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 RM\((1,m)\) 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].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 admits a diagonal transversal gate in the \(\nu\)th level of the Clifford hierarchy [7; Prop. 8].
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
- \([[3k + 8, k, 2]]\) triorthogonal code
- \([[49,1,5]]\) triorthogonal code
- \([[2^r-1,1,3]]\) simplex code — \([[2^r-1,1,3]]\) simplex codes come from RM\((1,m=r)\) codes, which are \((r-1)\)-even [10,11], 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 [9; Sec. VI.B].
Cousins
- Divisible code — The \(X\)-type stabilizers of a level-\(\nu\) quantum divisible code form a \(\nu\)-even linear binary code.
- \([2^m,m+1,2^{m-1}]\) First-order RM code — Quantum divisible codes can be constructed out of first-order RM\((1,m)\) codes [6].
- Concatenated qubit 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
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- S. Bravyi and A. Cross, “Doubled Color Codes”, (2015) arXiv:1509.03239
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- J. Hu, Q. Liang, and R. Calderbank, “Designing the Quantum Channels Induced by Diagonal Gates”, Quantum 6, 802 (2022) arXiv:2109.13481 DOI
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- P. A. Mishchenko and K. Xagawa, “Secure multi-party quantum computation based on triply-even quantum error-correcting codes”, (2022) arXiv:2206.04871
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- J. Haah, “Towers of generalized divisible quantum codes”, Physical Review A 97, (2018) arXiv:1709.08658 DOI
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- R. J. McEliece, “On periodic sequences from GF(q)”, Journal of Combinatorial Theory, Series A 10, 80 (1971) DOI
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- R. J. McEliece, “Weight congruences for p-ary cyclic codes”, Discrete Mathematics 3, 177 (1972) DOI
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- K. Betsumiya and A. Munemasa, “On triply even binary codes”, Journal of the London Mathematical Society 86, 1 (2012) arXiv:1012.4134 DOI
Page edit log
- Jingzhen Hu (2022-05-04) — most recent
- Victor V. Albert (2022-05-04)
Cite as:
“Quantum divisible code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_divisible