arXiv:2106.15552 · Journal of Physics A Mathematical and Theoretical
Variational quantum eigensolver for SU( N ) fermions
Mirko Consiglio, Wayne Jordan Chetcuti, Carlos Bravo-Prieto, Sergi Ramos-Calderer, Anna Minguzzi, José Ignacio Latorre, Luigi Amico A, Tony J. G. Apollaro
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.
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.
Variational quantum eigensolvers (VQE) are one of the possible tasks that can be performed in noisy intermediate-scale quantum (NISQ) computers. While one of the limitations of NISQ platforms is their restricted number of available qubits, the theory of compressable matchgate circuits can be used to circumvent this limitation for certain types of Hamiltonians. We show how VQE algorithms can be used to find the ground state of quadratic fermionic Hamiltonians, providing an expressible ansatz in a logarithmic number of qubits. In particular, for systems of n orbitals encoded to 2-local qubit models with nearest neighbour interactions, the ground state energy can be evaluated with O(logn) sets of measurements. This result is invariant of the dimensions in which the n sites are arranged.
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.
We introduce a general framework called neural-network- (NN) encoded variational quantum algorithms (VQAs), or NNVQA for short, to address the challenges of implementing VQAs on noisy intermediate-scale quantum (NISQ) computers. Specifically, NNVQA feeds input (such as parameters of a Hamiltonian) from a given problem to a neural network and uses its outputs to parameterize an ansatz circuit for the standard VQA. Combining the strengths of NN and parameterized quantum circuits, NNVQA can accelerate the training process of VQAs and handle a broad family of related problems with varying input parameters with the pretrained NN. To concretely illustrate the merits of NNVQA, we present results on a NN variational quantum eigensolver (VQE) for solving the ground state of parameterized XXZ spin models in one and two dimensions. Our results demonstrate that NNVQE is able to estimate the ground-state energies of parameterized Hamiltonians with high precision without fine tuning, and significantly reduce the overall training cost to estimate ground-state properties across the phases of the given Hamiltonian. We also employ an active learning strategy to further increase the training efficiency while maintaining prediction accuracy. These encouraging results demonstrate that NNVQAs offer an alternative hybrid quantum-classical paradigm to utilize NISQ resources for solving more realistic and challenging computational problems.
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.
The variational quantum eigensolver (VQE) is a hybrid algorithm that has the potential to provide a quantum advantage in practical chemistry problems that are currently intractable on classical computers. VQE trains parameterized quantum circuits using a classical optimizer to approximate the eigenvalues and eigenstates of a given Hamiltonian. However, VQE faces challenges in task-specific design and machine-specific architecture, particularly when running on noisy quantum devices. This can have a negative impact on its trainability, accuracy, and efficiency, resulting in noisy quantum data. We propose variational denoising, an unsupervised learning method that employs a parameterized quantum neural network to improve the solution of VQE by learning from noisy VQE outputs. Our approach can significantly decrease energy estimation errors and increase fidelities with ground states compared to noisy input data for the H2 , LiH, and BeH2 molecular Hamiltonians and the transverse field Ising model. Surprisingly, it only requires noisy data for training. Variational denoising can be integrated into quantum hardware, increasing its versatility as an end-to-end quantum processing for quantum data. Published by the American Physical Society 2024