Quantum parity code (QPC)[1–3] 

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

Encoding

Encoders for a recursively concatenated QPCs are related to quantum trees [4–6] and tree tensor networks [7].Linear-optical encoding [8].

Decoding

Teleportation-based QEC [9].

Threshold

All optical scheme using QPCs concatenated with either Steane or Golay codes [10].

Realizations

The \([[m^2,1,m]]\) codes for \(m\leq 7\) have been realized in trapped-ion quantum devices [11].QPCs have been discussed independently in the context of superconducting circuits [12; Eq. (1)][13; Eqs. (8-10)], and aspects of such designs have been realized in experiments [14].

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 \(m_2\).

Parents

Children

Cousins

  • Bacon-Shor code — Bacon-Shor codes reduce to QPCs when all \(X\)-type gauge generators are fixed [15; pg. 6].
  • Majorana stabilizer code — QPCs for \(m_1=m_2\) can be conveniantly expressed in terms of mutually commuting Majorana operators [16].
  • Constant-excitation (CE) code — QPCs for even \(m_1\) can be made into CE codes by a Pauli transformation (e.g., \(XIXI\cdots XI\)) applied to each block of \(m_1\) qubits.
  • Amplitude-damping (AD) code — An \([[8,1,2]]\) QPC correcting a single AD error is equivalent to a concatenation of the \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) (constant-excitation) subcode of the \([[4,2,2]]\) code with the dual-rail code [3,17,18]. More generally, an \([[m^2,1,m]]\) QPC corrects \(m-1\) AD errors [19].
  • Raussendorf-Bravyi-Harrington (RBH) cluster-state code — QPCs can be concatenated with RBH codes [20].
  • Dual-rail quantum code — An \([[8,1,2]]\) QPC correcting a single AD error is equivalent to a concatenation of the \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) (constant-excitation) subcode of the \([[4,2,2]]\) code with the dual-rail code [3,17,18]. More generally, an \([[m^2,1,m]]\) QPC corrects \(m-1\) AD errors [19].
  • Two-mode binomial code — Two-mode binomial codes can be concatenated with repetition codes to yield bosonic analogues of QPCs [21].
  • Concatenated GKP code — GKP codes have been concatenated with QPCs [22].
  • Asymmetric quantum code — QPC parameters against bit- and phase-noise can be tuned.
  • \([[4,2,2]]\) Four-qubit code — The \(\{|\overline{00}\rangle,|\overline{01}\rangle\}\) \([[4,1,2]]\) subcode is the smallest QPC, i.e., a concatenation of a two-qubit bit-flip with a two-qubit phase-flip repetition code. An \([[8,1,2]]\) QPC correcting a single AD error is equivalent to a concatenation of the \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) (constant-excitation) subcode of the \([[4,2,2]]\) code with the dual-rail code [3,17,18].
  • 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]
B. Ferté and X. Cao, “Solvable Model of Quantum-Darwinism-Encoding Transitions”, Physical Review Letters 132, (2024) arXiv:2305.03694 DOI
[5]
S. A. Yadavalli and I. Marvian, “Noisy Quantum Trees: Infinite Protection Without Correction”, (2024) arXiv:2306.14294
[6]
G. M. Sommers, D. A. Huse, and M. J. Gullans, “Dynamically generated concatenated codes and their phase diagrams”, (2024) arXiv:2409.13801
[7]
A. J. Ferris and D. Poulin, “Tensor Networks and Quantum Error Correction”, Physical Review Letters 113, (2014) arXiv:1312.4578 DOI
[8]
A. J. F. Hayes, A. Gilchrist, and T. C. Ralph, “Loss-tolerant operations in parity-code linear optics quantum computing”, Physical Review A 77, (2008) arXiv:0707.0903 DOI
[9]
S. Muralidharan, J. Kim, N. Lütkenhaus, M. D. Lukin, and L. Jiang, “Ultrafast and Fault-Tolerant Quantum Communication across Long Distances”, Physical Review Letters 112, (2014) arXiv:1310.5291 DOI
[10]
A. J. F. Hayes, H. L. Haselgrove, A. Gilchrist, and T. C. Ralph, “Fault tolerance in parity-state linear optical quantum computing”, Physical Review A 82, (2010) arXiv:0908.3932 DOI
[11]
N. H. Nguyen, M. Li, A. M. Green, C. Huerta Alderete, Y. Zhu, D. Zhu, K. R. Brown, and N. M. Linke, “Demonstration of Shor Encoding on a Trapped-Ion Quantum Computer”, Physical Review Applied 16, (2021) arXiv:2104.01205 DOI
[12]
B. Douçot, M. V. Feigel’man, L. B. Ioffe, and A. S. Ioselevich, “Protected qubits and Chern-Simons theories in Josephson junction arrays”, Physical Review B 71, (2005) arXiv:cond-mat/0403712 DOI
[13]
B. Douçot and L. B. Ioffe, “Physical implementation of protected qubits”, Reports on Progress in Physics 75, 072001 (2012) DOI
[14]
S. Gladchenko, D. Olaya, E. Dupont-Ferrier, B. Douçot, L. B. Ioffe, and M. E. Gershenson, “Superconducting nanocircuits for topologically protected qubits”, Nature Physics 5, 48 (2008) arXiv:0802.2295 DOI
[15]
M. Li, D. Miller, M. Newman, Y. Wu, and K. R. Brown, “2D Compass Codes”, Physical Review X 9, (2019) arXiv:1809.01193 DOI
[16]
S. B. Bravyi and A. Yu. Kitaev, “Fermionic Quantum Computation”, Annals of Physics 298, 210 (2002) arXiv:quant-ph/0003137 DOI
[17]
G. Alber, Th. Beth, Ch. Charnes, A. Delgado, M. Grassl, and M. Mussinger, “Stabilizing Distinguishable Qubits against Spontaneous Decay by Detected-Jump Correcting Quantum Codes”, Physical Review Letters 86, 4402 (2001) arXiv:quant-ph/0103042 DOI
[18]
Y. Ouyang, “Avoiding coherent errors with rotated concatenated stabilizer codes”, npj Quantum Information 7, (2021) arXiv:2010.00538 DOI
[19]
R. Duan, M. Grassl, Z. Ji, and B. Zeng, “Multi-error-correcting amplitude damping codes”, 2010 IEEE International Symposium on Information Theory (2010) arXiv:1001.2356 DOI
[20]
S.-H. Lee, S. Omkar, Y. S. Teo, and H. Jeong, “Parity-encoding-based quantum computing with Bayesian error tracking”, npj Quantum Information 9, (2023) arXiv:2207.06805 DOI
[21]
M. Bergmann and P. van Loock, “Quantum error correction against photon loss using NOON states”, Physical Review A 94, (2016) arXiv:1512.07605 DOI
[22]
K. Fukui, T. Matsuura, and N. C. Menicucci, “Efficient Concatenated Bosonic Code for Additive Gaussian Noise”, Physical Review Letters 131, (2023) arXiv:2102.01374 DOI
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Zoo Code ID: 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
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@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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