In this paper, we present a quantum algorithm that enables us to perform first-principles molecular dynamics. In this algorithm, the optimum of wavefunctions and atomic structures is given by a root of a system of polynomial equations derived from the first principles of quantum mechanics. The computational steps consist of symbolic and quantum parts. First, the symbolic computation prepares a set of matrices whose eigenvalues are the roots of the given equation. Second, the quantum computation evaluates the eigenvectors and eigenvalues. In a preceding work, the authors presented another algorithm directed to the same end, and its deficiency is the complexity of the symbolic computations, which would be enormous in the worst cases. In this study, we try to reduce the cost of symbolic computation, utilizing the latest achievements of classical algorithms. Finally, we show how to transform this classical algorithm into a quantum one.
In this paper, we present a quantum algorithm that enables us to perform first-principles molecular dynamics. In this algorithm, the optimum of wavefunctions and atomic structures is given by a root of a system of polynomial equations derived from the first principles of quantum mechanics. The computational steps consist of symbolic and quantum parts. First, the symbolic computation prepares a set of matrices whose eigenvalues are the roots of the given equation. Second, the quantum computation evaluates the eigenvectors and eigenvalues. In a preceding work, the authors presented another algorithm directed to the same end, and its deficiency is the complexity of the symbolic computations, which would be enormous in the worst cases. In this study, we try to reduce the cost of symbolic computation, utilizing the latest achievements of classical algorithms. Finally, we show how to transform this classical algorithm into a quantum one.
This article shows a proof of concept regarding the feasibility of ab initio molecular simulation, wherein the wavefunctions and the positions of nuclei are simultaneously determined by the quantum algorithm, as is realized by the so-called Car-Parrinello method by classical computing. The approach used in this article is of a hybrid style, which shall be realized by future fault-tolerant quantum computer. First, the basic equations are approximated by polynomials. Second, those polynomials are transformed to a specific form, wherein all variables (representing the wavefunctions and the atomic coordinates) are given by the transformations acting on a linear space of monomials with finite dimension, and the unknown variables could be determined as the eigenvalues of those transformation matrices. Third, the eigenvalues are determined by quantum phase estimation. Following these three steps, namely, symbolic, numeric, and quantum steps, we can determine the optimized electronic and atomic structures of molecules.
Eigenproblem arises in a large number of disciplines of sciences and engineering. Combining quantum algorithms to solve linear differential equations, quantum singular value estimation and quantum phase estimation, we propose quantum algorithms to compute the eigenvalues of diagonalizable matrices of two types: matrices that only have real eigenvalues and normal matrices. The quantum algorithms return a quantum state such that the first register stores eigenvalues and the second register stores the corresponding eigenvectors. For normal matrices, the complexity to obtain this quantum state is dominated by the complexity to perform quantum singular value estimation. For the other case, the complexity is determined by the solving of a linear system of differential equations. Under certain conditions (e.g. sparse), the complexities are polylog on the size n of the given matrix. If we perform measurements, then we can obtain all the eigenvalues classically. If M is s sparse, then the complexity is O(sn∥M∥maxκ2/ϵ2), where κ is the condition number of the matrix of eigenvectors.
Simulations of chemical dynamics are a powerful means for understanding chemistry. However, classical computers struggle to simulate many chemical processes, especially non-adiabatic ones, where the Born-Oppenheimer approximation breaks down. Quantum computers could simulate quantum-chemical dynamics more efficiently than classical computers, but there is currently no complete quantum algorithm for calculating dynamical observables to within a known error. Here, we develop an efficient, end-to-end quantum algorithm for simulating chemical dynamics that avoids all uncontrolled approximations (including the Born-Oppenheimer approximation) and whose error is bounded subject to mild assumptions. To do so, we treat the nuclei and the electrons on an equal footing and simulate the full molecular wavefunction on a momentum-space grid in first quantization, including all algorithmic steps: initial-state preparation, time evolution using qubitization, and measurement of chemical observables such as reaction yields and rates. Our work gives the first algorithm for quantum simulation of chemistry whose end-to-end complexity achieves sublinear scaling in the size of the grid. We achieve this by developing an exponentially faster method for initial-state-preparation. Photochemistry is a likely early application of our algorithm and we estimate resources required for end-to-end simulations of non-adiabatic dynamics of atmospherically important molecules. Classically intractable photochemical computations could be performed using resources comparable to those required for other chemical applications of quantum computing.
We present a quantum algorithm that analyzes time series data simulated by a quantum differential equation solver. The proposed algorithm is a quantum version of a dynamic mode decomposition algorithm used in diverse fields such as fluid dynamics, molecular dynamics, and epidemiology. Our quantum algorithm can also compute matrix eigenvalues and eigenvectors by analyzing the corresponding linear dynamical system. Our algorithm handles a broad range of matrices, particularly those with complex eigenvalues. The complexity of our quantum algorithm is O ( poly log N ) for an N -dimensional system. This is an exponential speedup over known classical algorithms with at least O ( N ) complexity. Thus, our quantum algorithm is expected to enable high-dimensional dynamical systems analysis and large matrix eigenvalue decomposition, intractable for classical computers. Published by the American Physical Society 2024
Quantum computation appears to offer significant advantages over classical computation and this has generated a tremendous interest in the field. In this thesis we study the application of quantum computers to computational problems in science and engineering, and to combinatorial optimization problems. We outline the results below. Algorithms for scientific computing require modules, i.e., building blocks, implementing elementary numerical functions that have well-controlled numerical error, are uniformly scalable and reversible, and that can be implemented efficiently. We derive quantum algorithms and circuits for computing square roots, logarithms, and arbitrary fractional powers, and derive worst-case error and cost bounds. We describe a modular approach to quantum algorithm design as a first step towards numerical standards and mathematical libraries for quantum scientific computing. A fundamental but computationally hard problem in physics is to solve the time-independent Schrödinger equation. This is accomplished by computing the eigenvalues of the corresponding Hamiltonian operator. The eigenvalues describe the different energy levels of a system. The cost of classical deterministic algorithms computing these eigenvalues grows exponentially with the number of system degrees of freedom. The number of degrees of freedom is typically proportional to the number of particles in a physical system. We show an efficient quantum algorithm for approximating a constant number of low-order eigenvalues of a Hamiltonian using a perturbation approach. We apply this algorithm to a special case of the Schrödinger equation and show that our algorithm succeeds with high probability, and has cost that scales polynomially with the number of degrees of freedom and the reciprocal of the desired accuracy. This improves and extends earlier results on quantum algorithms for estimating the ground state energy. We consider the simulation of quantum mechanical systems on a quantum computer. We show a novel divide and conquer approach for Hamiltonian simulation. Using the Hamiltonian structure, we can obtain faster simulation algorithms. Considering a sum of Hamiltonians we split them into groups, simulate each group separately, and combine the partial results. Simulation is customized to take advantage of the properties of each group, and hence yield refined bounds to the overall simulation cost. We illustrate our results using the electronic structure problem of quantum chemistry, where we obtain significantly improved cost estimates under mild assumptions. We turn to combinatorial optimization problems. An important open question is whether quantum computers provide advantages for the approximation of classically hard combinatorial problems. A promising recently proposed approach of Farhi et al. is the Quantum Approximate Optimization Algorithm (QAOA). We study the application of QAOA to the Maximum Cut problem, and derive analytic performance bounds for the lowest circuit-depth realization, for both general and special classes of graphs. Along the way, we develop a general procedure for analyzing the performance of QAOA for other problems, and show an example demonstrating the difficulty of obtaining similar results for greater depth. We show a generalization of QAOA and its application to wider classes of combinatorial optimization problems, in particular, problems with feasibility constraints. We introduce the Quantum Alternating Operator Ansatz, which utilizes more general unitary operators than the original QAOA proposal. Our framework facilitates low-resource implementations for many applications which may be particularly suitable for early quantum computers. We specify design criteria, and develop a set of results and tools for mapping diverse problems to explicit quantum circuits. We derive constructions for several important prototypical problems including Maximum Independent Set, Graph Coloring, and the Traveling Salesman problem, and show appealing resource cost estimates for their implementations.
This paper explores the potential of quantum simulation for advancing molecular dynamics (MD) modeling, addressing limitations in current computational methods due to computational cost and inability to accurately represent quantum effects. We propose a novel quantum-enhanced MD algorithm that utilizes superposition and entanglement to simulate molecular dynamics, aiming to achieve accurate modeling of complex quantum phenomena and predict material properties. The paper details the algorithm's core principles, mechanism, and potential applications, highlighting the advantages over traditional MD simulations.