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.
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.
The variational quantum eigensolver (VQE) is a hybrid algorithm that has the potential to provide a quantum advantage in practical chemistry problems that are currently intractable on classical computers. VQE trains parameterized quantum circuits using a classical optimizer to approximate the eigenvalues and eigenstates of a given Hamiltonian. However, VQE faces challenges in task-specific design and machine-specific architecture, particularly when running on noisy quantum devices. This can have a negative impact on its trainability, accuracy, and efficiency, resulting in noisy quantum data. We propose variational denoising, an unsupervised learning method that employs a parameterized quantum neural network to improve the solution of VQE by learning from noisy VQE outputs. Our approach can significantly decrease energy estimation errors and increase fidelities with ground states compared to noisy input data for the H2 , LiH, and BeH2 molecular Hamiltonians and the transverse field Ising model. Surprisingly, it only requires noisy data for training. Variational denoising can be integrated into quantum hardware, increasing its versatility as an end-to-end quantum processing for quantum data. Published by the American Physical Society 2024
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.
New approaches into computational quantum chemistry can be developed through the use of quantum computing. While universal, fault-tolerant quantum computers are still not available, and we want to utilize today's noisy quantum processors. One of their flagship applications is the variational quantum eigensolver (VQE)-an algorithm for calculating the minimum energy of a physical Hamiltonian. In this study, we investigate how various types of errors affect the VQE and how to efficiently use the available resources to produce precise computational results. We utilize a simulator of a noisy quantum device, an exact statevector simulator, and physical quantum hardware to study the VQE algorithm for molecular hydrogen. We find that the optimal method of running the hybrid classical-quantum optimization is to: (i) allow some noise in intermediate energy evaluations, using fewer shots per step and fewer optimization iterations, but ensure a high final readout precision; (ii) emphasize efficient problem encoding and ansatz parametrization; and (iii) run all experiments within a short time-frame, avoiding parameter drift with time. Nevertheless, current publicly available quantum resources are still very noisy and scarce/expensive, and even when using them efficiently, it is quite difficult to perform trustworthy calculations of molecular energies.
In recent times, a variety of hybrid quantum-classical algorithms have been developed that aim to calculate the ground state energies of molecular systems on Noisy Intermediate-Scale Quantum (NISQ) devices. Albeit the utilization of shallow depth circuits in these algorithms, the optimization of ansatz parameters necessitates a substantial number of quantum measurements, leading to prolonged runtimes on the scantly available quantum hardware. Through our work, we lay the general foundation for an interdisciplinary approach that can be used to drastically reduce the dependency of these algorithms on quantum infrastructure. We showcase these pertinent concepts within the framework of the recently developed Projective Quantum Eigensolver (PQE) that involves iterative optimization of the nonlinearly coupled parameters through repeated residue measurements on quantum hardware. We demonstrate that one may perceive such a nonlinear optimization problem as a collective dynamic interplay of fast and slow relaxing modes. As such, the synergy among the parameters is exploited using an on-the-fly supervised machine learning protocol that dynamically casts the PQE optimization into a smaller subspace by reducing the effective degrees of freedom. We demonstrate analytically and numerically that our proposed methodology ensures a drastic reduction in the number of quantum measurements necessary for the parameter updates without compromising the characteristic accuracy. Furthermore, the machine learning model may be tuned to capture the noisy data of NISQ devices, and thus the predicted energy is shown to be resilient under a given noise model.
As a hybrid quantum-classical algorithm, the variational quantum eigensolver is widely applied in quantum chemistry simulations, especially in computing the electronic structure of complex molecular systems. However, on existing noisy intermediate-scale quantum devices, some factors such as quantum decoherence, measurement errors, and gate operation imprecisions are unavoidable. To overcome these challenges, this study proposes an efficient noise-mitigating variational quantum eigensolver for accurate computation of molecular ground state energies in noisy environments. We design the quantum circuit with reference to the structure of matrix product states and utilize it to pre-train the circuit parameters, which ensures circuit stability and mitigates fluctuations caused by initialization. We also employ zero-noise extrapolation to mitigate quantum noise and combine it with neural networks to improve the accuracy of the noise-fitting function, which significantly eliminates noise interference. Furthermore, we implement an intelligent grouping strategy for measuring Hamiltonian Pauli strings, which not only reduces measurement errors but also improves sampling efficiency. We perform numerical simulations to solve the ground state energy of the H4molecule by using MindSpore Quantum framework, and the results demonstrate that our algorithm can constrain noise errors within the range of O(10−2)∼O(10−1), outperforming mainstream variational quantum eigensolvers. This work provides a new strategy for high-precision quantum chemistry calculations on near-term noisy quantum hardware.
The Variational Quantum Eigensolver (VQE) is one of the most promising algorithms for current quantum devices. It employs a classical optimizer to iteratively update the parameters of a variational quantum circuit in order to search for the ground state of a given Hamiltonian. The efficacy of VQEs largely depends on the optimizer employed. Recent studies suggest that Quantum Natural Gradient Descent (QNG) can achieve faster convergence than vanilla gradient descent (VG), but at the cost of additional quantum resources to estimate Fubini-Study metric tensor in each optimization step. The Fubini-Study metric tensor used in QNG is related to the entire quantum state space and does not incorporate information about the target Hamiltonian. To take advantage of the structure of the Hamiltonian and address the limitation of additional computational cost in QNG, we propose Hamiltonian-aware Quantum Natural Gradient Descent (H-QNG). In H-QNG, we propose to use the Riemannian pullback metric induced from the lower-dimensional subspace spanned by the Hamiltonian terms onto the parameter space. We show that H-QNG inherits the desirable features of both approaches: the low quantum computational cost of VG and the reparameterization-invariance of QNG. We also validate its performance through numerical experiments on molecular Hamiltonians, showing that H-QNG achieves faster convergence to chemical accuracy while requiring fewer quantum computational resources.