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

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.
  • \([[5,1,3]]\) perfect code — A fault-tolerant \(T\) gate on the five-qubit code can be obtained by concatenating with particular quantum divisible codes.
  • Steane \([[7,1,3]]\) code — A fault-tolerant \(T\) gate on the Steane code can be obtained by concatenating with particular quantum divisible codes.

Zoo code information

Internal code ID: quantum_divisible

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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} }
Permanent link:
https://errorcorrectionzoo.org/c/quantum_divisible

References

[1]
Jingzhen Hu, Qingzhong Liang, and Robert Calderbank, “Divisible Codes for Quantum Computation”. 2204.13176

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/quantum_divisible.yml.