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arXiv:2109.08420 · Physical Review A

Using gradient-based algorithms to determine ground-state energies on a quantum computer

Tomislav Piskor, Jan-Michael Reiner, Sebastian Zanker, Nicolas Vogt, Michael Marthaler, Frank K. Wilhelm, F. G. Eich

Published Jun 8, 2022• 5 citations
arXivHTMLPDF

Abstract

Variational algorithms are promising candidates to be implemented on near-term quantum computers. In our work, we investigate the variational Hamiltonian ansatz (VHA), where a parametrized trial state of the quantum mechanical wave function is optimized to obtain the ground-state energy. In the VHA, the trial state is given by a noninteracting reference state modified by unitary rotations using generators that are part of the Hamiltonian describing the system. The lowest energy is obtained by optimizing the angles of those unitary rotations. A standard procedure to optimize the variational parameters is to use gradient-based algorithms. However, shot noise and the intrinsic noise of the quantum device affect the evaluation of the required gradients. We study how different methods for obtaining the gradient, specifically the finite-difference and the parameter-shift rule, are affected by shot noise and the noise of the quantum computer. To this end, we simulate a simple quantum circuit, as well as the two-site and six-site Hubbard models.

DOI
10.1103/physreva.105.062415

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Variational algorithms are promising candidates to be implemented on near-term quantum computers. In our work, we investigate the variational Hamiltonian ansatz (VHA), where a parametrized trial state of the quantum mechanical wave function is optimized to obtain the ground-state energy. In the VHA, the trial state is given by a noninteracting reference state modified by unitary rotations using generators that are part of the Hamiltonian describing the system. The lowest energy is obtained by optimizing the angles of those unitary rotations. A standard procedure to optimize the variational parameters is to use gradient-based algorithms. However, shot noise and the intrinsic noise of the quantum device affect the evaluation of the required gradients. We study how different methods for obtaining the gradient, specifically the finite-difference and the parameter-shift rule, are affected by shot noise and the noise of the quantum computer. To this end, we simulate a simple quantum circuit, as well as the two-site and six-site Hubbard models.
Abstract A key goal of digital quantum computing is the simulation of fermionic systems such as molecules or the Hubbard model. Unfortunately, for present and near-future quantum computers the use of quantum error correction schemes is still out of reach. Hence, the finite error rate limits the use of quantum computers to algorithms with a low number of gates. The variational Hamiltonian ansatz (VHA) has been shown to produce the ground state in good approximation in a manageable number of steps. Here we study explicitly the effect of gate errors on its performance. The VHA is inspired by the adiabatic quantum evolution under the influence of a time-dependent Hamiltonian, where the—ideally short—fixed Trotter time steps are replaced by variational parameters. The method profits substantially from quantum variational error suppression, e.g. unitary quasi-static errors are mitigated within the algorithm. We test the performance of the VHA when applied to the Hubbard model in the presence of unitary control errors on quantum computers with realistic gate fidelities.
One of the most promising applications for near term quantum computers is the simulation of physical quantum systems, particularly many-electron systems in chemistry and condensed matter physics. In solid state physics, finding the correct symmetry broken ground state of an interacting electron system is one of the central challenges. The Variational Hamiltonian Ansatz (VHA), a variational hybrid quantum-classical algorithm especially suited for finding the ground state of a solid state system, will in general not prepare a broken symmetry state unless the initial state is chosen to exhibit the correct symmetry. In this work, we discuss three variations of the VHA designed to find the correct broken symmetry states close to a transition point between different orders. As a test case we use the two-dimensional Hubbard model where we break the symmetry explicitly by means of external fields coupling to the Hamiltonian and calculate the response to these fields. For the calculation we simulate a gate-based quantum computer and also consider the effects of dephasing noise on the algorithms. We find that two of the three algorithms are in good agreement with the exact solution for the considered parameter range. The third algorithm agrees with the exact solution only for a part of the parameter regime, but is more robust with respect to dephasing compared to the other two algorithms.
The opportunities afforded by near-term quantum computers to calculate the ground-state properties of small molecules depend on the structure of the computational ansatz as well as the errors induced by device noise. Here we investigate the behavior of these noisy quantum circuits using numerical simulations to estimate the accuracy and fidelity of the prepared quantum states relative to the ground truth obtained by conventional means. We implement several different types of ansatz circuits derived from unitary coupled cluster theory for the purposes of estimating the ground-state energy of sodium hydride using the variational quantum eigensolver algorithm. We show how relative error in the energy and the fidelity scale with the levels of gate-based noise, the internuclear configuration, the ansatz circuit depth, and the parameter optimization methods.
The preparation of quantum states using short quantum circuits is one of the most promising near-term applications of small quantum computers, especially if the circuit is short enough and the fidelity of gates high enough that it can be executed without quantum error correction. Such quantum state preparation can be used in variational approaches, optimizing parameters in the circuit to minimize the energy of the constructed quantum state for a given problem Hamiltonian. For this purpose we propose a simple-to-implement class of quantum states motivated by adiabatic state preparation. We test its accuracy and determine the required circuit depth for a Hubbard model on ladders with up to 12 sites (24 spin orbitals), and for small molecules. We find that this ansatz converges faster than previously proposed schemes based on unitary coupled clusters. While the required number of measurements is astronomically large for quantum chemistry applications to molecules, applying the variational approach to the Hubbard model (and related models) is found to be far less demanding and potentially practical on small quantum computers. We also discuss another application of quantum state preparation using short quantum circuits, to prepare trial ground states of models faster than using adiabatic state preparation.
Abstract The opportunities afforded by near-term quantum computers to calculate the ground-state properties of small molecules depend on the structure of the computational ansatz as well as the errors induced by device noise. Here we investigate the behavior of these noisy quantum circuits using numerical simulations to estimate the accuracy and fidelity of the prepared quantum states relative to the ground truth obtained by conventional means. We implement several different types of ansatz circuits derived from unitary coupled cluster theory for the purposes of estimating the ground-state energy of sodium hydride using the variational quantum eigensolver algorithm. We show how relative error in the energy and the fidelity scale with the levels of gate-based noise, the internuclear configuration, the ansatz circuit depth, and the parameter optimization methods.
Classical optimizers play a crucial role in determining the accuracy and convergence of variational quantum algorithms; leading algorithms use a near-term quantum computer to solve the ground state properties of molecules, simulate dynamics of different quantum systems, and so on. In the literature, many optimizers, each having its own architecture, have been employed expediently for different applications. In this work, we consider a few popular and efficacious optimizers and assess their performance in variational quantum algorithms for applications in quantum chemistry in a realistic noisy setting. We benchmark the optimizers with critical analysis based on quantum simulations of simple molecules, such as hydrogen, lithium hydride, beryllium hydride, water, and hydrogen fluoride. The errors in the ground state energy, dissociation energy, and dipole moment are the parameters used as yardsticks. All the simulations were carried out with an ideal quantum circuit simulator, a noisy quantum circuit simulator, and finally a noisy simulator with noise embedded from the IBM Cairo quantum device to understand the performance of the classical optimizers in ideal and realistic quantum environments. We used the standard unitary coupled cluster ansatz for simulations, and the number of qubits varied from two starting from the hydrogen molecule to ten qubits in hydrogen fluoride. Based on the performance of these optimizers in the ideal quantum circuits, the conjugate gradient, limited-memory Broyden-Fletcher-Goldfarb-Shanno bound, and sequential least squares programming optimizers are found to be the best-performing gradient-based optimizers. While constrained optimization by linear approximation (COBYLA) and Powell's conjugate direction algorithm for unconstrained optimization (POWELL) perform most efficiently among the gradient-free methods, in noisy quantum circuit conditions, simultaneous perturbation stochastic approximation, POWELL, and COBYLA are among the best-performing optimizers.