CoChat
Sign Up
Cited by 276Quantum Computing Algorithms and ArchitectureQuantum Information and CryptographyQuantum many-body systems

Quantum generalisation of feedforward neural networks

Open access
Wan, Kwok Ho, Dahlsten, Oscar, Kristjánsson, Hlér, Gardner, Robert, Kim, M. S.·2017·npj Quantum Information
DOI 10.1038/s41534-017-0032-4
Open this paper in CoChat — free accountOpen PDF

Abstract

We propose a quantum generalisation of a classical neural network. The classical neurons are firstly rendered reversible by adding ancillary bits. Then they are generalised to being quantum reversible, i.e., unitary (the classical networks we generalise are called feedforward, and have step-function activation functions). The quantum network can be trained efficiently using gradient descent on a cost function to perform quantum generalisations of classical tasks. We demonstrate numerically that it can: (i) compress quantum states onto a minimal number of qubits, creating a quantum autoencoder, and (ii) discover quantum communication protocols such as teleportation. Our general recipe is theoretical and implementation-independent. The quantum neuron module can naturally be implemented photonically.

AI snapshot

Written from the paper’s full text.

TL;DR

The paper proposes a quantum generalisation of classical feedforward neural networks by making classical neurons reversible with ancillary bits and then unitary, allowing coherent processing of quantum inputs. It numerically demonstrates a quantum autoencoder and a network that discovers the teleportation protocol, and sketches a photonic implementation.

Key findings

  • The quantum network can be trained by global gradient descent on a cost function based on local Pauli expectation values, and the number of weights grows polynomially in the number of neurons.
  • As a quantum autoencoder, it compresses quantum states onto fewer qubits: for inputs \((|00\rangle+|11\rangle)/\sqrt{2}\) and \((|00\rangle-|11\rangle)/\sqrt{2}\), the cost function converged to zero, and qualitatively identical convergence was obtained for other orthogonal 2-qubit inputs and for 3 input qubits with 1 bottleneck qubit.
  • For two non-orthogonal inputs and a 1-qubit bottleneck, the cost function does not converge to zero, but training yields an approximately compressing unitary.
  • The network discovered the teleportation protocol: after training, testing on 1000 states randomly picked from the Haar measure gave a cost-function distribution with mean \(5.0371 \times 10^{-4}\) and standard deviation \(1.7802 \times 10^{-4}\), described as effectively zero.

Methods

  • Theoretical construction: classical feedforward neurons with Heaviside activation are first made reversible by adding ancillary bits, then generalised to unitary quantum neurons; classical copying is replaced by a general 2-qubit “fan-out” unitary \(U_F\).
  • Training: global gradient descent on a quantum cost function using local Pauli expectation values, with parameters \(\alpha_{j_1,\dots,j_N}\) in the unitary parametrisation; numerical simulations were run for the autoencoder and teleportation tasks.
  • Photonic implementation: a neuron module is designed using the Cerf-Adami-Kwiat protocol, encoding one qubit in polarisation and others in spatial modes, with variable beamsplitters, phase shifters, and polarisation rotators.

Limitations

  • The paper states it does not prove that convergence to zero cost, where possible, will always take a number of iterations growing polynomially in the number of neurons.
  • The paper states that classical simulation of quantum systems is generally inefficient, so efficiency claims refer to physical quantum implementation.
  • The paper states the single-photon photonic scheme does not scale well because the number of spatial modes grows exponentially in the number of qubits; it also notes that omitting ancillary qubits can sacrifice classical-mode performance.
  • Evident from the text: the demonstrations are numerical simulations, and the data availability statement says this is a theoretical paper with no experimental data beyond the described numerical simulation data.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1038/s41534-017-0032-4

Cites

54 of 54 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 24 cited sources checked · 0 supported · 0 flagged

  • Findingreasoning

    In the reversible generalisation, the output is a function of the inputs and a discrete set of permutation matrices, not of continuous weights.

    “In the reversible generalisation, where each neuron is replaced by a permutation matrix, we find that the output is no longer a function of the inputs and continuous weights, but rather of the inputs and a discrete set of permutation matrices.”
    Quote in the paperSubstantial claim
  • Findingnone given

    Entanglement between different neurons is needed to perform the autoencoder task.

    “However when investigating applications of the network we realised that there are scenarios (the autoencoder in particular) where entanglement between different neurons is needed to perform the task.”
    Quote in the paperSubstantial claim
  • Findingown data

    The quantum network can compress quantum states onto a minimal number of qubits, creating a quantum autoencoder.

    “We demonstrate numerically that it can: (i) compress quantum states onto a minimal number of qubits, creating a quantum autoencoder”
    Quote in the paperSubstantial claim
  • Findingreasoning

    The number of evaluations of the cost function for a given iteration of gradient descent grows polynomially in the number of neurons.

    “Thus the number of evaluations of the cost function for a given iteration of the gradient descent grows polynomially in the number of neurons.”
    Quote in the paperSubstantial claim
  • Findingown data

    In training with a variety of two possible orthogonal input states including superposition states, the cost function of the quantum autoencoder converged towards zero through global gradient descent in every case, starting with uniformly randomised weights.

    “In training with a variety of two possible orthogonal input states including superposition states, the cost function of the quantum autoencoder converged towards zero through global gradient descent in every case, starting with uniformly randomised weights, α j1;:::;jN 2 À1; 1 ½ .”
    Quote in the paperSubstantial claim
  • Classified as probably not a substantial claim. 19
  • May combine two claims. 14

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Findingnone given

    A simpler and less general form of U_N has been sufficient for the tasks discussed in this paper.

    “A simpler and less general form of U N has been sufficient for the tasks discussed in this paper:”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingnone given

    For two non-orthogonal inputs and a 1-qubit bottleneck, the cost function will not converge to zero, but training results in an approximately compressing unitary.

    “For two non-orthogonal inputs and a 1-qubit bottleneck the cost-function will not converge to zero as is to be expected, but the training rather results in an approximately compressing unitary.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingown data

    Figure 3b shows the convergence of the cost function during training, simulated on a classical computer.

    “Figure 3b shows the convergence of the cost function during training, simulated on a classical computer.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S4 (based).

    Findingnone given

    The number of weights grows polynomially in the number of neurons.

    “It is important to note that the number of weights grow polynomially in the number of neurons.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    The cost function can be seen to converge to zero, showing that the network has learned to compress the input state onto one qubit and then later recreate the input state.

    “The cost function can be seen to converge to zero, showing that the network has learned to compress the input state onto one qubit and then later recreate the input state.”
    • Also stated in: § Quantum photonics neuron module · plain
      “The cost function can be seen to converge to zero, showing that the network has learned to compress the input state onto one qubit and then later recreate the input state.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    The network now teleports any qubit state: picking 1000 states at random from the Haar measure gives a cost function distribution with mean 5.0371 × 10 -4 and standard deviation 1.7802 × 10 -4 , which is effectively zero.

    “The network now teleports any qubit state: picking 1000 states at random from the Haar measure (uniform distribution over the Bloch sphere) gives a cost function distribution with mean 5.0371 × 10 -4 and standard deviation 1.7802 × 10 -4 , which is effectively zero”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisecitation

    Any unitary matrix U can be expressed as U = e^{iH}, where H is a Hermitian matrix, and such matrices can be written as linear combinations of tensor products of the Pauli matrices and the identity.

    “Given that any unitary matrix U can be expressed as U = e iH , where H is a Hermitian matrix, 20 and that such matrices can be written as linear combinations of tensor products of the Pauli matrices and the identity, it follows that a general N-qubit unitary can be expressed as”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [20] Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information (Cambridge University Press, 2000).

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning

    The unitaries can be parametrised by continuous variables, which allows the application of gradient descent.

    “This means that the unitaries can be parametrised by continuous variables, which once again allows the application of gradient descent.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisenone given

    For a gate with n inputs and outputs, there exist an infinite number of unitaries, in contrast with the discrete set of permutation matrices.

    “However, in the generalisation to unitaries, for a gate with n inputs and outputs, there exist an infinite number of unitaries, in contrast with the discrete set of permutation matrices.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisereasoning

    Any irreversible classical computation can be recovered as a special case from reversible classical computation, and reversible classical computation from quantum computation.

    “The generalisation recipe summarised in Fig. 1 also illustrates how any irreversible classical computation can be recovered as a special case from reversible classical computation (by ignoring the dummy and copied bits), which in turn can be recovered as a special case from quantum computation.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationnone given

    In the classical mode of operation, where the total density matrix state is diagonal in the computational basis, only σ3 will have non-zero expectation, and the cost function becomes the same as in the classical case up to a simple transformation.

    “We note in the classical mode of operation, where the total density matrix state is diagonal in the computational basis, only σ 3 will have non-zero expectation, and the cost function becomes the same as in the classical case (Eq.( 2)) up to a simple transformation.”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Interpretationnone given

    The classical networks correspond to a particular parameter regime for the quantum networks.

    “The comparison of classical and quantum neural networks is well-defined within our set-up, as the classical networks correspond to a particular parameter regime for the quantum networks.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationnone given

    The same scheme can be used to make quantum generalisations of networks whose generalised neurons have more inputs/outputs and connections.

    “The same scheme can be used to make quantum generalisations of networks whose generalised neurons have more inputs/outputs and connections.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S4 (based). May combine two claims.

    Interpretationnone given

    The networks are strict generalisations of the classical networks, providing a clear framework for comparing the power of quantum and classical neural networks.

    “Moreover they are strict generalisations of the classical networks, providing a clear framework for comparing the power of quantum and classical neural networks.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Premisenone given

    The number of neurons in a given hidden layer for a classical neuron will not exceed j_max.

    “The number of neurons in a given hidden layer for a classical neuron will not exceed j max .”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S4 (based).

    Interpretationnone given

    If one wishes the number of free parameters of a neuron to grow no more than polynomially in the number of inputs, one needs to restrict the unitary.

    “If one wishes the number of free parameters of a neuron to grow no more than polynomially in the number of inputs, one needs to restrict the unitary.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationnone given

    One can force the compressed output to be diagonal in a particular basis by adding an extra term to the cost function.

    “One can force the compressed output to be diagonal in a particular basis by adding an extra term to the cost-function (e.g., desiring the expectation values of Pauli X and Y to be zero in the case of a single qubit will push the network to give an output diagonal in the Z-basis).”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationnone given

    The quantum neuron module can naturally be implemented photonically.

    “The quantum neuron module can naturally be implemented photonically.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisecitation

    The C-A-K scheme and related ones are experimentally viable, theoretically clean and can implement any unitary on a single photon spread out over spatial modes.

    “This scheme and related ones such as in refs.41, 42 are experimentally viable, theoretically clean and can implement any unitary on a single photon spread out over spatial modes.”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [41] Reck, M., Zeilinger, A., Bernstein, H. J. & Bertani, P. Experimental realization of any discrete unitary operator. Phys. Rev. Lett. 73, 58–61 (1994). source

      Not checked in the free run.

    • cites [42] Clements, W. R., Humphreys, P. C., Metcalf, B. J., Kolthammer, W. S. & Walmsley, I. A. An optimal design for universal multiport interferometers. Preprint at https://arxiv.org/abs/1603.08788 (2016). source

      Not checked in the free run.

  • May combine two claims.

    Premisecitation

    Quantum photonics is an attractive platform for QIP: it has room temperature operation, the possibility of robust miniaturisation through photonic integrated circuits; in general it harnesses the highly developed optical fibre-related technology for QIP purposes.

    “This is an attractive platform for QIP: it has room temperature operation, the possibility of robust miniaturisation through photonic integrated circuits; in general it harnesses the highly developed optical fibre-related technology for QIP purposes.38”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [38] Rudolph, T. Why I am optimistic about the silicon-photonic route to quantum computing. Preprint at https://arxiv.org/abs/1607.08535 (2016). source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S4 (based).

    Interpretationreasoning

    The tuning can be naturally implemented in the proposed photonic platform.

    “A final motivation for choosing this platform is that the tuning can be naturally implemented, as detailed below.”
    Quote in the paperNot a substantial claim

  • Classified as probably not a substantial claim.

    Interpretationreasoning

    The networks generalise classical neural networks to the quantum case in a similar sense to how quantum computing generalises classical computing.

    “The networks generalise classical neural networks to the quantum case in a similar sense to how quantum computing generalises classical computing.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    Optical implementations have been viewed as optimal for neural networks, in the classical case, due to the low design cost of adding multiple connections (as light passes through light without interacting).

    “Moreover optical implementations have been viewed as optimal for neural networks, in the classical case, due to the low design cost of adding multiple connections (as light passes through light without interacting).39”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [39] Rojas, R. Neural Networks (Springer, 1996). source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Interpretationreasoning

    Omitting ancillary qubits sacrifices performance in the classical mode of operation, and the network may not be as good as a classical network with the same number of neurons for all tasks.

    “Then one would sacrifice performance in the classical mode of operation, and the network may not be as good as a classical network with the same number of neurons for all tasks.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationreasoning

    The non-monotonic decrease is to be expected as we are varying the input states.

    “The non-monotonic decrease is to be expected as we are varying the input states.”
    • Also stated in: § Quantum photonics neuron module · plain
      “The non-monotonic decrease is to be expected as we are varying the input states.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    This circuit can be made more robust and miniaturised using silicon or silica optical waveguides.

    “This circuit can be made more robust and miniaturised using silicon or silica optical waveguides.38”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [38] Rudolph, T. Why I am optimistic about the silicon-photonic route to quantum computing. Preprint at https://arxiv.org/abs/1607.08535 (2016). source

      Not checked in the free run.

  • May combine two claims.

    Premisecitation

    Conventionally phase shifters built upon the electro-optic effect are known to work in the megahertz region and have extremely low loss.

    “Conventionally phase shifters built upon the electro-optic effect are known to work in the megahertz region and have extremely low loss.38”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [38] Rudolph, T. Why I am optimistic about the silicon-photonic route to quantum computing. Preprint at https://arxiv.org/abs/1607.08535 (2016). source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    Silicon waveguides are birefringent.

    “this may have additional difficulties in that silicon waveguides are birefringent.44”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [44] Humphreys, P. C. et al. Strain-optic active control for quantum integrated photonics. Opt. Express 22, 21719–21726 (2014). source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisecitation

    Silicon or silica optical waveguides have been extensively used to control spatial modes and recently also polarisation.

    “They have been extensively used to control spatial modes and recently also polarisation.45”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [45] Sansoni, L. et al. Polarization entangled state measurement on a chip. Phys. Rev. Lett. 105, 200503 (2010). source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    The standard teleportation protocol allows only classical communication of 2 bits.

    “The standard teleportation protocol allows only classical communication of 2 bits 20 ; this is enforced by only allowing two connections, which are dephased in the Z-basis (D).”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [20] Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information (Cambridge University Press, 2000).

      Not checked in the free run.

30 of 30

  • The paper relies on this without support.

    Simulation of quantum systems with a classical computer is, with the best known methods, in general inefficient.

    “Note also that the statements about the efficiency of the training procedure refer to the physical implementation with quantum technology: the simulation of quantum systems with a classical computer is, with the best known methods, in general inefficient.”

  • The paper relies on this without support.

    Several labs can implement the phase shifting via heaters or the electro-optic effect.

    “Several labs can implement the phase shifting via heaters or the electro-optic effect.”
2 of 2

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • The quantum kernel trick has been demonstrated experimentally using an ensemble of nuclear spins.

    “Finally, the quantum kernel trick, which means that the dimensions of the quantumenhanced feature space are larger than the number of data sets, has been demonstrated experimentally by using an ensemble nuclear spins [162].”

    From Variational quantum algorithms (2021) · cites this paper as [162]

    The Quantum neural networks section describes a teleportation-training result simulated on a classical computer, while the Quantum photonics neuron module section proposes an optical design. Searches for “quantum kernel trick” and “ensemble of nuclear spins” found no passage in our copy addressing the claimed experimental demonstration. (LLM, conf=0.76)

    Experimental — accuracy not yet measured

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • Two different references have the same arXiv identifier

    Minor

    References 23 and 36 give the same arXiv identifier for works with different authors and titles.

1 of 1

Cites

Flagged first, then by citations

Research works printed but not linked: 1

  1. Hedemann, S. R. Hyperspherical parameterization of unitary matrices. Preprint at https://arxiv.org/abs/1303.5904 (2013). · arXiv

Cited by

Sorted by citations

Related

Authors
Wan, K. H., Dahlsten, O., Kristjánsson, H., Gardner, R., & Kim, M. S.
Affiliations
  • London Institute for Mathematical Sciences (GB)
  • Southern University of Science and Technology (CN)
  • Imperial College London (GB)
Corresponding author
Dahlsten, Oscar
ORCID
Venue
npj Quantum Information
Publisher
Springer Science and Business Media LLC
Published
2017-09-14
Volume
3
Issue
1
ISSN
2056-6387 (ISSN-L)
DOI
10.1038/s41534-017-0032-4
OpenAlex
W2560386163
Semantic Scholar
d9cfa31d9de66527d719c1511fb2d41b04121dee
Cited by
276
References
52
Citations by year
2026: 7 · 2025: 25 · 2024: 30 · 2023: 38 · 2022: 42 · 2021: 47 · 2020: 41 · 2019: 24 · 2018: 19 · 2017: 3
FWCI
14.19
Type
journal article
Language
en
Open access
Licence
Locations
  • npj Quantum Information (publishedVersion, cc-by)
  • arXiv (Cornell University) (submittedVersion)
  • DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion)
  • Spiral (Imperial College London) (submittedVersion, cc-by)
  • arXiv (Cornell University)
Funders
  • Leverhulme Trust (RPG-2014-055)
  • Royal Society
  • European Commission (618074)
  • Engineering and Physical Sciences Research Council (EP/K034480, EP/K034480/1, K034480/1)
  • Division of Mathematical Sciences
Topics
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
Keywords
quantum neural networks, feedforward neural networks, quantum autoencoders, quantum state compression, quantum teleportation, quantum communication protocols, photonics, gradient descent learning

About these checks

Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

Claims read by deepseek-v4.1-flash under definition version 3 (extract-v3) on 2026-10-10.

Classified by jev-1.13-20260917 (gold-guideline-v0.2) on 2026-10-10.

Statistics and arithmetic are recomputed, not judged.

Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.

What counts as a claim ›How we measure confidence ›How we check this ›

Are you an author, or do you see a problem with this page? Request a review