Description
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}
Protection
Has distance \(d=\min(m_1,m_2)\).
Realizations
The \([[m^2,1,m]]\) codes for \(m\leq 7\) have been realized in trapped-ion quantum devices [5].
Notes
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].
Parents
Children
- 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 — Shor's code is part of the sub-family of \([[m^2,1,m]]\) QPC codes.
Cousins
- Concatenated quantum code — A QPC is a concatenation of a phase-flip repetition code with a bit-flip repetition code.
- Bacon-Shor code — Bacon-Shor codes reduce to QPCs for a particular gauge configuration.
- 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
References
- [1]
- E. Knill, R. Laflamme, and G. Milburn, “Efficient Linear Optics Quantum Computation”, (2000) arXiv:quant-ph/0006088
- [2]
- T. C. Ralph, A. J. F. Hayes, and A. Gilchrist, “Loss-Tolerant Optical Qubits”, Physical Review Letters 95, (2005) arXiv:quant-ph/0501184 DOI
- [3]
- E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics”, Nature 409, 46 (2001) DOI
- [4]
- D. Bacon and A. Casaccino, “Quantum Error Correcting Subsystem Codes From Two Classical Linear Codes”, (2006) arXiv:quant-ph/0610088
- [5]
- N. H. Nguyen et al., “Demonstration of Shor Encoding on a Trapped-Ion Quantum Computer”, Physical Review Applied 16, (2021) arXiv:2104.01205 DOI
- [6]
- S. Muralidharan et al., “Ultrafast and Fault-Tolerant Quantum Communication across Long Distances”, Physical Review Letters 112, (2014) arXiv:1310.5291 DOI
- [7]
- S. B. Bravyi and A. Yu. Kitaev, “Fermionic Quantum Computation”, Annals of Physics 298, 210 (2002) arXiv:quant-ph/0003137 DOI
Page edit log
- Victor V. Albert (2021-12-31) — most recent
- Xinyuan Zheng (2021-12-20)
Cite as:
“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/quantum_parity.yml.