Classical postprocessing for the unitary block-optimization scheme to reduce the effect of noise on the optimization of variational quantum eigensolvers
Variational quantum eigensolvers (VQEs) are a promising approach for finding the classically intractable ground state of a Hamiltonian. The unitary block-optimization scheme (UBOS) is a state-of-the-art VQE method that works by sweeping over gates and finding optimal parameters for each gate in the environment of other gates. UBOS improves the convergence time to the ground state by an order of magnitude over stochastic gradient descent. It nonetheless suffers in both rate of convergence and final converged energies in the face of highly noisy expectation values coming from shot noise. Here we develop two classical postprocessing techniques that improve UBOS especially when measurements have large shot noise. Using Gaussian process regression, we generate artificial augmented data using original data from the quantum computer to reduce the overall error when solving for the improved parameters. Using double robust optimization plus rejection, we prevent outlying data which are atypically noisy from resulting in a particularly erroneous single optimization step, thereby increasing robustness against noisy measurements. Combining these techniques further reduces the final relative error that UBOS reaches by a factor of 3 without adding additional quantum measurement or sampling overhead. This work further demonstrates that developing techniques that use classical resources to postprocess quantum measurement results can significantly improve VQE algorithms.
Variational quantum eigensolvers (VQEs) are a promising approach for finding the classically intractable ground state of a Hamiltonian. The unitary block-optimization scheme (UBOS) is a state-of-the-art VQE method that works by sweeping over gates and finding optimal parameters for each gate in the environment of other gates. UBOS improves the convergence time to the ground state by an order of magnitude over stochastic gradient descent. It nonetheless suffers in both rate of convergence and final converged energies in the face of highly noisy expectation values coming from shot noise. Here we develop two classical postprocessing techniques that improve UBOS especially when measurements have large shot noise. Using Gaussian process regression, we generate artificial augmented data using original data from the quantum computer to reduce the overall error when solving for the improved parameters. Using double robust optimization plus rejection, we prevent outlying data which are atypically noisy from resulting in a particularly erroneous single optimization step, thereby increasing robustness against noisy measurements. Combining these techniques further reduces the final relative error that UBOS reaches by a factor of 3 without adding additional quantum measurement or sampling overhead. This work further demonstrates that developing techniques that use classical resources to postprocess quantum measurement results can significantly improve VQE algorithms.
Variational Quantum Eigensolvers (VQE) are a promising approach for finding the classically intractable ground state of a Hamiltonian. The Unitary Block Optimization Scheme (UBOS) is a state-of-the-art VQE method which works by sweeping over gates and finding optimal parameters for each gate in the environment of other gates. UBOS improves the convergence time to the ground state by an order of magnitude over Stochastic Gradient Descent (SGD). It nonetheless suffers in both rate of convergence and final converged energies in the face of highly noisy expectation values coming from shot noise. Here we develop two classical post-processing techniques which improve UBOS especially when measurements have large noise. Using Gaussian Process Regression (GPR), we generate artificial augmented data using original data from the quantum computer to reduce the overall error when solving for the improved parameters. Using Double Robust Optimization plus Rejection (DROPR), we prevent outlying data which are atypically noisy from resulting in a particularly erroneous single optimization step thereby increasing robustness against noisy measurements. Combining these techniques further reduces the final relative error that UBOS reaches by a factor of three without adding additional quantum measurement or sampling overhead. This work further demonstrates that developing techniques which use classical resources to post-process quantum measurement results can significantly improve VQE algorithms.
Variational quantum algorithms are a promising hybrid framework for solving chemistry and physics problems with broad applicability to optimization as well. They are particularly well suited for noisy intermediate-scale quantum computers. In this paper we describe the unitary block optimization scheme (UBOS) and apply it to two variational quantum algorithms: the variational quantum eigensolver (VQE) and variational time evolution. The goal of VQE is to optimize a classically intractable parameterized quantum wave function to target a physical state of a Hamiltonian or solve an optimization problem. UBOS is an alternative to other VQE optimization schemes with a number of advantages, including fast convergence, less sensitivity to barren plateaus, the ability to tunnel through some local minima, and no hyperparameters to tune. We additionally describe how UBOS applies to real and imaginary time-evolution (TUBOS).
Variational Quantum Eigensolvers (VQEs) are a powerful class of hybrid quantum-classical algorithms designed to approximate the ground state of a quantum system described by its Hamiltonian. VQEs hold promise for various applications, including lattice field theory. However, the inherent noise of Noisy Intermediate-Scale Quantum (NISQ) devices poses a significant challenge for running VQEs as these algorithms are particularly susceptible to noise, e.g., measurement shot noise and hardware noise. In a recent work, it was proposed to enhance the classical optimization of VQEs with Gaussian Processes (GPs) and Bayesian Optimization, as these machine-learning techniques are well-suited for handling noisy data. In these proceedings, we provide additional insights into this new algorithm and present further numerical experiments. In particular, we examine the impact of hardware noise and error mitigation on the algorithm's performance. We validate the algorithm using classical simulations of quantum hardware, including hardware noise benchmarks, which have not been considered in previous works. Our numerical experiments demonstrate that GP-enhanced algorithms can outperform state-of-the-art baselines, laying the foundation for future research on deploying these techniques to real quantum hardware and lattice field theory setups.
While quantum computers are a very promising tool for the far future, in their current state of the art they remain limited both in size and quality. This has given rise to hybrid quantum-classical algorithms, where the quantum device performs only a small but vital part of the overall computation. Among these, variational quantum algorithms (VQAs), which combine a classical optimization procedure with quantum evaluation of a cost function, have emerged as particularly promising. However, barren plateaus and ill-conditioned optimization landscapes remain among the primary obstacles faced by VQAs, often leading to unstable convergence and high sensitivity to initialization. Motivated by this challenge, we investigate whether a purely classical remedy, standard L2 squared-norm regularization, can systematically stabilize hybrid quantum-classical optimization. Specifically, we augment the Variational Quantum Eigensolver (VQE) objective with a quadratic penalty proportional to the squared norm of the parameters, without modifying the quantum circuit or measurement process. Across all tested Hamiltonians, H2, LiH, and the Random Field Ising Model (RFIM), we observe improved performance over a broad window of the regularization strength. Our large-scale numerical results demonstrate that classical regularization provides a robust, system-independent mechanism for mitigating VQE instability, enhancing the reliability and reproducibility of variational quantum optimization without altering the underlying quantum circuit.
Abstract The variational quantum eigensolver (VQE) is a hybrid quantum–classical algorithm for preparing ground states in the current era of noisy devices. The classical component of the algorithm requires a large number of measurements on intermediate parameter values that are typically discarded. However, intermediate steps across many calculations can contain valuable information about the relationship between the quantum circuit parameters, resultant measurements, and noise specific to the device. This information would be useful when analysing families of Hamiltonians such as for potential energy landscapes for molecular interaction with catalytic surfaces or confinement in metal–organic-frameworks. In this work, we use supervised machine learning on the intermediate parameter and measurement data to amortise prediction of optimal final parameters across families of Hamiltonians. Our technique optimises parameters leading to chemically accurate ground state energies much faster than conventional techniques. It requires significantly fewer iterations and simultaneously shows the ability to find noise-aware angles if trained on noisy devices even with noise drift. We demonstrate this technique on IBM quantum devices by predicting ground state energies of H 2 for one and two qubits; H 3 for three qubits; and HeH + for four qubits where it finds optimal angles using only modelled data for training. Our technique amortises classical optimisation across multiple VQE runs, resulting in significantly faster overall classical optimisation without additional quantum overhead compared to running VQE.
Variational quantum algorithms (VQAs) offer the most promising path to obtaining quantum advantages via noisy intermediate-scale quantum (NISQ) processors. Such systems leverage classical optimization to tune the parameters of a parameterized quantum circuit (PQC). The goal is minimizing a cost function that depends on measurement outputs obtained from the PQC. Optimization is typically implemented via stochastic gradient descent (SGD). On NISQ computers, gate noise due to imperfections and decoherence affects the stochastic gradient estimates by introducing a bias. Quantum error mitigation (QEM) techniques can reduce the estimation bias without requiring any increase in the number of qubits, but they in turn cause an increase in the variance of the gradient estimates. This work studies the impact of quantum gate noise on the convergence of SGD for the variational eigensolver (VQE), a fundamental instance of VQAs. The main goal is ascertaining conditions under which QEM can enhance the performance of SGD for VQEs. It is shown that quantum gate noise induces a nonzero error-floor on the convergence error of SGD (evaluated with respect to a reference noiseless PQC), which depends on the number of noisy gates, the strength of the noise, as well as the eigenspectrum of the observable being measured and minimized. In contrast, with QEM, any arbitrarily small error can be obtained. Furthermore, for error levels attainable with or without QEM, QEM can reduce the number of required iterations, but only as long as the quantum noise level is sufficiently large, and a sufficiently large number of measurements is allowed at each SGD iteration. Numerical examples for a max-cut problem corroborate the main theoretical findings.