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

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


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].


Efficient decoder [3].




  • 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]


D. Bacon and A. Casaccino, “Quantum Error Correcting Subsystem Codes From Two Classical Linear Codes”, (2006) arXiv:quant-ph/0610088
M. Li and T. J. Yoder, “A Numerical Study of Bravyi-Bacon-Shor and Subsystem Hypergraph Product Codes”, (2020) arXiv:2002.06257
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
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
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
D. Poulin, “Stabilizer Formalism for Operator Quantum Error Correction”, Physical Review Letters 95, (2005) arXiv:quant-ph/0508131 DOI
A. Nemec and A. Klappenecker, “Encoding classical information in gauge subsystems of quantum codes”, International Journal of Quantum Information 20, (2022) DOI
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

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“Subsystem hypergraph product (SHP) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.
@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={} }
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“Subsystem hypergraph product (SHP) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.