This paper explores the potential of Variational Quantum Eigensolvers (VQEs) for simulating classical dynamical systems. Traditional numerical methods for simulating dynamical systems often struggle with complex systems exhibiting sparse interactions or requiring high computational resources. We propose a novel algorithm that utilizes the optimization capabilities of VQEs to approximate the time evolution of these systems. The core idea is to map the system's state vector onto a qubit representation and then iteratively minimize the energy functional derived from the dynamical system's equations, effectively simulating the system's evolution. We specifically target systems with sparse interactions, where the optimization landscape is expected to be more amenable to VQE's strengths. This approach offers a potentially scalable solution for simulating dynamical systems on near-term quantum computers. This work demonstrates the feasibility of employing VQEs as a tool for tackling previously intractable problems in fields such as molecular dynamics, fluid dynamics, and chaos theory.
This paper explores the potential of Variational Quantum Eigensolvers (VQEs) for simulating classical dynamical systems. Traditional numerical methods for simulating dynamical systems often struggle with complex systems exhibiting sparse interactions or requiring high computational resources. We propose a novel algorithm that utilizes the optimization capabilities of VQEs to approximate the time evolution of these systems. The core idea is to map the system's state vector onto a qubit representation and then iteratively minimize the energy functional derived from the dynamical system's equations, effectively simulating the system's evolution. We specifically target systems with sparse interactions, where the optimization landscape is expected to be more amenable to VQE's strengths. This approach offers a potentially scalable solution for simulating dynamical systems on near-term quantum computers. This work demonstrates the feasibility of employing VQEs as a tool for tackling previously intractable problems in fields such as molecular dynamics, fluid dynamics, and chaos theory.
Quantum mechanics has introduced a new theoretical framework for the study of molecules, enabling the prediction of properties and dynamics through the solution of the Schrödinger equation applied to these systems. However, solving this equation is computationally expensive, which has led to the development of various mathematical frameworks and computational methods designed to balance the available resources with the desired level of accuracy. In particular, quantum computers have emerged as a promising technology with the potential to address these problems more efficiently in the coming decades, whether through reductions in memory, time, and energy consumption, i. e., reductions in computational complexity or by enhancing precision. This research field is known as Quantum Simulation. Given the current technological limitations of quantum computers, Variational Quantum Algorithms (VQAs), especially the Variational Quantum Eigensolver (VQE), stand out as a feasible approach to demonstrating advantages over classical methods in the near term. This feasibility arises from their lower demand for quantum gates and the reduced depth of the circuits required for their implementation. In this work, we present the application of quantum mechanics to the study of molecules, provide an introduction to the fundamentals of quantum computing, and explore the integration of these fields by employing the VQE in molecular simulations. Finally, we discuss the spatial and temporal complexity associated with the algorithm, highlighting its implications and challenges.
The Variational Quantum Eigensolver (VQE) is a key algorithm for near-term quantum computers, yet its performance is often limited by the classical optimization of circuit parameters. We propose using the velocity Verlet algorithm, inspired by classical molecular dynamics, to address this challenge. By introducing an inertial "velocity" term, our method efficiently explores complex energy landscapes. We compare its performance against standard optimizers on H2 and LiH molecules. For H2, our method achieves chemical accuracy with fewer quantum circuit evaluations than L-BFGS-B. For LiH, it attains the lowest final energy, demonstrating its potential for high-accuracy VQE simulations.
Variational Quantum Eigensolver (VQE) has emerged as a prominent hybrid quantum-classical algorithm for near-term quantum computing, particularly in applications like quantum machine learning. However, the performance of VQE is significantly constrained by the limitations of classical optimization methods employed to minimize the expectation value of the Hamiltonian. Traditional VQE approaches often suffer from slow convergence and sensitivity to initial conditions, particularly when dealing with complex, high-dimensional problems encountered in machine learning scenarios. This work introduces a novel adaptive quantum computing framework for VQE optimization, designed to dynamically adjust the quantum circuit parameters during the optimization process. By continuously monitoring the optimization progress and intelligently modifying the parameters, we aim to accelerate convergence and improve the robustness of VQE algorithms for machine learning tasks. The core idea is to shift from a static, fixed-parameter VQE to a dynamic, adaptive one. We explore various adaptive control strategies and demonstrate their potential to significantly reduce the number of iterations required to reach a solution, leading to improved computational efficiency and enabling VQE to tackle more complex machine learning problems within the constraints of current quantum hardware. The research focuses on the theoretical underpinnings and the design principles of this adaptive approach, laying the foundation for future implementations and experimental validation.
Abstract Variational Quantum Eigensolver (VQE) addresses challenges that conventional computation struggles to overcome. However, VQE lacks the ability to autonomously trend towards the direction of quantum advantage, leading to unnecessary consumption of time and resources during parameter optimization. In this work, we present a Simulating Dynamic VQE (D-VQE) that incorporates time evolution for achieving desired simulation accuracy and parameter optimization. Introducing causal cones for Parameterized Quantum Circuits (PQC) and establishing dynamic judgment rules. Consequently, we can reduce the overstacking of components in PQC ansatz. Based on the requirements of precision and circuit scale, a reasonable operator pool is set to deal with the parameters. Through numerical experiments, our method effectively simulates the search for the ground state in the transverse-field Ising model, achieving a relative energy error below 10^(-3) and improving ground state approximation capability. Moreover, in the context of optimized Boltzmann machine learning, the Kullback-Leibler divergence (KLD) value converges to 0.03. These results prove that our work provides an effective and feasible consideration for the improvement of variational quantum algorithms.
Variational quantum algorithms (VQAs) hold significant promise for solving complex optimization problems in quantum computing. However, existing VQA approaches often suffer from premature convergence, struggling to effectively explore the high-dimensional optimization landscape and frequently settling into local minima. This paper introduces a novel strategy, Variational Quantum Evolution (VQE), designed to overcome these limitations. VQE utilizes a quantum-classical hybrid approach, continuously evolving the parameters of a quantum circuit through a simulated stochastic evolutionary process. A quantum process operator is employed to generate a range of quantum states, while a classical feedback loop, informed by the quantum measurements, dynamically adjusts the circuit parameters to promote exploration and escape local optima. The core innovation lies in integrating quantum evolution directly within the variational framework, harnessing quantum superposition and interference to drive a more robust and adaptive optimization trajectory. The proposed method offers a potential pathway to unlocking the full capabilities of VQAs by enabling a more thorough exploration of the solution space. We demonstrate, theoretically, the advantages of this approach and outline the key components required for its implementation.
Abstract The variational quantum eigensolver (VQE) is a method that uses a hybrid quantum-classical computational approach to find eigenvalues of a Hamiltonian. VQE has been proposed as an alternative to fully quantum algorithms such as quantum phase estimation (QPE) because fully quantum algorithms require quantum hardware that will not be accessible in the near future. VQE has been successfully applied to solve the electronic Schrödinger equation for a variety of small molecules. However, the scalability of this method is limited by two factors: the complexity of the quantum circuits and the complexity of the classical optimization problem. Both of these factors are affected by the choice of the variational ansatz used to represent the trial wave function. Hence, the construction of an efficient ansatz is an active area of research. Put another way, modern quantum computers are not capable of executing deep quantum circuits produced by using currently available ansatzes for problems that map onto more than several qubits. In this review, we present recent developments in the field of designing efficient ansatzes that fall into two categories—chemistry–inspired and hardware–efficient—that produce quantum circuits that are easier to run on modern hardware. We discuss the shortfalls of ansatzes originally formulated for VQE simulations, how they are addressed in more sophisticated methods, and the potential ways for further improvements.