Quantum Neural Networks

• Quantum Neural Networks are models that merge quantum computational processes with neural architectures, enabling representations of quantum amplitudes and complex learning dynamics.
• These networks span a diverse range of architectures, including parameterized quantum circuits, open-system models, and continuous-variable frameworks, each leveraging unique quantum properties.
• Key challenges include reconciling nonlinear, dissipative neural dynamics with linear quantum evolution, as well as addressing scalability and measurement stochasticity in NISQ devices.

Quantum neural networks (QNNs) denote a heterogeneous class of models at the intersection of quantum computing and neural computation. In the literature, the term ranges from layered parameterized quantum circuits and quantum-classical hybrids to open-system neuron models, continuous-variable variational circuits, dissipative networks that learn quantum channels, and even a path-integral interpretation in which a QNN is a representation of the amplitude of an arbitrary quantum process (Ezhov, 2021). Across these formulations runs a persistent technical issue: neural models emphasize nonlinear, dissipative, and often attractor-based dynamics, whereas closed quantum dynamics are linear and unitary (Schuld et al., 2014). Reviews of the area therefore treat QNNs not as a single architecture but as a family of proposals that use quantum states, unitary or nonunitary transformations, and measurement as the computational substrate (Kwak et al., 2021).

1. Definitions and conceptual scope

The meaning of “quantum neural network” has shifted substantially. One line of work distinguishes an early-2000s view, in which a QNN was described vaguely as a field combining classical neurocomputing with quantum computing, from a modern machine-learning view that identifies a QNN as a model or machine-learning algorithm combining quantum computing with artificial neural networks. Against both, "On quantum neural networks" proposes a broader definition: a QNN is a representation of the amplitude of an arbitrary quantum process, with the network structure identified with the sum over paths in the Feynman path integral (Ezhov, 2021). In that formulation,

$$A=\sum_{\text{paths }j}\exp\!\left(\frac{i}{\hbar}S_j\right),$$

and the amplitude is interpreted as the output of a two-layer wide neural network whose hidden units correspond to paths and whose activation is the phase factor $\exp(iS/\hbar)$.

A different foundational tradition frames the problem by asking what a meaningful QNN should satisfy. "The quest for a Quantum Neural Network" formulates three requirements: associative memory or neural input-output behavior, genuine neural-network structure, and genuine quantum-theoretic content. It introduces the term “quron” for a qubit used as a neural unit and argues that many proposals satisfy only subsets of these requirements (Schuld et al., 2014). The same work identifies the central obstacle as the mismatch between nonlinear, dissipative neural dynamics and linear, reversible quantum evolution.

In the parameterized-circuit literature, the term is narrower. "Quantum Neural Networks: Concepts, Applications, and Challenges" defines a quantum neuron as a parameterized quantum operation, typically built from rotation gates and entangling gates, and defines a QNN as a layered composition of such blocks acting on encoded input data and read out by measurement (Kwak et al., 2021). In this usage, the network state is often written as

$$|\Psi(\boldsymbol{x},\boldsymbol{\theta})\rangle = U(\boldsymbol{\theta})\,V(\boldsymbol{x})\,|0\rangle^{\otimes n},$$

with measurement and classical post-processing supplying the effective nonlinearity absent from unitary evolution.

These definitions are not equivalent. Some identify QNNs with variational quantum models for supervised learning, some with open-system neuron dynamics, and some with a more general representational structure already implicit in quantum mechanics. The field is therefore best understood as a cluster of related research programs rather than a settled formal category.

2. Architectural families

Several distinct architectural families recur in the literature.

Family	Core construction	Representative sources
Variational and hybrid circuits	Encoded data evolved by layered parameterized unitaries; outputs extracted by measurement or classical post-processing	(Kwak et al., 2021, Nakaji et al., 2021, Petitzon, 2022, Boneberg et al., 27 Apr 2026)
Open-system and “soft” neurons	Mixed-state neurons updated by controlled Kraus channels or classically controlled single-qubit operations	(Chen, 2018, Zhou et al., 2023)
Continuous-variable and dissipative networks	Gaussian plus non-Gaussian CV layers; ancilla-unitary-trace completely positive maps	(Killoran et al., 2018, Beer, 2022)
Specialized training-oriented models	Annealer-trained Ising encodings, band-limited quantum perceptrons, quantum-deformed threshold layers	(Abel et al., 2022, Heidari et al., 2022, Bondesan et al., 2020)

A large portion of the recent literature uses parameterized quantum circuits. In the standard review formulation, a layer consists of parameterized single-qubit rotations and entangling gates, stacked into a trainable ansatz (Kwak et al., 2021). Variants differ mainly in how data are inserted and how measurements are used. "New quantum neural network designs" replaces the usual separation between feature map and variational block by a merged data-dependent circuit $U(\theta,X)$ and then post-processes measured $\langle Z_i\rangle$ values with a small classical neural network (Petitzon, 2022). "Quantum-enhanced neural networks in the neural tangent kernel framework" instead fixes a quantum feature extractor based on local random unitaries sampled from unitary 2-designs and trains only the classical network that follows the measured quantum features (Nakaji et al., 2021). "Getting large-scale quantum neural networks ready for quantum hardware" uses a layered circuit on a rectangular lattice, with local gates determined by Hamiltonian and jump-operator coefficients and an output loss based on an order parameter, especially magnetization (Boneberg et al., 27 Apr 2026).

A second family makes openness and decoherence central rather than incidental. "Quantum Neural Network and Soft Quantum Computing" models each neuron as a logical qubit in a noisy open system, with neuron-neuron couplings implemented by controlled Kraus operations (Chen, 2018). The target neuron undergoes a quantum channel only when the presynaptic signal equals $1$, and the network evolution is time ordered because the channels may not commute. "Quantum Neural Network for Quantum Neural Computing" develops a closely related feedforward model built only from classically controlled single-qubit operations, measurements, and local bias superoperators; each neuron outputs a classical bit after measurement, and the model is designed to avoid exponential memory growth by propagating local single-qubit states rather than a global $N$-qubit state (Zhou et al., 2023).

A third family is explicitly quantum-native in its state space. "Continuous-variable quantum neural networks" constructs layers from interferometers, squeezers, displacements, and local non-Gaussian gates such as the cubic phase gate or Kerr gate (Killoran et al., 2018). The Gaussian part plays the role of an affine map in phase space, while the non-Gaussian gate supplies the analogue of an activation function. "Quantum neural networks" develops dissipative quantum neural networks (DQNNs) in which each layer is a completely positive map obtained by tensoring in fresh ancillas, applying a unitary, and tracing out the previous layer (Beer, 2022).

Specialized architectures target distinct computational substrates. "Completely Quantum Neural Networks" maps an entire neural-network training problem into a quadratic Ising Hamiltonian for a quantum annealer by binary-encoding the trainable parameters, polynomially approximating the activation, and reducing higher-order binary polynomials to quadratic form with auxiliary variables (Abel et al., 2022). "Toward Physically Realizable Quantum Neural Networks" builds feed-forward QNNs from band-limited quantum perceptrons acting on at most $k$ qubits, so that each perceptron has at most $4^k$ Pauli coefficients (Heidari et al., 2022). "Quantum Deformed Neural Networks" recasts binary threshold layers through quantum phase estimation, with the weighted sum represented as an eigenvalue of a quantum observable and the threshold realized by reading the most significant ancilla bit of a phase-estimation register (Bondesan et al., 2020).

3. Training and optimization

The dominant training pattern in variational QNNs is hybrid classical-quantum optimization. A classical optimizer updates circuit parameters, while the quantum device prepares states, applies the ansatz, and returns measurement statistics from which a loss is computed (Kwak et al., 2021). The review literature explicitly mentions mean squared error and cross-entropy losses, as well as Adam-based optimization and parameter-shift-style derivatives for gates generated by Pauli operators (Kwak et al., 2021).

Recent work emphasizes that finite-shot estimation is not a minor implementation detail but part of the optimization problem itself. In the hardware-oriented lattice model of (Boneberg et al., 27 Apr 2026), the loss is defined from repeated measurements of an output-layer observable, usually magnetization $\hat m^\alpha$, and the standard deviation of the shot-based estimator scales as $\exp(iS/\hbar)$0, where $\exp(iS/\hbar)$1 is the number of shots and $\exp(iS/\hbar)$2 the number of sites in a layer. The paper trains via minibatch finite-difference gradients and a Nadam-like update rule on the Hamiltonian and jump-operator coefficients, thereby treating noisy objective evaluations as the native regime rather than an afterthought.

A different response to physical constraints appears in "Toward Physically Realizable Quantum Neural Networks." That paper argues that many earlier QNN training protocols rely on repeated measurements of several copies of each sample, which raises difficulties because unknown quantum states cannot be cloned and measurement outcomes are stochastic. Its solution is a randomized quantum stochastic gradient descent procedure that estimates a single gradient coordinate from one use of the sample plus an ancilla-assisted measurement circuit, proves that the estimator is unbiased, and establishes convergence in expectation (Heidari et al., 2022).

Not all QNN training is hybrid. "Completely Quantum Neural Networks" performs the entire optimization on a quantum annealer. The loss is encoded into an Ising Hamiltonian

$\exp(iS/\hbar)$3

and training consists of a single annealing event whose ground state corresponds to the optimal network parameters (Abel et al., 2022). By contrast, "Learning to learn with quantum neural networks via classical neural networks" keeps the QNN itself variational but trains a classical LSTM meta-optimizer to generate good parameter initializations for QAOA and VQE instances, substantially reducing the number of subsequent local-optimization queries needed to reach a target accuracy (Verdon et al., 2019).

Dissipative models often use different objectives because they learn quantum processes rather than classical labels. DQNNs are trained on pairs of quantum input and desired output states, and the supervised objective is the average fidelity between the network output and the target output state (Beer, 2022). The update rule has a back-propagation-like structure in which forward-propagated states and backward-propagated adjoint states meet in a local commutator expression that defines a perceptron update matrix.

Some papers are explicit that training is not their primary concern. The path-integral formulation treats the QNN chiefly as a representation framework for quantum amplitudes rather than a trainable machine-learning model (Ezhov, 2021). The soft quantum computing model similarly discusses learning as a time-dependent patterning of controlled quantum channels but does not define a loss function, gradient-based optimization, or a complete training protocol (Chen, 2018).

4. Expressivity, kernels, open-system dynamics, and foundational claims

Several papers study QNNs through formal properties rather than benchmark performance. In the NTK-based hybrid model, the infinite-width limit of the classical post-processing network yields a Gaussian process whose covariance depends on a projected quantum kernel built from overlaps of local reduced density matrices. In that regime, the quantum neural tangent kernel is time independent, training dynamics become linear, and positive definiteness of the kernel gives global convergence for mean-squared loss (Nakaji et al., 2021). This line of work treats the quantum part as a structured feature extractor rather than as the trainable core of the model.

Universality results appear in multiple formalisms. The continuous-variable framework proves that Gaussian gates together with any single non-Gaussian gate form a universal gate set for CV quantum computation, so layered CV QNNs inherit universality (Killoran et al., 2018). DQNNs establish universality differently: because each layer is an ancilla-unitary-trace completely positive map, and because the architecture can simulate standard two-qubit universal circuits, DQNNs are capable of universal quantum computation (Beer, 2022).

Open-system dynamics are not merely a nuisance in several proposals. The large-scale lattice model of (Boneberg et al., 27 Apr 2026) is constructed so that, in the small-$\exp(iS/\hbar)$4 limit, its layerwise propagation approximates a Lindblad evolution

$\exp(iS/\hbar)$5

with coherent and dissipative contributions determined by local Hamiltonians and jump operators. The paper argues that this close link to Markovian open many-body dynamics can create attractors and nontrivial decision boundaries in output order-parameter space. In the soft quantum computing model, controlled Kraus operations generate quantum discord from mixed states and produce genuinely order-dependent dynamics when the induced channels do not commute (Chen, 2018).

The literature also contains more radical theoretical proposals. The path-integral view identifies the hidden layer with all possible histories, the activation with $\exp(iS/\hbar)$6, and the final amplitude with a linear summation over hidden units, thereby interpreting even cosmological actions as QNN activations (Ezhov, 2021). "Quantum Neural Networks -- Computational Field Theory and Dynamics" formalizes quantum artificial neural networks as discrete-time quantum dynamical systems with unitary neural maps, neural activity field operators, local Hamiltonians, and fluctuating local entropies; in its two-neuron recurrent simulations, the resulting dynamics display fixed-point, periodic, quasiperiodic, and edge-of-chaos signatures (Gonçalves, 2022).

Physical realizability has generated its own theory. The band-limited perceptron framework proves that if each quantum perceptron acts on at most $\exp(iS/\hbar)$7 qubits, then the total parameter count is $\exp(iS/\hbar)$8, where $\exp(iS/\hbar)$9 is the number of perceptrons, and derives a convergence bound of order $$|\Psi(\boldsymbol{x},\boldsymbol{\theta})\rangle = U(\boldsymbol{\theta})\,V(\boldsymbol{x})\,|0\rangle^{\otimes n},$$0 for its randomized training scheme (Heidari et al., 2022). This line of work treats no-cloning, measurement stochasticity, and data efficiency as foundational constraints on QNN design rather than as secondary engineering issues.

5. Applications and empirical evidence

Reported applications span classical-data classification, quantum-state classification, generative modeling, autoencoding, device characterization, and medical imaging.

Setting	Representative model	Reported result
Annealer-trained binary classification	Fully quantum annealing pipeline	Circles 100%, Quadrants 100%, Bands 92%, ttbar 78% AUC (Abel et al., 2022)
Single-qubit “soft” feedforward model	Soft quantum perceptron / SQFNN	XOR 100% test accuracy after the first epoch; moons 99%; MNIST {3,8} 89.67% (Zhou et al., 2023)
Quantum-generated data	qcNN in NTK framework	For $$|\Psi(\boldsymbol{x},\boldsymbol{\theta})\rangle = U(\boldsymbol{\theta})\,V(\boldsymbol{x})\,|0\rangle^{\otimes n},$$1, regression final cost $$|\Psi(\boldsymbol{x},\boldsymbol{\theta})\rangle = U(\boldsymbol{\theta})\,V(\boldsymbol{x})\,|0\rangle^{\otimes n},$$2; classification final cost 0.251, versus 0.503 for qNN and 0.364 for cNN (Nakaji et al., 2021)
Continuous-variable QNN	CV variational circuit	Fraud-detection AUC 0.963; tetromino generation average fidelity 98.4%; autoencoder fidelities about 99.5% (Killoran et al., 2018)
Medical image classification	qNN and qOrthNN	PneumoniaMNIST test AUC/ACC up to 0.91/0.87; RetinaMNIST up to 0.82/0.80 in reported hardware-linked settings (Mathur et al., 2021)
Merged data-dependent circuit	Combined qnn / combined qnn + cnn	Generated dataset: combined qnn 91.4% accuracy versus 81.4% for featureVar.; banknote: combined qnn + cnn 87.6% versus 88.4% for classical net (Petitzon, 2022)
Nonlinear quantum neuron hardware test	Discrete ReLU circuit	100% accuracy on simulator, about 70% on ibmq_rome, about 60% on ibmq_santiago (Yan et al., 2020)

These results are method-specific and not directly comparable across papers, but they demonstrate the breadth of tasks under the QNN label. Some studies focus on classical tabular or image data, as in the annealer, merged-circuit, and medical-imaging works (Abel et al., 2022, Petitzon, 2022, Mathur et al., 2021). Others target quantum-native tasks. DQNNs are trained on quantum input-output pairs and are proposed for characterizing unknown or untrusted quantum devices, with successful implementations on actual quantum computers reported in the thesis-length treatment (Beer, 2022). The large-scale physics-informed lattice model demonstrates 550-qubit simulations with $$|\Psi(\boldsymbol{x},\boldsymbol{\theta})\rangle = U(\boldsymbol{\theta})\,V(\boldsymbol{x})\,|0\rangle^{\otimes n},$$3 sites and $$|\Psi(\boldsymbol{x},\boldsymbol{\theta})\rangle = U(\boldsymbol{\theta})\,V(\boldsymbol{x})\,|0\rangle^{\otimes n},$$4 layers, using 5000-shot output-magnetization measurements to train a classifier for quantum states (Boneberg et al., 27 Apr 2026).

Application-specific architectural conclusions also vary. In the hybrid NTK study, the quantum encoder plus classical wide network outperforms both a fully classical neural network and a fully quantum neural network on a quantum data-generating process (Nakaji et al., 2021). In medical imaging, the reported conclusion is more cautious: the techniques show promise, but the experiments also expose the limitations of current quantum hardware (Mathur et al., 2021).

6. Limitations, controversies, and research directions

The most persistent controversy concerns what should count as a QNN at all. One influential review concluded that no existing proposal fully exploited both the advantages of quantum physics and the defining properties of neural networks, largely because of the conflict between nonlinear, dissipative neural dynamics and linear, unitary quantum evolution (Schuld et al., 2014). Later work has not eliminated this plurality; instead, it has expanded it. Some approaches move toward classical machine-learning practice by treating QNNs as variational circuits, some embrace open-system dynamics and decoherence, and some reinterpret the concept so broadly that the path integral itself becomes a neural architecture (Ezhov, 2021).

Scalability remains a central technical difficulty. Reviews of quantum deep learning emphasize barren plateaus, noise, circuit-depth limits, and classical-data encoding overhead on NISQ hardware (Kwak et al., 2021). Physically realizable training adds further constraints: unknown quantum samples cannot be cloned, and measurement makes the loss stochastic, so repeated-copy gradient estimation is problematic both physically and statistically (Heidari et al., 2022). Annealer-based approaches face a different bottleneck: qubit count and limited connectivity sharply constrain network size once embedding overhead is included (Abel et al., 2022).

The empirical literature also tempers claims of advantage. The soft quantum computing model explicitly makes no universal computing claim and does not provide a strong complexity-theoretic advantage beyond comparisons to DQC1-like settings (Chen, 2018). The medical-imaging study reports promise but also clear hardware limitations (Mathur et al., 2021). The broad review literature is similarly explicit that universality or expressive power does not by itself establish practical quantum advantage; task-by-task comparison is required (Kwak et al., 2021).

Current research directions therefore split along several axes. One axis pursues better hardware alignment: physics-informed open-system circuits, noise-aware ansätze, local-gate architectures, and order-parameter readouts designed for current devices (Boneberg et al., 27 Apr 2026). Another axis pursues stronger physical consistency: band-limited perceptrons that respect no-cloning and measurement stochasticity, or open and dissipative models in which attractor-like behavior is not artificially imposed (Heidari et al., 2022, Schuld et al., 2014). A third axis broadens the task class: DQNNs augmented with graph structure, DQNN-based generative adversarial models, larger and deeper annealer-trained networks, finer parameter discretization, and more efficient encodings (Beer, 2022, Abel et al., 2022). Hybrid classical-quantum learning, error mitigation, and task-specific inductive bias remain recurring themes across these directions (Kwak et al., 2021).

In aggregate, QNN research is less a single method than a technically diverse attempt to understand how quantum states, measurements, open-system dynamics, and trainable architectures can be organized into learning systems. Its unifying question is not whether a neural network can be made quantum in one canonical way, but which computational roles genuinely benefit from quantum structure, under which physical constraints, and with what notion of “neural” retained.