Subsystem hypergraph product (SHP) code[1,2] 

Also known as Subsystem generalized Shor code, Bacon-Casaccino subsystem code.

Description

A CSS subsystem version of the generalized Shor code that has the same parameters as the subspace version, but requires fewer stabilizer measurements, resulting in a simpler error recovery routine. The code can also be thought of as a subsystem version of an HGP code because two such codes reduce to an HGP code upon gauge fixing [2; Sec. III]. The code can be obtained from a generalized Shor code by removing certain stabilizers that do no affect the code distance.

The \(X\)- and \(Z\)-type gauge generators of this CSS \([[n_1n_2,k_1k_2,\min(d_1,d_2)]]\) code correspond to rows of the following two respective matrices, \begin{align} \begin{split} G_{X}&=H_{1}\otimes I_{n_{2}}\\ G_{Z}&=I_{n_{1}}\otimes H_{2}~, \end{split} \tag*{(1)}\end{align} where \(H_{1,2}\) are the parity-check matrices of two binary linear codes, \(C_1 = [n_1, k_1, d_1]\) and \(C_2 = [n_2, k_2, d_2]\) [2].

Decoding

Efficient decoder [3].

Parents

Child

Cousins

  • Hypergraph product (HGP) code — Two SHP codes can be gauge-fixed to yield an HGP code [2; Sec. III]. The SHP and HGP code constructions yield the same dimension and minimum distance, but the former does not yield QLDPC codes; see [5; pg. 18].
  • Generalized Shor code — In a \([[n_1n_2, k_1k_2, min(d_1, d_2)]]\) generalized Shor code, error correction is achieved by measuring \((n_1−k_1)n_2+(n_2−k_2)\) stabilizer generators [6]. The SHP code achieves the same degree of correctability, but requires only \((n_1−k_1)k_2+k_1(n_2−k_2)\) stabilizer measurements.
  • Bravyi-Bacon-Shor (BBS) code — SHP and BBS codes have been numerically compared [2].
  • Subsystem QECC — Classical information can also be encoded in subsystem codes using their gauge qubits [7].
  • Bravyi-Bacon-Shor (BBS) code — The BBS code construction can utilize different classical codes in different rows and columns of \(A\), while the subsystem construction does not; see [8; pg. 4]

References

[1]
D. Bacon and A. Casaccino, “Quantum Error Correcting Subsystem Codes From Two Classical Linear Codes”, (2006) arXiv:quant-ph/0610088
[2]
M. Li and T. J. Yoder, “A Numerical Study of Bravyi-Bacon-Shor and Subsystem Hypergraph Product Codes”, (2020) arXiv:2002.06257
[3]
P. K. Sarvepalli, A. Klappenecker, and M. Rotteler, “New decoding algorithms for a class of subsystem codes and generalized shor codes”, 2009 IEEE International Symposium on Information Theory (2009) DOI
[4]
W. Zeng and L. P. Pryadko, “Minimal distances for certain quantum product codes and tensor products of chain complexes”, Physical Review A 102, (2020) arXiv:2007.12152 DOI
[5]
J.-P. Tillich and G. Zemor, “Quantum LDPC Codes With Positive Rate and Minimum Distance Proportional to the Square Root of the Blocklength”, IEEE Transactions on Information Theory 60, 1193 (2014) arXiv:0903.0566 DOI
[6]
D. Poulin, “Stabilizer Formalism for Operator Quantum Error Correction”, Physical Review Letters 95, (2005) arXiv:quant-ph/0508131 DOI
[7]
A. Nemec and A. Klappenecker, “Encoding classical information in gauge subsystems of quantum codes”, International Journal of Quantum Information 20, (2022) DOI
[8]
S. Bravyi, “Subsystem codes with spatially local generators”, Physical Review A 83, (2011) arXiv:1008.1029 DOI
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Zoo Code ID: subsystem_quantum_parity

Cite as:
“Subsystem hypergraph product (SHP) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/subsystem_quantum_parity
BibTeX:
@incollection{eczoo_subsystem_quantum_parity, title={Subsystem hypergraph product (SHP) code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/subsystem_quantum_parity} }
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“Subsystem hypergraph product (SHP) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/subsystem_quantum_parity

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/subsystem/qldpc/homological/subsystem_quantum_parity.yml.