Quantum divisible code

Quantum divisible code[1]

Description

Consider a CSS code whose \(Z\)-stabilizers are determined by the dual of a classical \([n, k_1]\) linear binary code \(C_1\), and whose \(X\)-stabilizers are determined by a classical \([n, k_2]\) binary code \(C_2 \subset C_1\). This code is quantum divisible if 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, 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.

Protection

Distance \(d\) is upper bounded by the two classical codes that determine the CSS code.

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. 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.

Parent

Calderbank-Shor-Steane (CSS) stabilizer code

Cousins

Divisible code — Quantum divisible codes are constructed via the CSS construction using a divisible linear binary code.
Reed-Muller (RM) code — Quantum divisible codes can be constructed out of first-order RM codes.
Quantum Reed-Muller code — Quantum RM codes can be derived using a procedure that yields sufficient conditions for a CSS code to admit a given transversal diagonal logical gate. Quantum divisible codes are derived in a similar procedure, but one that yields necessary and sufficient conditions.
Triorthogonal code — Triorthogonal codes can be derived using a procedure that yields sufficient conditions for a CSS code to admit a given transversal diagonal logical gate. Quantum divisible codes are derived in a similar procedure, but one that yields necessary and sufficient conditions.
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.
Five-qubit perfect code — A fault-tolerant \(T\) gate on the five-qubit code can be obtained by concatenating with particular quantum divisible codes.
\([[7,1,3]]\) Steane code — A fault-tolerant \(T\) gate on the Steane code can be obtained by concatenating with particular quantum divisible codes.

References

[1]: J. Hu, Q. Liang, and R. Calderbank, “Divisible Codes for Quantum Computation”, (2022) arXiv:2204.13176

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qubits/small_distance/quantum_divisible.yml.