Quantum divisible code[13] 

Description

A level-\(\nu\) quantum divisible code is a CSS code whose \(X\)-type stabilizers form a \(\nu\)-even linear binary code in the symplectic representation and which admits a transversal gate at the \(\nu\)th level of the Clifford hierarchy. A CSS code is doubly-even (triply-even) if all \(X\)-type stabilizer generators have weight divisible by four (eight), i.e., if they form a doubly-even (triply-even) linear binary code. Doubly-even codes can yield a transversal \(S\) gate, while triply-even codes yield a transversal \(T\) gate for odd \(n\) [4].

More formally [5,6], a quantum divisible code is a CSS code defined from two linear binary codes \(C_{1,2}\) such that all weights in \(C_2\) share a common divisor \(\Delta > 1\), and all weights in each coset of \(C_2\) in \(C_1\) are congruent to \(\Delta\).

For example [6], if \(C_2\) is the first-order Reed-Muller code, and \(C_1/ C_2\) consists of quadratic forms with a bounded rank, then \([[n = 2m − 1, 1 \leq k \leq 1 + \sum_{i=1}^{m-4}(m − i), d = 3]]\) is a family of quantum divisible codes.

Transversal Gates

Doubly-even codes can yield a transversal \(S\) gate, while triply-even codes yield a transversal \(T\) gate for odd \(n\) [4]. More generally, if the \(X\)-type stabilizers of a CSS code form an \(\nu\)-even classical code, and if all \(X\)-type logicals are \((\nu-1)\)-even, then the code admit a diagonal transversal gate in the \(\nu\)th level of the Clifford hierarchy [7; Prop. 8]. Necessary and sufficient conditions for a CSS code to yield such a gate have been formulated [5; Thm. 9][6; Thm. 5].

Gates

The \([[2m − 1, 1 \leq k \leq 1 + \sum_{i=1}^{m-4}(m − i), 3]]\) quantum divisible code family can serve as outer codes of either the five-qubit \([[5,1,3]]\) or Steane \([[7,1,3]]\) code to realize a \(T\) gate on the inner code. For example, when \(m=5\) (\(m=6\)), the resulting \([[31,5,3]]\) (\([[63,7,3]]\)) code yields the \(T\) gate on the inner five-qubit (Steane) code.

Fault Tolerance

The \(T\) gate realized by concatenating members of the \([[2m − 1, 1 \leq k \leq 1 + \sum_{i=1}^{m-4}(m − i), 3]]\) quantum divisible code family with either the five-qubit \([[5,1,3]]\) or Steane \([[7,1,3]]\) code is fault-tolerant and does not require magic-state distillation [6]. The gate is performed on the inner five-qubit/Steane code and does require encoding and decoding algorithms to pass between the inner and outer codes.

Realizations

Triply-even codes can yield secure multi-party quantum computation [8].

Parent

  • Generalized quantum divisible code — Generalized level-\(\nu\) quantum divisible codes reduce to quantum level-\(\nu\) divisible codes when \(t\) is a vector with \(\pm 1\) entries. The classical code formed by their \(X\)-type stabilizer generator matrix is \(\nu\)-even [9; pg. 10]. Both types of codes realize transversal gates outside of the Clifford group.

Children

Cousins

  • Divisible code — The \(X\)-type stabilizers of a level-\(\nu\) quantum divisible code form a \(\nu\)-even linear binary code.
  • Reed-Muller (RM) code — Quantum divisible codes can be constructed out of first-order RM codes [6].
  • Concatenated quantum code — A fault-tolerant \(T\) gate on the five-qubit or Steane code can be obtained by concatenating with particular quantum divisible codes [6].
  • Five-qubit perfect code — A fault-tolerant \(T\) gate on the five-qubit code can be obtained by concatenating with particular quantum divisible codes [6].
  • \([[7,1,3]]\) Steane code — A fault-tolerant \(T\) gate on the Steane code can be obtained by concatenating with particular quantum divisible codes [6].
  • Doubled color code — Doubled color codes are subsystem codes constructed using a generalization of the doubling transformation [12] that combines doubly-even linear binary codes to make triply-even codes. The doubling transformation is a special case of level lifting (from two to three) [9; Sec. VI.D].

References

[1]
A. J. Landahl and C. Cesare, “Complex instruction set computing architecture for performing accurate quantum \(Z\) rotations with less magic”, (2013) arXiv:1302.3240
[2]
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
[3]
J. Haah and M. B. Hastings, “Codes and Protocols for DistillingT, controlled-S, and Toffoli Gates”, Quantum 2, 71 (2018) arXiv:1709.02832 DOI
[4]
S. Bravyi and A. Cross, “Doubled Color Codes”, (2015) arXiv:1509.03239
[5]
J. Hu, Q. Liang, and R. Calderbank, “Designing the Quantum Channels Induced by Diagonal Gates”, Quantum 6, 802 (2022) arXiv:2109.13481 DOI
[6]
J. Hu, Q. Liang, and R. Calderbank, “Divisible Codes for Quantum Computation”, (2022) arXiv:2204.13176
[7]
C. Vuillot and N. P. Breuckmann, “Quantum Pin Codes”, IEEE Transactions on Information Theory 68, 5955 (2022) arXiv:1906.11394 DOI
[8]
P. A. Mishchenko and K. Xagawa, “Secure multi-party quantum computation based on triply-even quantum error-correcting codes”, (2022) arXiv:2206.04871
[9]
J. Haah, “Towers of generalized divisible quantum codes”, Physical Review A 97, (2018) arXiv:1709.08658 DOI
[10]
R. J. McEliece, “On periodic sequences from GF(q)”, Journal of Combinatorial Theory, Series A 10, 80 (1971) DOI
[11]
R. J. McEliece, “Weight congruences for p-ary cyclic codes”, Discrete Mathematics 3, 177 (1972) DOI
[12]
K. Betsumiya and A. Munemasa, “On triply even binary codes”, Journal of the London Mathematical Society 86, 1 (2012) arXiv:1012.4134 DOI
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Zoo Code ID: quantum_divisible

Cite as:
“Quantum divisible code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_divisible
BibTeX:
@incollection{eczoo_quantum_divisible, title={Quantum divisible code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/quantum_divisible} }
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“Quantum divisible code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_divisible

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