Quantum parity code (QPC)[13] 

Also known as Subspace Shor code.

Description

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 [4].

Notes

Non-determinisitic linear-optical encoding [3] 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 [5].

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 — The Shor code is part of the sub-family of \([[m^2,1,m]]\) QPC codes.

Cousins

  • Bacon-Shor code — Bacon-Shor codes reduce to QPCs when all \(X\)-type gauge generators are fixed [6; pg. 6].
  • 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 — The \([[4,1,2]]\) subcode \(\{|\overline{00}\rangle,|\overline{01}\rangle\}\) is the smallest member of the sub-family of \([[m^2,1,m]]\) QPC codes.
  • Compass code — The Shor-density compass code family interpolates between Bacon-Shor codes and QPCs.

References

[1]
P. W. Shor, “Scheme for reducing decoherence in quantum computer memory”, Physical Review A 52, R2493 (1995) DOI
[2]
E. Knill, R. Laflamme, and G. Milburn, “Efficient Linear Optics Quantum Computation”, (2000) arXiv:quant-ph/0006088
[3]
T. C. Ralph, A. J. F. Hayes, and A. Gilchrist, “Loss-Tolerant Optical Qubits”, Physical Review Letters 95, (2005) arXiv:quant-ph/0501184 DOI
[4]
N. H. Nguyen et al., “Demonstration of Shor Encoding on a Trapped-Ion Quantum Computer”, Physical Review Applied 16, (2021) arXiv:2104.01205 DOI
[5]
S. Muralidharan et al., “Ultrafast and Fault-Tolerant Quantum Communication across Long Distances”, Physical Review Letters 112, (2014) arXiv:1310.5291 DOI
[6]
M. Li et al., “2D Compass Codes”, Physical Review X 9, (2019) arXiv:1809.01193 DOI
[7]
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.), 2023. https://errorcorrectionzoo.org/c/quantum_parity
BibTeX:
@incollection{eczoo_quantum_parity, title={Quantum parity code (QPC)}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/quantum_parity} }
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“Quantum parity code (QPC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/quantum_parity

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