Parameterized Quantum Circuits (PQC)

• Parameterized Quantum Circuits are quantum architectures with tunable parameters that integrate data encoding and variational ansatz layers.
• They use hybrid quantum–classical optimization methods, including gradient-based and gradient-free approaches, to efficiently train parameters.
• PQC applications span quantum machine learning, chemistry simulations, and combinatorial optimization by balancing expressibility and hardware limitations.

A parameterized quantum circuit (PQC) is a quantum circuit architecture whose gates are controlled by a set of tunable, continuous parameters. PQCs form the backbone of hybrid quantum–classical algorithms—particularly variational quantum algorithms (VQAs) and quantum machine learning (QML) models—where quantum operations are interleaved with classical training and optimization routines. PQCs encompass both data-encoding layers, which map classical information into quantum states, and variational “ansatz” layers, which are iteratively adjusted to solve a range of tasks from generative modeling to combinatorial optimization, regression, and quantum chemistry simulations.

1. Fundamental Structure and Principles

A PQC is typically constructed as a sequence of parameterized unitary gates, interleaved with entangling operations. The overall action of a PQC with $N$ qubits can be represented as

$$U(\bm{\theta}) = U_L(\bm{\theta}_L)\cdots U_1(\bm{\theta}_1)U_0(\bm{\theta}_0)$$

where each $U_\ell(\bm{\theta}_\ell)$ consists of a layer of parameterized single-qubit gates—often of the form $R_{X/Y/Z}(\theta) = \exp(-i\,\theta\,P/2)$, with $P$ a Pauli operator—followed by an entangling layer (e.g., CNOT or CZ gates). The parameter vector $\bm{\theta} = (\theta_1,\dots,\theta_M)$ is typically optimized during training.

A hybrid learning algorithm prepares quantum states

$$|\psi(\bm{x}, \bm{\theta})\rangle = U_{\text{var}}(\bm{\theta}) U_{\phi}(\bm{x}) |0\rangle$$

where $U_{\phi}(\bm{x})$ is a data-encoding unitary (feature map) and $U_{\text{var}}(\bm{\theta})$ is the variational circuit. Outputs are derived by measuring suitable observables $M$: $$\langle M \rangle = \langle \psi(\bm{x}, \bm{\theta}) | M | \psi(\bm{x}, \bm{\theta}) \rangle$$ which, after post-processing, yield the prediction or generative output.

2. Optimization and Training Methodologies

Optimization of PQCs proceeds via hybrid quantum–classical loops. Loss functions $L(\bm{\theta})$—such as mean squared error for regression or Kullback–Leibler divergence for generative modeling—are minimized by updating parameters according to classical routines. Three main optimization methodologies are prominent:

• Gradient-Based Methods: Parameter-shift rule allows analytic estimation of derivatives:

$$\frac{\partial}{\partial\theta_j}\langle M \rangle = \frac{\langle M \rangle_{\theta_j + \pi/2} - \langle M \rangle_{\theta_j - \pi/2}}{2}$$

enabling classical optimizers like Adam or stochastic gradient descent (SGD) to be used.

• Gradient-Free Methods: Sequential optimizers such as Rotosolve, Free-Axis Selection (Fraxis), and Free-Quaternion Selection (FQS) operate by sweeping over individual gates, updating either their parameters by closed-form solutions (e.g., fitting to sinusoidal expectation value forms as in Rotosolve) or optimizing over entire rotation axes/quaternions via eigenvalue problems (Watanabe et al., 2021, Pankkonen et al., 10 Jul 2025).
• Hybrid Optimization Schemes: Recent advances combine the strengths of different optimizers, initiating with fast single-parameter optimizers (e.g., Rotosolve), then switching—using cost function-based triggers—to more expressive methods (e.g., FQS). Criteria for switching include early stopping based on cost improvement thresholds or running averages (Pankkonen et al., 9 Oct 2025).

Gate-freezing strategies, which temporarily halt updates to parameters that change little between iterations, further improve resource allocation and convergence (Pankkonen et al., 10 Jul 2025).

3. Expressibility, Entanglement, and Circuit Design

The expressibility of a PQC quantifies its ability to cover the Hilbert space of quantum states. This is formalized by measuring the divergence (typically KL divergence) between the distribution of fidelities of states generated by the PQC and the Haar distribution: $$\text{Expr} = D_{\text{KL}}(P_{\mathcal{C}}(F) \parallel P_{\text{Haar}}(F))$$ Low divergence indicates high expressibility, which is necessary (but not sufficient) for universality in variational and ML tasks (Liu et al., 2 Aug 2024, Azad et al., 2022).

Gate composition crucially determines expressibility. Statistical and machine learning analyses consistently show that:

• Single-qubit rotational gates (especially RX and RY) positively enhance expressibility.
• CNOT gates and other entanglers are necessary for introducing non-trivial quantum correlations, but excessive use can decrease expressibility.
• Expressibility Saturation: As the number of layers/gates increases, expressibility saturates—additional layers confer marginal benefit beyond some threshold (Liu et al., 2 Aug 2024).

Selection of data-encoding strategies and circuit architectures using automated searches (e.g., genetic algorithms (Ding et al., 2022), Bayesian optimization (Benítez-Buenache et al., 17 Apr 2024)) can lead to architectures that balance expressibility, trainability, and robustness.

4. Bayesian and Ancilla-Enhanced Circuit Frameworks

Standard PQCs can be enhanced by incorporating ancillary qubits, enabling more flexible architectures:

• Bayesian Quantum Circuits (BQC): By adding ancillary qubits to encode explicit prior distributions, BQC architectures can realize generative models that learn both the likelihood $p(x|\lambda, \theta)$ and the prior $p(\lambda|\gamma)$, overcoming typical issues such as mode contraction and enhancing fidelity in generative and semi-supervised learning tasks (Du et al., 2018, Du et al., 2018).
• Ancilla-Driven and Post-Selection-Enriched Circuits: These architectures allow simulation of post-IQP circuits, expanding expressive power beyond that of multilayer PQCs (MPQCs) alone. The ability to represent distributions not efficiently simulable by classical neural networks (unless the polynomial hierarchy collapses) is formally established via tensor network and complexity-theoretic connections (Du et al., 2018).

5. Robustness, Verification, and Hardware Adaptation

Practical deployment of PQCs on NISQ processors entails dealing with noise, decoherence, and device-specific constraints:

• Noise-Aware and Hardware-Adapted Training: Incorporating real device error models (T$_1$, T$_2$, gate errors, connectivity) during PQC training produces circuits that retain high fidelity under temporal variations (Alam et al., 2019). Bayesian optimization frameworks (BPQCO) further tailor circuit architectures to hardware-specific transpilation and error profiles, either by online evaluation in noisy environments or via circuit complexity penalization (Benítez-Buenache et al., 17 Apr 2024).
• Pulse-Level PQC Design: Direct manipulation of control pulses for implementing two-qubit entanglers (e.g., cross-resonance) mitigates decoherence by reducing state preparation times while maintaining trainability, even if overall expressibility is reduced—often beneficial for avoiding barren plateaus (Ibrahim et al., 2022).
• Equivalence Checking: Efficient verification of compiled/optimized PQCs is essential. Canonical tensor decision diagrams (TDDs), extended to encode symbolic parameter dependence as trigonometric polynomials, enable scalable equivalence checking without requiring parameter instantiation, an advance for circuit compilation and error mitigation workflows (Hong et al., 29 Apr 2024).

6. Theoretical and Algorithmic Limits

PQC models have been proven universal approximators for functions in $L^p$, $C^0$, and Sobolev $H^k$ spaces, provided sufficient depth and expressive architectures. Data normalization (favoring inputs scaled to $[-\pi/2,\pi/2]$) and loss functions incorporating derivatives (Sobolev-inspired) play a critical role in approximation quality and generalization (Manzano et al., 2023). The information-theoretic analysis reveals a severe exponential bottleneck: training based on sample queries (i.e., real device shots) conveys exponentially little information as the parameter count increases, whereas oracle-like evaluation queries (unrealistic for real hardware) would provide full information in a single evaluation (Dolzhkov et al., 2019).

Recent advances leveraging quantum gradient descent directly in the Hilbert space, and enhanced circuit synthesis steps, address the exponential vanishing of classical parameter gradients, efficiently circumventing barren plateaus and adapting the circuit architecture dynamically (Li et al., 30 Sep 2024).

7. Applications and Emerging Directions

PQCs have demonstrated concrete value in:

• Quantum Machine Learning: Supervised learning (via variational classifiers, quantum kernel methods), generative modeling (QCBM, BQC), and reinforcement learning with evolved architectures (Ding et al., 2022, Du et al., 2018).
• Quantum Chemistry and Physics: VQE for ground state problems, adaptive loop gas circuits for topological phases, and learning molecular properties from compact feature encodings (Sun et al., 2022, Jones et al., 10 Jul 2025).
• Optimization and Combinatorics: Variational circuits for hard optimization problems, leveraging gradient-free and hybrid training routines for robust convergence in the presence of noise (Pankkonen et al., 9 Oct 2025, Pankkonen et al., 10 Jul 2025).

Open challenges include mitigating the impact of barren plateaus, developing further instance- and hardware-dependent circuit design methodologies, extending fast expressibility estimation techniques (GNN-based predictors) (Aktar et al., 2023, Aktar et al., 13 May 2024), and automating architecture selection via evolutionary and Bayesian search strategies. The interplay of expressibility, entanglement, and hardware-constrained trainability remains a central focus for achieving scalable quantum advantage in practical settings.