\([[2^D,D,2]]\) hypercube quantum code[1,2][3; Exam. 3] 

Also known as Hyperoctahedron code, Hyperoctahedron color code.

Description

Member of a family of codes defined by placing qubits on a \(D\)-dimensional hypercube, \(Z\)-stabilizers on all two-dimensional faces, and an \(X\)-stabilizer on all vertices. These codes realize gates at the \((D-1)\)-st level of the Clifford hierarchy. It can be generalized to a \([[4^D,D,4]]\) error-correcting family [4]. Various other concatenations give families with increasing distance (see cousins).

Protection

The code detects a single general error but has an \(X\)-distance \(d_X = 4\).

Transversal Gates

CZ, CCZ, and generalized \(CZ\) gates at the \((D-1)\)-st level of the Clifford hierarchy [2][5; Exam. 6.10]. CNOT and SWAP gates can be realized by qubit permutations [4].

Notes

Degree-\(D\) instantaneous quantum polynomial (IQP) circuits [6] can be realized on hypercube quantum codes in a hardware-efficient way [4].

Parents

  • Ball color code — \([[2^D,D,2]]\) hypercube quantum codes can be thought of as small color codes defined on balls constructed from hyperoctahedra [3; Exam. 3], or on lattices with no bulk qubits and cubic boundaries [1,2].
  • Quantum Reed-Muller code — \([[2^D,D,2]]\) hypercube quantum codes are special cases of the \([[2^m,{m \choose r}, 2^r]]\) quantum Reed-Muller codes for \(m=D\) and \(r=1\) [8–10][7; Exam. 8].
  • XP stabilizer code — The \(D\)th hypercube quantum code can be viewed as an XP stabilizer code with precision \(N = 2^D\) [5; Exam. 6.10].
  • Self-complementary quantum code — A basis of hypercube quantum codewords of the form \(|c\rangle+|\overline{c}\rangle\) can be obtained via the qubit CSS codeword construction since their sole \(X\)-type stabilizer generator acts on all qubits.

Children

Cousins

  • Hypercube code — \([[2^D,D,2]]\) hypercube quantum code qubits are placed on vertices of a \(D\)-cube.
  • Quantum repetition code — The hypercube quantum code can be concatenated with a two-qubit quantum repetition code to yield a \([[2^{D+1},D,4]]\) error-detecting code family [4].
  • Homological code — The hypercube quantum code can be concatenated with a distance-two \(D\)-dimensional surface code to yield a \([[2^D(2^D+1),D,4]]\) error-correcting code family that admits a transversal implementation of the logical \(C^{D-1}Z\) gate [4].
  • Concatenated qubit code — The hypercube quantum code can be concatenated with a two-qubit quantum repetition code to yield a \([[2^{D+1},D,4]]\) error-detecting code family [4]. It can also be concatenated with a distance-two \(D\)-dimensional surface code to yield a \([[2^D(2^D+1),D,4]]\) error-correcting code family that admits a transversal implementation of the logical \(C^{D-1}Z\) gate [4].

References

[1]
A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015) arXiv:1503.02065 DOI
[2]
E. Campbell, “The smallest interesting colour code,” Online available at https://earltcampbell.com/2016/09/26/the-smallest-interesting-colour-code/ (2016), accessed on 2019-12-09.
[3]
M. Vasmer and A. Kubica, “Morphing Quantum Codes”, PRX Quantum 3, (2022) arXiv:2112.01446 DOI
[4]
D. Hangleiter, M. Kalinowski, D. Bluvstein, M. Cain, N. Maskara, X. Gao, A. Kubica, M. D. Lukin, and M. J. Gullans, “Fault-tolerant compiling of classically hard IQP circuits on hypercubes”, (2024) arXiv:2404.19005
[5]
M. A. Webster, B. J. Brown, and S. D. Bartlett, “The XP Stabiliser Formalism: a Generalisation of the Pauli Stabiliser Formalism with Arbitrary Phases”, Quantum 6, 815 (2022) arXiv:2203.00103 DOI
[6]
M. J. Bremner, A. Montanaro, and D. J. Shepherd, “Average-Case Complexity Versus Approximate Simulation of Commuting Quantum Computations”, Physical Review Letters 117, (2016) arXiv:1504.07999 DOI
[7]
N. Rengaswamy, R. Calderbank, M. Newman, and H. D. Pfister, “On Optimality of CSS Codes for Transversal T”, IEEE Journal on Selected Areas in Information Theory 1, 499 (2020) arXiv:1910.09333 DOI
[8]
E. T. Campbell and M. Howard, “Unified framework for magic state distillation and multiqubit gate synthesis with reduced resource cost”, Physical Review A 95, (2017) arXiv:1606.01904 DOI
[9]
E. T. Campbell and M. Howard, “Unifying Gate Synthesis and Magic State Distillation”, Physical Review Letters 118, (2017) arXiv:1606.01906 DOI
[10]
J. Haah and M. B. Hastings, “Codes and Protocols for DistillingT, controlled-S, and Toffoli Gates”, Quantum 2, 71 (2018) arXiv:1709.02832 DOI
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Zoo Code ID: hypercube_quantum

Cite as:
“\([[2^D,D,2]]\) hypercube quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/hypercube_quantum
BibTeX:
@incollection{eczoo_hypercube_quantum, title={\([[2^D,D,2]]\) hypercube quantum code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/hypercube_quantum} }
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“\([[2^D,D,2]]\) hypercube quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/hypercube_quantum

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