Here is a list of all codes that belong to the quantum-into-classical domain.

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  1. BPSK c-q code
    Coherent-state c-q binary code encoding into two coherent states \(|\pm\alpha\rangle\) for complex \(\alpha\). A shifted version, with codewords \(\{|0\rangle,|\alpha\rangle\}\), is called binary amplitude modulation (BAM), The three-state subcode \(\{|\alpha,\alpha\rangle,|-\alpha,\alpha\rangle,|\alpha,-\alpha\rangle\}\) of two-mode BPSK is called the single-degeneracy code [1].
  2. Bosonic c-q code
    Bosonic code designed for transmission of classical information through non-classical channels.
  3. Classical-quantum (c-q) code
    Code designed specifically for transmission of classical information through non-classical channels, e.g., quantum channels, hybrid quantum-classical channels, or channels with classical inputs and quantum outputs. Such codes include maps from a classical alphabet into a quantum Hilbert space.
  4. Coherent FSK (CFSK) c-q code[2,3]
    Coherent-state c-q code encoding into coherent states that are frequency-shifted with certain initial relative phase.
  5. Coherent-state c-q code
    Bosonic c-q code whose codewords form a constellation of coherent states. Encodes real numbers into coherent states for transmission over a quantum channel and decoding with a quantum-enhanced receiver.
  6. Concatenated c-q code
    A c-q code constructed out of two classical or quantum codes for the purposes of transmission of classical information over quantum channels.
  7. Hadamard BPSK c-q code[1]
    Multimode coherent-state c-q code that is a concatenation of a Hadamard code with a BPSK c-q code. Its codewords are \(n\)-mode coherent states whose components \(\pm\alpha\) are arranged according to rows of a Hadamard matrix.
  8. Lechner-Hauke-Zoller (LHZ) code[4,5] a.k.a. Lechner-Hauke-Zoller (LHZ) parity code.
    LDPC c-q code designed to convert the long-range interactions of a quantum annealer into local constraints. The code maps the bits of a classical Ising model with all-to-all \(D\)-body interactions into one on a \(D\)-dimensional lattice. An extension maps more general models onto the same lattice [6].
  9. Niset-Andersen-Cerf code[7]
    Coherent-state c-q code encoding two-mode coherent states \(\{|\alpha\rangle, |\beta\rangle\}\) into four modes such that the complex values \((\alpha,\beta)\) are recoverable after a single-mode erasure. There are two variations of the storage procedure: a deterministic protocol that offers recovery against a single mode erasure, and a probabalistic that can protect against multiple errors with post selection. This code is effectively protecting classical information stored in \((\alpha,\beta)\) using quantum operations.
  10. On-off keyed (OOK) c-q code[8]
    Coherent-state c-q binary code whose encoding is either in the vacuum \(|0\rangle\) or in a nonzero coherent state \(|\alpha\rangle\).
  11. PPM c-q code[9]
    A \(q\)-PPM c-q code is a coherent-state c-q code whose \(j\)th codeword corresponds to a tensor-product state of zero-amplitude coherent states at all modes except mode \(j\). For example, a 3-PPM encoding corresponds to the three-mode states \(|\alpha\rangle|0\rangle|0\rangle\), \(|0\rangle|\alpha\rangle|0\rangle\), and \(|0\rangle|0\rangle|\alpha\rangle\) for some complex \(\alpha\). The dual of a PPM code is obtained by the exchange \(0\leftrightarrow\alpha\).
  12. PSK c-q code[10]
    Coherent-state c-q \(q\)-ary code whose \(j\)th codeword corresponds to a coherent state whose phase is the \(j\)th multiple of \(2\pi/q\). These states are also called geometrically uniform states (GUS) [11].
  13. Polar c-q code[12,13]
    Polar code adapted to transmit classical information over channels with classical inputs and quantum outputs.
  14. Qubit c-q code
    Qubit code designed for transmission of classical information in the form of bits through non-classical channels.

References

[1]
S. Guha, “Structured Optical Receivers to Attain Superadditive Capacity and the Holevo Limit”, Physical Review Letters 106, (2011) arXiv:1101.1550 DOI
[2]
I. A. Burenkov, O. V. Tikhonova, and S. V. Polyakov, “Quantum receiver for large alphabet communication”, Optica 5, 227 (2018) arXiv:1802.08287 DOI
[3]
I. A. Burenkov, M. V. Jabir, A. Battou, and S. V. Polyakov, “Time-Resolving Quantum Measurement Enables Energy-Efficient, Large-Alphabet Communication”, PRX Quantum 1, (2020) DOI
[4]
W. Lechner, P. Hauke, and P. Zoller, “A quantum annealing architecture with all-to-all connectivity from local interactions”, Science Advances 1, (2015) DOI
[5]
F. Pastawski and J. Preskill, “Error correction for encoded quantum annealing”, Physical Review A 93, (2016) arXiv:1511.00004 DOI
[6]
K. Ender, R. ter Hoeven, B. E. Niehoff, M. Drieb-Schön, and W. Lechner, “Parity Quantum Optimization: Compiler”, Quantum 7, 950 (2023) arXiv:2105.06233 DOI
[7]
J. Niset, U. L. Andersen, and N. J. Cerf, “Experimentally Feasible Quantum Erasure-Correcting Code for Continuous Variables”, Physical Review Letters 101, (2008) arXiv:0710.4858 DOI
[8]
R. L. Cook, P. J. Martin, and J. M. Geremia, “Optical coherent state discrimination using a closed-loop quantum measurement”, Nature 446, 774 (2007) DOI
[9]
J. Chen, J. L. Habif, Z. Dutton, R. Lazarus, and S. Guha, “Optical codeword demodulation with error rates below the standard quantum limit using a conditional nulling receiver”, Nature Photonics 6, 374 (2012) arXiv:1111.4017 DOI
[10]
F. E. Becerra, J. Fan, and A. Migdall, “Photon number resolution enables quantum receiver for realistic coherent optical communications”, Nature Photonics 9, 48 (2014) DOI
[11]
Y. C. Eldar and G. D. Forney, “On quantum detection and the square-root measurement”, IEEE Transactions on Information Theory 47, 858 (2001) DOI
[12]
M. M. Wilde and S. Guha, “Polar Codes for Classical-Quantum Channels”, IEEE Transactions on Information Theory 59, 1175 (2013) arXiv:1109.2591 DOI
[13]
R. Nasser and J. M. Renes, “Polar Codes for Arbitrary Classical-Quantum Channels and Arbitrary cq-MACs”, IEEE Transactions on Information Theory 64, 7424 (2018) arXiv:1701.03397 DOI