## Description

Stub.

## Protection

There exist approximate versions of the Knill-Laflamme conditions that can be used to determine the degree to which a code is an approximate error-correcting code [6]. Various sufficient criteria have also been derived; see for instance refs. [1][4][7][8]. See Ref. [9] for bounds on approximate code length, size, and distance.

## Encoding

Given a decoder, an encoding that yields the optimal entanglement fidelity can be obtained by solving a semi-definite program [10][3] (see also Ref. [11]).Variational quantum circuit encoder [12].

## Decoding

Given an encoding, a decoder that yields the optimal entanglement fidelity can be obtained by solving a semi-definite program [10][3] (see also Ref. [11]).The Petz recovery map (a.k.a. the transpose map) [13][14], a quantum channel determined by the codespace and noise channel, recovers information perfectly for strictly correctable noise and yields an infidelity of recovery that is at most twice away from the infidelity of the best possible recovery [15]. The infidelity of a modified Petz recovery map can be bounded using relative entropies between uncorrupted and corrupted code states on countably infinite Hilbert spaces [16].

## Notes

See review [17].

## Parent

## Children

- Matrix-model code — Matrix-model codes approximately protect against gauge-invariant errors in the large-mode limit.
- \(U(d)\)-covariant approximate erasure code
- W-state code — The W-state code approximately protects against a single erasure while allowing for a universal transversal set of gates.
- Eigenstate thermalization hypothesis (ETH) code — ETH codes approximately protect against erasures in the thermodynamic limit.
- Matrix-product state (MPS) code — MPS codes approximately protect against erasures in the thermodynamic limit.
- Singleton-bound approaching AQECC

## Cousins

- Renormalization group (RG) cat code — RG cat codes approximately protect against displacements that represent ultraviolet coherent operators.
- Qudit-into-oscillator code — Approximate QEC techniques of finding the entanglement fidelity can be adapted to bosonic codes with a finite-dimensional codespace [18].
- Square-lattice GKP code — Square-lattice GKP codes approximately protect against photon loss [19][18][20].
- Gottesman-Kitaev-Preskill (GKP) code — Approximate error-correction offered by GKP codes yields achievable rates that are a constant away from the capacity of Guassian loss channels [21][22][23][20].
- Covariant code — Normalizable constructions of infinite-dimensional \(G\)-covariant codes for continuous \(G\) are approximately error-correcting.
- Local Haar-random circuit code
- \([[4,2,2]]\) CSS code — \([[4,1,2]]\) subcodes \(\{|\overline{00}\rangle,|\overline{10}\rangle\}\) [1] and \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) [24] approximately correct a single amplitude damping error.
- Approximate secret-sharing code — Secret-sharing codes approximately correct errors on up to \(\lfloor (n-1)/2 \rfloor\) errors.
- GNU permutation-invariant code — GNU codes protect approximately against amplitude damping errors.

## References

- [1]
- D. W. Leung et al., “Approximate quantum error correction can lead to better codes”, Physical Review A 56, 2567 (1997) arXiv:quant-ph/9704002 DOI
- [2]
- A. Y. Kitaev, “Quantum computations: algorithms and error correction”, Russian Mathematical Surveys 52, 1191 (1997) DOI
- [3]
- M. Reimpell and R. F. Werner, “Iterative Optimization of Quantum Error Correcting Codes”, Physical Review Letters 94, (2005) arXiv:quant-ph/0307138 DOI
- [4]
- C. Crepeau, D. Gottesman, and A. Smith, “Approximate Quantum Error-Correcting Codes and Secret Sharing Schemes”, (2005) arXiv:quant-ph/0503139
- [5]
- C. Bény, “Conditions for the approximate correction of algebras”, (2009) arXiv:0907.4207
- [6]
- C. Bény and O. Oreshkov, “General Conditions for Approximate Quantum Error Correction and Near-Optimal Recovery Channels”, Physical Review Letters 104, (2010) arXiv:0907.5391 DOI
- [7]
- F. G. S. L. Brandão et al., “Quantum Error Correcting Codes in Eigenstates of Translation-Invariant Spin Chains”, Physical Review Letters 123, (2019) arXiv:1710.04631 DOI
- [8]
- P. Faist et al., “Continuous Symmetries and Approximate Quantum Error Correction”, Physical Review X 10, (2020) arXiv:1902.07714 DOI
- [9]
- S. T. Flammia et al., “Limits on the storage of quantum information in a volume of space”, Quantum 1, 4 (2017) arXiv:1610.06169 DOI
- [10]
- K. Audenaert and B. De Moor, “Optimizing completely positive maps using semidefinite programming”, Physical Review A 65, (2002) arXiv:quant-ph/0109155 DOI
- [11]
- A. S. Fletcher, “Channel-Adapted Quantum Error Correction”, (2007) arXiv:0706.3400
- [12]
- C. Cao et al., “Quantum variational learning for quantum error-correcting codes”, Quantum 6, 828 (2022) arXiv:2204.03560 DOI
- [13]
- D. Petz, “Sufficient subalgebras and the relative entropy of states of a von Neumann algebra”, Communications in Mathematical Physics 105, 123 (1986) DOI
- [14]
- D. PETZ, “SUFFICIENCY OF CHANNELS OVER VON NEUMANN ALGEBRAS”, The Quarterly Journal of Mathematics 39, 97 (1988) DOI
- [15]
- H. Barnum and E. Knill, “Reversing quantum dynamics with near-optimal quantum and classical fidelity”, (2000) arXiv:quant-ph/0004088
- [16]
- M. Junge et al., “Universal Recovery Maps and Approximate Sufficiency of Quantum Relative Entropy”, Annales Henri Poincaré 19, 2955 (2018) arXiv:1509.07127 DOI
- [17]
- A. Jayashankar and P. Mandayam, “Quantum Error Correction: Noise-adapted Techniques and Applications”, (2022) arXiv:2208.00365
- [18]
- V. V. Albert et al., “Performance and structure of single-mode bosonic codes”, Physical Review A 97, (2018) arXiv:1708.05010 DOI
- [19]
- B. M. Terhal and D. Weigand, “Encoding a qubit into a cavity mode in circuit QED using phase estimation”, Physical Review A 93, (2016) arXiv:1506.05033 DOI
- [20]
- K. Noh, V. V. Albert, and L. Jiang, “Quantum Capacity Bounds of Gaussian Thermal Loss Channels and Achievable Rates With Gottesman-Kitaev-Preskill Codes”, IEEE Transactions on Information Theory 65, 2563 (2019) arXiv:1801.07271 DOI
- [21]
- J. Harrington and J. Preskill, “Achievable rates for the Gaussian quantum channel”, Physical Review A 64, (2001) arXiv:quant-ph/0105058 DOI
- [22]
- K. Sharma et al., “Bounding the energy-constrained quantum and private capacities of phase-insensitive bosonic Gaussian channels”, New Journal of Physics 20, 063025 (2018) arXiv:1708.07257 DOI
- [23]
- M. Rosati, A. Mari, and V. Giovannetti, “Narrow bounds for the quantum capacity of thermal attenuators”, Nature Communications 9, (2018) arXiv:1801.04731 DOI
- [24]
- 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

## Page edit log

- Victor V. Albert (2022-08-12) — most recent
- Philippe Faist (2022-07-15)
- Victor V. Albert (2021-12-05)
- Manasi Shingane (2021-12-05)

## Cite as:

“Approximate quantum error-correcting code (AQECC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/approximate_qecc