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arXiv:2304.05262 · Quantum Science and Technology

Sequence of penalties method to study excited states using VQE

Rodolfo Carobene, Stefano Barison, Andrea Giachero

Published May 2, 2023• 4 citations
arXivHTMLPDF

Abstract

Abstract We propose an extension of the variational quantum eigensolver (VQE) that leads to more accurate energy estimations and can be used to study excited states. The method is based on the introduction of a sequence of increasing penalties in the cost function. This approach does not require circuit modifications and thus can be applied with no additional depth cost. Through numerical simulations, we show that we are able to produce variational states with desired physical properties, such as total spin and charge. We assess its performance both on classical simulators and on currently available quantum devices, calculating the potential energy curves of small molecular systems in different physical configurations. Finally, we compare our method to the original VQE and to another extension, obtaining a better agreement with exact simulations for both energy and targeted physical quantities.

DOI
10.1088/2058-9565/acd1a9

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Abstract We propose an extension of the variational quantum eigensolver (VQE) that leads to more accurate energy estimations and can be used to study excited states. The method is based on the introduction of a sequence of increasing penalties in the cost function. This approach does not require circuit modifications and thus can be applied with no additional depth cost. Through numerical simulations, we show that we are able to produce variational states with desired physical properties, such as total spin and charge. We assess its performance both on classical simulators and on currently available quantum devices, calculating the potential energy curves of small molecular systems in different physical configurations. Finally, we compare our method to the original VQE and to another extension, obtaining a better agreement with exact simulations for both energy and targeted physical quantities.
The variational quantum eigensolver (VQE) is a promising algorithm to compute eigenstates and eigenenergies of a given quantum system that can be performed on a near-term quantum computer.Obtaining eigenstates and eigenenergies in a specific symmetry sector of the system is often necessary for practical applications of the VQE in various fields ranging from high-energy physics to quantum chemistry.It is common to add a penalty term in the cost function of the VQE to calculate such a symmetry-resolving energy spectrum; however, systematic analysis of the effect of the penalty term has been lacking, and the use of the penalty term in the VQE has not been justified rigorously.In this paper, we investigate two major types of penalty terms for the VQE that were proposed in previous studies.We show that a penalty term of one of the two types works properly in that eigenstates obtained by the VQE with the penalty term reside in the desired symmetry sector.We further give a convenient formula to determine the magnitude of the penalty term, which may lead to faster convergence of the VQE.Meanwhile, we prove that the other type of penalty term does not work for obtaining the target state with the desired symmetry in a rigorous sense and even gives completely wrong results in some cases.We finally provide numerical simulations to validate our analysis.Our results apply to general quantum systems and lay the theoretical foundation for the use of the VQE with the penalty terms to obtain the symmetry-resolving energy spectrum of the system, which fuels the application of a near-term quantum computer.
High Resolution Image Download MS PowerPoint Slide The recent developments of quantum computing present novel potential pathways for quantum chemistry as the scaling of the computational power of quantum computers could be harnessed to naturally encode and solve electronic structure problems. Theoretically exact quantum algorithms for chemistry have been proposed (e.g., quantum phase estimation), but the limited capabilities of current noisy intermediate-scale quantum devices motivated the development of less demanding hybrid algorithms. In this context, the variational quantum eigensolver (VQE) algorithm was successfully introduced as an effective method to compute the ground-state energies of small molecules. This study investigates the folded spectrum (FS) method as an extension of the VQE algorithm for the computation of molecular excited states. It provides the possibility of directly computing excited states around a selected target energy using the same or quantum circuit as for the ground-state calculation. Inspired by the variance-based methods from the quantum Monte Carlo literature, the FS method minimizes the energy variance, thus, in principle, requiring a computationally expensive squared Hamiltonian to be applied. We alleviate this potentially poor scaling by employing a Pauli grouping procedure to identify sets of commuting Pauli strings that can be evaluated simultaneously. This allows for a significant reduction in the computational cost. We applied the FS-VQE method to small molecules (H 2, LiH), obtaining all electronic excited states with chemical accuracy on ideal quantum simulators. Furthermore, we explore the application of quantum error mitigation techniques, demonstrating improved energy accuracy on noisy simulators compared with simulations without mitigation.
Quantum mechanics has introduced a new theoretical framework for the study of molecules, enabling the prediction of properties and dynamics through the solution of the Schrödinger equation applied to these systems. However, solving this equation is computationally expensive, which has led to the development of various mathematical frameworks and computational methods designed to balance the available resources with the desired level of accuracy. In particular, quantum computers have emerged as a promising technology with the potential to address these problems more efficiently in the coming decades, whether through reductions in memory, time, and energy consumption, i. e.\textit{i. e.}i. e., reductions in computational complexity or by enhancing precision. This research field is known as Quantum Simulation. Given the current technological limitations of quantum computers, Variational Quantum Algorithms (VQAs), especially the Variational Quantum Eigensolver (VQE), stand out as a feasible approach to demonstrating advantages over classical methods in the near term. This feasibility arises from their lower demand for quantum gates and the reduced depth of the circuits required for their implementation. In this work, we present the application of quantum mechanics to the study of molecules, provide an introduction to the fundamentals of quantum computing, and explore the integration of these fields by employing the VQE in molecular simulations. Finally, we discuss the spatial and temporal complexity associated with the algorithm, highlighting its implications and challenges.
Calculations of molecular spectral properties, like photodissociation rates and absorption bands, rely on knowledge of the excited state energies of the molecule of interest. Protocols based on the variational quantum eigensolver (VQE) are promising candidates to calculate such energies on emerging noisy intermediate scale quantum (NISQ) computers. The successful implementation of these protocols on NISQ computers, relies on ansatze that can accurately approximate the molecular states and that can be implemented by shallow quantum circuits. In this paper, we introduce the ``excited iterative qubit-excitation-based variational quantum eigensolver'' (e-IQEB-VQE) protocol to calculate molecular excited state energies. The e-IQEB-VQE constructs efficient problem-tailored ansatze by iteratively appending evolutions of qubit excitation operators. The e-IQEB-VQE is an adaptation of the IQEB-VQE protocol, designed to be less dependent on the choice of an initial reference state. We perform classical numerical simulations for LiH and BeH2_22​ to benchmark the performance of the IQEB-VQE. We demonstrate that the e-IQEB-VQE can construct highly accurate ansatze that require at least an order of magnitude fewer CNOTCNOTCNOTs than standard fixed UCC ansatze, such as the UCCSD and the GUCCSD.
Variational Quantum Algorithms (VQAs) are iterative algorithms suited to implementation on current-era quantum devices. VQAs employ classical optimization to minimize cost functions evaluated on quantum circuits. However, the extent to which VQAs manage noise is often insufficient for quantum chemistry applications. One method of improving VQAs is through accurate ansatz initialization. The CAFQA (Clifford Ansatz For Quantum Accuracy) protocol runs a discrete search through a classically simulatable subset of the entire state space to find a desirable initialization. Prior work has evaluated CAFQA applied to the Variational Quantum Eigensolver (VQE), a VQA that computes grounds states of a Hamiltonian. Motivated by CAFQA's success, we propose Excited-CAFQA initialization for Variational Quantum Deflation (VQD), a quantum algorithm that extends VQE by allowing the computation of excited states. VQD recursively computes excited states, by constraining the kth state to be orthogonal to the previous k-1 computed energy states via a penalty term appended to the standard VQE cost function. Just as with VQE, the VQD cost function can be efficiently computed classically for the states considered in the discrete CAFQA search, allowing for the discrete CAFQA optimizer to find good initial parameters for each energy level computation. Preliminary evaluation shows that Excited-CAFQA achieves 90 to 99+% accuracy across a variety of bond lengths and excited states for H2 and HeH+ molecular systems.
A programmable quantum device that has a large number of qubits without fault-tolerance has emerged recently. Variational quantum eigensolver (VQE) is one of the most promising ways to utilize the computational power of such devices to solve problems in condensed matter physics and quantum chemistry. As the size of the current quantum devices is still not large for rivaling classical computers at solving practical problems, Fujii et al. proposed a method called ``Deep VQE'', which can provide the ground state of a given quantum system with the smaller number of qubits by combining the VQE and the technique of coarse graining [K. Fujii, K. Mitarai, W. Mizukami, and Y. O. Nakagawa, arXiv:2007.10917]. In this paper, we extend the original proposal of Deep VQE to obtain the excited states and apply it to quantum chemistry calculation of a periodic material, which is one of the most impactful applications of the VQE. We first propose a modified scheme to construct quantum states for coarse graining in Deep VQE to obtain the excited states. We also present a method to avoid a problem of meaningless eigenvalues in the original Deep VQE without restricting variational quantum states. Finally, we classically simulate our modified Deep VQE for quantum chemistry calculation of a periodic hydrogen chain as a typical periodic material. Our method reproduces the ground-state energy and the first-excited-state energy with the errors up to O(1)O(1)%O(1) despite the decrease in the number of qubits required for the calculation by two or four compared with the naive VQE. Our result will serve as a beacon for tackling quantum chemistry problems with classically-intractable sizes by smaller quantum devices in the near future.