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arXiv:2408.05406 · IEEE International Conference on Quantum Computing and Engineering (QCE)

Efficient Quantum Gradient and Higher-Order Derivative Estimation Via Generalized Hadamard Test

Dantong Li, Dikshant Dulal, Mykhailo Ohorodnikov, Hanrui Wang, Yongshan Ding

Published Aug 30, 2025• 1 citation
arXivHTMLPDF

Abstract

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 the$k$-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.

DOI
10.1109/qce65121.2025.00024

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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.
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.
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.
Training parameterised quantum circuits (PQCs) on quantum hardware is bottlenecked by the measurement cost of gradient estimation, which under the parameter-shift rule scales linearly in the number of trainable parameters and dominates the total shot budget of training at scale. In this work, we propose a framework of forward gradient estimators for PQCs, based on the forward mode of automatic differentiation, that yields an unbiased estimator of the gradient by averaging a freely tunable number of random directional derivatives and recovers SPSA, random coordinate descent, and the parameter-shift rule as limiting cases, with no ancilla qubits or controlled-gate overhead. We prove that stochastic quantum forward gradient descent converges under standard assumptions, with an explicit second-moment expansion that interpolates between the single-direction extreme of SPSA and the full-gradient extreme of parameter-shift. Within this framework we derive QUIVER (Quantum Iterative V-adaptive Estimator Rule), an adaptive optimiser for parameterised circuits whose update rule follows from a closed-form minimum measurement-cost allocation. We show numerically that forward gradients train Hamming-weight-preserving orthogonal quantum neural networks with up to 60 qubits and 1770 parameters on the ECG5000 and MNIST datasets orders of magnitude more efficiently than the parameter-shift rule. We also demonstrate that our proposed QUIVER optimiser can outperform iCANS and gCANS measurement-frugal optimisers on optimisation problems using the quantum approximate optimisation algorithm and quantum simulation with the variational quantum eigensolver.
Parameterized quantum circuits (PQCs) play an essential role in the application of variational quantum algorithms (VQAs) in noisy intermediate-scale quantum (NISQ) devices. The PQCs are a leading candidate to achieve a quantum advantage in NISQ devices and have already been applied in various domains such as quantum chemistry, quantum machine learning, combinatorial optimization, and many others. There is no single definitive way to optimize PQCs. The most commonly used methods are based on computing the gradient via the parameter-shift rule to use classical gradient descent (GD) optimizers like Adam, stochastic GD, and others. In addition, sequential single-qubit optimizers have been proposed, such as Rotosolve, Free-Axis Selection (Fraxis), Free-Quaternion Selection (FQS), and hybrid algorithms from the aforementioned optimizers. We further develop hybrid algorithms than those represented in the previous work by drawing inspiration from the early stopping method used in classical machine learning. The switch between the optimizers depends on the previous cost function values compared to the previous ones. We introduce two new hybrid algorithms that are more robust and scalable, and they outperform previous hybrid methods in terms of convergence towards the global minima across various cost functions. In addition, we find that they are feasible for NISQ devices with different noise profiles.
Abstract Many near-term quantum computing algorithms are conceived as variational quantum algorithms, in which parameterized quantum circuits are optimized in a hybrid quantum–classical setup. Examples are variational quantum eigensolvers, quantum approximate optimization algorithms as well as various algorithms in the context of quantum-assisted machine learning. A common bottleneck of any such algorithm is constituted by the optimization of the variational parameters. A popular set of optimization methods work on the estimate of the gradient, obtained by means of circuit evaluations. We will refer to the way in which one can combine these circuit evaluations as gradient rules. This work provides a comprehensive picture of the family of gradient rules that vary parameters of quantum gates individually. The most prominent known members of this family are the parameter shift rule (PSR) and the finite differences method. To unite this family, we propose a generalized PSR that expresses all members of the aforementioned family as special cases, and discuss how all of these can be seen as providing access to a linear combination of exact first- and second-order derivatives. We further prove that a PSR with one non-shifted evaluation and only one shifted circuit evaluation does not exist, and introduce a novel perspective for approaching new gradient rules.
Parameterized quantum circuits (PQCs) are pivotal components of variational quantum algorithms (VQAs), which represent a promising pathway to quantum advantage in noisy intermediate-scale quantum (NISQ) devices. PQCs enable flexible encoding of quantum information through tunable quantum gates and have been successfully applied across domains such as quantum chemistry, combinatorial optimization, and quantum machine learning. Despite their potential, PQC performance on NISQ hardware is hindered by noise, decoherence, and the presence of barren plateaus, which can impede gradient-based optimization. To address these limitations, we propose novel methods for improving gradient-free optimizers Rotosolve, Fraxis, and FQS, incorporating information from previous parameter iterations. Our approach conserves computational resources by reallocating optimization efforts toward poorly optimized gates, leading to improved convergence. The experimental results demonstrate that our techniques consistently improve the performance of various optimizers, contributing to more robust and efficient PQC optimization.