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\([[2m,2m-2,2]]\) error-detecting code[1–3]

Alternative names: Iceberg code, \([[2m,2m-2,2]]\) quantum parity code.

Description

Self-complementary CSS code for \(m\geq 2\) with generators \(\{XX\cdots X, ZZ\cdots Z\} \) acting on all \(2m\) physical qubits. The code is constructed via the CSS construction from an SPC code and a repetition code [4; Sec. III]. This is the highest-rate distance-two code when an even number of qubits is used [5].

Admits a basis such that each codeword is a superposition of a computational basis state labeled by an even-weight bitstring \(b\) and a state labeled by the negation of \(b\). Its all-zero logical state is a conventional GHZ state. Removing the \(Z\)-type generator expands the number of codewords to all combinations of bitstrings and their negations, yielding a code with \(k=2m-1\) [6].

All of its automorphisms lie in the Clifford group [7; Thm. 13].

Protection

Detects a single-qubit error.

Encoding

Adaptive constant-depth circuit with geometrically local gates and measurements throughout [8,9].

Transversal Gates

Transveral CNOT gates can be performed by first teleporting qubits into different code blocks [2; Sec. VII].For \(2m\) being a multiple of four, this code houses a transversal representation of the single-qubit Clifford group [10; Sec. 3.1]. Its \(n\)-block version houses a transversal representation of the \(n\)-qubit Clifford group [10; Secs. 3.1 and 3.4]. The commutant of transversal representations of the Clifford group consists of qubit permutations and projections onto the \([[2m,2m-2,2]]\) error-detecting code [11].

Gates

Logical SWAP gates can be performed fault tolerantly using an ancilla qubit [2; Sec. VII].Universal set of gates, each of which is supported on two qubits [12].Fault-tolerant Clifford Trotter circuits that are linear in \(k\) using flag qubits via a solve-and-stitch algorithm and application of a logical identity circuit [13].

Decoding

The \([[2m,2m-2,2]]\) error-detecting code [14] and its relative the code with single stabilizer \(XX\cdots X\) [15] admit autonomous QEC against single AD errors.

Fault Tolerance

Logical SWAP gates can be performed fault tolerantly using an ancilla qubit [2; Sec. VII].Two-qubit fault-tolerant state preparation, error detection and projective measurements [16] (see also [12]).CNOT and Hadamard gates using only two extra qubits and four-qubit fault-tolerant \(CCZ\) gate [17].Fault-tolerant Clifford Trotter circuits that are linear in \(k\) using flag qubits via a solve-and-stitch algorithm and application of a logical identity circuit [13].Weak fault tolerance: any single gate error can be detected by measuring stabilizers and utilizing extra ancillas [18].

Realizations

Trapped-ion devices: the \(m=5\) code has been realized on a 12-qubit device by Quantinuum [12]. The QAOA algorithm has been realized on the \(m=18\) code using 510 two-logical-qubit gates [19].

Notes

See description of the code in Ref. [20].The code solves [21] the mean king's measurement problem [22].The code is useful for entanglement distillation [23].The code is used in a fault-tolerant implementation of the QAOA algorithm [24].

Cousins

  • Single parity-check (SPC) code— The \([[2m,2m-2,2]]\) error-detecting code is constructed via the CSS construction from an SPC code and its dual repetition code [4; Sec. III].
  • Repetition code— The \([[2m,2m-2,2]]\) error-detecting code is constructed via the CSS construction from an SPC code and its dual repetition code [4; Sec. III].
  • Truncated trihexagonal (4.6.12) color code— The \([[2m,2m-2,2]]\) error-detecting code for \(m=4\) is a color code defined on a single octagon of the 6.6.6 or 4.6.12 tilings.
  • Amplitude-damping (AD) code— The \([[2m,2m-2,2]]\) error-detecting code [14] and its relative the code with single stabilizer \(XX\cdots X\) [15] admit autonomous QEC against single AD errors.
  • \([[4,2,2]]_{G}\) four group-qudit code— The four group-qudit code can be extended to the \([[2m,2m-2,2]]_{G}\) group-qudit code [25; Sec. VIII]. The latter reduces to the \([[2m,2m-2,2]]\) error-detecting code for \(G=\mathbb{Z}_2\).
  • Jump code— The subcode of the \([[2m,2m-2,2]]\) error-detecting code consisting of codewords labeled by weight-\(m\) bitstrings is a \(((2m,\frac{1}{2}{2m \choose m},1))_{m}\) optimal jump code [26][27; Corr. 9].
  • Hybrid stabilizer code— The \([[2m+1,2m+2:1,2]]\) hybrid stabilizer code [28] (extendable to modular qudits [29]) is closely related to the \([[2m,2m-2,2]]\) error-detecting code.
  • Qubit stabilizer code— Stabilizer states on \(n\) qubits form complex projective 3-designs [30], while the Clifford group is a unitary 3-design [31,32]. The \([[2m,2m-2,2]]\) code for \(2m\) being a multiple of four obstructs the Clifford group from being a 4-design [10].
  • Trapezoid subsystem code— The trapezoid code family can be obtained from the \([[2m,2m-2,2]]\) error-detecting code by using some logical qubits as gauge qubits and imposing a two-dimensional qubit geometry [33].

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
Quantum multi-dimensional parity-check (QMDPC) codeCSS QLDPC Stabilizer Qubit Hamiltonian-based Concatenated quantum Small-distance block quantum QECC Quantum
Parents
Quantum multi-dimensional parity-check (QMDPC) code
The \([[2m,2m-2,2]]\) error-detecting code is a 1D QMDPC.
Quantum maximum-distance-separable (MDS) codeQECC Quantum
The \([[2m,2m-2,2]]\) error-detecting code forms one of the two families qubit quantum MDS codes [34].
Ball color codeColor Generalized homological-product QLDPC CSS Stabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum
The \([[2m,2m-2,2]]\) error-detecting code is a ball color code [35; Sec. III.A].
Hermitian qubit codeStabilizer Hamiltonian-based Qubit QECC Quantum
The \([[2m,2m-2,2]]\) error-detecting code is Hermitian [36; Table 6].
Self-complementary qubit codeAmplitude-damping (AD) AQECC Qubit QECC Quantum
\([[2m,2m-2,2]]\) error-detecting code
Children
\([[4,2,2]]\) Four-qubit code
The \([[2m,2m-2,2]]\) error-detecting code for \(m=2\) reduces to the \([[4,2,2]]\) code.
\([[6,4,2]]\) error-detecting code
The \([[2m,2m-2,2]]\) error-detecting code for \(m=3\) reduces to the \([[6,4,2]]\) error-detecting code.

References

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Zoo Code ID: iceberg

Cite as:
“\([[2m,2m-2,2]]\) error-detecting code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/iceberg
BibTeX:
@incollection{eczoo_iceberg, title={\([[2m,2m-2,2]]\) error-detecting code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/iceberg} }
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“\([[2m,2m-2,2]]\) error-detecting code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/iceberg

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/iceberg.yml.