Parameterized Quantum Circuits

• Parameterized quantum circuits are quantum circuits with adjustable parameters that enable versatile state transformations, simulation of complex quantum phenomena, and function approximation.
• They employ advanced optimization strategies, including parameter-shift rules and interpolation methods, to enhance convergence in variational quantum algorithms amid noisy hardware conditions.
• Their adaptive design and automated architecture search facilitate robust verification, high expressibility, and efficient implementation in quantum machine learning and optimization tasks.

Parameterized quantum circuits (PQCs) are a foundational construct in contemporary quantum information science, underpinning hybrid quantum–classical algorithms, quantum machine learning, quantum optimization, and the simulation of quantum many-body systems. A PQC consists of a quantum circuit composed of gates whose actions are determined by a set of continuous parameters, enabling the circuit to realize a broad family of quantum transformations and, by extension, probability distributions or functional mappings. The paper of parameterized quantum circuits encompasses their expressive power, optimization strategies, structural design, resource requirements, applications, and the intrinsic connections to the computational complexity of quantum and classical systems.

1. Mathematical Structure and Expressive Power

A parameterized quantum circuit is typically defined as a unitary transformation $U(\theta)$ acting on $N$ qubits, where $\theta$ is a real vector of trainable parameters. The standard composition is

$$U(\theta) = U_L(\theta_L) \cdots U_2(\theta_2) U_1(\theta_1)$$

with each $U_l(\theta_l)$ a parametric (often, single-qubit or local two-qubit) unitary gate. The output state is given by $$|\psi_\theta\rangle = U(\theta)|0\rangle^{\otimes N}$$. Observables measured on this state yield functions of the parameters: $$f_M(\theta) = \langle \psi_\theta | M | \psi_\theta \rangle$$.

The expressive power of PQCs, as discussed in (Du et al., 2018), is rigorously quantified relative to classical generative models via the ability to efficiently represent quantum states and probability distributions. Multilayer PQCs (MPQCs), built from a polynomial number of parameterized single-qubit and controlled gates, can generate quantum states exhibiting volume-law entanglement and can efficiently simulate matrix product states (MPS) with large bond dimension $D$ using $O(\mathrm{poly}(\log D))$ parameter blocks. The resulting hierarchy is: $$\text{MPQC} > \text{DBM} > \text{long-range RBM} > \text{TPQC} > \text{short-range RBM}$$ where MPQC is strictly more expressive unless the polynomial hierarchy collapses. Notably, MPQCs can simulate instantaneous quantum polynomial (IQP) circuit distributions, which are conjecturally hard to reproduce with any classical neural network model.

A central analytic tool is entanglement entropy $\mathcal{S}(\rho_A) \sim \log(D)$ across partitions: PQCs rapidly increase the accessible bond dimension $D$ with depth and connectivity. Consequently, PQCs can efficiently represent distributions and states that are not accessible to classical neural networks with comparable resource constraints.

2. Parameter Optimization and Quantum Circuit Calculus

Optimizing PQC parameters is central to their application in variational algorithms and quantum machine learning. The loss (or cost) function is

$$J(\theta) = \frac{1}{m} \sum_{i=1}^m (y_i - \mathrm{expectation}(\mathrm{PQC}(x_i, \theta): Q_T))^2$$

where $x_i$ are data inputs, $y_i$ labels, and $Q_T$ is the measured qubit. For efficient gradient computation, parameter–shift rules are employed for gates of the form $\exp(-i\theta_j H_j)$, with

$$\frac{\partial}{\partial \theta_j} \langle M \rangle_\theta = \frac{1}{2} \left[ \langle M \rangle_{\theta + \frac{\pi}{2} e_j} - \langle M \rangle_{\theta - \frac{\pi}{2} e_j} \right]$$

where $e_j$ is the unit vector in parameter space. For more complex Hamiltonians with additional eigenvalue spacings, generalizations are provided (Vidal et al., 2018), leveraging Fourier analysis of the expectation function $f(t)$, exploiting its $2\pi$-periodicity and reconstructing derivatives via evaluations at $2n+1$ points determined by the support $D$ of the eigenvalue differences.

The optimal interpolation-based coordinate descent (OICD) method (Lai et al., 6 Mar 2025) reconstructs a trigonometric surrogate of the cost function along each parameter by evaluating at carefully chosen nodes (ideally $\frac{2\pi}{2r+1}$-equidistant), obtaining robust updates via argmin operations that are less sensitive to stochastic error and more global than local gradient-based approaches. Classical minimization over the reconstructed surrogate brings faster convergence, particularly in noisy settings.

3. Circuit Structure, Expressibility, and Resource Scaling

Expressibility of a PQC is defined as its ability to uniformly cover the Hilbert space, frequently quantified via the Kullback-Leibler divergence between the distribution of fidelities $P_{\text{PQC}}(F)$ over circuit outputs and the corresponding Haar measure $P_{\text{Haar}}(F)$: $$\mathrm{Expr} = D_{KL}(P_{\text{PQC}}(F) \| P_{\text{Haar}}(F))$$ Low values indicate high expressibility, implying the circuit's outputs mimic random states drawn from the Haar ensemble. Empirical studies (Hubregtsen et al., 2020, Liu et al., 2 Aug 2024) show a strong correlation (Pearson coefficient $|\rho| \sim 0.7$) between expressibility and classification accuracy in quantum machine learning, while entangling capability plays a less significant predictive role.

Connectivity and gate composition are key: circuits with RX and RY rotations and moderate numbers of CNOT gates maximize expressibility, while overuse of CNOTs is detrimental. Expressibility saturates with increasing depth or gate count, indicating diminishing returns with very deep circuits (Liu et al., 2 Aug 2024).

Topology influences complexity convergence: ring and linear-connected circuits reach Haar–like complexity and entanglement with significantly fewer gates than universal random circuits built from discrete sets (G3) (Correr et al., 29 May 2024). Majorization—via Lorenz curve fluctuations—provides a complementary view of complexity.

Effective quantum dimension $G_C(\theta)$, defined via the number of nonzero eigenvalues of the quantum Fisher information (QFI) matrix, quantifies the number of independent directions in Hilbert space accessible via infinitesimal parameter shifts (Haug et al., 2021). The quantum geometry (QFI) also enables construction of the quantum natural gradient, which can outperform standard gradients, though both may suffer vanishing in deep circuits (the barren plateau phenomenon).

4. Adaptive and Automated Circuit Design

Automated PQC architecture search is enabled by optimization strategies such as BPQCO (Bayesian Parameterized Quantum Circuit Optimization) (Benítez-Buenache et al., 17 Apr 2024), which employs Bayesian optimization (e.g., TPE) to select both circuit topology and parameterization. The circuit design is formalized via: a feature map, an ansatz, entanglement matrices $\mathcal{E}$, and a gate selection matrix $\mathcal{P}$. The combinatorial search space can be navigated efficiently, and the resulting architectures are more robust to hardware noise when designed using error-aware and multi-objective cost functions.

Structure optimization methods such as Rotoselect (Ostaszewski et al., 2019) (per-gate rotation-axis selection) and free-axis selection (Fraxis) (Watanabe et al., 2021) further expand expressibility by allowing arbitrary or continuous selection of single-qubit rotation axes, beyond fixed predefined rotations about $X$, $Y$, or $Z$. Fraxis, in particular, finds optimal axes via a closed-form solution to a system of equations built from experimentally accessible expectation values, leading to improved convergence in VQE, MaxCut, and other variational algorithms.

5. Noise, Hardware Constraints, and Practical Implementation

Variational quantum algorithms using PQCs are particularly affected by noise and temporal fluctuations in qubit metrics (T1, T2, gate error rates). Training a circuit without incorporating these fluctuations often results in poor generalization over time. A noise-aware, fully classical training regime (Alam et al., 2019) incorporating temporally averaged noise models leads to substantial improvements in predictive fidelity and robustness, with reported performance gains up to 42.5% on NISQ platforms.

At the hardware level, pulse-level access can produce PQCs with custom, faster, and more robust two-qubit entangling gates (for example, optimized cross-resonance gates on superconducting devices) (Ibrahim et al., 2022). Such pulse-optimized ansätze show reduced state preparation time (a 2×-3× improvement), maintain high expressibility, and can outperform standard circuits in VQE and combinatorial optimization in real hardware experiments.

Automated circuit generators such as Q-gen (Mao et al., 26 Jul 2024) facilitate the systematic investigation and benchmarking of PQCs at scale, enabling analysis of structure, complexity, and performance across numerous application-relevant algorithms through systematic parameterization of both circuit width and algorithm-specific features.

6. Universality and Function Approximation Properties

Parameterized quantum circuits are provably universal approximators of a wide class of functions and distributions under expectation value sampling (Barthe et al., 15 Feb 2024, Manzano et al., 2023). Given any continuous target $g : [0,1]^M \rightarrow [-1,1]$, a one-qubit PQC with a fixed observable $O$ and appropriate $U(x)$ can satisfy $$|\langle 0 | U^\dagger(x) O U(x) | 0 \rangle - g(x)| < \epsilon$$ for all $x$.

This universality extends to the approximation of functions in $L^p$, $C^0$, and Sobolev $H^k$ spaces, with error rates and generalization properties connected (via classic results like Fejér’s theorem) to the underlying structure of the PQC's Fourier-like expansion (Manzano et al., 2023). These features are especially relevant for applications to physics-informed learning and differential equation solvers, where minimizing a derivative-sensitive empirical risk ensures both function and derivative approximations.

Resource bounds for universality relate minimal qubit number, observable spectral norm, and measurement resources (Barthe et al., 15 Feb 2024). For "product encoding," the number of qubits grows linearly with output dimension $M$; for "dense encoding," the qubit cost grows logarithmically but at the expense of increased observable norm and sampling overhead.

7. Verification and Information-Theoretic Perspectives

Verification of PQC compilation or transpilation is addressed via symbolic tensor decision diagrams (S-TDDs) (Hong et al., 29 Apr 2024), which compactly represent PQCs with parameterized gate symbols and enable equivalence checking across all parameter values. This symbolic approach efficiently detects equivalence or inequivalence, outperforming prior ZX-calculus-based methods and providing a critical tool for correct circuit deployment on hardware.

Training PQCs is constrained by information-theoretic limits: while exact expectation evaluation queries convey complete information about the objective function, sample queries (i.e., single circuit runs yielding $\pm 1$) contribute exponentially little information per query (Dolzhkov et al., 2019). As a result, training based on sample queries requires an exponential number of runs in the number of parameters—an insight that underpins the empirical challenges of scaling PQC-based variational algorithms.

8. Advanced Optimization, Trainability, and Future Directions

Recent advances introduce quantum gradient approaches that operate over the Hilbert space (quantum gradient operator $\mathcal{D}$ acting on $|z\rangle$), circumventing cases of vanishing classical parameter gradients (barren plateaus) and enabling effective state evolution even when the parameter landscape appears flat (Li et al., 30 Sep 2024).

Adaptive circuit expansion—adding expressivity via additional layers only when the cost indicator ($c = 1 - |\langle z' | U(\theta)|0\rangle|^2$) signals the need—paired with reinforcement learning-based circuit synthesis, leads to improved sample efficiency and robustness against gradient vanishing, with demonstrated success in MaxCut and polynomial optimization tasks.

Optimization remains sensitive to the barren plateau problem, scaling of parameter dimension and entangling gate choice, and the structure of the cost function landscape. Initialization strategies (e.g., scaling parameters from zero to random values) and pruning redundant parameters using QFI analysis are key in balancing expressivity and trainability (Haug et al., 2021).

Practical design of PQCs for VQAs and QML increasingly relies on automated, noise-aware, and hardware-adapted circuit search, as well as theoretical insights from quantum geometry, entanglement theory, and statistical learning analysis.

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The paper of parameterized quantum circuits thus sits at the intersection of quantum information theory, computational complexity, machine learning, and experimental physics. Rigorous comparison with classical models, precise optimization strategies, architectural flexibility, and resource-efficient universality results place PQCs at the forefront of current efforts to harness near-term quantum devices for computational advantage.