Variational quantum eigensolver (VQE) is an appealing candidate for the application of near-term quantum computers. A technique introduced in [Higgot et al., Quantum 3, 156 (2019)], which is named variational quantum deflation (VQD), has extended the ability of the VQE framework for finding excited states of a Hamiltonian. However, no method to evaluate transition amplitudes between the eigenstates found by the VQD without using any costly Hadamard-test-like circuit has been proposed despite its importance for computing properties of the system such as oscillator strengths of molecules. Here we propose a method to evaluate transition amplitudes between the eigenstates obtained by the VQD avoiding any Hadamard-test-like circuit. Our method relies only on the ability to estimate overlap between two states, so it does not restrict to the VQD eigenstates and applies for general situations. To support the significance of our method, we provide a comprehensive comparison of three previously proposed methods to find excited states with numerical simulation of three molecules (lithium hydride, diazene, and azobenzene) in a noiseless situation and find that the VQD method exhibits the best performance among the three methods. Finally, we demonstrate the validity of our method by calculating the oscillator strength of lithium hydride in numerical simulations with shot noise. Our results illustrate the superiority of the VQD to find excited states and widen its applicability to various quantum systems.
Variational quantum eigensolver (VQE) is an appealing candidate for the application of near-term quantum computers. A technique introduced in [Higgot et al., Quantum 3, 156 (2019)], which is named variational quantum deflation (VQD), has extended the ability of the VQE framework for finding excited states of a Hamiltonian. However, no method to evaluate transition amplitudes between the eigenstates found by the VQD without using any costly Hadamard-test-like circuit has been proposed despite its importance for computing properties of the system such as oscillator strengths of molecules. Here we propose a method to evaluate transition amplitudes between the eigenstates obtained by the VQD avoiding any Hadamard-test-like circuit. Our method relies only on the ability to estimate overlap between two states, so it does not restrict to the VQD eigenstates and applies for general situations. To support the significance of our method, we provide a comprehensive comparison of three previously proposed methods to find excited states with numerical simulation of three molecules (lithium hydride, diazene, and azobenzene) in a noiseless situation and find that the VQD method exhibits the best performance among the three methods. Finally, we demonstrate the validity of our method by calculating the oscillator strength of lithium hydride in numerical simulations with shot noise. Our results illustrate the superiority of the VQD to find excited states and widen its applicability to various quantum systems.
Variational quantum eigensolver (VQE) is an appealing candidate for the application of near-term quantum computers. A technique introduced in [Higgot et al., Quantum 3, 156 (2019)], which is named variational quantum deflation (VQD), has extended the ability of the VQE framework for finding excited states of a Hamiltonian. However, no method to evaluate transition amplitudes between the eigenstates found by the VQD without using any costly Hadamard-test-like circuit has been proposed despite its importance for computing properties of the system such as electronic oscillator strengths of molecules. Here we propose a method to evaluate transition amplitudes between the eigenstates obtained by the VQD avoiding any Hadamard-test-like circuit. Our method relies only on the ability to estimate overlap between two states, so it does not restrict to the VQD eigenstates and applies for general situations. To support the significance of our method, we provide a comparison of three previously proposed methods to find excited states with numerical simulation of three molecules (lithium hydride, diazene, and azobenzene) and find that the VQD method exhibits the best performance among the three methods within the scope of noiseless simulation. Finally, we demonstrate the validity of our method by calculating the oscillator strength of lithium hydride, comparing results from numerical simulations and real-hardware experiments on the cloud enabled quantum computer IBMQ Rome. Our results illustrate the superiority of the VQD to find excited states and widen its applicability to various quantum systems.
Within the evolving domain of quantum computational chemistry, the Variational Quantum Eigensolver (VQE) has been developed to explore not only the ground state but also the excited states of molecules. In this study, we compare the performance of Variational Quantum Deflation (VQD) and Subspace-Search Variational Quantum Eigensolver (SSVQE) methods in determining the low-lying excited states of LiH. Our investigation reveals that while VQD exhibits a slight advantage in accuracy, SSVQE stands out for its efficiency, allowing the determination of all low-lying excited states through a single parameter optimization procedure. We further evaluate the effectiveness of optimizers, including Gradient Descent (GD), Quantum Natural Gradient (QNG), and Adam optimizer, in obtaining LiH's first excited state, with the Adam optimizer demonstrating superior efficiency in requiring the fewest iterations. Moreover, we propose a novel approach combining Folded Spectrum VQE (FS-VQE) with either VQD or SSVQE, enabling the exploration of highly excited states. We test the new approaches for finding all three H4's excited states. Folded Spectrum SSVQE (FS-SSVQE) can find all three highly excited states near −1.0 Ha with only one optimizing procedure, but the procedure converges slowly. In contrast, although Folded spectrum VQD (FS-VQD) gets highly excited states with individual optimizing procedures, the optimizing procedure converges faster.
Abstract Variational Quantum Eigensolver (VQE) and its extension, Variational Quantum Deflation (VQD), have emerged as promising algorithms for computing ground and excited state energy eigenvalues of Hamiltonians, particularly in quantum physics, chemistry, and materials science. Recently, the VQE and VQD algorithms were implemented to calculate the electronic band structure described by the tight-binding Hamiltonian. Despite their success for simple models, scalability remains a challenge due to the need for a large number of circuit executions, limited qubit coherence times, and low gate fidelities. In this work, we implemented the VQE and VQD algorithms for tight-binding models of diamond and zincblende crystal structures (Sn, C, Si, Ge, AlP, AlAs, AlSb, and GaP) using a quantum computer simulator. We investigate the scalability challenges, focusing on the significant overhead caused by a large number of quantum circuit executions required to find the energy eigenvalues. Our analysis highlights the substantial amount of both measurement and optimization overheads in VQD, particularly as the number of excited states increases. Based on our findings, we critically discuss the implications of implementing VQD on current noisy intermediate-scale quantum (NISQ) devices.
The variational quantum eigensolver (VQE) is an algorithm to find eigenenergies and eigenstates of systems in quantum chemistry and quantum many-body physics.The VQE is one of the most promising applications of near-term quantum devices to investigate such systems.Here we propose an extension of the VQE to calculate the nonadiabatic couplings of molecules in quantum chemical systems and Berry's phase in quantum many-body systems.Both quantities play an important role in understanding the properties of a system beyond the naive adiabatic picture, e.g., nonadiabatic dynamics and topological phase of matter.We provide quantum circuits and classical postprocessings to calculate the nonadiabatic couplings and Berry's phase.Specifically, we show that the evaluation of the nonadiabatic couplings reduces to that of expectation values of observables while that of Berry's phase also requires one additional Hadamard test.Furthermore, we simulate the photodissociation dynamics of a lithium fluoride molecule using the nonadiabatic couplings evaluated on a real quantum device.Our proposal widens the applicability of the VQE and the possibility of near-term quantum devices to study molecules and quantum many-body systems.
The calculation of excited state energies of electronic structure Hamiltonians has many important applications, such as the calculation of optical spectra and reaction rates. While low-depth quantum algorithms, such as the variational quantum eigenvalue solver (VQE), have been used to determine ground state energies, methods for calculating excited states currently involve the implementation of high-depth controlled-unitaries or a large number of additional samples. Here we show how overlap estimation can be used to deflate eigenstates once they are found, enabling the calculation of excited state energies and their degeneracies. We propose an implementation that requires the same number of qubits as VQE and at most twice the circuit depth. Our method is robust to control errors, is compatible with error-mitigation strategies and can be implemented on near-term quantum computers.
Determining the spectral structure of molecular Hamiltonians is one of the most important and challenging research fields in quantum chemistry. In the near future, such a problem may be solvable by means of quantum simulation. However, constrained by a handful of qubits and noisy quantum gates available in the near-term quantum devices, hybrid quantum-classical algorithms based on variational methods such as variational quantum eigensolver have become a preferable choice. Here we propose an interesting variational quantum algorithm for obtaining the excited states of many-body Hamiltonians. To obtain the kth excited states, the previous variational quantum deflation algorithm requires the use of k different variational circuits, which is now reduced to only two circuits with our algorithm. The feasibility of the algorithm was numerically tested on the hydrogen molecule with and without random noise, where the chemical accuracy is reached for both cases. This algorithm is expected to significantly reduce the resource requirements of quantum simulation.