Subsystem QPC[1] 


Subsystem version of the QPC which has the same parameters as the subspace version, but requires significantly fewer stabilizer measurements, resulting in a much simpler error recovery routine.

This \([[n_1n_2,k_1k_2,\min(d_1,d_2)]]\) code can be defined [2] via the CSS construction applied to two binary linear codes, \(C_X\) and \(C_Z\), satisfying \(C_X^{\perp}\subset C_Z\). These codes are in turn constructed from two more binary linear codes, \(C_1 = [n_1, k_1, d_1]\) and \(C_2 = [n_2, k_2, d_2]\), with parity-check matrices \(H_{1,2}\) and generator matrices \(G_{1,2}\), respectively. The parity-check matrices of \(C_X\) and \(C_Z\) are then \begin{align} \begin{split} H_X &= H_1 \otimes I_{n_2}\\ H_Z &= G_1 \otimes H_2~. \end{split} \tag*{(1)}\end{align}

Based on the above construction, the Hilbert space on \(n_1n_2\) qubits can be decomposed into a multiple direct sums of multiple tensor products of Hilbert spaces of lower dimensions, as outlined in [3].


In a \([[n_1n_2, k_1k_2, min(d_1, d_2)]]\) QPC, error correction is achieved by measuring \((n_1−k_1)n_2+(n_2−k_2)\) stabilizer generators [4]. The subsystem QPC achieves the same degree of correctability, but requires only \((n_1−k_1)k_2+k_1(n_2−k_2)\) stabilizer measurements.





D. Bacon and A. Casaccino, “Quantum Error Correcting Subsystem Codes From Two Classical Linear Codes”, (2006) arXiv:quant-ph/0610088
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. Bacon, “Operator quantum error-correcting subsystems for self-correcting quantum memories”, Physical Review A 73, (2006) arXiv:quant-ph/0506023 DOI
D. Poulin, “Stabilizer Formalism for Operator Quantum Error Correction”, Physical Review Letters 95, (2005) arXiv:quant-ph/0508131 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

Cite as:
“Subsystem QPC”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
  title={Subsystem QPC},
  booktitle={The Error Correction Zoo},
  editor={Albert, Victor V. and Faist, Philippe},
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Cite as:

“Subsystem QPC”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.