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 [4,5] and tree tensor networks [6].
Decoding
Teleportation-based QEC [7].
Threshold
All optical scheme using QPCs concatenated with either Steane or Golay codes [8].
Realizations
The \([[m^2,1,m]]\) codes for \(m\leq 7\) have been realized in trapped-ion quantum devices [9].QPCs have been discussed independently in the context of superconducting circuits [10; Eq. (1)][11; Eqs. (8-10)], and aspects of such designs have been realized in experiments [12].
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
- Generalized Shor code
- Group-based QPC — A \([[m_1 m_2,1,\min(m_1,m_2)]]_G\) group-based QPC reduces to a QPC for \(G=\mathbb{Z}_2\).
- Quantum Lego code — Encoders for a recursively concatenated QPCs are related to quantum trees [4,5] and tree tensor networks [6].
Children
- Quantum repetition code — A \([[m_1 m_2,1,\min(m_1,m_2)]]\) QPC reduces 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 \([[m^2,1,m]]\) QPCs.
Cousins
- Bacon-Shor code — Bacon-Shor codes reduce to QPCs when all \(X\)-type gauge generators are fixed [13; pg. 6].
- Majorana stabilizer code — QPCs for \(m_1=m_2\) can be conveniantly expressed in terms of mutually commuting Majorana operators [14].
- 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 [3,15,16]. More generally, an \([[m^2,1,m]]\) QPC corrects \(m-1\) AD errors [17].
- Raussendorf-Bravyi-Harrington (RBH) cluster-state code — QPCs can be concatenated with RBH codes [18].
- 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 [3,15,16]. More generally, an \([[m^2,1,m]]\) QPC corrects \(m-1\) AD errors [17].
- Two-mode binomial code — Two-mode binomial codes can be concatenated with repetition codes to yield bosonic analogues of QPCs [19].
- Concatenated GKP code — GKP codes have been concatenated with QPCs [20].
- 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 [3,15,16].
- Compass code — The Shor-density compass code family interpolates between Bacon-Shor codes and QPCs.
References
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- 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
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- B. Ferté and X. Cao, “Solvable Model of Quantum-Darwinism-Encoding Transitions”, Physical Review Letters 132, (2024) arXiv:2305.03694 DOI
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- S. A. Yadavalli and I. Marvian, “Noisy Quantum Trees: Infinite Protection Without Correction”, (2024) arXiv:2306.14294
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- A. J. Ferris and D. Poulin, “Tensor Networks and Quantum Error Correction”, Physical Review Letters 113, (2014) arXiv:1312.4578 DOI
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- S. Muralidharan et al., “Ultrafast and Fault-Tolerant Quantum Communication across Long Distances”, Physical Review Letters 112, (2014) arXiv:1310.5291 DOI
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- A. J. F. Hayes et al., “Fault tolerance in parity-state linear optical quantum computing”, Physical Review A 82, (2010) arXiv:0908.3932 DOI
- [9]
- N. H. Nguyen et al., “Demonstration of Shor Encoding on a Trapped-Ion Quantum Computer”, Physical Review Applied 16, (2021) arXiv:2104.01205 DOI
- [10]
- B. Douçot et al., “Protected qubits and Chern-Simons theories in Josephson junction arrays”, Physical Review B 71, (2005) arXiv:cond-mat/0403712 DOI
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- B. Douçot and L. B. Ioffe, “Physical implementation of protected qubits”, Reports on Progress in Physics 75, 072001 (2012) DOI
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- S. Gladchenko et al., “Superconducting nanocircuits for topologically protected qubits”, Nature Physics 5, 48 (2008) arXiv:0802.2295 DOI
- [13]
- M. Li et al., “2D Compass Codes”, Physical Review X 9, (2019) arXiv:1809.01193 DOI
- [14]
- S. B. Bravyi and A. Yu. Kitaev, “Fermionic Quantum Computation”, Annals of Physics 298, 210 (2002) arXiv:quant-ph/0003137 DOI
- [15]
- G. Alber et al., “Stabilizing Distinguishable Qubits against Spontaneous Decay by Detected-Jump Correcting Quantum Codes”, Physical Review Letters 86, 4402 (2001) arXiv:quant-ph/0103042 DOI
- [16]
- Y. Ouyang, “Avoiding coherent errors with rotated concatenated stabilizer codes”, npj Quantum Information 7, (2021) arXiv:2010.00538 DOI
- [17]
- R. Duan et al., “Multi-error-correcting amplitude damping codes”, 2010 IEEE International Symposium on Information Theory (2010) arXiv:1001.2356 DOI
- [18]
- S.-H. Lee et al., “Parity-encoding-based quantum computing with Bayesian error tracking”, npj Quantum Information 9, (2023) arXiv:2207.06805 DOI
- [19]
- M. Bergmann and P. van Loock, “Quantum error correction against photon loss using NOON states”, Physical Review A 94, (2016) arXiv:1512.07605 DOI
- [20]
- 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
Page edit log
- Victor V. Albert (2023-11-14) — most recent
- Victor V. Albert (2021-12-31)
- Xinyuan Zheng (2021-12-20)
Cite as:
“Quantum parity code (QPC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/quantum_parity