Variational Quantum Circuits (VQCs)

• Variational Quantum Circuits (VQCs) are hybrid quantum-classical models characterized by parameterized quantum gates optimized via classical algorithms.
• They employ structured ansatz with rotations and entangling gates to achieve exponential Hilbert space representation using fewer parameters than classical networks.
• Optimized with gradient-based methods and strategies to mitigate barren plateaus, VQCs enable robust performance in tasks like simulation, classification, and reinforcement learning.

Variational Quantum Circuits (VQCs) are hybrid quantum-classical computational models in which the parameters of quantum gates are optimized, typically via classical algorithms, to approximate functions or prepare states relevant to tasks such as machine learning, optimization, and quantum simulation. VQCs underpin many NISQ-era quantum algorithms by combining quantum sampling or transformation with efficient classical parameter updates, enabling expressive modeling with resource constraints characteristic of near-term hardware.

1. Circuit Structure, Expressivity, and Parameterization

A VQC typically consists of a sequence of parameterized quantum gates—single-qubit rotations (such as $R_x(\theta)$, $R_y(\theta)$, $R_z(\theta)$) and multi-qubit entangling gates (like CNOT or CZ)—applied to an initial state. The generic form is:

$U(\theta) = \prod_{l=1}^L U_l(\theta_l) W_l$

where $U_l$ are parameterized layers (possibly including general unitaries $$R(\alpha,\beta,\gamma) = R_z(\alpha) R_y(\beta) R_z(\gamma)$$), and $W_l$ are fixed gates (often entanglers) (Chen et al., 2019).

In contrast to classical neural networks that require large numbers of weights (often $10^3$–$10^6$ for competitive deep learning tasks), representative quantum circuits can offer an exponential Hilbert space using only $n$ qubits, and thus require orders of magnitude fewer parameters for comparable expressive power—e.g., only 28 parameters in a VQC may suffice for effective policy approximation in a 16-state frozen lake environment (Chen et al., 2019), and quantum classifiers using amplitude encoding for $M$-dimensional data require only $n=\lceil \log_2 M \rceil$ qubits (Miyahara et al., 2021).

The expressivity of VQCs is determined both by the choice of ansatz (i.e., template circuit design) and by the encoding of inputs. General frameworks such as the unitary kernel method (UKM) allow direct optimization over the space of all unitaries, yielding a performance upper bound relative to ansatz-constrained models (Miyahara et al., 2021).

2. Data Encoding and Quantum Information Representation

Efficient input encoding is central. Quantum data encoding schemes include:

• Computational Basis Encoding: Binary representation of discrete states, where each classical bit $b_i$ is mapped onto single-qubit rotations with $\theta_i = \pi b_i$ and $\varphi_i = \pi b_i$, as in $R_x(\theta_i) R_z(\varphi_i) \vert 0\rangle$, achieving sparse, parameter-efficient state mapping (Chen et al., 2019).
• Amplitude Encoding: For $M$-dimensional data, prepare a normalized superposition directly reflecting the classical vector in the amplitude of basis states,

$$\vert \psi^{(\mathrm{in})}(x) \rangle = \sum_{j=1}^{2^n} \tilde{x}_j \vert j \rangle$$

which achieves logarithmic scaling in qubit count (Miyahara et al., 2021).

By mapping classical inputs into exponentially large Hilbert spaces with a linear or logarithmic number of qubits, VQCs achieve compression and flexibility unmatched by classical parameterizations.

3. Training, Differentiability, and Optimization

VQCs are typically trained via gradient-based optimization. The measured value of an observable $O$ after quantum evolution defines an objective function:

$$f(\theta) = \mathrm{tr}[ O U(\theta) \vert \psi_0\rangle \langle \psi_0 \vert U(\theta)^\dagger ]$$

with optimization aiming to minimize a loss such as mean-square error (MSE) between output and target (Chen et al., 2019).

Quantum gradients are obtained via parameter-shift rules, which respect quantum no-cloning constraints. For single-parameter gates:

$$\frac{\partial}{\partial \theta_i} f(\theta) = \frac{1}{2}[f(\theta_i + s) - f(\theta_i - s)]$$

with $s = \pi/2$ for Pauli rotations (Huembeli et al., 2020).

To address the challenges from non-cloning and control flow (e.g., measurement-based branches), formal frameworks based on additive quantum programs use ancilla qubits and controlled rotation gadgets so that auto-differentiation can be extended to imperative quantum programs, including those with conditionals and bounded loops (Zhu et al., 2020).

State-of-the-art optimization strategies for VQCs include:

• Use of Hessian information to dynamically adapt learning rates (e.g., $\eta = 1/\lambda_{\max}$ where $\lambda_{\max}$ is the largest Hessian eigenvalue), thus accelerating convergence and addressing flat regions known as barren plateaus (Huembeli et al., 2020).
• Regularization mechanisms such as data-informed parameter initialization and Gaussian noise diffusion to preserve gradient variance and avoid saddle points (Zhuang et al., 2 May 2024).
• Evolutionary, gradient-free methods—mutation-only or recombination—are especially useful in reinforcement learning settings to avoid barren plateaus and vanishing gradients (Kölle et al., 30 Jul 2024).

4. Barren Plateaus: Optimization Obstruction and Remediation

Barren plateaus (BPs) are regimes where the gradient variance decreases exponentially with the number of qubits or circuit depth, preventing effective training:

$$\operatorname{Var}[\partial C] \leq F(N), \quad F(N) = o(1/b^N),\; b>1$$

(Cunningham et al., 25 Jul 2024). Causes of BPs include excessive circuit expressivity, too-randomized initializations (e.g., compilation to unitary 2-designs), and nonlocal cost observables.

Mitigation strategies, classified into five categories (Cunningham et al., 25 Jul 2024), include:

1. Initialization-based: Set parameter distributions or circuit segments (e.g., to identity) to yield nonzero starting gradients.
2. Optimization-based: Employ layer-wise or local-batch training, adaptive learning rates, or gradient-free methods to navigate flat landscapes.
3. Model-based: Architectures that suppress expressibility (e.g., tree tensor networks, residual skip connections), or restrict parameter domains.
4. Regularization-based: Penalize entanglement or use noise-injection (e.g., Gaussian noise diffusion after updates) to encourage exploration.
5. Measurement-based: Post-selection or additional metrics that amplify observability of gradient changes.

For continuous-variable (CV) VQCs, barren plateau severity depends non-trivially on the number of bosonic modes ($M$) and circuit energy ($E$); gradient variance decays as $1/E^{M\nu}$, with $\nu=1$ (shallow) or $2$ (deep) (Zhang et al., 2023). Proper matching of circuit initialization energy to the target state is an effective mitigation.

5. Practical Realizations and Applications

VQCs have been deployed as policy approximators or function approximators in reinforcement learning, regression, and classification. Key findings include:

• Deep Q-learning routines (e.g., experience replay, target networks) can be mapped to VQC-based architectures with the quantum circuit outputting Q-values via qubit expectation values, using only a fraction of classical model parameters (Chen et al., 2019).
• Proof-of-principle experiments in discrete environments (frozen lake), and resource allocation (cognitive radio channels), show that VQCs realize near-optimal policies with dramatically fewer parameters and robust noise tolerance, verifiable on real NISQ hardware (Chen et al., 2019).
• Quantum phase estimation (QPE) subroutines may be replaced by trained VQCs of much shorter depth, substantially reducing circuit noise and rendering quantum algorithms on real hardware more feasible (Liu et al., 2023).
• Hybrid quantum-classical models, especially for scientific prediction tasks such as density functional theory for liquid silicon, benefit from quantum-enhanced nonlinear readout modules; VQC-augmented models achieve state-of-the-art accuracy for molecular dynamics simulations (Willow et al., 6 Aug 2025).
• VQC parameter efficiency is consistently demonstrated across benchmarks: e.g., for supervised and RL tasks, VQCs match classical neural networks' performance using only $\sim 37\%$ of the parameters (Kölle et al., 9 Apr 2025).

6. Model Compression, Verification, and Security

Advancements in model compression, verification, and watermarking have expanded VQC practicality:

• Lottery Ticket Hypothesis (LTH): VQCs possess sparse “winning ticket” subnetworks—reachable via magnitude-based or evolutionary pruning—that match the performance of the full model while using as little as 26% (weak LTH) or 45% (strong LTH, obtained without training) of the original parameters. Winning tickets reduce susceptibility to barren plateaus and demand for quantum hardware resources (Kölle et al., 14 Sep 2025).
• Formal Verification: Abstract interpretation frameworks for VQCs (using interval-based reachability analyses) enable the certification of classifier robustness to adversarial perturbations. This method rigorously bounds the maximum input perturbation $\varepsilon$ such that model predictions remain invariant, providing formal guarantees for quantum machine learning systems (Assolini et al., 14 Jul 2025).
• Intellectual Property Protection: The BVQC watermarking scheme embeds a “backdoor” watermark as a loss constraint, ensuring ownership can be proven with negligible impact on regular performance and strong resilience to circuit recompilation (Chu et al., 3 Aug 2025).

7. Outlook and Evolution

Strategies such as quality diversity (QD) optimization—balancing circuit objective, sparsity, and gate diversity—have accelerated the discovery of robust, expressive VQC architectures, outperforming gradient-based QAOA and evolutionary QNEAT in combinatorial optimization benchmarks (Zorn et al., 11 Apr 2025). As hardware improves, the interplay of architectural advances, scalable training, and formal methods positions VQCs as a leading tool for quantum machine learning, offering unique advantages in parameter efficiency, expressivity, and problem generalizability.

These developments collectively underscore that VQCs, leveraging judicious architectural design, encoding, and optimization, yield parameter-efficient, scalable, and robust quantum models, conducive to both near-term and longer-term quantum computational advances.