Generalized Shor code

Generalized Shor code[1,2]

Description

Qubit CSS code constructed by concatenating two classical codes in a way the generalizes the Shor and quantum parity codes.

This \([[n_1n_2,k_1k_2,\min(d_1,d_2)]]\) code can be defined [3] via the CSS construction applied to two binary linear codes, \(C_X\) and \(C_Z\), satisfying \(C_X^{\perp}\subset C_Z\). These codes are in turn constructed from two more binary linear codes, \(C_1 = [n_1, k_1, d_1]\) and \(C_2 = [n_2, k_2, d_2]\), with parity-check matrices \(H_{1,2}\) and generator matrices \(G_{1,2}\), respectively. The parity-check matrices of \(C_X\) and \(C_Z\) are then \begin{align} \begin{split} H_X &= H_1 \otimes I_{n_2}\\ H_Z &= G_1 \otimes H_2~. \end{split} \tag*{(1)}\end{align}

Based on the above construction, the Hilbert space on \(n_1n_2\) qubits can be decomposed into a multiple direct sums of multiple tensor products of Hilbert spaces of lower dimensions, as outlined in [1].

Protection

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

Decoding

Efficient decoder [4].

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 [6] 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 [7].

Parent

Qubit CSS code

Child

Quantum parity code (QPC)

Cousin

Subsystem hypergraph product (SHP) code — In a \([[n_1n_2, k_1k_2, min(d_1, d_2)]]\) generalized Shor code, error correction is achieved by measuring \((n_1−k_1)n_2+(n_2−k_2)\) stabilizer generators [8]. The SHP code achieves the same degree of correctability, but requires only \((n_1−k_1)k_2+k_1(n_2−k_2)\) stabilizer measurements.

References

[1]: D. Bacon, “Operator quantum error-correcting subsystems for self-correcting quantum memories”, Physical Review A 73, (2006) arXiv:quant-ph/0506023 DOI
[2]: D. Bacon and A. Casaccino, “Quantum Error Correcting Subsystem Codes From Two Classical Linear Codes”, (2006) arXiv:quant-ph/0610088
[3]: J.-P. Tillich and G. Zemor, “Quantum LDPC Codes With Positive Rate and Minimum Distance Proportional to the Square Root of the Blocklength”, IEEE Transactions on Information Theory 60, 1193 (2014) arXiv:0903.0566 DOI
[4]: P. K. Sarvepalli, A. Klappenecker, and M. Rotteler, “New decoding algorithms for a class of subsystem codes and generalized shor codes”, 2009 IEEE International Symposium on Information Theory (2009) DOI
[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]: T. C. Ralph, A. J. F. Hayes, and A. Gilchrist, “Loss-Tolerant Optical Qubits”, Physical Review Letters 95, (2005) arXiv:quant-ph/0501184 DOI
[7]: S. Muralidharan et al., “Ultrafast and Fault-Tolerant Quantum Communication across Long Distances”, Physical Review Letters 112, (2014) arXiv:1310.5291 DOI
[8]: D. Poulin, “Stabilizer Formalism for Operator Quantum Error Correction”, Physical Review Letters 95, (2005) arXiv:quant-ph/0508131 DOI

Page edit log

Victor V. Albert (2023-11-14) — most recent

Cite as:

“Generalized Shor code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/generalized_shor

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