lu-Shor code— The encoder-respecting form of both the Steane and Khesin-Lu-Shor codes is the graph of a hypercube [7].
\([[10,1,2]]\) Vasmer-Kubica code— A fault-tolerant universal gate set can be obtained via code switching between the Steane code and the \([[10,1,2]]\) code [33].
\([[4,2,2]]\) Four-qubit code— The Steane code can be built from two \([[4,2,2]]\) codes in the quantum Lego code framework [62]. Ref. [64] also introduces a \(C_4\)/Steane concatenated code, obtained by concatenating the \([[4,2,2]]\) code with the Steane code, as an underlying code for further concatenation with quantum Hamming codes.
\([[5,1,2]]\) rotated surface code— The \([[5,1,2]]\) rotated surface code can be obtained by morphing the Steane code [65].
\(((7,2,3))\) Pollatsek-Ruskai code— The Pollatsek-Ruskai code can be continuously deformed to the Steane code [66].
Quantum divisible code— A fault-tolerant \(T\) gate on the Steane code can be obtained by concatenating with particular quantum divisible codes [67].
Raussendorf-Bravyi-Harrington (RBH) cluster-state code— Concatenation of the RBH code with small codes such as the \([[2,1]]\) repetition code, \([[4,1,1,2]]\) subsystem code, or Steane code can improve thresholds [68].
\([[15,1,3]]\) quantum RM 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 [28]. A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[15,1,3]]\) code [29–32,69]. 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 [70,71].

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

Qubit QLDPC codeStabilizer Hamiltonian-based QECC Quantum

Color codeCSS Stabilizer Hamiltonian-based Qubit QECC Quantum

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

Honeycomb (6.6.6) color codeGeneralized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum

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 RM code QLDPC CSS Stabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum

\([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming codeQuantum RM code Qubit CSS Stabilizer Hamiltonian-based Small-distance block quantum QECC Quantum

\([[2^{2r-1}-1,1,2^r-1]]\) quantum punctured RM codeQuantum RM code CSS Stabilizer Hamiltonian-based Qubit QECC Quantum

Hermitian qubit codeStabilizer Hamiltonian-based Qubit QECC Quantum

The Steane code is Hermitian [4; Table 6].

Cyclic quantum codeQECC Quantum

The Steane code is equivalent to a cyclic code via qubit permutations [55; Exam. 1].

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

The Steane code is a qubit quantum QR code [26,72].

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; a \([15,3]\) syndrome-measurement code beats five-fold repeated syndrome extraction at the same measurement cost [26,73].

Finite-geometry (FG) qubit QLDPC codeQLDPC CSS Stabilizer Hamiltonian-based Qubit 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 [74; Def. 4.9].

Group-representation codeQECC Quantum

The Steane code is a group-representation code with \(G\) being the \(2O\) subgroup of \(SU(2)\) [75].

Concatenated Steane codeCSS Stabilizer Hamiltonian-based Qubit Concatenated quantum HQECC 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.