The DOI matches its Crossref record; 54 of the 54 references it cites were found in a registry or an index; 1 of the 24 sources its claims cite were checked for support.
Quantum generalisation of feedforward neural networks
Open accessAbstract
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.
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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.
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Claims the paper makes
24 Own interpretation · 20 Not checked yet · 12 Own result · 9 No basis given · 1 Supported
Every source the paper cites for this claim was checked and supports it.
In the paper
“We make the individual neurons reversible and then naturally generalise them to being quantum reversible (unitary).This resolves the classical-quantum clash mentioned above from ref. 24.”
§ Quantum generalisation of feedforward neural networks > INTRODUCTION
What the sources say
- Supportedcites [24] Schuld, M., Sinayskiy, I. & Petruccione, F. The quest for a quantum neural network. Quant. Inf. Process. 13, 25672586 (2014). source
The Introduction (p. 2) describes a fundamental difference between nonlinear, dissipative neural dynamics and linear, unitary quantum dynamics; Requirements for a QNN model (p. 9) calls their integration a fundamental challenge. The citing authors present reversible neurons as their own proposed resolution. They attribute the clash, not that resolution, to this source. (LLM, conf=0.92)
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“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.”
§ Quantum generalisation of feedforward neural networks > Quantum neural networks
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“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.”
§ Quantum generalisation of feedforward neural networks > Quantum neural networks
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“We demonstrate numerically that it can: (i) compress quantum states onto a minimal number of qubits, creating a quantum autoencoder”
§ Quantum generalisation of feedforward neural networks > Abstract
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“Thus the number of evaluations of the cost function for a given iteration of the gradient descent grows polynomially in the number of neurons.”
§ Quantum generalisation of feedforward neural networks > Quantum neural networks
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“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 .”
§ Quantum generalisation of feedforward neural networks > Quantum neural networks
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“As can be seen, the training was found to be successful, i.e., the cost function converged towards zero.”
§ Quantum generalisation of feedforward neural networks > Quantum neural networks
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“A fully trained network can teleport the state ψ j i (from Alice) to the output port of qubit 6 (to Bob).”
§ Quantum generalisation of feedforward neural networks > Quantum neural networks
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“Qualitatively identical graphs of the cost function converging to 0 were also obtained for other examples of 2 orthogonal input states, including for the case of 3 input qubits and 1 bottleneck qubit”
- Also stated in: § Quantum photonics neuron module · plain
“Qualitatively identical graphs of the cost function converging to 0 were also obtained for other examples of 2 orthogonal input states, including for the case of 3 input qubits and 1 bottleneck qubit”
§ Quantum generalisation of feedforward neural networks > Quantum neural networks
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In the paper
“Figure 2c shows the network learning to compress in the case of two possible inputs: 00 j i þ 11 j i ð Þ = ffiffi ffi 2 p and 00 j i À 11 j i ð Þ = ffiffi ffi 2 p .”
§ Quantum generalisation of feedforward neural networks > Quantum neural networks
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- Probably not a substantial claim. 19
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“A simpler and less general form of U N has been sufficient for the tasks discussed in this paper:”
§ Quantum generalisation of feedforward neural networks > Quantum neural networks
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“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.”
§ Quantum generalisation of feedforward neural networks > Quantum neural networks
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“Figure 3b shows the convergence of the cost function during training, simulated on a classical computer.”
§ Quantum generalisation of feedforward neural networks > Quantum neural networks
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“It is important to note that the number of weights grow polynomially in the number of neurons.”
§ Quantum generalisation of feedforward neural networks > Quantum neural networks
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“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.”
§ Quantum generalisation of feedforward neural networks > Quantum neural networks
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In the paper
“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”
§ Quantum generalisation of feedforward neural networks > Quantum photonics neuron module
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“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”
§ Quantum generalisation of feedforward neural networks > Quantum neural networks
What the sources say
- Not checked yetcites [20] Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information (Cambridge University Press, 2000).
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“This means that the unitaries can be parametrised by continuous variables, which once again allows the application of gradient descent.”
§ Quantum generalisation of feedforward neural networks > Quantum neural networks
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“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.”
§ Quantum generalisation of feedforward neural networks > Quantum neural networks
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“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.”
§ Quantum generalisation of feedforward neural networks > Quantum neural networks
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“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.”
§ Quantum generalisation of feedforward neural networks > Quantum neural networks
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“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.”
§ Quantum generalisation of feedforward neural networks > Quantum vs. classical
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“The same scheme can be used to make quantum generalisations of networks whose generalised neurons have more inputs/outputs and connections.”
§ Quantum generalisation of feedforward neural networks > Scaling to bigger networks
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In the paper
“Moreover they are strict generalisations of the classical networks, providing a clear framework for comparing the power of quantum and classical neural networks.”
§ Quantum generalisation of feedforward neural networks > INTRODUCTION
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“The number of neurons in a given hidden layer for a classical neuron will not exceed j max .”
§ Quantum generalisation of feedforward neural networks > Quantum neural networks
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In the paper
“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.”
§ Quantum generalisation of feedforward neural networks > Scaling to bigger networks
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In the paper
“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).”
§ Quantum generalisation of feedforward neural networks > Quantum neural networks
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“The quantum neuron module can naturally be implemented photonically.”
§ Quantum generalisation of feedforward neural networks > Abstract
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In the paper
“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.”
§ Quantum generalisation of feedforward neural networks > Quantum photonics neuron module
What the sources say
- Not checked yetcites [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 yetcites [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
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“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”
§ Quantum generalisation of feedforward neural networks > Quantum photonics neuron module
What the sources say
- Not checked yetcites [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
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“A final motivation for choosing this platform is that the tuning can be naturally implemented, as detailed below.”
§ Quantum generalisation of feedforward neural networks > Quantum photonics neuron module
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“The networks generalise classical neural networks to the quantum case in a similar sense to how quantum computing generalises classical computing.”
§ Quantum generalisation of feedforward neural networks > INTRODUCTION
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“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”
§ Quantum generalisation of feedforward neural networks > Quantum photonics neuron module
What the sources say
- Not checked yetcites [39] Rojas, R. Neural Networks (Springer, 1996). source
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In the paper
“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.”
§ Quantum generalisation of feedforward neural networks > Quantum vs. classical
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“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.”
§ Quantum generalisation of feedforward neural networks > Quantum photonics neuron module
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In the paper
“This circuit can be made more robust and miniaturised using silicon or silica optical waveguides.38”
§ Quantum generalisation of feedforward neural networks > Quantum photonics neuron module
What the sources say
- Not checked yetcites [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
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In the paper
“Conventionally phase shifters built upon the electro-optic effect are known to work in the megahertz region and have extremely low loss.38”
§ Quantum generalisation of feedforward neural networks > Quantum photonics neuron module
What the sources say
- Not checked yetcites [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
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In the paper
“this may have additional difficulties in that silicon waveguides are birefringent.44”
§ Quantum generalisation of feedforward neural networks > Quantum photonics neuron module
What the sources say
- Not checked yetcites [44] Humphreys, P. C. et al. Strain-optic active control for quantum integrated photonics. Opt. Express 22, 21719–21726 (2014). source
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“They have been extensively used to control spatial modes and recently also polarisation.45”
§ Quantum generalisation of feedforward neural networks > Quantum photonics neuron module
What the sources say
- Not checked yetcites [45] Sansoni, L. et al. Polarization entangled state measurement on a chip. Phys. Rev. Lett. 105, 200503 (2010). source
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- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
In the paper
“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).”
§ Quantum generalisation of feedforward neural networks > Quantum photonics neuron module
What the sources say
- Not checked yetcites [20] Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information (Cambridge University Press, 2000).
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
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.”
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 check wasn’t confident either way, so it gives no verdict — never disputed.
In the citing paper
“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]
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.
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Quantum generalisation of feedforward neural networks > References
“23. Wiebe, N., Kapoor, A. & Svore, K. M. Quantum deep learning. Preprint at https:// arxiv.org/abs/1303.5904 (2015).”
§ Quantum generalisation of feedforward neural networks > References
“36. Hedemann, S. R. Hyperspherical parameterization of unitary matrices. Preprint at https://arxiv.org/abs/1303.5904 (2013).”
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On compression rate of quantum autoencoders: Control design, numerical and experimental realization
Ma, H., Huang, C.-J., Chen, C., Dong, D., Wang, Y., Wu, R.-B., & Xiang, G. (2022) · Automatica · doi:10.1016/j.automatica.2022.110659 · Cited by 26
Defending Against Byzantine Attacks in Quantum Federated Learning
Xia, Q., Tao, Z., & Li, Q. (2021) · International Conference on Mobility, Sensing and Networking (MSN) · doi:10.1109/msn53354.2021.00035 · Cited by 28
The impact of cost function globality and locality in hybrid quantum neural networks on NISQ devices
Kashif, M., & Al-Kuwari, S. M. (2023) · Machine Learning Science and Technology · doi:10.1088/2632-2153/acb12f · Cited by 33
Metadata
- 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
- Venue
- npj Quantum Information
- Published
- 2017-09-14
- Volume
- 3
- Issue
- 1
- ISSN
- 2056-6387 (ISSN-L)
- 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
- gold
- doi.org
- 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 › Physics and Astronomy › Atomic and Molecular Physics, and Optics › 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.
A language model drew the claims from the paper on 2026-10-10, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-10-10 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).
Cited sources are checked with a language model’s help (gpt-6-sol).
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.
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