Quantum MLP Neural Networks

• Multilayer Perceptron-based Quantum Neural Networks are quantum or hybrid architectures that transpose classical MLP operations into quantum circuits for efficient state manipulation.
• They employ trainable quantum components such as variational circuits, adiabatic gates, and re-uploading layers to boost model expressivity and computational speed.
• Empirical studies demonstrate notable speedups, noise resilience, and improved performance using designs like QBMLP, QMLP, DeepQMLP, and hybrid quantum-classical models.

to=arxiv_search 天天中彩票如何 大发快三是不是json {"query":"multilayer perceptron quantum neural networks perceptron multilayer QMLP DeepQMLP quantum perceptron", "max_results": 10, "sort_by":"submittedDate"} to=arxiv_search 天天中彩票为什么json {"query":"\"Quantum Multilayer Perceptron\" OR DeepQMLP OR VQC-MLPNet OR \"multilayer perceptron\" quantum neural network", "max_results": 10, "sort_by":"relevance"} to=search_arxiv qq天天中彩票json {"query":"multilayer perceptron quantum neural networks", "max_results": 5} Multilayer perceptron-based quantum neural networks are quantum or hybrid learning architectures that transpose the multilayer perceptron template—weighted aggregation, hidden-layer composition, activation, and supervised parameter adaptation—into quantum states, parameterized circuits, measurements, adiabatic gates, or annealing-based samplers. The literature includes direct quantum perceptrons, amplitude-encoded quantum multilayer perceptrons, backpropagating qubit models such as the quantum backpropagation multilayer perceptron (QBMLP), variational-circuit analogues such as QMLP and DeepQMLP, and hybrid systems in which quantum circuits complement or generate the parameters of a classical MLP (Kairon et al., 2020, Shao, 2018, Chu et al., 2022, Alam et al., 2022, Qi et al., 12 Jun 2025).

1. Early formulations and the emergence of quantum MLP analogues

Early work framed the quantum perceptron as a direct analogue of the classical neuron while varying the source of nonlinearity and the representation of weights. The “Autonomous Quantum Perceptron Neural Network” introduced a single quantum neuron with self-adaptive activation operators; in the reported benchmarks, the XOR function was solved in 1 iteration, the quantum NOT and Hadamard gates were also solved in one iteration, and an overlapped-classes task achieved approximately $97.7\%$ accuracy after only one iteration (Sagheer et al., 2013). “Simulating a perceptron on a quantum computer” then proposed a phase-estimation-based perceptron that imitates the classical step activation function with resource scaling $\mathcal{O}(n)$, and explicitly described the model as stackable into feed-forward quantum neural networks (Schuld et al., 2014).

A different line of work sought algebraic fidelity to the classical perceptron. “Quantum perceptron over a field” defined the quantum perceptron over a finite field by replacing classical addition and multiplication with unitary operators,

$$\mathcal{P}|a\rangle|b\rangle|c\rangle = |a\rangle|b\rangle|c \oplus (a \odot b)\rangle,$$

$$\mathcal{S}|a\rangle|b\rangle|c\rangle = |a\rangle|b\rangle|c \oplus (a \oplus b)\rangle,$$

and stated that, when weights are in the computational basis, the model acts identically to a classical perceptron (Silva et al., 2016). The same paper also presented the Superposition based Architecture Learning algorithm (SAL), which searches over architectures and weights in superposition and uses a non-linear quantum operator to recover a satisfactory configuration (Silva et al., 2016).

The first explicit quantum model for multilayer perceptrons in this corpus is “A Quantum Model for Multilayer Perceptron,” which encoded vectors as amplitudes,

$$|x\rangle = \frac{1}{\|x\|}\sum_{i=0}^{n-1} x_i |i\rangle,$$

and proposed state-preparation procedures for hidden-layer outputs together with online and batch learning algorithms based on the parallel swap test, controlled rotations, and linear combination of unitaries (Shao, 2018). The paper stated that single-layer output preparation can be reduced from classical $O(mn)$ to quantum $O(\log(mn)/\epsilon)$, and that its online and batch procedures achieve at least quadratic speedup or even exponential speedup over classical algorithms (Shao, 2018).

2. Quantum realizations of perceptron and MLP primitives

In direct qubit-based MLP analogues, classical features are typically mapped to quantum amplitudes and transformed by gate-parameterized weights. QBMLP is a representative example. Its inputs are first fuzzified or squashed to $[0,1]$, then represented as

$|x\rangle = \cos(x)|0\rangle + \sin(x)|1\rangle,$

with weights modeled as rotation gates $R(\theta_i)$ and bias encoded as $\mathcal{O}(n)$0. The layer transformation is

$\mathcal{O}(n)$1

and the output is extracted from quantum amplitudes rather than from a conventional classical nonlinearity. The measured prediction is

$\mathcal{O}(n)$2

The paper describes a layered structure that mimics classical MLPs with an input layer, configurable hidden layers, and a single output neuron (Kairon et al., 2020).

Other constructions realize the perceptron more directly in hardware or in oracle form. “Direct implementation of a perceptron in superconducting circuit quantum hardware” demonstrated an adiabatic controlled gate in fixed-frequency superconducting transmon qubits coupled by a flux-tunable coupler. In this setting, activation steepness is controlled by the adiabatic gate length $\mathcal{O}(n)$3, the weight is controlled by the ZZ coupling $\mathcal{O}(n)$4, and the bias is controlled by the final drive frequency $\mathcal{O}(n)$5. The output qubit takes the form

$\mathcal{O}(n)$6

with $\mathcal{O}(n)$7, so the gate acts as a continuous quantum generalization of a classical threshold unit (Pechal et al., 2021). By contrast, “Implementing An Artificial Quantum Perceptron” represented both inputs and weights as quantum states prepared by unitary oracles $\mathcal{O}(n)$8 and $\mathcal{O}(n)$9, and used the inner product $$\mathcal{P}|a\rangle|b\rangle|c\rangle = |a\rangle|b\rangle|c \oplus (a \odot b)\rangle,$$0 as the core computation. Its training rule was explicitly bit-flip-based rather than gradient-based, and the paper reported simulations for 2, 3, and 4 qubits while noting the absence of explicit bias terms and the restriction of inputs to $$\mathcal{P}|a\rangle|b\rangle|c\rangle = |a\rangle|b\rangle|c \oplus (a \odot b)\rangle,$$1 (Hathidara et al., 2024).

Taken together, these models preserve the perceptron abstraction—inputs, weights, bias, activation, and output—but distribute those roles across distinct quantum mechanisms: amplitude geometry, adiabatic control, measurement probabilities, or oracle-defined state overlap.

3. Multilayer depth, re-uploading, and hybrid MLP architectures

Recent work increasingly treats the quantum component as one layer or submodule within a broader multilayer architecture. QMLP is a fully quantum analogue of a classical MLP in which each input feature is encoded by a single-qubit $$\mathcal{P}|a\rangle|b\rangle|c\rangle = |a\rangle|b\rangle|c \oplus (a \odot b)\rangle,$$2 rotation, variational blocks combine parameterized single-qubit rotations with parameterized two-qubit entangling gates such as $$\mathcal{P}|a\rangle|b\rangle|c\rangle = |a\rangle|b\rangle|c \oplus (a \odot b)\rangle,$$3 or $$\mathcal{P}|a\rangle|b\rangle|c\rangle = |a\rangle|b\rangle|c \oplus (a \odot b)\rangle,$$4, and Re-Uploading Units (RUUs) reapply data between variational blocks to enrich nonlinearity. Qubits are measured in the $$\mathcal{P}|a\rangle|b\rangle|c\rangle = |a\rangle|b\rangle|c \oplus (a \odot b)\rangle,$$5-basis, and the results are passed to a classical fully connected layer. On the 10-class MNIST dataset, the paper stated that QMLP increases inference accuracy by $$\mathcal{P}|a\rangle|b\rangle|c\rangle = |a\rangle|b\rangle|c \oplus (a \odot b)\rangle,$$6 with 2 times fewer quantum gates and 3 times reduced parameters relative to prior art (Chu et al., 2022).

DeepQMLP pushes the MLP analogy further by stacking shallow quantum neural networks so that each QNN acts as a hidden layer. The Pauli-$$\mathcal{P}|a\rangle|b\rangle|c\rangle = |a\rangle|b\rangle|c \oplus (a \odot b)\rangle,$$7 expectation values produced by one QNN are rescaled and angle-encoded into the next QNN, and the final quantum features are sent to a classical dense layer with softmax output. The architecture was designed to trade one deep parametric quantum circuit for multiple shallow ones, thereby reducing susceptibility to decoherence and gate errors. Under noise, the paper reported up to $$\mathcal{P}|a\rangle|b\rangle|c\rangle = |a\rangle|b\rangle|c \oplus (a \odot b)\rangle,$$8 lower loss and $$\mathcal{P}|a\rangle|b\rangle|c\rangle = |a\rangle|b\rangle|c \oplus (a \odot b)\rangle,$$9 higher accuracy during inference than QMLP (Alam et al., 2022).

Hybridization can also be parallel rather than sequential. “Parallel Hybrid Networks” routed the same input simultaneously into a classical MLP and a variational quantum circuit, then linearly combined their outputs,

$$\mathcal{S}|a\rangle|b\rangle|c\rangle = |a\rangle|b\rangle|c \oplus (a \oplus b)\rangle,$$0

The paper argued that the VQC learns a smooth sinusoidal foundation while the classical perceptrons fill non-harmonic gaps, and reported improved optimality on periodic datasets with additional noise (Kordzanganeh et al., 2023).

A more recent hybrid design, VQC-MLPNet, uses a variational quantum circuit to generate the weights of part of a classical MLP. First-layer weight vectors are amplitude-encoded,

$$\mathcal{S}|a\rangle|b\rangle|c\rangle = |a\rangle|b\rangle|c \oplus (a \oplus b)\rangle,$$1

transformed by a parameterized VQC, and then used as quantum-enhanced hidden-layer weights in

$$\mathcal{S}|a\rangle|b\rangle|c\rangle = |a\rangle|b\rangle|c \oplus (a \oplus b)\rangle,$$2

Only a subset of the MLP weights is quantum-generated; the remaining parameters are classical. The paper emphasized that, once training is complete, inference proceeds entirely classically (Qi et al., 12 Jun 2025).

These designs indicate a shift from “fully quantum MLP” formulations toward modular depth: shallow quantum blocks, repeated encoding, classical post-processing, and quantum-generated parameterizations.

4. Training regimes and learning algorithms

Training procedures in MLP-based quantum neural networks vary more widely than their forward models. QBMLP follows the classical feedforward/backpropagation template at a high level: it performs feedforward computation and error backpropagation, uses differentiable activation behavior through $$\mathcal{S}|a\rangle|b\rangle|c\rangle = |a\rangle|b\rangle|c \oplus (a \oplus b)\rangle,$$3, and updates parameters by gradients with respect to an error or cost function. The paper also noted a specifically quantum complication: backpropagation must account for both real and imaginary components associated with quantum states (Kairon et al., 2020).

Several papers depart from gradient descent. “A Derivative-free Method for Quantum Perceptron Training in Multi-layered Neural Networks” defined training instances by outer products,

$$\mathcal{S}|a\rangle|b\rangle|c\rangle = |a\rangle|b\rangle|c \oplus (a \oplus b)\rangle,$$4

used singular value decomposition to render $$\mathcal{S}|a\rangle|b\rangle|c\rangle = |a\rangle|b\rangle|c \oplus (a \oplus b)\rangle,$$5 unitary, and proposed the update

$$\mathcal{S}|a\rangle|b\rangle|c\rangle = |a\rangle|b\rangle|c \oplus (a \oplus b)\rangle,$$6

The paper stated that the resulting computational efficiency is devoid of the number of layers in the network and gave the speedup estimate

$$\mathcal{S}|a\rangle|b\rangle|c\rangle = |a\rangle|b\rangle|c \oplus (a \oplus b)\rangle,$$7

for the SVD component (Khan et al., 2020).

Search-based training is another recurrent theme. “Development of Quantum Circuits for Perceptron Neural Network Training, Based on the Principles of Grover’s Algorithm” cast weight selection as an oracle-based search problem. For a perceptron with two binary inputs and two weights, the oracle marks weights satisfying $$\mathcal{S}|a\rangle|b\rangle|c\rangle = |a\rangle|b\rangle|c \oplus (a \oplus b)\rangle,$$8 or $$\mathcal{S}|a\rangle|b\rangle|c\rangle = |a\rangle|b\rangle|c \oplus (a \oplus b)\rangle,$$9, and Grover iterations amplify the marked states with

$$|x\rangle = \frac{1}{\|x\|}\sum_{i=0}^{n-1} x_i |i\rangle,$$0

The paper described this as extendable to multilayer perceptrons by adding more weight registers and more complex oracles (Pronin et al., 2021).

Quantum annealing supplies a different training substrate. “Training Multilayer Perceptrons by Sampling with Quantum Annealers” proved an equivalence between a one-hidden-layer sigmoid MLP trained with binary cross-entropy and an energy-based model trained by maximum conditional likelihood. The energy function is

$$|x\rangle = \frac{1}{\|x\|}\sum_{i=0}^{n-1} x_i |i\rangle,$$1

and the annealer is used as a sampling engine for the model term in the gradient. The reported implementation used the D-Wave Advantage System on binary image classification tasks with $$|x\rangle = \frac{1}{\|x\|}\sum_{i=0}^{n-1} x_i |i\rangle,$$2 inputs, $$|x\rangle = \frac{1}{\|x\|}\sum_{i=0}^{n-1} x_i |i\rangle,$$3 hidden units, and $$|x\rangle = \frac{1}{\|x\|}\sum_{i=0}^{n-1} x_i |i\rangle,$$4 output (Yang et al., 2023).

The most ambitious search formulation is SAL, which initializes architecture and weight registers in uniform superposition, evaluates all configurations in parallel over the training set, and uses a non-linear quantum operator to recover a configuration meeting a performance threshold. The paper gives overall cost $$|x\rangle = \frac{1}{\|x\|}\sum_{i=0}^{n-1} x_i |i\rangle,$$5 and describes SAL as the first learning algorithm to determine neural network architectures in polynomial time (Silva et al., 2016).

5. Expressivity, nonlinearity, and conceptual limitations

The central theoretical dispute in this area concerns what gives a quantum MLP genuine representational depth. A teacher-student comparison of quantum perceptron and quantum neural network structures found that simply adding gates, entanglers, ancillas, or deferred measurement does not substantially improve expressive power if the data are encoded only once. The paper’s key finding was that expressive power is determined predominantly by the number of distinct data encodings, not by measurement-induced non-linearity or deferred measurement alone. Hidden units become useful only when paired with repeated or nonlinear data embeddings (Aikaterini et al., 2021).

A sharper limitation was formalized in “Exploiting the equivalence between quantum neural networks and perceptrons,” which proved an exact mapping from standard QNNs to classical perceptrons acting on $$|x\rangle = \frac{1}{\|x\|}\sum_{i=0}^{n-1} x_i |i\rangle,$$6. For binary classification,

$$|x\rangle = \frac{1}{\|x\|}\sum_{i=0}^{n-1} x_i |i\rangle,$$7

is always a real quadratic form in $$|x\rangle = \frac{1}{\|x\|}\sum_{i=0}^{n-1} x_i |i\rangle,$$8, hence equivalent to a perceptron on the lifted feature map. The paper argued that a QNN with amplitude encoding cannot express the Boolean parity function for $$|x\rangle = \frac{1}{\|x\|}\sum_{i=0}^{n-1} x_i |i\rangle,$$9, that such models capture only a vanishingly small fraction of Boolean functions as $O(mn)$0 grows, and that several popular embeddings exhibit weak or unhelpful inductive bias. It then proposed two ways forward: using QNNs to generate classical DNN-inspired kernels, and constructing layered non-linear QNNs with intermediate measurements that are provably fully expressive on Boolean data (Mingard et al., 2024).

One response to these expressivity bottlenecks is to enrich the perceptron potential itself. “Quantum neural networks with multi-qubit potentials” generalized the neural potential from pairwise terms to

$O(mn)$1

allowing a single perceptron to implement XOR and enabling one-layer realizations of CNOT, Toffoli, and Fredkin gates without hidden layers or ancillary qubits required by standard pairwise-only constructions (Ban et al., 2021). This suggests that some “depth” in quantum MLPs can be traded for richer interaction structure within a neuron.

A distinct controversy surrounds physical realizability. SAL relies on a non-linear quantum operator, and the paper itself notes that such non-linearity is controversial in the physics literature (Silva et al., 2016). Accordingly, the field contains both hardware-realizable proposals and more speculative models that are best interpreted as theoretical constructions.

6. Empirical demonstrations, hardware results, and current outlook

Empirical work spans regression, image classification, quantum-device benchmarking, and domain-specific hybrid modeling. The QBMLP study compared a 3-3-1 QBMLP and a continuous-variable quantum neural network on COVID-19 outbreak prediction for India and the USA using confirmed cases, deaths, recoveries, and days passed as features. Both quantum models reportedly outperformed classical artificial neural networks; two-tailed $O(mn)$2-tests indicated that the difference between QBMLP and CVQNN was not statistically significant, although QBMLP was marginally better for confirmed cases and CVQNN was slightly better for deaths (Kairon et al., 2020).

Hardware implementation has progressed from simulation to direct physical realization. In superconducting circuits, the quantum perceptron gate was experimentally shown to provide tunable activation functions, tunable weights through ZZ coupling, and tunable bias through microwave drive frequency, while also generating entanglement when inputs are in superposition. The paper reported process fidelities of approximately $O(mn)$3 to $O(mn)$4, but also identified connectivity, cross-talk, weight range, and adiabatic speed limits as barriers to multilayer scaling (Pechal et al., 2021).

On noisy classification benchmarks, QMLP and DeepQMLP were designed explicitly for NISQ constraints. QMLP emphasized error-tolerant input embedding and parameterized two-qubit entanglers on 10-class MNIST (Chu et al., 2022), whereas DeepQMLP emphasized stacked shallow circuits and reported greater resilience to device noise than a single deeper QNN layer (Alam et al., 2022). VQC-MLPNet extended the empirical range to semiconductor quantum-dot charge-state classification and genomic transcription factor binding site prediction, where it reported loss $O(mn)$5 and accuracy $O(mn)$6 on clean quantum-dot data, loss $O(mn)$7 and accuracy $O(mn)$8 on noisy quantum-dot data, and loss $O(mn)$9 with accuracy $O(\log(mn)/\epsilon)$0 on transcription factor binding site prediction, while maintaining resilience under realistic IBM quantum noise simulations (Qi et al., 12 Jun 2025).

Annealing-based training offers another application route. The D-Wave study demonstrated supervised training of binary image classifiers on small subsets of MNIST and Fashion-MNIST by exploiting the equivalence between one-hidden-layer MLPs and conditional energy-based models, thereby treating the quantum annealer as a sampling engine rather than as a circuit-based hidden layer (Yang et al., 2023).

The current literature therefore supports a differentiated view. Multilayer perceptron-based quantum neural networks are not a single architecture but a family of constructions: direct quantum neurons, layered variational circuits, repeated-encoding models, hardware-level perceptron gates, annealing-assisted MLP training, and hybrids in which quantum circuits serve as feature generators or parameter generators. The strongest theoretical critiques apply to shallow perceptron-equivalent QNNs on classical data, while the most promising positive results come from architectures that introduce repeated data uploadings, multi-qubit interactions, shallow-but-stacked modules, or tightly constrained quantum-classical division of labor (Mingard et al., 2024, Aikaterini et al., 2021, Qi et al., 12 Jun 2025).