\([[2^r-1,1,3]]\) simplex code

\([[2^r-1,1,3]]\) simplex code[1–3]

Alternative names: \([[2^r-1,1,3]]\) quantum RM code.

Description

Member of color-code code family constructed with a punctured first-order RM\((1,m=r)\) \([2^r-1,r+1,2^{r-1}-1]\) code and its even subcode for \(r \geq 3\). Each code transversally implements a diagonal gate at the \((r-1)\)st level of the Clifford hierarchy [3,4]. Each code is a color code defined on a simplex in \(r-1\) dimensions [5,6], where qubits are placed on the vertices, edges, and faces as well as on the simplex itself.

Transversal Gates

Each code transversally implements a diagonal gate at the \((r-1)\)st level of the Clifford hierarchy in the form of a \(Z\)-rotation by angle \(-\pi/2^{r-1}\) [3,4]. These are the smallest distance-three qubit stabilizer codes with such a (strongly) transversal gate [7].

Fault Tolerance

Fault-tolerant syndrome extraction circuits using flag qubits [8].

Cousins

\(k\)-orthogonal code— \([[2^r-1,1,3]]\) simplex codes are \((r-1)\)-orthogonal [7; Lemma 2].
\([2^m,m+1,2^{m-1}]\) First-order RM code— The \([[2^r-1,1,3]]\) simplex code is constructed with a punctured first-order RM code and its even subcode.
Simplex spherical code— Each \([[2^r-1,1,3]]\) simplex code is a color code whose qubits are placed on the vertices, edges, and faces of an \((r-1)\)-simplex [5,6].
Binary dihedral PI code— The \(((2^{r-1}+3,2,3))\) family of binary dihedral PI codes realizes the same (strongly) transversal gates as the \([[2^r-1,1,3]]\) quantum Reed-Muller codes, but require fewer qubits in almost all cases.

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

Generalized homological-product qubit CSS codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum

Quantum rainbow code

Quantum pin codeStabilizer Hamiltonian-based Qubit QECC Quantum

Quantum Reed-Muller codeCSS Stabilizer Hamiltonian-based QECC Quantum

Parents

Quantum Reed-Muller code

\([[2^r-1,1,3]]\) simplex codes are special cases of the \([[\sum_{i=w+1}^m \binom{m}{i}, \sum_{i=0}^{w} \binom{m}{i}, \sum_{i=w+1}^{r+1} \binom{r+1}{i}]]\) quantum RM codes for \(w=r=0\) and \(m=r-1\) [9; Thm. 1].

Color codeGeneralized homological-product QLDPC CSS Stabilizer Qubit Hamiltonian-based QECC Quantum

Each \([[2^r-1,1,3]]\) simplex code is a color code defined on a simplex in \(r-1\) dimensions [5,6].

XP stabilizer codeQubit QECC Quantum

Each \([[2^r-1,1,3]]\) simplex code can be viewed as an XP stabilizer code with precision \(N = 2^{r-2}\) [10; Exam. 6.4].

Quantum divisible codeCSS Stabilizer Qubit Hamiltonian-based QECC Quantum

\([[2^r-1,1,3]]\) simplex codes come from RM\((1,m=r)\) codes, which are \((r-1)\)-even [11,12], and admit transversal gates at levels of the Clifford hierarchy. Building a tower of generalized divisible codes by starting with the Steane code yields the \([[2^r-1,1,3]]\) simplex codes [13; Sec. VI.B].

Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum

\([[2^r-1,1,3]]\) simplex code

Children

\([[7,1,3]]\) Steane code

\([[15,1,3]]\) quantum Reed-Muller code

References

[1]: D. Gottesman and I. L. Chuang, “Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations”, Nature 402, 390 (1999) arXiv:quant-ph/9908010 DOI
[2]: S. Bravyi and A. Kitaev, “Universal quantum computation with ideal Clifford gates and noisy ancillas”, Physical Review A 71, (2005) arXiv:quant-ph/0403025 DOI
[3]: B. Zeng, H. Chung, A. W. Cross, and I. L. Chuang, “Local unitary versus local Clifford equivalence of stabilizer and graph states”, Physical Review A 75, (2007) arXiv:quant-ph/0611214 DOI
[4]: S. X. Cui, D. Gottesman, and A. Krishna, “Diagonal gates in the Clifford hierarchy”, Physical Review A 95, (2017) arXiv:1608.06596 DOI
[5]: H. Bombin, “Gauge Color Codes: Optimal Transversal Gates and Gauge Fixing in Topological Stabilizer Codes”, (2015) arXiv:1311.0879
[6]: B. J. Brown, N. H. Nickerson, and D. E. Browne, “Fault-tolerant error correction with the gauge color code”, Nature Communications 7, (2016) arXiv:1503.08217 DOI
[7]: S. Koutsioumpas, D. Banfield, and A. Kay, “The Smallest Code with Transversal T”, (2022) arXiv:2210.14066
[8]: C. Chamberland and M. E. Beverland, “Flag fault-tolerant error correction with arbitrary distance codes”, Quantum 2, 53 (2018) arXiv:1708.02246 DOI
[9]: M. B. Hastings and J. Haah, “Distillation with Sublogarithmic Overhead”, Physical Review Letters 120, (2018) arXiv:1709.03543 DOI
[10]: 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
[11]: R. J. McEliece, “On periodic sequences from GF(q)”, Journal of Combinatorial Theory, Series A 10, 80 (1971) DOI
[12]: R. J. McEliece, “Weight congruences for p-ary cyclic codes”, Discrete Mathematics 3, 177 (1972) DOI
[13]: J. Haah, “Towers of generalized divisible quantum codes”, Physical Review A 97, (2018) arXiv:1709.08658 DOI

Page edit log

Victor V. Albert (2022-11-15) — most recent

Cite as:

“\([[2^r-1,1,3]]\) simplex code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/diagonal_clifford

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/rm/diagonal_clifford.yml.