## Description

Stub.

## Encoding

Given a decoder, an encoding that yields the optimal entanglement fidelity can be obtained by solving a semi-definite program [7][3] (see also Ref. [8]).

## Decoding

Given an encoding, a decoder that yields the optimal entanglement fidelity can be obtained by solving a semi-definite program [7][3] (see also Ref. [8]).The Petz recovery map (a.k.a. the transpose map) [9][10], 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 [11]. 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 [12].

## Parent

## Child

- Eigenstate thermalization hypothesis (ETH) code — ETH codes approximately protect against erasures in the thermodynamic limit.

## Cousins

- 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.
- Gottesman-Kitaev-Preskill (GKP) code — GKP codes approximately protect against photon loss [13][14][15].
- Local Haar-random circuit code
- Multi-mode GKP code — Approximate error-correction offered by GKP codes yields achievable rates that are a constant away from the capacity of Guassian loss channels [16][17][18][15].
- Qudit-into-oscillator code — Approximate QEC techniques of finding the entanglement fidelity can be adapted to bosonic codes with a finite-dimensional codespace [14].
- \([[4,2,2]]\) CSS code — \([[4,1,2]]\) subcodes \(\{|\overline{00}\rangle,|\overline{10}\rangle\}\) [1] and \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) [19] approximately correct a single amplitude damping error.

## Zoo code information

## References

- [1]
- D. W. Leung et al., “Approximate quantum error correction can lead to better codes”, Physical Review A 56, 2567 (1997). DOI; quant-ph/9704002
- [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). DOI; quant-ph/0307138
- [4]
- Claude Crepeau, Daniel Gottesman, and Adam Smith, “Approximate Quantum Error-Correcting Codes and Secret Sharing Schemes”. quant-ph/0503139
- [5]
- Cédric Bény, “Conditions for the approximate correction of algebras”. 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). DOI; 0907.5391
- [7]
- K. Audenaert and B. De Moor, “Optimizing completely positive maps using semidefinite programming”, Physical Review A 65, (2002). DOI; quant-ph/0109155
- [8]
- Andrew S. Fletcher, “Channel-Adapted Quantum Error Correction”. 0706.3400
- [9]
- D. Petz, “Sufficient subalgebras and the relative entropy of states of a von Neumann algebra”, Communications in Mathematical Physics 105, 123 (1986). DOI
- [10]
- D. PETZ, “SUFFICIENCY OF CHANNELS OVER VON NEUMANN ALGEBRAS”, The Quarterly Journal of Mathematics 39, 97 (1988). DOI
- [11]
- H. Barnum and E. Knill, “Reversing quantum dynamics with near-optimal quantum and classical fidelity”. quant-ph/0004088
- [12]
- M. Junge et al., “Universal Recovery Maps and Approximate Sufficiency of Quantum Relative Entropy”, Annales Henri Poincaré 19, 2955 (2018). DOI; 1509.07127
- [13]
- B. M. Terhal and D. Weigand, “Encoding a qubit into a cavity mode in circuit QED using phase estimation”, Physical Review A 93, (2016). DOI; 1506.05033
- [14]
- V. V. Albert et al., “Performance and structure of single-mode bosonic codes”, Physical Review A 97, (2018). DOI; 1708.05010
- [15]
- 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). DOI; 1801.07271
- [16]
- J. Harrington and J. Preskill, “Achievable rates for the Gaussian quantum channel”, Physical Review A 64, (2001). DOI; quant-ph/0105058
- [17]
- 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). DOI; 1708.07257
- [18]
- M. Rosati, A. Mari, and V. Giovannetti, “Narrow bounds for the quantum capacity of thermal attenuators”, Nature Communications 9, (2018). DOI; 1801.04731
- [19]
- G. Alber et al., “Stabilizing Distinguishable Qubits against Spontaneous Decay by Detected-Jump Correcting Quantum Codes”, Physical Review Letters 86, 4402 (2001). DOI; quant-ph/0103042

## 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