Parameterized Quantum Circuits

• Parameterized Quantum Circuits (PQCs) are quantum circuits with tunable gates that enable variational algorithms for machine learning, optimization, and simulation.
• They utilize sequential layers of parameterized rotations and entangling gates to generate expressive quantum states, akin to neural networks in classical computing.
• Hybrid quantum-classical optimization, using gradient-based and gradient-free methods, robustly trains PQCs even in noisy, resource-constrained environments.

A parameterized quantum circuit (PQC) is a quantum circuit composed of both fixed and parameter-dependent (i.e., tunable) unitary gates, enabling variational approaches to quantum computing. PQCs serve as the foundational model for hybrid quantum-classical algorithms on noisy intermediate-scale quantum (NISQ) devices and are instrumental in quantum machine learning, quantum simulation, optimization, and quantum algorithm design. Their centrality lies in their ability to create highly expressive quantum states via adjustable parameters, effectively making them the quantum analog of neural networks for a range of data-centric and generative tasks.

1. Formal Structure and Expressive Power

A PQC is typically described by applying a parameterized unitary $U(\theta)$ to an initial state $|0\rangle^{\otimes n}$, producing a variational quantum state $$|\Psi(\theta)\rangle = U(\theta)|0\rangle^{\otimes n}$$. Here, the vector $\theta \in \mathbb{R}^m$ specifies the continuous gate parameters. The mathematical output of interest is usually the measurement outcome statistics, expectation values of observables, or the full probability distribution over computational basis states: $$q(X = \mathbf{x}) = |\langle \mathbf{x} | \Psi(\theta) \rangle|^2.$$ Certain PQCs (for example, multilayer PQCs) have provable expressive advantages over classical neural network generative models, such as deep Boltzmann machines, especially when simulating distributions with volume-law entanglement; this is only efficiently achievable by classical models if the polynomial hierarchy collapses (Du et al., 2018).

When equipped with a polynomially scaling number of blocks parameterized by $O(\text{poly}(n))$, MPQCs can efficiently represent probability distributions that are intractable for classical networks, including those generated by instantaneous quantum polynomial (IQP) circuits. Ancillary qubits and post-selection further enhance the expressive power, supporting models (such as Bayesian Quantum Circuits) capable of encoding and inferring rich probabilistic structures (Du et al., 2018).

2. Design Patterns and Circuit Architecture

PQCs are constructed from sequential layers comprising single-qubit rotations (typically parameterized as $U_j(\theta_j) = \exp(-i \theta_j P_j / 2)$, with $P_j$ a Pauli string), fixed gates, and entangling operations (such as CNOT, CZ). The generic design may include:

• Data Encoding Blocks ($U_{\text{enc}}(x)$): Map classical data vectors to quantum states, e.g., via feature-wise $R_y(\phi(x_j))$.
• Variational Ansatz Blocks: Repeated layers of parameterized single-qubit gates and entangling gates, sometimes arranged as ring or all-to-all connectivity (Jones et al., 10 Jul 2025).
• Ancillary Qubits and Controls: Used for Bayesian modeling, prior encoding, and post-selection (Du et al., 2018, Du et al., 2018).

Hybrid frameworks leverage classical optimization to update circuit parameters based on quantum measurements (Benedetti et al., 2019). Advanced variants such as Fraxis use free-axis selection for single-qubit gates, optimizing both the rotation axis and angle in a continuous parameter space, with the optimal axis determined via eigenvalue problems or linear systems (Watanabe et al., 2021). The use of pulse-level control for entangling gates (custom cross-resonance pulses) is shown to significantly reduce circuit execution time and enhance trainability on actual devices (Ibrahim et al., 2022).

3. Training and Optimization Algorithms

Training PQCs entails optimizing a cost function—often the expectation value of a Hamiltonian or a supervised/regression objective—over the parameter space. Common methodologies include:

• Gradient-Based Optimization: Classical optimizers (Adam, stochastic GD) leveraging gradients via the parameter-shift rule:

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

• Gradient-Free Methods: Sequential single-qubit optimizers (e.g., Rotosolve, Fraxis) and hybrid algorithms that switch strategies based on early-stopping or convergence criteria, allowing for resource-efficient and robust optimization in the presence of barren plateaus and noise (Pankkonen et al., 9 Oct 2025).
• Information-Theoretic Barriers: The sample efficiency of PQC optimization can be limited; a single binary outcome from a circuit run provides exponentially little information relative to the number of qubits, necessitating aggregation over many runs or novel gradient strategies, including quantum gradient techniques (Dolzhkov et al., 2019, Li et al., 30 Sep 2024).
• Robust and Noise-Aware Training: Incorporation of realistic hardware noise models—including amplitude damping, dephasing, and depolarization—directly into the objective function can result in up to 42% fidelity gain in practical NISQ deployment (Alam et al., 2019). Bayesian optimization methods (e.g., BPQCO) can efficiently explore the discrete and categorical PQC design space conditioned on target hardware characteristics, supporting robust circuit discovery under noisy or resource-constrained environments (Benítez-Buenache et al., 17 Apr 2024).

4. Applications Across Quantum Computing

PQCs underpin a wide range of NISQ-era quantum algorithms:

• Quantum Machine Learning: Serve as both generative models (quantum circuit Born machines, quantum GANs) and discriminative models (quantum classifiers, kernel methods), leveraging hybrid quantum–classical loops for optimization (Benedetti et al., 2019).
• Quantum Generative Modeling: PQCs learn $p(x)$ as the measurement distribution; Bayesian Quantum Circuits enable explicit mixture modeling with priors represented by ancillary qubits, addressing issues such as mode contraction, spurious samples, and efficient Bayesian updating (Du et al., 2018).
• Quantum Chemistry and Physics: PQCs, used as ansätze in variational quantum eigensolvers (VQE), accurately represent ground and excited states of complex Hamiltonians, including the toric code in topologically ordered matter (Sun et al., 2022, Jones et al., 10 Jul 2025).
• Quantum Optimization and Reinforcement Learning: PQCs are exploited for combinatorial optimization (QAOA) and as the neural module in quantum reinforcement learning. Automated search (e.g., QRL-NAS) adaptively optimizes PQC structure, outperforming fixed designs (Son et al., 1 Jul 2025).

5. Approximation Theory and Generalization Capabilities

The approximation capacity of PQCs is mathematically established for broad function classes:

• Universal Approximation: PQCs with sufficient depth can approximate continuous functions, $L^p$-integrable functions, and $H^k$ Sobolev spaces, with explicit error bounds as a function of qubits, layers, and parameters (Manzano et al., 2023, Yu et al., 2023).
• Explicit Circuit Constructions: Data re-uploading and localized Taylor approximations can be encoded within PQC architectures, yielding non-asymptotic approximation error bounds. For suitable classes of functions, PQCs can require exponentially fewer parameters than comparable deep ReLU neural networks to achieve a fixed error (Yu et al., 2023).
• Generalization: Empirical risk minimization with Sobolev-type loss functions (including derivative information) yields generalization guarantees in the continuum limit, providing a theoretical foundation for PQCs in scientific and engineering domains (e.g., quantum PDE solvers) (Manzano et al., 2023).

6. Circuit Structure, Expressibility, and Resource Scaling

The expressibility of a PQC—the circuit's ability to uniformly cover Hilbert space—is tightly connected to its structure:

• Gate-Type Impact: Expressibility is enhanced by increasing the number of RX and RY gates, with CNOT count requiring careful tuning to avoid reduced expressibility and greater optimization difficulty. Beyond a certain layer/depth, expressibility saturates, exhibiting diminishing returns with continued gate addition (Liu et al., 2 Aug 2024).
• Resource Trade-offs: Circuit width (qubit number), depth (number of layers), and parameter count are explicitly linked to the functional approximation error. For fixed function smoothness, PQC constructions can achieve the same accuracy with exponentially fewer parameters compared to deep neural networks in high dimensions (Yu et al., 2023).
• Hardware-Efficient and Noise-Resilient Designs: Customization of gate implementations at the pulse level and adaptive circuit design (e.g., BPQCO, hybrid optimizers, gate-freezing strategies) lead to significant gains in noise robustness and resource efficiency on actual hardware (Ibrahim et al., 2022, Benítez-Buenache et al., 17 Apr 2024, Pankkonen et al., 10 Jul 2025, Pankkonen et al., 9 Oct 2025).

7. Verification, Analysis, and Practical Considerations

The practical deployment of PQCs requires methods for verification, scalability, and adaptation:

• Equivalence Checking: Canonical tensor decision diagram representations (S-TDD) for PQCs enable efficient, parameter-independent equivalence checking of compiled vs. ideal circuits, outperforming ZX-calculus-based or SMT approaches especially in large circuits (Hong et al., 29 Apr 2024).
• Loss Landscapes and Mode Connectivity: PQCs exhibit loss landscapes with multiple low-loss valleys (mode connectivity), allowing for flexible transitions between solutions and improved noise resilience; optimizers leveraging geometric or quantum natural gradients further facilitate convergence (Hamilton et al., 2021).
• Adaptive and Automated Design: NAS approaches integrated in reinforcement learning or general PQC architecture search (BPQCO, QRL-NAS) automate circuit topology optimization conditioned on both the data and hardware specifics (Benítez-Buenache et al., 17 Apr 2024, Son et al., 1 Jul 2025).
• Benchmarks and Quantum Advantage: PQCs have established empirical superiority in representative quantum generative and classification benchmarks (e.g., Bars-and-Stripes dataset, quantum chemistry testbeds) and theoretically proven complexity-theoretic advantages unless unlikely polynomial hierarchy collapses occur (Du et al., 2018, Du et al., 2018, Jones et al., 10 Jul 2025).

Summary Table: PQC Key Properties

Property	Description/Implication	Reference
Expressive Power	Surpasses classical neural networks for generative modeling with volume-law entanglement	(Du et al., 2018)
Universal Approx.	Approximates $C^0$, $L^p$, $H^k$ functions, with explicit qubit and depth scaling	(Manzano et al., 2023, Yu et al., 2023)
Gate-Type Effects	RX/RY increase expressibility; excessive CNOTs can hinder it; expressibility saturates	(Liu et al., 2 Aug 2024)
Training Limitation	Sample queries convey exponentially less information than evaluation in PQC training	(Dolzhkov et al., 2019)
Noise/Hardware	BPQCO, pulse-optimized, and gate-freezing PQCs enhance robustness and efficiency	(Ibrahim et al., 2022, Benítez-Buenache et al., 17 Apr 2024, Pankkonen et al., 10 Jul 2025)

Parameterised quantum circuits provide a flexible and theoretically robust modeling and computational paradigm for quantum algorithms, machine learning, and simulation, with current advances focused on optimizing their architecture, expressibility, noise resilience, and verifiability for scalable near- and long-term quantum applications.