Quantum parity code (QPC)[13] 


Also called a generalized Shor code [4]. A \([[m_1 m_2,1,\min(m_1,m_2)]]\) CSS code family obtained from concatenating an \(m_1\)-qubit phase-flip repetition code with an \(m_2\)-qubit bit-flip repetition code. Logical codewords are \begin{align} \begin{split} |\overline{0}\rangle&=\frac{1}{2^{m_2/2}}\left(|0\rangle^{\otimes m_1}+|1\rangle^{\otimes m_1}\right)^{\otimes m_2}\\ |\overline{1}\rangle&=\frac{1}{2^{m_2/2}}\left(|0\rangle^{\otimes m_1}-|1\rangle^{\otimes m_1}\right)^{\otimes m_2}~. \end{split} \tag*{(1)}\end{align}


Has distance \(d=\min(m_1,m_2)\).


The \([[m^2,1,m]]\) codes for \(m\leq 7\) have been realized in trapped-ion quantum devices [5].


Non-determinisitic linear-optical encoding [2] whose success probability \(P_{E}\) is determined by the efficiency \(\eta\) of the photonic encoding circuit. A threshold \(\eta > 0.82 \) exists for the efficiency, above which \(P_{E}\to 1\) as \(m_1\to\infty\) given particular \(m_2\).Studied in the context of error-corrected quantum repeaters [6].



  • Quantum repetition code — A \([[m_1 m_2,1,\min(m_1,m_2)]]\) QPC is a concatenation of a \(m_1\) bit-flip and a \(m_2\) phase-flip repetition codes, reducing to a repetition code when \(m_1\) or \(m_2\) is one.
  • \([[9,1,3]]\) Shor code — The Shor code is part of the sub-family of \([[m^2,1,m]]\) QPC codes.


  • Majorana stabilizer code — QPCs for \(m_1=m_2\) can be conveniantly expressed in terms of mutually commuting Majorana operators [7].
  • \([[4,2,2]]\) CSS code — \([[4,1,2]]\) subcode \(\{|\overline{00}\rangle,|\overline{01}\rangle\}\) is the smallest member of the sub-family of \([[m^2,1,m]]\) QPC codes.
  • Subsystem QPC


E. Knill, R. Laflamme, and G. Milburn, “Efficient Linear Optics Quantum Computation”, (2000) arXiv:quant-ph/0006088
T. C. Ralph, A. J. F. Hayes, and A. Gilchrist, “Loss-Tolerant Optical Qubits”, Physical Review Letters 95, (2005) arXiv:quant-ph/0501184 DOI
E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics”, Nature 409, 46 (2001) DOI
D. Bacon and A. Casaccino, “Quantum Error Correcting Subsystem Codes From Two Classical Linear Codes”, (2006) arXiv:quant-ph/0610088
N. H. Nguyen et al., “Demonstration of Shor Encoding on a Trapped-Ion Quantum Computer”, Physical Review Applied 16, (2021) arXiv:2104.01205 DOI
S. Muralidharan et al., “Ultrafast and Fault-Tolerant Quantum Communication across Long Distances”, Physical Review Letters 112, (2014) arXiv:1310.5291 DOI
S. B. Bravyi and A. Yu. Kitaev, “Fermionic Quantum Computation”, Annals of Physics 298, 210 (2002) arXiv:quant-ph/0003137 DOI
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Zoo Code ID: quantum_parity

Cite as:
“Quantum parity code (QPC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/quantum_parity
  title={Quantum parity code (QPC)},
  booktitle={The Error Correction Zoo},
  editor={Albert, Victor V. and Faist, Philippe},
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“Quantum parity code (QPC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/quantum_parity

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qubits/stabilizer/quantum_parity.yml.