Here is a list of all codes that belong to the classical-quantum (c-q) domain.

[Jump to code graph excerpt]

[ShowHide short descriptions]
  1. BPSK c-q modulation format a.k.a. BPSK c-q modulation code, BPSK c-q modulation scheme, BPSK c-q signaling format.
    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 a.k.a. Bosonic c-q modulation format, Bosonic c-q modulation scheme, Bosonic c-q modulation code, Bosonic c-q signaling format.
    Bosonic code designed for transmission of classical information through non-classical channels. Encodes classical symbols into bosonic quantum states for transmission over a quantum channel and decoding with a quantum-enhanced receiver. This entry includes bosonic c-q modulation formats and is distinct from a classical modulation scheme, which maps classical symbols into classical electromagnetic signals for transmission over classical channels. A bosonic c-q modulation format instead treats the transmitted signals as quantum states and allows the receiver to use quantum measurements.
  3. Classical-quantum (c-q) code
    Code designed specifically for transmission of classical information through non-classical channels, e.g., quantum channels, hybrid classical-quantum 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 modulation format[2,3] a.k.a. Coherent FSK (CFSK) c-q modulation code, Coherent FSK (CFSK) c-q modulation scheme, Coherent FSK (CFSK) c-q signaling format.
    Coherent-state c-q code encoding into coherent states that are frequency-shifted with certain initial relative phase.
  5. Coherent-state c-q modulation format a.k.a. Coherent-state c-q modulation code, Coherent-state c-q modulation scheme, Coherent-state c-q signaling format.
    Bosonic c-q code whose codewords form a constellation of coherent states. Encodes classical symbols 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. EA mixed-alphabet Reed-Solomon c-q code[4] a.k.a. Mixed-alphabet Reed-Solomon EACC code, Mixed-alphabet RS entanglement-assisted classical code.
    Entanglement-assisted c-q code obtained from a mixed-alphabet Reed-Solomon construction over \(\mathbb{F}_q\) and \(\mathbb{F}_{q^2}\). A codeword of an \([n,k,d;c]_q\) code consists of \(n-c\) symbols transmitted directly over \(q\)-dimensional quantum systems and \(c\) symbols transmitted through super-dense coding using \(c\) pre-shared maximally entangled qudit pairs.
  8. Entanglement-assisted (EA) c-q code a.k.a. Entanglement-assisted classical communication (EACC) code, Entanglement-assisted classical code.
    Classical-quantum code whose encoding and decoding utilize pre-shared entanglement between sender and receiver. The sender encodes classical information into quantum systems sent through a quantum channel, while the receiver decodes using the channel outputs together with retained halves of pre-shared entangled states.
  9. Fock-state OOK c-q modulation format[5] a.k.a. Number-state OOK c-q modulation format, Fock-state OOK c-q modulation code, Fock-state OOK c-q modulation scheme, Fock-state OOK c-q signaling format, Single-photon OOK c-q modulation format, Single-rail c-q code.
    Bosonic c-q on-off keying (OOK) modulation format whose binary alphabet consists of the vacuum state \(|0\rangle\) and the single-photon Fock state \(|1\rangle\) of one mode. More generally, the nonzero OOK symbol can be a number state or a mixture of adjacent number states. Fock-state OOK with photon-number detection was analyzed as a nonclassical alternative to coherent-state OOK for photon-efficient communication [5].
  10. Hadamard BPSK c-q modulation format[1] a.k.a. Hadamard BPSK c-q modulation code, Hadamard BPSK c-q modulation scheme, Hadamard BPSK c-q signaling format.
    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.
  11. Lechner-Hauke-Zoller (LHZ) code[6,7] 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 pairwise couplings of a fully connected classical Ising model into local fields together with local parity constraints on a lattice of physical qubits. An extension maps more general models onto the same lattice [8].
  12. Niset-Andersen-Cerf code[9]
    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 probabilistic one that can protect against multiple errors with post-selection. This code effectively protects classical information stored in \((\alpha,\beta)\) using quantum operations.
  13. On-off keyed (OOK) c-q modulation format[10] a.k.a. On-off keyed (OOK) c-q modulation code, On-off keyed (OOK) c-q modulation scheme, On-off keyed (OOK) c-q signaling format.
    Coherent-state c-q binary code whose encoding is either in the vacuum \(|0\rangle\) or in a nonzero coherent state \(|\alpha\rangle\).
  14. PSK c-q modulation format[11] a.k.a. PSK c-q modulation code, PSK c-q modulation scheme, PSK c-q signaling format.
    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) [12].
  15. Polar c-q code[13,14]
    Polar code adapted to transmit classical information over channels with classical inputs and quantum outputs.
  16. Pulse-position (PPM) c-q modulation format[15] a.k.a. Pulse-position (PPM) c-q modulation code, Pulse-position (PPM) c-q modulation scheme, Pulse-position (PPM) c-q signaling format.
    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 \(0\leftrightarrow\alpha\).
  17. Qubit c-q code
    A qubit code designed for transmission of classical information in the form of bits through non-classical channels.
  18. Squeezed-coherent BPSK c-q modulation format[16] a.k.a. Displaced-squeezed BPSK c-q modulation format, Squeezed-state BPSK c-q modulation code, Squeezed-state BPSK c-q modulation scheme, Squeezed-state BPSK c-q signaling format, Two-photon coherent-state BPSK c-q modulation format.
    Bosonic c-q binary modulation format whose codewords are antipodal displaced-squeezed states, i.e., states of the form \(D(\pm\alpha)S(\zeta)|0\rangle\) for a common squeezing parameter \(\zeta\). The format was originally formulated using the term two-photon coherent states (TCS), an early name for squeezed states.
List (domain): All codes in Classical-quantum Domain.

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]
T. Prasad and M. Grassl, “Codes for entanglement-assisted classical communication”, npj Quantum Information 11, (2025) arXiv:2310.19774 DOI
[5]
B. Erkmen, B. Moision, S. J. Dolinar, K. M. Birnbaum, and D. Divsalar, “Approaching the ultimate limits of communication efficiency with a photon-counting detector”, 2012 Information Theory and Applications Workshop 420 (2012) DOI
[6]
W. Lechner, P. Hauke, and P. Zoller, “A quantum annealing architecture with all-to-all connectivity from local interactions”, Science Advances 1, (2015) DOI
[7]
F. Pastawski and J. Preskill, “Error correction for encoded quantum annealing”, Physical Review A 93, (2016) arXiv:1511.00004 DOI
[8]
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
[9]
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
[10]
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
[11]
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
[12]
Y. C. Eldar and G. D. Forney, “On quantum detection and the square-root measurement”, IEEE Transactions on Information Theory 47, 858 (2001) DOI
[13]
M. M. Wilde and S. Guha, “Polar Codes for Classical-Quantum Channels”, IEEE Transactions on Information Theory 59, 1175 (2013) arXiv:1109.2591 DOI
[14]
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
[15]
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
[16]
J. Shapiro, H. Yuen, and A. Mata, “Optical communication with two-photon coherent states–Part II: Photoemissive detection and structured receiver performance”, IEEE Transactions on Information Theory 25, 179 (1979) DOI