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

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

The single-qubit Clifford group [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].

Decoding

Shor error correction fidelity calculation [13,14].

Fault Tolerance

A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[15,1,3]]\) code [15–18].A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[10,1,2]]\) code [19].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 [20,21] or with no extra qubits [22]. The depth of syndrome extraction circuits can be lowered by using past syndrome values [23].

Realizations

Trapped-ion devices: seven-qubit device in Blatt group [24]. Ten-qubit QCCD device by Quantinuum [25] 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 [26]). Fault-tolerant universal two-qubit gate set using T injection by Monz group [27]. 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 [28]; 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 [29]. Fault-tolerant universal gate set via code switching between the Steane code and the \([[10,1,2]]\) code [19]. 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 [30]. The quantum Fourier transform on three code blocks [31]. Fault-tolerant transversal and lattice-surgery state teleportation protocols as well as Knill error correction [32]. Rains shadow enumerators have been measured [33].Rydberg atom arrays: Lukin group [34]; 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 [35].

Parents

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].
  • \([[10,1,2]]\) CSS code — A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[10,1,2]]\) code [19].
  • \([[4,2,2]]\) 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 [15–18,50]. 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].

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.