We present and analyze large-scale simulation results of a hybrid quantum-classical variational method to calculate the ground state energy of the anti-ferromagnetic Heisenberg model. Using a massively parallel universal quantum computer simulator, we observe that a low-depth-circuit ansatz advantageously exploits the efficiently preparable Néel initial state, avoids potential barren plateaus, and works for both one- and two-dimensional lattices. The analysis reflects the decisive ingredients required for a simulation by comparing different ansätze, initial parameters, and gradient-based versus gradient-free optimizers. Extrapolation to the thermodynamic limit accurately yields the analytical value for the ground state energy, given by the Bethe ansatz. We predict that a fully functional quantum computer with 100 qubits can calculate the ground state energy with a relatively small error.
We present and analyze large-scale simulation results of a hybrid quantum-classical variational method to calculate the ground state energy of the anti-ferromagnetic Heisenberg model. Using a massively parallel universal quantum computer simulator, we observe that a low-depth-circuit ansatz advantageously exploits the efficiently preparable Néel initial state, avoids potential barren plateaus, and works for both one- and two-dimensional lattices. The analysis reflects the decisive ingredients required for a simulation by comparing different ansätze, initial parameters, and gradient-based versus gradient-free optimizers. Extrapolation to the thermodynamic limit accurately yields the analytical value for the ground state energy, given by the Bethe ansatz. We predict that a fully functional quantum computer with 100 qubits can calculate the ground state energy with a relatively small error.
The variational quantum eigensolver (VQE) is a hybrid quantum classical algorithm designed for current and near-term quantum devices. Despite its initial success, there is a lack of understanding involving several of its key aspects. There are problems with VQE that forbid a favorable scaling towards quantum advantage. In order to alleviate the problems, we propose and extensively test a quantum annealing inspired heuristic that supplements VQE. The improved VQE enables an efficient initial state-preparation mechanism, in a recursive manner, for a quasidynamical unitary evolution. We conduct an in-depth scaling analysis of finding the ground-state energies with increasing lattice sizes of the Heisenberg model, employing simulations of up to 40 qubits that manipulate the complete state vector. In addition to systematically finding the ground-state energy, we observe that it avoids barren plateaus, escapes local minima, and works with low-depth circuits. For the current devices, we further propose a benchmarking toolkit using a mean-field model and test it on IBM Q devices. Realistic gate execution times estimate a longer computational time to complete the same computation on a fully functional error-free quantum computer than on a quantum computer emulator implemented on a classical computer. However, our proposal can be expected to help accurate estimations of the ground-state energies beyond 50 qubits when the complete state vector can no longer be stored on a classical computer, thus enabling quantum advantage.
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.
The variational quantum eigensolver (VQE) is a hybrid quantum-classical algorithm designed for current and near-term quantum devices. Despite its initial success, there is a lack of understanding involving several of its key aspects. There are problems with VQE that forbid a favourable scaling towards quantum advantage. In order to alleviate the problems, we propose and extensively test a quantum annealing inspired heuristic that supplements VQE. The improved VQE enables an efficient initial state preparation mechanism, in a recursive manner, for a quasi-dynamical unitary evolution. We conduct an in-depth scaling analysis of finding the ground state energies with increasing lattice sizes of the Heisenberg model, employing simulations of up to 40 qubits that manipulate the complete state vector. For the current devices, we further propose a benchmarking toolkit using a mean-field model and test it on IBM Q devices. The improved VQE avoids barren plateaus, exits local minima, and works with low-depth circuits. Realistic gate execution times estimate a longer computational time to complete the same computation on a fully functional error-free quantum computer than on a quantum computer emulator implemented on a classical computer. However, our proposal can be expected to help accurate estimations of the ground state energies beyond 50 qubits when the complete state vector can no longer be stored on a classical computer, thus enabling quantum advantage.
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.
Simulating the Hubbard model is of great interest to a wide range of applications within condensed matter physics, however its solution on classical computers remains challenging in dimensions larger than one. The relative simplicity of this model, embodied by the sparseness of the Hamiltonian matrix, allows for its efficient implementation on quantum computers, and for its approximate solution using variational algorithms such as the variational quantum eigensolver. While these algorithms have been shown to reproduce the qualitative features of the Hubbard model, their quantitative accuracy in terms of producing true ground state energies and other properties, and the dependence of this accuracy on the system size and interaction strength, the choice of variational ansatz, and the degree of spatial inhomogeneity in the model, remains unknown. Here we present a rigorous classical benchmarking study, demonstrating the potential impact of these factors on the accuracy of the variational solution of the Hubbard model on quantum hardware, for systems with up to 32 qubits. We find that even when using the most accurate wavefunction ansätze for the Hubbard model, the error in its ground state energy and wavefunction plateaus for larger lattices, while stronger electronic correlations magnify this issue. Concurrently, spatially inhomogeneous parameters and the presence of off-site Coulomb interactions only have a small effect on the accuracy of the computed ground state energies. Our study highlights the capabilities and limitations of current approaches for solving the Hubbard model on quantum hardware, and we discuss potential future avenues of research.
The variational approach is a cornerstone of computational physics, considering both conventional and quantum computing computational platforms. The variational quantum eigensolver algorithm aims to prepare the ground state of a Hamiltonian exploiting parametrized quantum circuits that may offer an advantage compared to classical trial states used, for instance, in quantum Monte Carlo or tensor network calculations. While, traditionally, the main focus has been on developing better trial circuits, we show that the algorithm's success, if optimized within stochastic gradient descent (SGD) or quantum natural gradient descent (QNGD), crucially depends on other parameters such as the learning rate, the number Ns of measurements to estimate the gradient components, and the Hamiltonian gap $\mathrm{\ensuremath{\Delta}}$. Within the standard SGD or QNGD, we first observe the existence of a finite Ns value below which the optimization is impossible, and the energy variance resembles the behavior of the specific heat in second-order phase transitions. Second, when Ns is above such threshold level, and learning is possible, we develop a phenomenological model that relates the fidelity of the state preparation with the optimization hyperparameters and $\mathrm{\ensuremath{\Delta}}$. More specifically, we observe that the computational resources scale as $1/{\mathrm{\ensuremath{\Delta}}}^{2}$, and we propose a symmetry enhancement of the variational ansatz as a way to increase the closing gap. We test our understanding on several instances of two-dimensional frustrated quantum magnets, which are believed to be the most promising candidates for near-term quantum advantage through variational quantum simulations.