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

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

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 ([5], Fig. 10.14).

Transversal Gates

All single-qubit Clifford gates, which realize the \(2O\) binary octahedral subgroup of \(SU(2)\) [6,7].

Gates

Fault-tolerant logical zero and magic state preparation [8]. Magic-state preparation converts unbiased noise into biased noise [9].Pieceable fault-tolerant CCZ gate [10].

Fault Tolerance

A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[15,1,3]]\) code [1114].A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[10,1,2]]\) code [15].Fault-tolerant logical zero and magic state preparation [8]. Magic-state preparation converts unbiased noise into biased noise [9].Pieceable fault-tolerant CCZ gate [10].Syndrome measurement can be done with ancillary flag qubits [16,17] or with no extra qubits [18]. The depth of syndrome extraction circuits can be lowered by using past syndrome values [19].

Realizations

Trapped-ion devices: seven-qubit device in Blatt group [20]. Ten-qubit QCCD device by Quantinuum [21] 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 [22]). Fault-tolerant universal two-qubit gate set using T injection by Monz group [23]. 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 [24]; 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 [25]. Fault-tolerant universal gate set via code switching between the Steane code and the \([[10,1,2]]\) code [15]. 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 [26]. The quantum Fourier transform on three code blocks [27]. Fault-tolerant transversal and lattice-surgery state teleportation protocols as well as Knill error correction [28].Rydberg atom arrays: Lukin group [29]; 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 [30].

Parents

Cousins

References

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Zoo Code ID: steane

Cite as:
\([[7,1,3]]\) Steane code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/steane
BibTeX:
@incollection{eczoo_steane, title={\([[7,1,3]]\) Steane code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/steane} }
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\([[7,1,3]]\) 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.