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Chinese Physics B

Variational quantum eigensolvers by variance minimization

Dan-Bo Zhang, Bin-Lin Chen, Zhan-Hao Yuan, Tao Yin

Published Aug 18, 2022• 42 citations
Publisher

Abstract

The original variational quantum eigensolver (VQE) typically minimizes energy with hybrid quantum-classical optimization that aims to find the ground state. Here, we propose a VQE based on minimizing energy variance and call it the variance-VQE, which treats the ground state and excited states on the same footing, since an arbitrary eigenstate for a Hamiltonian should have zero energy variance. We demonstrate the properties of the variance-VQE for solving a set of excited states in quantum chemistry problems. Remarkably, we show that optimization of a combination of energy and variance may be more efficient to find low-energy excited states than those of minimizing energy or variance alone. We further reveal that the optimization can be boosted with stochastic gradient descent by Hamiltonian sampling, which uses only a few terms of the Hamiltonian and thus significantly reduces the quantum resource for evaluating variance and its gradients.

DOI
10.1088/1674-1056/ac8a8d

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•Jan 1, 2023•0 citations
The original variational quantum eigensolver (VQE) typically minimizes energy with hybrid quantum-classical optimization that aims to find the ground state. Here, we propose a VQE based on minimizing energy variance and call it the variance-VQE, which treats the ground state and excited states on the same footing, since an arbitrary eigenstate for a Hamiltonian should have zero energy variance. We demonstrate the properties of the variance-VQE for solving a set of excited states in quantum chemistry problems. Remarkably, we show that optimization of a combination of energy and variance may be more efficient to find low-energy excited states than those of minimizing energy or variance alone. We further reveal that the optimization can be boosted with stochastic gradient descent by Hamiltonian sampling, which uses only a few terms of the Hamiltonian and thus significantly reduces the quantum resource for evaluating variance and its gradients.
Variational quantum eigensolver(VQE) typically minimizes energy with hybrid quantum-classical optimization, which aims to find the ground state. Here, we propose a VQE by minimizing energy variance, which is called as variance-VQE(VVQE). The VVQE can be viewed as an self-verifying eigensolver for arbitrary eigenstate by designing, since an eigenstate for a Hamiltonian should have zero energy variance. We demonstrate properties and advantages of VVQE for solving a set of excited states with quantum chemistry problems. Remarkably, we show that optimization of a combination of energy and variance may be more efficient to find low-energy excited states than those of minimizing energy or variance alone. We further reveal that the optimization can be boosted with stochastic gradient descent by Hamiltonian sampling, which uses only a few terms of the Hamiltonian and thus significantly reduces the quantum resource for evaluating variance and its gradients.
The zero-energy variance principle can be exploited in variational quantum eigensolvers for solving general eigenstates but its capacity for obtaining a specified eigenstate, such as ground state, is limited as all eigenstates are of zero energy variance. We propose a variance-based variational quantum eigensolver for solving the ground state by searching in an enlarged space of wavefunction and Hamiltonian. With a mutual variance-Hamiltonian optimization procedure, the Hamiltonian is iteratively updated to guild the state towards to the ground state of the target Hamiltonian by minimizing the energy variance in each iteration. We demonstrate the performance and properties of the algorithm with numeral simulations. Our work suggests an avenue for utilizing guided Hamiltonian in hybrid quantum-classical algorithms.
A variational quantum eigensolver (VQE) optimizes parametrized eigenstates of a Hamiltonian on a quantum processor by updating parameters with a classical computer. Such a hybrid quantum-classical optimization serves as a practical way to leverage classical algorithms in exploiting the power of near-term quantum devices. Here, aiming to solve a group of Hamiltonians more efficiently, we develop an extension of the conventional VQE. A snake algorithm is incorporated to couple optimizing processes for VQEs of different Hamiltonians by gradient descent. Such a so-called collective VQE (CVQE) is applied to simulate molecules with varied bond lengths for demonstration. Numeral simulations show that the CVQE exhibits clear collective behavior in the optimization process of updating parameters. Remarkably, the CVQE tends to avoid a single VQE task to be trapped in the local minimum. The collective optimization utilizes intrinsic relations between related tasks and may inspire advanced hybrid quantum-classical algorithms for solving practical problems with current quantum technologies.
The variational quantum eigensolver (VQE) typically optimizes variational parameters in a quantum circuit to prepare eigenstates for a quantum system. Its application to many problems may involve a group of Hamiltonians, e.g., a Hamiltonian of a molecule is a function of the nuclear configurations. In this paper we incorporate derivatives of the Hamiltonian into the VQE and develop some hybrid quantum-classical algorithms, which explore both Hamiltonian and wave-function spaces for solving quantum chemistry problems more efficiently. For geometry optimization, we propose a mutual gradient-descent algorithm that updates the parameters of the Hamiltonian and wave function alternately, which can give a rapid convergence towards equilibrium structures of molecules as demonstrated in numerical simulations. Moreover, to speed up the calculation of the energy potential surface, we establish differential equations that govern how the optimized variational parameters of the wave function change with the intrinsic parameters of the Hamiltonian. Our study suggests a direction for a hybrid quantum-classical algorithm for solving quantum systems more efficiently by considering spaces of both the Hamiltonian and the wave function.
The ability of near-term quantum computers to represent classically-intractable quantum states has brought much interest in using such devices for estimating the ground and excited state energies of fermionic Hamiltonians. The usefulness of such near-term techniques, generally based on the Variational Quantum Eigensolver (VQE), however, is limited by device noise and the need to perform many circuit repetitions. This paper addresses these challenges by generalizing VQE to consider wavefunctions in a subspace spanned by classically tractable states and states that can be prepared on a quantum computer. The manuscript shows how the ground and excited state energies can be estimated using such "classical-boosting" and how this approach can be combined with VQE Hamiltonian decomposition techniques. Unlike existing VQE approaches, the sensitivity to sampling error and device noise approaches zero in the limit where the classically tractable states are able to describe an eigenstate. A detailed analysis of the measurement requirements in the simplest case, where a single computational basis state is used to boost conventional VQE, shows that the ground-state energy estimation of several closed-shell homonuclear diatomic molecules can be accelerated by a factor of approximately 10-1000. The analysis also shows that the measurement reduction of such single basis state boosting, relative to conventional VQE, can be estimated using only the overlap between the ground state and the computational basis state used for boosting.
The variational quantum eigensolver (VQE) is a hybrid quantum-classical algorithm used to find the ground state of a Hamiltonian using variational methods. It has a wide range of potential applications, from quantum chemistry to lattice gauge theories in the Hamiltonian formulation. VQE relies on quantum computers to evaluate the energy of the system in terms of circuit parameters, and it minimizes this parametrized energy with a classical optimization routine. This work describes a Bayesian optimization (BO) algorithm specifically designed to minimize the parametrized energy obtained with a quantum computer. BO based on Gaussian process regression (GPR) is an algorithm for finding the global minimum of a black-box cost function, e.g. the energy, with a very low number of iterations even when using data affected by statistical noise. Furthermore, the GPR procedure developed for this work proved to be very versatile as we also used it to compute discrete integral transforms of noisy data. In particular, this procedure was used to reconstruct parton distribution functions from lattice QCD data.