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arXiv:2407.15629

Concurrent VQE for Simulating Excited States of the Schwinger Model

Yibin Guo, Takis Angelides, Karl Jansen, Stefan Kühn

Published Jul 22, 2024• 2 citations
arXivHTMLPDF

Abstract

This work explores the application of the concurrent variational quantum eigensolver (cVQE) for computing excited states of the Schwinger model. By designing suitable ansatz circuits utilizing universal SO(4) or SO(8) qubit gates, we demonstrate how to efficiently obtain the lowest two, four, and eight eigenstates with one, two, and three ancillary qubits for both vanishing and non-vanishing background electric field cases. Simulating the resulting quantum circuits classically with tensor network techniques, we demonstrate the capability of our approach to compute the two lowest eigenstates of systems with up to $\mathcal{O}(100)$ qubits. Given that our method allows for measuring the low-lying spectrum precisely, we also present a novel technique for estimating the additive mass renormalization of the lattice based on the energy gap. As a proof-of-principle calculation, we prepare the ground and first-excited states with one ancillary and four physical qubits on quantum hardware, demonstrating the practicality of using the cVQE to simulate excited states.

DOI
10.48550/arxiv.2407.15629

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This work explores the application of the concurrent variational quantum eigensolver (cVQE) for computing excited states of the Schwinger model. By designing suitable ansatz circuits utilizing universal SO(4) or SO(8) qubit gates, we demonstrate how to efficiently obtain the lowest two, four, and eight eigenstates with one, two, and three ancillary qubits for both vanishing and non-vanishing background electric field cases. Simulating the resulting quantum circuits classically with tensor network techniques, we demonstrate the capability of our approach to compute the two lowest eigenstates of systems with up to O(100)\mathcal{O}(100)O(100) qubits. Given that our method allows for measuring the low-lying spectrum precisely, we also present a novel technique for estimating the additive mass renormalization of the lattice based on the energy gap. As a proof-of-principle calculation, we prepare the ground and first-excited states with one ancillary and four physical qubits on quantum hardware, demonstrating the practicality of using the cVQE to simulate excited states.
The Variational Quantum Eigensolver (VQE), as a hybrid quantum-classical algorithm, is an important tool for effective quantum computing in the current noisy intermediate-scale quantum (NISQ) era. However, the traditional hardware-efficient ansatz without taking into account symmetries requires more computational resources to explore the unnecessary regions in the Hilbert space. The conventional Subspace-Search VQE (SSVQE) algorithm, which can calculate excited states, is also unable to effectively handle degenerate states since the loss function only contains the expectation value of the Hamiltonian. In this study, the energy eigenstates of the one-dimensional Fermi-Hubbard model with two lattice sites and the two-dimensional Hubbard model with four lattice sites are calculated. By incorporating symmetries into the quantum circuits and loss function, we find that both the ground state and excited state calculations are improved greatly compared to the case without symmetries. The enhancement in excited state calculations is particularly significant. This is because quantum circuits that conserve the particle number are used, and appropriate penalty terms are added to the loss function, enabling the optimization process to correctly identify degenerate states. The results are verified through repeated simulations.
We propose an ansatz quantum circuit for the variational quantum eigensolver (VQE), suitable for exploring the phase structure of the multiflavor Schwinger model in the presence of a chemical potential. Our ansatz is capable of incorporating relevant model symmetries via constrains on the parameters, and can be implemented on circuit-based as well as measurement-based quantum devices. We show via classical simulation of the VQE that our ansatz is able to capture the phase structure of the model, and can approximate the ground state to a high level of accuracy. Moreover, we perform proof-of-principle simulations on superconducting, gate-based quantum hardware. Our results show that our approach is suitable for current gate-based quantum devices, and can be readily implemented on measurement-based quantum devices once available. Published by the American Physical Society 2024
We propose a momentum-space based variational quantum eigensolver (VQE) framework for simulating quasiparticle excitations in interacting quantum many-body systems on near-term quantum devices. Leveraging translational invariance and other symmetries of the Hamiltonian, we reconstruct the momentum-resolved quasiparticle excitation spectrum through targeted simulation of low-lying excited states using VQE. We construct a translationally symmetric variational ansatz designed to evolve a free-fermion particle-hole excited state with definite momentum qqq to an excited state of the interacting system at the same momentum, employing a fermionic fast Fourier transform (FFFT) circuit coupled to a Hamiltonian Variational Ansatz (HVA) circuit. Even though the particle number is not explicitly conserved in the variational ansatz, the correct quasiparticle state is reached by energetic optimization. We benchmark the performance of the proposed VQE implementation on the XXZ Hamiltonian, which maps onto the Tomonaga-Luttinger liquid in the fermionic representation. Our numerical results show that VQE can capture the low-lying excitation spectrum of the bosonic quasiparticle/two-spinon dispersion of this model at various interaction strengths. We estimate the renormalized velocity of the quasiparticles by calculating the slope of the dispersion near zero momentum using the VQE-optimized energies at different system sizes, and demonstrate that it closely matches theoretical results obtained from Bethe ansatz. Finally, we highlight extensions of our proposed VQE implementation to simulate quasiparticles in other interacting quantum many-body 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.
The state-of-the-art quantum computing hardware has entered the noisy intermediate-scale quantum (NISQ) era. Having been constrained by the limited number of qubits and shallow circuit depth, NISQ devices have nevertheless demonstrated the potential of applications on various subjects. One example is the variational quantum eigensolver (VQE) that was first introduced for computing ground states. Although VQE has now been extended to the study of excited states, the algorithms previously proposed involve a recursive optimization scheme which requires many extra operations with significantly deeper quantum circuits to ensure the orthogonality of different trial states. Here we propose a new algorithm to determine many low energy eigenstates simultaneously. By introducing ancillary qubits to purify the trial states so that they keep orthogonal to each other throughout the whole optimization process, our algorithm allows these states to be efficiently computed in one quantum circuit. Our algorithm reduces significantly the complexity of circuits and the readout errors, and enables flexible post-processing on the eigen-subspace from which the eigenpairs can be accurately determined. We demonstrate this algorithm by applying it to the transverse Ising model. By comparing the results obtained using this variational algorithm with the exact ones, we find that the eigenvalues of the Hamiltonian converge quickly with the increase of the circuit depth. The accuracies of the converged eigenvalues are of the same order, which implies that the difference between any two eigenvalues can be more accurately determined than the eigenvalues themselves.
Abstract We explore the first-order phase transition in the lattice Schwinger model in the presence of a topological θ-term by means of the variational quantum eigensolver (VQE). Using two different fermion discretizations, Wilson and staggered fermions, we develop parametric ansatz circuits suitable for both discretizations, and compare their performance by simulating classically an ideal VQE optimization in the absence of noise. The states obtained by the classical simulation are then prepared on the IBM's superconducting quantum hardware. Applying state-of-the art error-mitigation methods, we show that the electric field density and particle number, observables which reveal the phase structure of the model, can be reliably obtained from the quantum hardware. To investigate the minimum system sizes required for a continuum extrapolation, we study the continuum limit using matrix product states, and compare our results to continuum mass perturbation theory. We demonstrate that taking the additive mass renormalization into account is vital for enhancing the precision that can be obtained with smaller system sizes. Furthermore, for the observables we investigate we observe universality, and both fermion discretizations produce the same continuum limit.