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arXiv:2509.17822 · Physical Review A

Exact gradient for general cost functions in variational quantum algorithms

Jesus Urbaneja, Le Bin Ho

Published Apr 1, 2026• 1 citation
arXivHTMLPDF

Abstract

We present a unitary-based gradient formulation for variational quantum algorithms (VQAs) that applies to general differentiable cost function defined by a parameterized quantum circuit composed of Pauli-generated rotations. The gradient is obtained directly from the underlying unitary evolution, without assuming a specific expectation-value form of the cost function. The resulting expressions can be accessed on quantum hardware using the Hadamard and Hilbert-Schmidt tests. We demonstrate the method in variational quantum compilation, where it yields stable and accurate gradient estimates. This unitary-based framework therefore provides a broadly applicable and hardware-compatible tool for gradient evaluation in VQAs.

DOI
10.1103/6s2c-mxgx

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We present a unitary-based gradient formulation for variational quantum algorithms (VQAs) that applies to general differentiable cost function defined by a parameterized quantum circuit composed of Pauli-generated rotations. The gradient is obtained directly from the underlying unitary evolution, without assuming a specific expectation-value form of the cost function. The resulting expressions can be accessed on quantum hardware using the Hadamard and Hilbert-Schmidt tests. We demonstrate the method in variational quantum compilation, where it yields stable and accurate gradient estimates. This unitary-based framework therefore provides a broadly applicable and hardware-compatible tool for gradient evaluation in VQAs.
Variational quantum algorithms (VQAs) are commonly developed under the assumption that a parameterized circuit corresponds to a single, well-defined physical realization. In practice, this assumption is not satisfied. Differences in compilation, gate ordering, and hardware-level perturbations lead to multiple admissible circuit realizations, each associated with a distinct objective over the same parameter space. The resulting optimization problem is therefore not defined by a single loss function, but by a family of configuration-dependent objectives whose aggregate behaviour must be considered. This study formulates optimization explicitly at the level of such configuration families. A consistent extension of natural-gradient methods is developed in which both the descent direction and the local geometric structure are obtained through averaging across configurations. The resulting update preserves the structure of quantum natural gradient methods while ensuring alignment with the family-defined objective. The approach is evaluated using exact statevector simulations on transverse-field Ising and Heisenberg Hamiltonians under controlled variability. Across all tested regimes, the method produces lower average loss, substantially reduced dispersion across configurations, and improved worst-case performance relative to both configuration-specific optimization and gradient-averaged baselines. These results show that when the objective depends on a set of admissible circuit realizations, optimization must be constructed at the same level to obtain stable and reliable solutions.
In the context of Noisy Intermediate-Scale Quantum (NISQ) computing, parameterized quantum circuits (PQCs) are a promising paradigm for quantum sensing, optimal control, optimization, and machine learning on near-term quantum hardware. Gradient-based methods are crucial for analyzing PQCs and offer significant advantages in the convergence rates of Variational Quantum Algorithms (VQAs) compared to gradient-free methods. However, existing gradient estimation techniques-such as Finite Difference, Parameter Shift Rule, Hadamard Test, and Direct Hadamard Test-can yield suboptimal circuits for certain PQCs. To address these limitations, we introduce the Flexible Hadamard Test, which can invert the roles of ansatz generators and observables in first-order gradient estimation, enabling the use of measurement optimization techniques to improve circuit efficiency. For higher-order derivatives, we propose thekkk-fold Hadamard Test, which evaluates higher-order partial derivatives using a single quantum circuit. Additionally, we present Quantum Automatic Differentiation (QAD)-a unified gradient estimation framework that adaptively selects the most efficient method for each parameter in a PQC, departing from the standard practice of using a single method uniformly. Through extensive numerical experiments, we demonstrate that our proposed methods yield up to a linear-factor reduction in circuit execution cost for real VQA applications. These advances accelerate quantum algorithm training and provide practical utility in the NISQ era.
Optimization of unitary transformations in variational quantum algorithms benefits highly from efficient evaluation of cost function gradients with respect to amplitudes of unitary generators. We propose several extensions of the parameter-shift rule to formulating these gradients as linear combinations of expectation values for generators with general eigenspectra (i.e., with more than two eigenvalues). Our approaches are exact and do not use any auxiliary qubits; instead they rely on a generator eigenspectrum analysis. Two main directions in the parameter-shift-rule extensions are (1) polynomial expansion of the exponential unitary operator based on a limited number of different eigenvalues in the generator and (2) decomposition of the generator as a linear combination of low-eigenvalue operators (e.g., operators with only two or three eigenvalues). These techniques have a range of scalings for the number of needed expectation values with the number of generator eigenvalues from quadratic (for polynomial expansion) to linear and even log2{log}_{2}log2​ (for generator decompositions). This allowed us to propose efficient differentiation schemes for commonly used two-qubit transformations (e.g., match gates, transmon gates, and fSim gates) and ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{S}}^{2}$-conserving fermionic operators for the variational quantum eigensolver.
Gradient estimation is a central challenge in training parameterized quantum circuits (PQCs) for hybrid quantum-classical optimization and learning problems. This difficulty arises from several factors, including the exponential dimensionality of the Hilbert spaces and the information loss in quantum measurements. Existing estimators, such as finite difference and the parameter shift rule, often fail to adequately address these challenges for certain classes of PQCs. In this work, we propose a novel gradient estimation framework that leverages the underlying Lie algebraic structure of PQCs, combined with the Hadamard test. By analyzing the differential of the matrix exponential, we derive an expression for the gradient as a linear combination of expectation values obtained via Hadamard tests. The coefficients in this decomposition depend solely on the circuit's parameterization and can be estimated using state-of-the-art shadow tomography techniques. Hence, our approach enables efficient gradient estimation, requiring a number of measurement shots that scales logarithmically with the number of parameters, and with polynomial classical and quantum time. This is an exponential reduction in the measurement cost and a polynomial speed-up in time compared to existing works.
Gradient estimation is a central challenge in training parameterized quantum circuits (PQCs) for hybrid quantum-classical optimization and learning problems.This difficulty arises from several factors, including the exponential dimensionality of the Hilbert spaces and the information loss in quantum measurements.Existing estimators, such as finite difference and the parameter shift rule, often fail to adequately address these challenges for certain classes of PQCs.In this work, we propose a novel gradient estimation framework that leverages the underlying Lie algebraic structure of PQCs, combined with the Hadamard test.By analyzing the differential of the matrix exponential in Lie algebras, we derive an expression for the gradient as a linear combination of expectation values obtained via Hadamard tests.The coefficients in this decomposition depend solely on the circuit's parameterization and can be computed efficiently.Furthermore, these expectation values can be estimated using state-of-the-art shadow tomography techniques.Our approach enables efficient gradient estimation, requiring a number of measurement shots that scales logarithmically with the number of parameters, and with polynomial classical and quantum time.This is an exponential reduction in the measurement cost and a polynomial speed-up in time compared to existing works.
Many optimization methods for training variational quantum algorithms are based on estimating gradients of the cost function. Due to the statistical nature of quantum measurements, this estimation requires many circuit evaluations, which is a crucial bottleneck of the whole approach. We propose a new gradient estimation method to mitigate this measurement challenge and reduce the required measurement rounds. Within a Bayesian framework and based on the generalized parameter shift rule, we use prior information about the circuit to find an estimation strategy that minimizes expected statistical and systematic errors simultaneously. We demonstrate that this approach can significantly outperform traditional gradient estimation methods, reducing the required measurement rounds by up to an order of magnitude for a common QAOA setup. Our analysis also shows that an estimation via finite differences can outperform the parameter shift rule in terms of gradient accuracy for small and moderate measurement budgets.