\([[2^D,D,2]]\) hypercube quantum code[1,2][3; Exam. 3]
Alternative names: 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. The measured physical bit string can be post-processed into both a logical output string and stabilizer checks, enabling end-of-circuit error detection directly from classical samples [4].
Higher-distance generalizations include a \([[2^{2D},D,4]]\) hyperoctahedron family and a \([[2^D(2^D+1),D,4]]\) family built from distance-two \(D\)-dimensional toric/surface-code blocks [4]. Various other concatenations give families with increasing distance (see cousins).
In the color-code picture, they arise from hypercube-like lattices with no bulk qubits and opposite boundaries carrying the same color; after local Clifford disentangling, the transversal \(\widetilde{R_D}\) operator acts as a logical \(C^{D-1}Z\) gate on \(D\) decoupled distance-two toric/surface-code factors [1].
Protection
The code detects a single general error but has an \(X\)-distance \(d_X = 4\). In encoded IQP sampling, this allows error detection without intermediate measurements by postselecting on the final stabilizer data extracted from measurement samples [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; Ref. [4] proposes hypercube IQP (hIQP) circuits on a hypercube connectivity graph.For \(D=4\), Bell sampling on two copies of degree-\(4\) IQP circuits encoded in the \([[16,4,2]]\) member is proposed as an efficiently classically verifiable quantum-advantage experiment [4].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 \(D\) distance-two \(D\)-dimensional toric/surface-code blocks to yield a \([[2^D(2^D+1),D,4]]\) error-correcting 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 \(D\) distance-two \(D\)-dimensional toric/surface-code blocks 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].
Primary Hierarchy
Parents
\([[2^D,D,2]]\) hypercube quantum codes can be thought of as small ball codes constructed from hyperoctahedra [3; Exam. 3], or on lattices with no bulk qubits and cubic boundaries [1,2].
\([[2^D,D,2]]\) hypercube quantum codes are special cases of the \([[2^m,{m \choose r}, 2^r]]\) quantum RM codes for \(m=D\) and \(r=1\) [8–11][7; Exam. 8].
The \(D\)th hypercube quantum code can be viewed as an XP stabilizer code with precision \(N = 2^D\) [5; Exam. 6.10].
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.
\([[2^D,D,2]]\) hypercube quantum code
Children
The \([[4,2,2]]\) code is a hypercube code for \(D=2\).
The \([[8,3,2]]\) code is a hypercube code for \(D=3\).
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 Instantaneous Quantum Polynomial Circuits on Hypercubes”, PRX Quantum 6, (2025) arXiv:2404.19005 DOI
- [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
- [11]
- A. Gong and J. M. Renes, “Computation with quantum Reed-Muller codes and their mapping onto 2D atom arrays”, (2024) arXiv:2410.23263
Page edit log
- Victor V. Albert (2023-11-28) — most recent
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