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.
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.
Abstract The variational quantum eigensolver (VQE) is one of the most promising and widely used algorithms for exploiting the capabilities of current Noisy Intermediate-Scale Quantum (NISQ) devices. However, VQE algorithms suffer from a plethora of issues, such as barren plateaus, local minima, quantum hardware noise, and limited qubit connectivity, thus posing challenges for their successful deployment on hardware and simulators. In this work, we propose a VQE optimization strategy that builds upon recent advances in the literature, and exhibits very shallow circuit depths when applied to the specific system of interest, namely a model Hamiltonian representing a cuprate superconductor. These features make our approach a favorable candidate for generating good ground state approximations on current NISQ devices. Our findings illustrate the potential of VQE algorithmic development for leveraging the full capabilities of NISQ devices.
Quantum Phase Estimation (QPE) is the quantum algorithmic workhorse for computing ground state energies of quantum Hamiltonians with quantum computers. Ground state energy calculation of physical systems is perhaps the most promising use case for quantum computing in terms of scientific and commercial value with a plausible path to outperformance of classical alternatives. This path, however, hinges on the availability of initial states for QPE with significant overlap with the true ground state. Using extensive (classical) numerical computations, we study whether the NISQ-era algorithm VQE (Variational Quantum Eigensolver) could be used to efficiently prepare high-overlap states of disordered fully-connected anisotropic Heisenberg spin glass quantum Hamiltonians with up to 15 qubits. We find that (i) -- consistent with widely held, but rarely numerically illustrated beliefs -- VQE is generally unable to efficiently converge to the ground state for our Hamiltonians, which is a well-known issue with VQE due to a variety of factors including vanishing gradients and local minima; (ii) low energy states do not necessarily have large ground-state overlap, but there is typically a correlation between the two measures; (iii) adding more than three layers to the VQE ansatz neither improves overlap nor the energies found; and (iv) the best-found overlap scaling as a function of the Hamiltonian system size is not strongly exponentially decreasing, suggesting potential for VQE to be a heuristic state preparation algorithm for QPE.
Approximating the ground states of strongly interacting electron systems in quantum chemistry and condensed matter physics is expected to be one of the earliest applications of quantum computers. In this paper, we prepare highly accurate ground states of the Fermi-Hubbard model for small grids of up to six sites (12 qubits) by using an interpretable, adaptive variational quantum eigensolver (VQE) called ADAPT-VQE [H. R. Grimsley et al., Nat. Commun. 10, 3007 (2019)]. In contrast with nonadaptive VQE, this algorithm builds a system-specific ansatz by adding an optimal gate built from one-body or two-body fermionic operators at each step. We show this adaptive method outperforms the nonadaptive counterpart in terms of fewer variational parameters, short gate depth, and scaling with the system size. The fidelity and energy of the prepared state appear to improve asymptotically with ansatz depth. We also demonstrate the application of adaptive variational methods by preparing excited states and Green's functions using a proposed ADAPT-SSVQE algorithm. Lower depth, asymptotic convergence, noise tolerance of a variational approach [A. Peruzzo et al., Nat. Commun. 5, 4213 (2014); J. R. McLean et al., New J. Phys. 18, 023023 (2016); K. Sharma et al., New J. Phys. 22, 043006 (2020)], and a highly controllable, system-specific ansatz make the adaptive variational methods particularly well suited for noisy intermediate-scale quantum devices.
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.
Quantum computers have the potential to deliver speed-ups for solving certain important problems that are intractable for classical counterparts, making them a promising avenue for advancing modern computation. However, many quantum algorithms require deep quantum circuits, which are challenging to implement on current noisy devices. To address this limitation, variational quantum algorithms (VQAs) have been actively developed, enabling practical quantum computing in the noisy intermediate-scale quantum (NISQ) era. Among them, the variational quantum eigensolver (VQE) stands out as a leading approach for solving problems in quantum chemistry, many-body physics, and even integer factorization. The VQE algorithm can be implemented on various quantum hardware platforms, including photonic systems, quantum dots, trapped ions, neutral atoms, and superconducting circuits. In particular, photonic platforms offer several advantages: they operate at room temperature, exhibit low decoherence, and support multiple degrees of freedom, making them suitable for scalable, high-dimensional quantum computation. Here we present methodologies for realizing VQE on photonic systems, highlighting their potential for practical quantum computing. We first provide a theoretical overview of the VQE framework, focusing on the procedure for variationally estimating ground state energies. We then explore how photonic systems can implement these processes, showing that a wide variety of problems can be addressed using either multiple qubit states or a single qudit state.
Abstract Variational quantum algorithms aim at harnessing the power of noisy intermediate-scale quantum (NISQ) computers, by using a classical optimizer to train a parameterized quantum circuit to solve tractable quantum problems. The variational quantum eigensolver (VQE) is one of the aforementioned algorithms designed to determine the ground-state of many-body Hamiltonians. Here, we apply the VQE to study the ground-state properties of N -component fermions. With such knowledge, we study the persistent current of interacting SU( N ) fermions, which is employed to reliably map out the different quantum phases of the system. Our approach lays out the basis for a current-based quantum simulator of many-body systems that can be implemented on NISQ computers.