Parameterized quantum circuits as machine learning models

Marcello BenedettiErika LloydStefan H. SackMattia Fiorentini

article2019Quantum Science and Technology1,407 citations

Explains how parameterized quantum circuits operate as machine learning models, detailing their core components and practical deployment across supervised learning and generative tasks on near-term quantum hardware.

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Large-scale, fault-tolerant quantum computers remain years away due to physical noise and limited qubit counts. Currently available noisy intermediate-scale quantum devices cannot execute deep traditional algorithms, raising the practical question of how to deliver near-term computational value. The article evaluates hybrid quantum-classical machine learning architectures based on parameterized quantum circuits—quantum routines with adjustable parameters that can be iteratively trained alongside classical computers—to determine their viability for near-term applications.

The article synthesizes recent theoretical models, numerical simulations, and physical experiments across superconducting, trapped-ion, and photonic hardware platforms. In these hybrid systems, data is pre-processed classically, encoded into quantum states, transformed via parameterized quantum operations, and measured; the measurement outputs are then post-processed classically to compute objective functions and update circuit parameters in an iterative feedback loop.

The findings show that parameterized quantum circuits possess high expressive power, allowing them to map data into exponentially large feature spaces and model complex probability distributions with fewer parameters than standard classical networks. Analytical gradient methods, such as the parameter-shift rule, offer superior scaling and unbiased optimization compared to numerical finite-difference approaches, which require substantially more oracle evaluations. The framework demonstrates versatility across classical tasks, such as supervised classification and generative modeling, as well as native quantum tasks like quantum state tomography and circuit compilation. However, experimental deployments reveal major practical hurdles: random circuit designs often lead to vanishing gradients where optimization stalls, and hardware noise and statistical variance currently cause physical quantum models to perform significantly worse than idealized numerical simulations.

These results indicate that near-term hybrid systems offer a practical transitional architecture by shifting costly, classically intractable subroutines to quantum processors while offloading parameter optimization to classical machines. Achieving a practical advantage in production, however, depends heavily on overcoming optimization plateaus and hardware noise, rather than purely increasing qubit numbers. Furthermore, native quantum learning tasks—such as state compression and quantum algorithm compilation—represent the most immediate opportunity for quantum advantage because classical systems require exponentially scaling resources to model the same quantum states.

Organizations evaluating near-term quantum technologies should focus research on structured circuit designs, specialized classical optimizers, and automated hyperparameter tuning to mitigate barren plateaus and noise. Teams should prioritize open-source hybrid software frameworks to benchmark implementations consistently across hardware platforms. Continued investment should be paired with controlled pilot projects on domain-specific datasets, as practical utility remains bounded by experimental noise and current demonstrations remain limited to small-scale, proof-of-concept benchmarks.

Cambridge Quantum Computing Limited University College London demonstrations experimental demonstrations software parameterized quantum circuits quantum neural networks quantum classifiers

arXiv: 1906.07682

Paper: Supervised learning with quantum-enhanced feature spaces, Vojtech Havlicek et al. (2018). It establishes fundamental methods for supervised classification using quantum feature spaces and variational circuits on near-term hardware, providing essential empirical and conceptual foundations for parameterized quantum circuit models.
Paper: Barren plateaus in quantum neural network training landscapes, Jarrod R. McClean et al. (2018). It analyzes the barren plateau phenomenon in parameterized quantum circuits, identifying crucial trainability limitations and gradient vanishing behaviors that parameterized quantum circuit models must address.
Paper: Quantum machine learning, Jacob Biamonte et al. (2016). It offers an overarching foundational survey of quantum machine learning paradigms, linear algebra speedups, and hardware constraints that contextualize parameterized quantum circuits.

Paper: Variational quantum algorithms, M. Cerezo et al. (2020). It comprehensively expands the parameterized circuit paradigm into the broader variational quantum algorithm (VQA) framework, systematically addressing trainability, optimization strategies, and error mitigation across diverse applications.
Paper: Predicting many properties of a quantum system from very few measurements, Hsin-Yuan Huang et al. (2020). It introduces the classical shadow formalism to drastically reduce measurement overheads when characterizing quantum states and evaluating observables from parameterized quantum circuits.

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Abstract

Hybrid quantum-classical systems make it possible to utilize existing quantum computers to their fullest extent. Within this framework, parameterized quantum circuits can be regarded as machine learning models with remarkable expressive power. This Review presents the components of these models and discusses their application to a variety of data-driven tasks, such as supervised learning and generative modeling. With an increasing number of experimental demonstrations carried out on actual quantum hardware and with software being actively developed, this rapidly growing field is poised to have a broad spectrum of real-world applications.

Topics

demonstrations experimental demonstrations software supervised machine learning quantum gates frequency bands

Keywords

parameterized quantum circuits quantum neural networks quantum classifiers quantum autoencoders variational circuits quantum approximate optimization quantum kernel gate parameterizations

1. Introduction
2. Framework
2.1. The encoder circuit U x f( )
2.2. The variational circuit U q
2.3. Circuit learning
3. Applications
3.1. Supervised learning
3.2. Generative modeling
3.3. Quantum learning tasks
4. Outlook
Acknowledgments
References

Knowls

Knowl 1 — Parameter-Shift Rule for Analytical Gradient Estimation
equation
For a parameterized quantum circuit $U_{J:1}(\boldsymbol{\theta}) = U_J(\theta_J) \cdots U_1(\theta_1)$ where each parameter-dependent unitary gate is generated by an $n$ -qubit Pauli tensor product $P_j \in \{I, X, Y, Z\}^{\otimes n}$ in the form $U_j(\theta_j) = \exp\left(-i \frac{\theta_j}{2} P_j\right)$ , the exact analytical derivative of an expectation value $\langle M_k \rangle_{\boldsymbol{\theta}} = \text{tr}\left( M_k U_{J:1}(\boldsymbol{\theta}) |0\rangle\langle 0|^{\otimes n} U_{J:1}^\dagger(\boldsymbol{\theta}) \right)$ of a Hermitian observable $M_k$ with respect to circuit parameter $\theta_j$ is given by:

$\frac{\partial \langle M_k \rangle_{\boldsymbol{\theta}}}{\partial \theta_j} = \frac{\langle M_k \rangle_{\boldsymbol{\theta} + \frac{\pi}{2} \mathbf{e}_j} - \langle M_k \rangle_{\boldsymbol{\theta} - \frac{\pi}{2} \mathbf{e}_j}}{2}$

where $\mathbf{e}_j$ denotes the $j$ -th unit vector in parameter space. This rule provides an unbiased gradient estimator evaluated on quantum hardware without requiring ancilla qubits or interference circuits, requiring exactly two circuit evaluations per parameter.
Knowl 2 — Hybrid Quantum-Classical Machine Learning Architecture with Parameterized Quantum Circuits
model/method
Parameterized quantum circuits (PQCs) serve as core predictive models within a hybrid quantum-classical pipeline comprising six integrated stages:
1. Classical pre-processing: A data point $\mathbf{x}$ sampled from data distribution $P_{\mathcal{D}}$ is transformed via a classical pre-processing function $\phi(\mathbf{x})$ into circuit control parameters.
2. Quantum state encoding: An encoder unitary $U_{\phi(\mathbf{x})}$ prepares a quantum state $|\psi(\mathbf{x})\rangle = U_{\phi(\mathbf{x})}|0\rangle^{\otimes n}$ on an $n$ -qubit register.
3. Variational circuit transformation: A parameterized unitary $U_{\boldsymbol{\theta}}$ , controlled by adjustable parameter vector $\boldsymbol{\theta}$ , acts on the state $|\psi(\mathbf{x})\rangle$ alongside an optional ancilla register of $m$ qubits initialized in $|0\rangle^{\otimes m}$ , generating the state $U_{\boldsymbol{\theta}} (U_{\phi(\mathbf{x})}|0\rangle^{\otimes n} \otimes |0\rangle^{\otimes m})$ .
4. Quantum measurement: A set of $K$ Hermitian observables $\{M_k\}_{k=1}^K$ is measured to estimate expectation values $\{\langle M_k \rangle_{\mathbf{x}, \boldsymbol{\theta}}\}_{k=1}^K$ .
5. Classical post-processing: The estimated expectation values are transformed via a classical function $f(\{\langle M_k \rangle_{\mathbf{x}, \boldsymbol{\theta}}\}_{k=1}^K, \mathbf{w})$ , parameterized by classical weights $\mathbf{w}$ , to yield model forecasts.
6. Classical optimization loop: A classical optimization algorithm evaluates a regularized loss $\mathcal{L}(\boldsymbol{\theta}, \mathbf{w})$ over training samples and iteratively updates $\boldsymbol{\theta}$ and $\mathbf{w}$ .
Knowl 3 — Quantum Circuit Born Machine
model/method
A Quantum Circuit Born Machine (QCBM) is an implicit generative model that represents discrete probability distributions over bitstrings $\mathbf{v} \in \{0, 1\}^n$ via the Born rule. The model sets the encoder circuit to the identity $U_{\phi(\mathbf{x})} = I$ and prepares a quantum state using a parameterized variational circuit $U_{\boldsymbol{\theta}}$ acting on $|0\rangle^{\otimes n}$ . Projective measurement in the computational basis using bitstring projectors $M_{\mathbf{v}} = |\mathbf{v}\rangle\langle\mathbf{v}|$ yields the probability distribution:

$q_{\boldsymbol{\theta}}(\mathbf{v}) = \text{tr}\left( M_{\mathbf{v}} U_{\boldsymbol{\theta}} |0\rangle\langle 0|^{\otimes n} U_{\boldsymbol{\theta}}^\dagger \right) = |\langle \mathbf{v} | U_{\boldsymbol{\theta}} |0\rangle^{\otimes n}|^2$

Because computing $q_{\boldsymbol{\theta}}(\mathbf{v})$ analytically is intractable for general circuits, samples $\mathbf{v} \sim q_{\boldsymbol{\theta}}$ are obtained directly via quantum measurement. Training over an empirical dataset $\mathcal{D} = \{\mathbf{v}^{(i)}\}_{i=1}^N$ drawn from unknown distribution $p$ is performed by minimizing statistical distances such as the Maximum Mean Discrepancy (MMD):

$D(p, q_{\boldsymbol{\theta}}) = \left\| \sum_{\mathbf{v}} p(\mathbf{v}) \phi(\mathbf{v}) - \sum_{\mathbf{v}} q_{\boldsymbol{\theta}}(\mathbf{v}) \phi(\mathbf{v}) \right\|^2$

where $\phi$ is a classical feature map and expectations are estimated directly from sample frequencies, enabling gradient-based optimization on discrete data without computing likelihoods.
Knowl 4 — Quantum Kernel Estimator and Variational Quantum Model Paradigms
model/method
Supervised learning with parameterized quantum circuits is structured into two distinct paradigms:
1. Quantum Kernel Estimator (QKE): Classical data points $\mathbf{x}$ are mapped to quantum states via an encoder circuit $U_{\phi(\mathbf{x})}|0\rangle^{\otimes n}$ . The quantum computer executes a SWAP test between state pairs to estimate the inner product kernel matrix $k(\mathbf{x}, \mathbf{x}') = |\langle 0|^{\otimes n} U_{\phi(\mathbf{x}')}^\dagger U_{\phi(\mathbf{x})} |0\rangle^{\otimes n}|^2$ . The model prediction is expressed via the representer theorem as $f(\mathbf{x}, \mathbf{w}) = \sum_{i=1}^N w_i k(\mathbf{x}, \mathbf{x}^{(i)})$ , where the weights $\mathbf{w}$ are trained purely on a classical computer using convex optimization (e.g., support vector machines).
2. Variational Quantum Model (VQM): Data is encoded via $U_{\phi(\mathbf{x})}$ and directly processed in quantum state space by applying a parameterized variational unitary $U_{\boldsymbol{\theta}}$ . Output forecasts are computed by classically post-processing expectation values $\{\langle M_k \rangle_{\mathbf{x}, \boldsymbol{\theta}}\}_{k=1}^K$ of measured observables. The parameter vector $\boldsymbol{\theta}$ parametrizes the quantum gates and is optimized iteratively via closed-loop quantum-classical algorithms.
Knowl 5 — Query Complexity of Analytical Gradients versus Finite Differences in Circuit Learning
theoretical result
For optimizing an $n$ -qubit parameterized quantum circuit within the vicinity of the optimum to achieve target precision $\epsilon$ in convex optimization settings:
- Circuit optimization utilizing the analytical gradient (e.g., via the parameter-shift rule) has an oracle query complexity scaling as:
$\mathcal{O}\left(\frac{n^2}{\epsilon}\right)$
- Circuit optimization using numerical finite difference gradient approximations requires an oracle query complexity of at least:
$\Omega\left(\frac{n^3}{\epsilon^2}\right)$

Analytical gradient evaluation provides both an unbiased gradient estimator and an asymptotic query complexity advantage over numerical finite difference approximations on quantum hardware.
Knowl 6 — Hadamard Test for Variational Circuit Gradient Estimation
model/method
For a variational circuit $U_{J:1}(\boldsymbol{\theta}) = U_J \cdots U_1$ with parameterized gates $U_j(\theta_j) = \exp\left(-i \frac{\theta_j}{2} P_j\right)$ generated by Pauli strings $P_j \in \{I, X, Y, Z\}^{\otimes n}$ , the partial derivative of an expectation value $\langle M_k \rangle_{\boldsymbol{\theta}}$ with respect to $\theta_j$ is given by:

$\frac{\partial \langle M_k \rangle_{\boldsymbol{\theta}}}{\partial \theta_j} = \text{Im}\left( \text{tr}\left( M_k U_{J:j+1} P_j U_{j:1} |0\rangle\langle 0|^{\otimes n} U_{J:1}^\dagger \right) \right)$

This quantity is estimated in a single quantum circuit using one additional ancilla qubit:
1. Initialize the target register in $|0\rangle^{\otimes n}$ and the ancilla qubit in $|0\rangle$ .
2. Apply a Hadamard gate $H$ to the ancilla.
3. Apply unitary sequence $U_{j:1}$ to the target register.
4. Apply a controlled- $P_j$ gate from the ancilla to the target register.
5. Apply the remaining unitary sequence $U_{J:j+1}$ to the target register.
6. Measure observable $M_k$ on the register, apply a Hadamard gate $H$ to the ancilla, and measure the ancilla Pauli- $Z$ observable.
The expectation value $\langle Z \rangle$ of the ancilla measurement equals the analytical derivative $\frac{\partial \langle M_k \rangle_{\boldsymbol{\theta}}}{\partial \theta_j}$ .
Knowl 7 — Barren Plateaus in Parameterized Quantum Circuit Optimization Landscapes
theoretical result
For parameterized quantum circuits $U(\boldsymbol{\theta})$ whose ansatzes are randomly initialized and form approximate 2-designs over the unitary group $U(2^n)$ , the variance of the gradient of the objective function $\mathcal{L}(\boldsymbol{\theta})$ vanishes exponentially with the number of qubits $n$ :

$\text{Var}_{\boldsymbol{\theta}}\left[ \frac{\partial \mathcal{L}(\boldsymbol{\theta})}{\partial \theta_j} \right] \in \mathcal{O}\left( \frac{1}{2^n} \right)$

This concentration of measure (governed by Lévy's lemma) creates an exponentially flat optimization landscape where the probability of observing a non-zero gradient above any threshold $\delta$ decreases exponentially in $n$ . Consequently, unbiased gradient estimators perform an ineffective random walk in parameter space when circuits are initialized randomly without structured ansatz constraints.
Knowl 8 — Quantum Generative Adversarial Networks
model/method
Quantum Generative Adversarial Networks (QGANs) formulate generative modeling as a two-player minimax game between a generator $G$ and a discriminator $D$ , where one or both players are parameterized quantum circuits:
- Classical data setting: A PQC generator (such as a QCBM) generates synthetic bitstrings $\mathbf{v}$ , while a classical or quantum discriminator $D$ evaluates sample distinguishability, supplying a differentiable surrogate loss to update the generator parameters.
- Quantum data setting: The generator implements a variational circuit $U_G(\boldsymbol{\theta})$ preparing quantum states $|\psi_G(\boldsymbol{\theta})\rangle = U_G(\boldsymbol{\theta})|0\rangle^{\otimes n}$ . The discriminator implements a variational circuit $U_D(\boldsymbol{\phi})$ that executes an optimal Helstrom measurement to distinguish $|\psi_G(\boldsymbol{\theta})\rangle$ from target quantum states $\rho_{\text{target}}$ . Alternating optimization drives the generator to minimize state distinguishability until the generated state approximates $\rho_{\text{target}}$ in trace distance.
Knowl 9 — Quantum Autoencoder for State Compression
model/method
A Quantum Autoencoder (QAE) performs lossy dimensionality reduction on an ensemble of $n$ -qubit quantum states $\rho_{\text{in}}$ to compress them into $k < n$ qubits:
1. Encoding: A parameterized encoder unitary $U_{\text{enc}}(\boldsymbol{\theta})$ transforms $\rho_{\text{in}}$ such that quantum information is concentrated into $k$ latent qubits, while $n - k$ trash qubits are disentangled.
2. Discarding: The $n - k$ trash qubits are traced out, leaving the compressed $k$ -qubit state.
3. Decoding: A decoder unitary $U_{\text{dec}}(\boldsymbol{\theta}) = U_{\text{enc}}^\dagger(\boldsymbol{\theta})$ acts jointly on the $k$ latent qubits and $n - k$ reference qubits initialized in $|0\rangle^{\otimes (n - k)}$ to produce reconstructed state $\rho_{\text{out}}$ .
4. Optimization: The parameter vector $\boldsymbol{\theta}$ is optimized to maximize state reconstruction fidelity $F(\rho_{\text{in}}, \rho_{\text{out}}) = \left( \text{tr}\sqrt{\sqrt{\rho_{\text{in}}} \rho_{\text{out}} \sqrt{\rho_{\text{in}}}} \right)^2$ , which is achieved on hardware by maximizing the overlap of the trash qubits with the fiducial state $|0\rangle^{\otimes (n - k)}$ .

Knowl 10 — Physical Implementations of Parameterized Quantum Circuit ML Models

data/table

The table below synthesizes early experimental demonstrations of parameterized quantum circuit machine learning models across physical superconducting (S), trapped ion (T), and photonic (P) quantum computing architectures:

Task	Model	Learning Algorithm	Qubits	Hardware Architecture
Classification	QKE	N/A	4	IBM Q5 Yorktown (S)
Classification	VQM	N/A	4	IBM Q5 Tenerife (S)
Classification	QKE, VQM	Gradient-based	2	IBM Q5 Yorktown (S)
Classification	Perceptron	Gradient-based	3	IBM Q5 Tenerife (S)
Generative	QCBM	N/A	4	Custom (T)
Generative	QCBM	Gradient-based	4	IBM Q20 Tokyo (S)
Generative	QCBM	Gradient-free	4	Custom (T)
Generative	QCBM	Gradient-based, Gradient-free	4	Rigetti 16Q-Aspen (S)
Generative	QCBM	Gradient-based	4	Rigetti 16Q-Aspen (S)
State learning	QGAN	Gradient-based	1	Custom (S)
State learning	QGAN	Gradient-based	3	IBM Q20 Poughkeepsie (S)
State learning	PAC	N/A	6	Custom (P)
Clustering	QAOA	Gradient-free	19	Rigetti 19Q-Acorn (S)
Compression	QAE	Gradient-free	3	Rigetti 8Q-Agave (S)
Learning parity with noise	Oracle	N/A	5	IBM Q5 Yorktown (S)

This dataset summarizes physical implementations executing supervised classification, generative distribution modeling, quantum state tomography, MaxCut clustering, and state compression across devices scaling from 1 to 19 qubits. Entries labeled N/A indicate experiments where circuit parameters were pre-trained via classical numerical simulation or derived analytically prior to deployment on quantum hardware.

Knowl 11 — Data Encoding Strategies for Parameterized Quantum Circuits
model/method
Classical feature vectors $\mathbf{x} \in \mathbb{R}^d$ are encoded into $n$ -qubit states $|\psi(\mathbf{x})\rangle = U_{\phi(\mathbf{x})}|0\rangle^{\otimes n}$ using four primary strategies:
1. Qubit / Angle Encoding: Each component $x_i$ of an input vector $\mathbf{x} \in [0, 1]^d$ parameterizes a single-qubit rotation on qubit $i$ , producing the unentangled product state $|\psi(\mathbf{x})\rangle = \bigotimes_{i=1}^d \left(\cos(\pi x_i / 2)|0\rangle + \sin(\pi x_i / 2)|1\rangle\right)$ . This requires $n = d$ qubits and $\mathcal{O}(1)$ circuit depth. Redundant assignment of individual features across multiple qubits introduces higher-order terms to fit nonlinear functions.
2. Amplitude Encoding: A normalized $2^n$ -dimensional vector $\mathbf{x}$ (with $\sum_{i=1}^{2^n} |x_i|^2 = 1$ ) is mapped to amplitudes of an $n$ -qubit state $|\psi(\mathbf{x})\rangle = \sum_{i=0}^{2^n - 1} x_i |i\rangle$ . This provides exponential memory compression ( $n = \log_2 d$ ), but generic states require circuit compilation depth scaling as $\mathcal{O}(2^n)$ .
3. Quantum Kitchen Sinks / Random Encoders: Inputs are pre-processed via randomized affine projections $\phi(\mathbf{x}) = A\mathbf{x} + \mathbf{b}$ that parameterize single-qubit rotations followed by entangling gates, approximating continuous kernel functions through random quantum features.
4. Classically Intractable Feature Maps: Encoder circuits structured as $U_{\phi(\mathbf{x})} = \exp\left(i \sum_{j,k} \phi_{j,k}(\mathbf{x}) Z_j Z_k\right) H^{\otimes n}$ mimic Boolean bent function hidden shift problems, producing kernel matrices that are conjectured to be classically intractable to compute.
Knowl 12 — Tensor Network-Inspired Parameterized Quantum Circuit Architectures
model/method
Tensor networks provide structured, scalable circuit designs that constrain multi-qubit entanglement to polynomial parameter complexity:
1. Tree Tensor Network (TTN): Unitary operations are organized in a hierarchical tree topology. Each layer applies two-qubit parameterized unitary gates to disjoint qubit pairs and traces out (discards) half of the qubits. This coarse-graining reduces the qubit register logarithmically with circuit depth.
2. Qubit-Efficient TTN: Instead of discarding qubits, the traced-out qubits are reinitialized to $|0\rangle$ and reused sequentially as inputs for subsequent unitary layers. This preserves the operational function of the TTN while minimizing physical qubit count on NISQ hardware at the cost of increased circuit depth.
3. Multi-Scale Entanglement Renormalization Ansatz (MERA): Disentangling unitary operations are interleaved with coarse-graining isometries to remove short-range entanglement at each scale, allowing the variational ansatz to capture long-range multi-scale correlations with depth scaling logarithmically in system size.

Coverage note — Omitted qualitative summaries of general open-source software libraries (such as Pennylane, Qiskit Aqua, and Yao) and high-level commentary on quantum compilation tools, as these represent reviews of external tooling rather than the paper's core conceptual contributions.

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Citation

MLA

Benedetti, M., et al. “Parameterized Quantum Circuits as Machine Learning Models”. Quantum Science and Technology, vol. 4, no. 4, 2019, p. 043001, https://doi.org/10.1088/2058-9565/ab4eb5.

APA

Benedetti, M., Lloyd, E., Sack, S., & Fiorentini, M. (2019). Parameterized quantum circuits as machine learning models. Quantum Science and Technology, 4(4), 043001. https://doi.org/10.1088/2058-9565/ab4eb5

Chicago

Benedetti, M., E. Lloyd, S. Sack, and M. Fiorentini. 2019. “Parameterized Quantum Circuits as Machine Learning Models”. Quantum Science and Technology 4 (4): 043001. https://doi.org/10.1088/2058-9565/ab4eb5.

Harvard

Benedetti, M. et al. (2019) “Parameterized quantum circuits as machine learning models”, Quantum Science and Technology, 4(4), p. 043001. Available at: https://doi.org/10.1088/2058-9565/ab4eb5.

Vancouver

1. Benedetti M, Lloyd E, Sack S, Fiorentini M (2019) Parameterized quantum circuits as machine learning models. Quantum Science and Technology 4:043001

BibTeX

@article{Benedetti_2019, title={Parameterized quantum circuits as machine learning models}, volume={4}, ISSN={2058-9565}, url={http://dx.doi.org/10.1088/2058-9565/ab4eb5}, DOI={10.1088/2058-9565/ab4eb5}, number={4}, journal={Quantum Science and Technology}, publisher={IOP Publishing}, author={Benedetti, Marcello and Lloyd, Erika and Sack, Stefan and Fiorentini, Mattia}, year={2019}, month=Nov, pages={043001} }

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Parameterized quantum circuits as machine learning models

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Knowls

Knowl 1 — Parameter-Shift Rule for Analytical Gradient Estimation

Knowl 2 — Hybrid Quantum-Classical Machine Learning Architecture with Parameterized Quantum Circuits

Knowl 3 — Quantum Circuit Born Machine

Knowl 4 — Quantum Kernel Estimator and Variational Quantum Model Paradigms

Knowl 5 — Query Complexity of Analytical Gradients versus Finite Differences in Circuit Learning

Knowl 6 — Hadamard Test for Variational Circuit Gradient Estimation

Knowl 7 — Barren Plateaus in Parameterized Quantum Circuit Optimization Landscapes

Knowl 8 — Quantum Generative Adversarial Networks

Knowl 9 — Quantum Autoencoder for State Compression

Knowl 10 — Physical Implementations of Parameterized Quantum Circuit ML Models

Knowl 11 — Data Encoding Strategies for Parameterized Quantum Circuits

Knowl 12 — Tensor Network-Inspired Parameterized Quantum Circuit Architectures

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