Sequential minimal optimization methods, such as the Rotosolve and the Nakanishi-Fujii-Todo algorithm (NFT), are widely used for Variational Quantum Eigensolvers (VQEs). These methods optimize one parameter direction at a time, requiring measurements at only a few locations along that direction. In the presence of measurement shot noise, however, their performance depends critically on the choice of measurement locations, and recent studies suggest that equidistant measurements are optimal. However, we often observe that equidistant measurements are not always optimal in practice. We argue that this discrepancy between theory and practice arises from the fact that two assumptions underlying previous analyses do not generally hold: (1) the absence of prior knowledge about the energy minimizer, and (2) the use of the uncertainty of the estimated energy as a proxy for optimization performance. In this paper, we develop a new theory for determining optimal measurement locations. First, we show that incorporating prior information about the minimizer is beneficial. Early in optimization, when little is known about the pivot, i.e., the current minimizer, equidistant measurements are indeed near-optimal, but as the prior belief sharpens the optimal locations move away from equidistant. Second, rather than analyzing the uncertainty of the estimated minimum energy, we study the uncertainty of the estimator of the minimizer itself, which leads to substantially different strategies. Based on this analysis, we propose Prior-informed Adaptive Shifts (PAS), a method that automatically adjusts measurement locations during optimization. Numerical experiments across different shot counts and problems validate our theoretical findings and demonstrate that PAS adaptively recovers whichever fixed shift is best in each regime without it being specified in advance.
Sequential minimal optimization methods, such as the Rotosolve and the Nakanishi-Fujii-Todo algorithm (NFT), are widely used for Variational Quantum Eigensolvers (VQEs). These methods optimize one parameter direction at a time, requiring measurements at only a few locations along that direction. In the presence of measurement shot noise, however, their performance depends critically on the choice of measurement locations, and recent studies suggest that equidistant measurements are optimal. However, we often observe that equidistant measurements are not always optimal in practice. We argue that this discrepancy between theory and practice arises from the fact that two assumptions underlying previous analyses do not generally hold: (1) the absence of prior knowledge about the energy minimizer, and (2) the use of the uncertainty of the estimated energy as a proxy for optimization performance. In this paper, we develop a new theory for determining optimal measurement locations. First, we show that incorporating prior information about the minimizer is beneficial. Early in optimization, when little is known about the pivot, i.e., the current minimizer, equidistant measurements are indeed near-optimal, but as the prior belief sharpens the optimal locations move away from equidistant. Second, rather than analyzing the uncertainty of the estimated minimum energy, we study the uncertainty of the estimator of the minimizer itself, which leads to substantially different strategies. Based on this analysis, we propose Prior-informed Adaptive Shifts (PAS), a method that automatically adjusts measurement locations during optimization. Numerical experiments across different shot counts and problems validate our theoretical findings and demonstrate that PAS adaptively recovers whichever fixed shift is best in each regime without it being specified in advance.
The Nakanishi Fujii Todo (NFT) algorithm, also known as Rotosolve, implements Sequential Minimal Optimization for Variational Quantum Eigensolvers (SMO-VQE) by exploiting the trigonometric dependence of the energy on individual circuit parameters. This enables analytical one-dimensional minimization using only a few , typically two, energy evaluations, but introduces bias in the estimated energy. Although performing additional measurements every few tens of iterations can mitigate bias accumulation, we find that such corrections often degrade optimization performance. In this paper, we analyze the origin and accumulation of bias during the SMO-VQE process. Specifically, we show that the bias can be accurately estimated without additional measurements. Furthermore, we find that bias correction destabilizes optimization along directions with small curvature, whereas the original biased estimator implicitly acts as a regularizer. Based on these insights, we propose a simple regularization method that implements error accumulation while maintaining unbiased energy estimation. The resulting algorithm consistently improves performance across different system sizes, circuit depths, target Hamiltonians, and measurement shots, with minimal hyperparameter tuning.
Variational Quantum Eigensolver (VQE) provides a powerful solution for approximating molecular ground state energies by combining quantum circuits and classical computers. However, estimating probabilistic outcomes on quantum hardware requires repeated measurements (shots), incurring significant costs as accuracy increases. Optimizing shot allocation is thus critical for improving the efficiency of VQE. Current strategies rely heavily on hand-crafted heuristics requiring extensive expert knowledge. This paper proposes a reinforcement learning (RL)-based approach that automatically learns shot assignment policies to minimize total measurement shots while achieving convergence to the minimum of the energy expectation in VQE. The RL agent assigns measurement shots across VQE optimization iterations based on the progress of the optimization. This approach reduces VQE's dependence on static heuristics and human expertise. When the RL-enabled VQE is applied to a small molecule, a shot reduction policy is learned. The policy demonstrates transferability across systems and compatibility with other wavefunction Ansätze. In addition to these specific findings, this work highlights the potential of RL for automatically discovering efficient and scalable quantum optimization strategies.
Parameter shift rules (PSRs) are key techniques for efficient gradient estimation in variational quantum eigensolvers (VQEs). In this paper, we propose its Bayesian variant, where Gaussian processes with appropriate kernels are used to estimate the gradient of the VQE objective. Our Bayesian PSR offers flexible gradient estimation from observations at arbitrary locations with uncertainty information and reduces to the generalized PSR in special cases. In stochastic gradient descent (SGD), the flexibility of Bayesian PSR allows the reuse of observations in previous steps, which accelerates the optimization process. Furthermore, the accessibility to the posterior uncertainty, along with our proposed notion of gradient confident region (GradCoRe), enables us to minimize the observation costs in each SGD step. Our numerical experiments show that the VQE optimization with Bayesian PSR and GradCoRe significantly accelerates SGD and outperforms the state-of-the-art methods, including sequential minimal optimization.
Abstract The adaptive variational quantum eigensolver (ADAPT-VQE) is a promising approach for quantum simulation in the noisy intermediate-scale quantum era, offering advantages over traditional VQE methods by reducing circuit depth and mitigating challenges in classical optimisation. Although ADAPT-VQE can require fewer total measurements than fixed ansätze for matched precision, the combined cost of repeated energy and gradient evaluations still leads to a substantial measurement overhead in practice, motivating strategies to reduce it further. In this work, we study two complementary strategies to reduce this cost using only standard projective Pauli measurements. First, we apply variance-based shot allocation (variance-minimised shot assignment and variance-preserved shot reduction (VPSR)) to both Hamiltonian energy and operator-gradient measurements, extending techniques previously studied only for energy estimation in standard VQE. VPSR achieves shot reductions of up to 43% for H 2 and 51% for LiH relative to uniform allocation while maintaining chemical accuracy. Second, we introduce a Pauli-measurement reuse protocol that stores outcomes from VQE optimisation and reuses them for gradient evaluation in the next ADAPT iteration. Across six molecular benchmarks (4–14 qubits), measurement reuse yields only marginal savings confined to the smallest systems and is effectively negligible for larger sizes, pivot-based grouping alone provides the dominant reduction and should be considered the practical baseline. These results characterise the interplay between shot allocation, measurement grouping, and cross-iteration reuse, offering practical guidance on when each strategy is most beneficial.
The variational quantum eigensolver (VQE) is a hybrid algorithm to find the minimum eigenvalue/vector of the given Hamiltonian by optimizing a parameterized quantum circuit (PQC) using a classical computer. Sequential optimization methods, which are often used in quantum circuit tensor networks, are popular for optimizing the parameterized gates of PQCs. In this paper, we focus on the case where the components to be optimized are single-qubit gates, in which the analytic optimization of a single-qubit gate is sequentially performed. The analytical solution is given by diagonalization of a matrix whose elements are computed from the expectation values of observables specified by a set of predetermined parameters, which we refer to as the parameter configurations. In this paper, we first show that the optimization accuracy significantly depends on the choice of parameter configurations owing to the statistical errors in the expectation values. We then identify a metric that quantifies the optimization accuracy of a parameter configuration for all possible statistical errors, named configuration overhead/cost or C-cost. We theoretically provide the lower bound of C-cost and show that, for the minimum size of parameter configurations, the lower bound is achieved if and only if the parameter configuration satisfies the so-called equiangular line condition. Finally, we provide numerical experiments demonstrating that the optimal parameter configuration exhibits the best result in several VQE problems. We hope that this general statistical methodology will enhance the efficacy of sequential optimization of PQCs for solving practical problems with near-term quantum devices.
Abstract The variational quantum eigensolver (VQE) is a fundamental algorithm in quantum computing, yet its theoretical understanding remains fragmented across fixed-ansatz and adaptive-circuit formulations, which are typically analyzed using different frameworks. As a result, a coherent geometric characterization of the VQE remains missing. In this paper, we establish a geometric analysis of VQE in terms of optimization landscape, initialization guarantee, and noise robustness. First, from a Riemannian geometric perspective, we study the optimization landscape via an ansatz-free product-unitary formulation over the unitary group, where each layer unitary is treated as an independent optimization variable, thereby capturing both fixed-ansatz and adaptive-circuit paradigms. For the single-unitary case, we establish explicit linear convergence guarantees for Riemannian gradient descent (RGD) under proper initialization and characterize the global landscape by proving that every critical point is either a global minimum or a strict saddle point. For the product-unitary case, we extend the convergence analysis of RGD and show that the convergence rate deteriorates polynomially with circuit depth, providing a convergence-theoretic explanation of the barren plateau phenomenon through the interplay between Hilbert-space geometry, target-unitary complexity, per-layer expressivity, and circuit depth. Second, we analyze initialization guarantees for the above convergence results. We consider a practical variational circuit composed of small-angle random Pauli rotations and prove that this initialization scheme satisfies the required geometric conditions with high probability. We further characterize how the resulting initialization error depends on the reference state, circuit depth, and random angles. Third, we study robustness under finite-shot measurements for VQE with Pauli-decomposed Hamiltonians. We show that RGD retains linear convergence up to a noise-dominated neighborhood of the global minimizer. Moreover, we extend the analysis to non-uniform measurement allocation under a fixed measurement budget and show that coefficient-adaptive allocation achieves strictly lower statistical error than uniform sampling. Together, these results provide a systematic geometric characterization of the optimization, initialization, and measurement behavior of VQE, offering new insights into its trainability and robustness.