Parameterised Quantum Circuits

• Parameterised Quantum Circuits are quantum architectures where gate parameters, often real-valued angles, are updated by a classical optimizer, enabling hybrid quantum-classical algorithms.
• Their structure combines parameterised rotations with fixed entangling gates, mapping to tensor networks and supporting universal function approximation and advanced generative modeling.
• Applications include variational algorithms, quantum machine learning, and Bayesian extensions, with ongoing research addressing optimization challenges such as barren plateaus.

A parameterised quantum circuit (PQC) is a quantum circuit architecture in which certain gates are specified by classical parameters, typically real-valued angles that are updated by a classical optimizer. PQCs are central to hybrid quantum-classical algorithms in the Noisy Intermediate-Scale Quantum (NISQ) era, especially in variational quantum algorithms (VQAs), quantum machine learning, and advanced generative modeling. PQCs are formally defined as compositions of parameterised rotations and entangling gates, with a classical–quantum feedback loop that enables dynamic adjustment of the parameters based on observed measurement outcomes.

1. Mathematical Structure and Expressive Power

A parameterised quantum circuit U(θ) typically takes the general form: $U(\vec{\theta}) = \prod_{j=1}^{L} U_j(\theta_j)$ where each $U_j(\theta_j)$ is a gate parametrised by $\theta_j$ (for example, a rotation $R_y(\theta_j)$ or a phase gate $Z(\theta_j)$). The circuit acts on an initial state (usually $|0\rangle^{\otimes n}$) and may also include non-parameterised (fixed) Clifford or entangling gates such as CNOT or CZ.

Expressive power of PQCs is rigorously differentiated from classical neural models. In generative modeling, multilayer PQCs (“MPQCs”) with polynomially many gates can efficiently represent distributions far beyond the reach of restricted Boltzmann machines (RBMs) or even deep Boltzmann machines (DBMs), under standard complexity assumptions (i.e., unless the polynomial hierarchy collapses) (Du et al., 2018). Entanglement scaling is key: MPQCs support states that satisfy a volume law for entanglement entropy, as opposed to area-law-constrained classical tensor networks. Theoretical analysis maps PQCs to tensor networks (e.g., matrix product states and string bond states), providing formal proofs for this expressive hierarchy. PQCs can also simulate instantaneous quantum polynomial-time circuits (IQP), further cementing their hardness for classical simulation.

2. Bayesian and Ancilla-Driven Extensions

In standard PQCs, the model directly defines a parameterised distribution over the amplitudes of quantum states, but assigning explicit control over latent or prior information is challenging. The Bayesian quantum circuit (BQC) paradigm (Du et al., 2018, Du et al., 2018) introduces a blockwise separation: $p(x) = \sum_\lambda p(x|\lambda) p(\lambda)$ Ancillary qubits (representing latent variables $\lambda$) are prepared in superpositions encoding the prior $p(\lambda)$: $$|\psi_{\text{anc}}\rangle = \sum_\lambda \sqrt{p(\lambda)}|\lambda\rangle$$ Conditional unitary blocks $U(\theta_\lambda)$ prepare $p(x|\lambda)$. This architecture decouples generation of global modes (via $p(\lambda)$) and local variation (via $p(x|\lambda)$).

This design overcomes foundational generative modeling bottlenecks:

• Mode contraction: Maintaining distinct modes by explicit latent-variable control.
• Unintended sample generation: Bayesian structure ensures generated samples are weighted by a principled prior.
• Direct and interpretable sampling: Sampling $\lambda$ then $x$ allows target distribution realization.

Numerical experiments (on the Rigetti Forest platform) demonstrate BQC’s ability to accurately model multimodal distributions, reduce mode collapse, and perform robust semi-supervised learning (Du et al., 2018, Du et al., 2018).

3. Quantum Geometry and Capacity

The quantum geometry of PQC parameter space is defined by the quantum Fisher information matrix (QFI or QFIM): $$\mathcal{F}_{ij}(\vec{\theta}) = \mathrm{Re}\left[\langle \partial_i \psi | \partial_j \psi \rangle - \langle \partial_i \psi | \psi \rangle\langle \psi | \partial_j \psi \rangle \right]$$ The effective quantum dimension $G_C(\vec{\theta})$ counts the number of independent state-space directions accessible by infinitesimal parameter updates, and is given by the rank of the QFI. Gate choice critically impacts capacity; for example, circuits with $\sqrt{\mathrm{iSWAP}}$ gates exhibit larger $G_C$ and favorable scaling compared to circuits using only CPHASE (Haug et al., 2021).

The quantum natural gradient (QNG): $$\nabla_Q E = \mathcal{F}^{-1} \frac{\partial E}{\partial \vec{\theta}}$$ adapts gradient directions to the true geometry. In shallow circuits, QNG steps can be much larger (and better-directed) than standard gradients. However, both can suffer from vanishing gradients (barren plateaus) as qubit number or layer depth increases.

The Natural PQC (NPQC) class (Haug et al., 2021) achieves Euclidean geometry at initialization: the QFI is proportional to the identity, and the circuit allows optimal, analytically computable learning rates and rapid early-stage convergence. The NPQC also enables parallel parameter estimation via basis measurements and achieves optimal sensitivity bound by the quantum Cramér–Rao bound for multi-parameter estimation.

4. Optimization, Initialization, and Scalability

Parameter initialization and circuit design are crucial to balancing expressiveness and trainability. Too many random layers push the circuit into regions with “barren plateaus” (vanishing gradients), while too little randomness reduces capacity (Haug et al., 2021). There exists a tunable regime—intermediate between trivial (all-zero) and fully random initialization—that preserves both high effective dimension and manageable gradient magnitude.

Redundant parameter pruning is recommended: by analyzing the QFI spectrum, one may iteratively remove rotation parameters along “null” eigenvector directions until only those contributing to the model state remain (Haug et al., 2021). Efficient algorithms for parameter count minimization are known only for PQCs composed of Clifford and unique parameterised phase gates (using ZX-calculus diagrammatic rewrite rules, “phase fusion”), while the general minimization problem is NP-hard (Wetering et al., 23 Jan 2024).

Pulse-level control and careful engineering of entanglers—notably CR(π/4) gates—can halve circuit duration and enhance trainability without substantial loss of expressibility (Ibrahim et al., 2022). Such optimizations are critical to mitigate noise and decoherence on present-day hardware.

Backpropagation (gradient estimation) is a recognized bottleneck: for generic circuits, parameter-shift methods scale linearly (or worse) in the number of parameters. For commuting-generator circuits, all gradients can be recovered from a single measurement in an appropriately diagonalized basis, yielding backpropagation-like scaling and reducing quantum resource requirements by orders of magnitude (Bowles et al., 2023).

5. Universality and Function Approximation

PQCs can approximate any function in $C^0$ (continuous functions), $L^p$ spaces, and Sobolev spaces $H^k$ for suitable architecture and normalization (Manzano et al., 2023). For generative modeling, they are universal in a strong sense: expectation value sampling models can approximate any bounded continuous multivariate distribution if the observables and state structure are appropriately chosen (Barthe et al., 15 Feb 2024). This universality holds via product-encoding (one qubit per output) or dense-encoding (logarithmic qubit count but exponentially large-norm observables), subject to trade-offs in resource scaling and measurement noise.

The universal approximation principle is underpinned by the fact that single-qubit circuits can approximate arbitrary continuous functions through amplitude encoding and observable measurement. Circuit-level universality emerges from stacking or entangling such local modules.

6. Applications and Advanced Workflows

PQCs underpin a wide range of quantum algorithms and machine learning protocols:

• In VQE and QAOA, PQCs serve as trainable ansätze whose parameters are optimized to minimize expectation values for chemistry, optimization, or spin-model Hamiltonians (Wetering et al., 23 Jan 2024, Ibrahim et al., 2022).
• In quantum generative modeling, both discrete (bars-and-stripes, BAS) and continuous-data distributions can be captured and sampled with high statistical fidelity—enabling tasks from semi-supervised learning to Bayesian inference (Du et al., 2018, Du et al., 2018, Barthe et al., 15 Feb 2024).
• Hybrid architectures integrate PQCs as quantum layers within classical deep-learning pipelines (e.g., in emotion recognition), delivering reduced parameter counts and improved generalization compared to classical-only networks (Rajapakshe et al., 21 Jan 2025).
• In natural language processing, full-document PQCs are automatically synthesized from syntactic diagrams (pregroup grammars) via compositional (category-theoretic) frameworks, as in DisCoCirc, preserving linguistic interpretability and enabling efficient NLP workflows on quantum hardware (Krawchuk et al., 19 May 2025).

Advanced diagrammatic and decision-diagram approaches are increasingly used for analysis, verification, and optimization of PQCs. Extensions of ZX-calculus now handle linear combinations (with explicit scalars), enabling systematic reasoning about observable expectation values and circuit identities (Stollenwerk et al., 2022, Hong et al., 29 Apr 2024). Symbolic tensor decision diagrams (S-TDDs), which operate over rings of trigonometric polynomials, yield canonical, efficiently minimizable representations of parameterised circuits suitable for automated equivalence verification.

7. Open Problems and Future Directions

• Gradient vanishing and barren plateaus remain a major challenge. Recent conic extensions—leveraging non-unitary updates and mid-circuit measurement with ancilla (LCU techniques)—allow optimization to escape flat regions via generalized eigenvalue solutions, substantially improving optimization landscapes in VQAs (Binkowski et al., 2023).
• Bayesian learning frameworks for circuit training (including Laplace priors and Langevin posterior sampling) promote robustness, efficient circuit dimension reduction, and effective avoidance of local minima in NISQ hardware deployments (Duffield et al., 2022).
• Resource-efficient circuit design is increasingly formalized around optimal trade-offs among expressibility, entanglement, and topological connectivity; highly connected topologies (linear, ring) reach Haar-random complexity more quickly than less connected or random circuit architectures, as measured by expressibility, entanglement, and majorization criteria (Correr et al., 29 May 2024).
• Software and benchmarking: Open-source modular generators (such as Q-gen) now provide parameterised circuit datasets for dozens of quantum algorithms, supporting both benchmarking and machine learning applications by standardizing input–output pairs for classifier training and circuit analysis (Mao et al., 26 Jul 2024).
• Equivalence checking and compilation tools bridge the gap between high-level PQC design and physical implementation, ensuring correctness even in the presence of compiler optimizations or hardware constraints (Hong et al., 29 Apr 2024, Wetering et al., 23 Jan 2024).

This technical landscape establishes parameterised quantum circuits as the core primitives for realising quantum advantage in NISQ-era computation, quantum-enhanced machine learning, and scalable quantum algorithm development.