Also known as \([[2^r-1,1,3]]\) 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 qubit stabilizer codes with such a (strongly) transversal gate [7].
Fault Tolerance
Fault-tolerant syndrome extraction circuits using flag qubits [8].
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 code — Each \([[2^r-1,1,3]]\) simplex code is a color code defined on a simplex in \(r-1\) dimensions [5,6].
- XP stabilizer code — 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 code — \([[2^r-1,1,3]]\) 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 block quantum code
Children
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 transversal gates as the \([[2^r-1,1,3]]\) quantum Reed-Muller codes, but require fewer qubits in almost all cases.
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