## 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

## Parents

- Subsystem lifted-product (SLP) code — SLP codes reduce to SHP codes when the lift is trivial.
- Subsystem homological product code — SP codes reduce to SHP codes when constructed from two classical codes instead of quantum CSS codes [4].

## 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

## Page edit log

- Victor V. Albert (2023-11-14) — most recent
- Sarah Meng Li (2022-02-21)
- Victor V. Albert (2022-02-21)

## 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