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arXiv:2106.02509 · APL Quantum

Efficient ground state preparation in variational quantum eigensolver with symmetry-breaking layers

Chae-Yeun Park

Published Feb 23, 2024• 11 citations
arXivHTMLPDF

Abstract

A variational quantum eigensolver (VQE) solves the ground state problem of a given Hamiltonian by finding the parameters of a quantum circuit Ansatz that minimizes the Hamiltonian expectation value. Among possible quantum circuit Ansätze, the Hamiltonian variational Ansatz (HVA) is widely studied for quantum many-body problems as the Ansatz with sufficiently large depth is theoretically guaranteed to express the ground state. However, since the HVA shares the same symmetry with the Hamiltonian, it is not necessarily good at finding symmetry-broken ground states that prevail in nature. In this paper, we systematically explore the limitations of the HVA for solving symmetry-broken systems and propose an alternative quantum circuit Ansatz with symmetry-breaking layers. With extensive numerical simulations, we show that the proposed Ansatz finds the ground state in depth significantly shorter than the bare HVA when the target Hamiltonian has symmetry-broken ground states.

DOI
10.1063/5.0186205

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A variational quantum eigensolver (VQE) solves the ground state problem of a given Hamiltonian by finding the parameters of a quantum circuit Ansatz that minimizes the Hamiltonian expectation value. Among possible quantum circuit Ansätze, the Hamiltonian variational Ansatz (HVA) is widely studied for quantum many-body problems as the Ansatz with sufficiently large depth is theoretically guaranteed to express the ground state. However, since the HVA shares the same symmetry with the Hamiltonian, it is not necessarily good at finding symmetry-broken ground states that prevail in nature. In this paper, we systematically explore the limitations of the HVA for solving symmetry-broken systems and propose an alternative quantum circuit Ansatz with symmetry-breaking layers. With extensive numerical simulations, we show that the proposed Ansatz finds the ground state in depth significantly shorter than the bare HVA when the target Hamiltonian has symmetry-broken ground states.
The variational quantum eigensolver (VQE), which has attracted attention as a promising application of noisy intermediate-scale quantum devices, finds a ground state of a given Hamiltonian by variationally optimizing the parameters of quantum circuits called Ans\"atze. Since the difficulty of the optimization depends on the complexity of the problem Hamiltonian and the structure of the Ans\"atze, it has been difficult to systematically analyze the performance of optimizers for the VQE. To resolve this problem, we propose a technique to construct a benchmark problem whose ground state is guaranteed to be achievable with a given Ansatz by using the idea of a parent Hamiltonian of a low-depth parametrized quantum circuit. We compare the convergence of several optimizers by varying the distance of the initial parameters from the solution and find that the converged energies showed a thresholdlike behavior depending on the distance. This work provides a systematic way to analyze optimizers for the VQE and contribute to the design of the Ansatz and its initial parameters.
Abstract 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. To help finding the correct broken symmetries in the thermodynamic limit methods that allow to determine the groundstate of large but finite interacting electron systems are very useful. 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 symmetry-breaking groundstate of a finite system 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.
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.
Quantum systems have historically been formidable to simulate using classical computational methods, particularly as the system size grows. In recent years, advancements in quantum computing technology have offered new opportunities for tackling complex quantum systems, potentially enabling the study and preparation of quantum states directly on quantum processors themselves. The Variational Quantum Eigensolver (VQE) algorithm is a system composed of a quantum circuit as well as a classical optimizer that can be used to efficiently prepare interesting many-body states on the current noisy intermediate-scale quantum (NISQ) devices. We assess the efficacy and scalability of VQE by preparing the ground states of the 1D generalized Heisenberg model, a pivotal model in understanding magnetic materials. We present an ansatz capable of preparing the ground states for all possible values of the coupling, including the critical states for the anisotropic XXZ model. This paper also aims to provide insights into the precision and time consumption involved in classical and optimized sampling approaches in the calculation of expectation values. In preparing the ground state for the Heisenberg models, this paper paves the way for more efficient quantum algorithms and contributes to the broader field of condensed matter physics.
Finding and probing the ground states of spin lattices, such as the antiferromagnetic Heisenberg model on the kagome lattice (KAFH), is a very challenging problem on classical computers and only possible for relatively small systems. We propose using the variational quantum eigensolver (VQE) to find the ground state of the KAFH on a quantum computer. We find efficient ansatz circuits and show how physically interesting observables can be measured efficiently. To investigate the expressiveness and scaling of our ansatz circuits we used classical, exact simulations of VQE for the KAFH for different lattices ranging from 8 to 24 qubits. We find that the fidelity with the ground state approaches one exponentially in the circuit depth for all lattices considered, except for a 24-qubit lattice with an almost degenerate ground state. We conclude that VQE circuits that are able to represent the ground state of the KAFH on lattices inaccessible to exact diagonalization techniques may be achievable on near-term quantum hardware. However, for large systems, circuits with many variational parameters are needed to achieve high fidelity with the ground state.
We propose a scheme to restore spatial symmetry of Hamiltonian in the variational-quantum-eigensolver (VQE) algorithm for which the quantum circuit structures used usually break the Hamiltonian symmetry. The symmetry-adapted VQE scheme introduced here simply applies the projection operator, which is Hermitian but not unitary, to restore the spatial symmetry in a desired irreducible representation of the spatial group. The entanglement of a quantum state is still represented in a quantum circuit but the nonunitarity of the projection operator is treated classically as postprocessing in the VQE framework. By numerical simulations for a spin-1/21/21/2 Heisenberg model on a one-dimensional ring, we demonstrate that the symmetry-adapted VQE scheme with a shallower quantum circuit can achieve significant improvement in terms of the fidelity of the ground state and has a great advantage in terms of the ground-state energy with decent accuracy, as compared to the non-symmetry-adapted VQE scheme. We also demonstrate that the present scheme can approximate low-lying excited states that can be specified by symmetry sectors, using the same circuit structure for the ground-state calculation.