Parameterized Quantum Circuits Overview

• Parameterized quantum circuits are architectures that use tunable continuous parameters to optimize state preparation and quantum machine learning tasks.
• They exhibit high expressive power by efficiently representing quantum many-body states with volume-law entanglement, surpassing classical generative models.
• Advanced techniques like ancillary qubits, Bayesian learning, and parameter pruning enhance their trainability and practical performance in quantum simulations.

A parameterized quantum circuit (PQC) is a quantum circuit architecture in which a set of gates depends explicitly on continuous parameters, typically serving as the variational degrees of freedom for optimization in hybrid quantum-classical algorithms. PQCs are central to quantum machine learning, variational quantum eigensolvers, quantum approximate optimization, quantum generative modeling, and advanced state preparation. Their significance derives from their ability to efficiently prepare highly entangled quantum states, serve as quantum function approximators with tunable expressive power, and be systematically optimized for various computational tasks.

1. Expressive Power and Tensor Network Embeddings

Parameterizations of quantum circuits can dramatically exceed the expressive power of comparable classical neural networks for generative tasks. Specifically, multilayer PQCs (MPQCs) and their tensor network variants (TPQCs) are proven to efficiently represent quantum many-body states exhibiting volume-law entanglement, where the entanglement entropy

$\mathcal{S} \sim N$

scales linearly with the number of qubits $N$ (Du et al., 2018). In contrast, classical models, notably restricted Boltzmann machines (RBMs) with short-range connectivity, are limited to area-law entanglement. This quantum advantage is further formalized via tensor network theory—MPQCs can efficiently simulate matrix product states (MPS) with bond dimension $D$ by employing $O(\operatorname{poly}(\log D))$ circuit blocks, leveraging controlled entangling gates such as CNOTs and arbitrary single-qubit rotations to realize general local tensors.

Moreover, a correspondence is established between MPQCs and instantaneous quantum polynomial-time (IQP) circuits, with the result that any IQP circuit (with $O(\operatorname{poly}(N))$ commuting gates on $N$ qubits) can be simulated by MPQCs with only $O(\operatorname{poly}(N))$ blocks. As IQP circuits generate distributions that are classically intractable unless the polynomial hierarchy collapses, this positions PQCs above deep Boltzmann machines (DBMs) and long-range RBMs in expressive capacity for generative modeling.

The expressive power hierarchy (from most to least expressive) derived in (Du et al., 2018) is:

• MPQCs
• DBMs
• Long-range RBMs
• TPQCs
• Short-range RBMs

2. Ancillary Qubits, Post-Selection, and Enhanced Expressivity

Ancillary qubits, when incorporated into PQC architectures with post-selection, further enhance expressivity. In the so-called “ancillary driven MPQC” (AD-MPQC) paradigm, blocks of the circuit are applied conditionally upon the measurement of the ancilla register in a specific state. This mechanism achieves the effect of “copy operations,” which reproduce or distribute information analogous to string bond states (SBS) or general tensor networks (GTNs) and is not naturally available in standard MPQCs.

The effective output distribution of the data qubits thus becomes conditioned on the ancillary state, thereby facilitating the implementation of more complex quantum models (such as post-IQP circuits) that exceed classical generative models in representational power (Du et al., 2018).

3. Bayesian Quantum Circuits and Learning Priors

A significant practical application enabled by PQCs—particularly the AD-MPQC architecture—is the Bayesian Quantum Circuit (BQC). In BQC, the parameterized circuit is engineered to learn both the prior and likelihood components of a generative model directly from data, which is especially valuable in semi-supervised and weakly supervised learning scenarios where class prior distributions are unknown (Du et al., 2018).

The construction consists of:

• A set of blocks $U(\bm{\gamma}^i)$ acting on ancillary qubits to establish the prior state $\ket{\Psi_A}$, where

$$q(\bm{\lambda}) = |\langle \bm{\lambda} | \Psi_A \rangle|^2.$$

• Conditional blocks $U(\bm{\theta}^j_{\lambda_i})$ applied to the data register per ancillary state $\ket{\lambda_i}$, encoding the likelihood $q(\bm{x}|\lambda_i)$.
• Measurement of the joint state

$$\ket{\Psi_{\bm{x},\bm{\lambda}}} = \sum_i \alpha_i \prod_{j=1}^L U(\bm{\theta}^j_{\lambda_i}) \ket{0}^{\otimes N} \otimes \ket{\lambda_i}$$

yields the joint and posterior probabilities via

$$q(\lambda_i|\bm{x}) = \frac{q(\bm{x},\lambda_i)}{\sum_j q(\bm{x},\lambda_j)}.$$

Numerical experiments using the Rigetti Forest platform demonstrated that BQC outperforms competing PQC-based generative models (e.g., DDQCL, QCBM) for the bars-and-stripes dataset, with accuracies up to 99.96% for $2\times2$ BAS and 98.65% for $3\times3$ images, and is capable of accurately learning unknown class priors even in simple semi-supervised settings.

4. Dimensional Expressivity, Redundant Parameter Pruning, and Quantum Geometry

Dimensional expressivity analysis quantifies the number of independent directions a PQC can access in Hilbert space as its parameters are varied (Funcke et al., 2020). This is operationalized by evaluating the (real) Jacobian of the circuit-output map, computing the rank of the corresponding Gram matrix $S = J^* J$, and systematically identifying parameters whose associated tangent vectors are linearly dependent. Redundant parameters are iteratively removed, enabling the design of maximally expressive ansätze with minimized parameter count and circuit depth.

The analysis can be implemented efficiently using a hybrid quantum–classical protocol: inner products required for Gram matrix elements are estimated via ancilla-assisted quantum circuits (e.g., Hadamard tests), and invertibility checks are carried out classically. This approach can also incorporate or remove symmetries—by constraining the ansatz to specific symmetry sectors or eliminating redundant global phase–generating parameters. Experimental validation on IBM Quantum hardware confirmed the efficacy of this strategy.

Relatedly, (Haug et al., 2021) formalizes “parameter dimension” $D_C$ (the number of independent parameters capable of affecting the quantum state), and introduces the “effective quantum dimension” $G_C(\theta)$, defined by the rank of the quantum Fisher information (QFI) matrix at point $\theta$. Redundant parameters are pruned via systematic analysis of the QFI spectrum, and circuits are initialized in a regime balancing large $G_C$ and robust gradient variance—helping avoid barren plateaus and maximize trainability.

5. Gate Structure, Topology, Expressibility, and Majorization Complexity

The expressibility of a PQC—how closely its output state distribution approximates the Haar measure—is contingent on its gate composition and connectivity topology. Expressibility is measured by the Kullback–Leibler divergence

$$\text{Expr} = D_\mathrm{KL}( P_\mathrm{PQC}(F) \,\|\, P_\mathrm{Haar}(F) )$$

of the fidelity distribution of output states to the analytical Haar distribution (Correr et al., 3 May 2024, Correr et al., 29 May 2024).

Key empirical results:

• Circuits composed predominantly of $R_X$ or $R_Y$ gates maximize expressibility, while extensive CNOT usage, though necessary for entanglement, may diminish it if overused (Liu et al., 2 Aug 2024).
• Expressibility saturates beyond a certain circuit depth or repetition, rendering additional layers or gates less effective in improving uniform coverage of the Hilbert space (Liu et al., 2 Aug 2024).
• Topologically, ring and linear connectivities among qubits foster more rapid convergence to Haar-random state properties (i.e., higher expressibility and entanglement) than star or unconnected layouts (Correr et al., 3 May 2024, Correr et al., 29 May 2024).
• The complexity of PQCs, beyond expressibility and mean entanglement, can be quantified by the majorization criterion, which analyzes the fluctuations of Lorenz curves associated with measurement probability distributions. PQCs with optimal topology achieve Haar-random statistical characteristics (in expressibility, entanglement, and majorization) using fewer gates than standard universal random circuits.

Integration of these diagnostics enables the systematic design of PQCs tailored for efficient state preparation, quantum simulation, or quantum machine learning—balancing expressibility, entanglement, and implementation cost.

6. Verification, Compilation, and Circuit Equivalence

Parameterization introduces significant challenges for compilation and verification. Compilation seeks to minimize the parameter count, especially relevant for variational algorithms and measurement-based quantum computing (MBQC), where each parameter may correspond to a distinct measurement or a variational degree of freedom (Wetering et al., 23 Jan 2024). For Clifford circuits augmented by unique parametrized Z [α] gates and under analytic (i.e., affine modulo $2\pi$) transformations, the only permissible parameter-reducing rewrites are additive fusions of adjacent phase gates. Circuit translation into the ZX-calculus supports efficient and optimal parameter minimization via established rewrite rules.

Equivalence checking between PQCs after optimization (e.g., during hardware compilation) is nontrivial since symbolic parameter dependencies must be addressed holistically. Decision diagram techniques extended to handle symbolic (parameterized) trigonometric polynomial weights—S-TDDs—enable canonical, structure-preserving representations that allow verification of equivalence (up to a global phase) without explicit parameter instantiation (Hong et al., 29 Apr 2024). Experiments confirm both the efficiency and robustness of this approach for circuits with up to hundreds of parameters and thousands of gates.

7. Applications, Limitations, and Future Directions

PQCs enable:

• Quantum advantage in generative modeling, including learning of complex, highly entangled or classically intractable distributions (Du et al., 2018, Barthe et al., 15 Feb 2024).
• Efficient variational eigensolvers, combinatorial optimization (via QAOA), and quantum machine learning models capable of L$^p$, uniform, and Sobolev-norm function approximation (Manzano et al., 2023).
• Advanced applications such as Bayesian learning (direct learning of priors), adaptive state preparation, and topological phase simulation (e.g., via weight-adjustable loop gas circuits for the toric code) (Sun et al., 2022).

Limitations include training bottlenecks caused by barren plateaus (exponentially vanishing gradients), which can be mitigated by careful ansatz selection, parameter initialization, dynamic circuit designs (with measurements and feedforward, as in (Deshpande et al., 8 Nov 2024)), and quantum gradient-based optimization blending quantum and classical search (Li et al., 30 Sep 2024).

Emerging research directions include:

• Hardware–software co-design for high-throughput parameterized circuit execution (Rajagopala et al., 5 Sep 2024).
• Construction of more expressive, trainable PQCs using linear combinations of unitaries (LCU) while maintaining polynomially lower-bounded gradient variance and enabling quantum speed-ups in expectation value estimation (Khatri et al., 27 Jun 2025).
• Adaptation of PQCs to kernel-based approaches in quantum reinforcement learning (Bossens et al., 11 Nov 2024).
• Systematic exploration of the interplay between expressibility, topology, and optimization landscape to tailor circuit architectures for specific quantum tasks.

These features collectively establish PQCs as a versatile and powerful paradigm in near-term quantum computing, with theoretical advantages in expressivity and practical adaptability for a broad array of quantum information processing tasks.