Quantum computers could potentially simulate the dynamics of systems such as polyatomic molecules on a much larger scale than classical computers. One of the central unresolved problems in developing quantum simulation algorithms has been constructing the necessary time evolution operators in enough detail for implementation in a quantum computer. This work presents a complete and very general solution to this problem: namely, a quantum computational algorithm for simulating an arbitrary Coulombic many-body system acting under the three-dimensional Schrödinger equation. We use the simplest discretized model and discuss detailed constructions of the operators involved. The algorithm is also implemented in MAT-LAB and tested against both analytical solutions and classical methods. Our algorithm can determine an arbitrary system’s dynamics rather than merely associated constants, as investigated in previous literature. It can hence be adapted for electronic structure calculations, chemical reaction simulations, etc. 1
Quantum computers could potentially simulate the dynamics of systems such as polyatomic molecules on a much larger scale than classical computers. One of the central unresolved problems in developing quantum simulation algorithms has been constructing the necessary time evolution operators in enough detail for implementation in a quantum computer. This work presents a complete and very general solution to this problem: namely, a quantum computational algorithm for simulating an arbitrary Coulombic many-body system acting under the three-dimensional Schrödinger equation. We use the simplest discretized model and discuss detailed constructions of the operators involved. The algorithm is also implemented in MAT-LAB and tested against both analytical solutions and classical methods. Our algorithm can determine an arbitrary system’s dynamics rather than merely associated constants, as investigated in previous literature. It can hence be adapted for electronic structure calculations, chemical reaction simulations, etc. 1
We describe a general quantum computational algorithm that simulates the time evolution of an arbitrary nonrelativistic, Coulombic many-body system in three dimensions, considering only spatial degrees of freedom. We use a simple discretized model of Schrödinger evolution in the coordinate representation and discuss detailed constructions of the operators necessary to realize the scheme of Wiesner and Zalka. The algorithm is simulated numerically for small test cases, and its outputs are found to be in good agreement with analytical solutions.
This paper explores the potential of quantum computing for simulating complex chemical reactions. Traditional classical computational methods often struggle with the inherent quantum mechanical nature of chemical systems, particularly when dealing with large molecules and intricate reaction pathways. This work proposes a novel quantum algorithm leveraging superposition and entanglement to directly simulate the time evolution of these systems. The core of the algorithm centers on a time-dependent Schrödinger equation (TDSE) implemented on a quantum computer. We outline the key elements, including the encoding of molecular states, the construction of a suitable Hamiltonian operator, and the execution of the quantum algorithm to obtain the final state of the system. The efficiency gains compared to classical methods, particularly for reactions involving significant quantum effects, are discussed. Furthermore, we highlight the potential impact of this technology on accelerating drug discovery, materials science, and fundamental chemical research. The algorithm's core relies on the ability of a quantum computer to perform a vast number of calculations simultaneously, offering a substantial advantage over classical computers when simulating quantum systems.
Abstract Digital quantum computers provide a computational framework for solving the Schrödinger equation for a variety of many‐particle systems. Quantum computing algorithms for the quantum simulation of these systems have recently witnessed remarkable growth, notwithstanding the limitations of existing quantum hardware, especially as a tool for electronic structure computations in molecules. In this review, we provide a self‐contained introduction to emerging algorithms for the simulation of Hamiltonian dynamics and eigenstates, with emphasis on their applications to the electronic structure in molecular systems. Theoretical foundations and implementation details of the method are discussed, and their strengths, limitations, and recent advances are presented. This article is categorized under: Quantum Computing > Algorithms Electronic Structure Theory > Ab Initio Electronic Structure Methods Quantum Computing > Theory Development
ConspectusSimulating molecular dynamics (MD) within a comprehensive quantum framework has been a long-standing challenge in computational chemistry. An exponential scaling of computational cost renders solving the time dependent Schrödinger equation (TDSE) of a molecular Hamiltonian, including both electronic and nuclear degrees of freedom (DOFs), as well as their couplings, infeasible for more than a few DOFs. In the Born-Oppenheimer (BO), or adiabatic, picture, electronic and nuclear parts of the wave function are decoupled and treated separately. Within this framework, the nuclear wave function evolves along potential energy surfaces (PESs) computed as solutions to the electronic Schrödinger equation parametrized in the nuclear DOFs. This approximation, together with increasingly elaborate numerical approaches to solve the nuclear time dependent Schrödinger equation (TDSE), enabled the treatment of up to a few dozens of degrees of freedom (DOFs). However, for particular applications, such as photochemistry, the BO approximation breaks down. In this regime of non-adiabatic dynamics, solving the full molecular problem including electron-nuclear couplings becomes essential, further increasing the complexity of the numerical solution. Although valuable methods such as multiconfigurational time-dependent Hartree (MCTDH) have been proposed for the solution of the coupled electron-nuclear dynamics, they remain hampered by an exponential scaling in the number of nuclear DOFs and by the difficulty of finding universal variational forms.In this Account, we present a perspective on novel quantum computational algorithms, aiming to alleviate the exponential scaling inherent to the simulation of many-body quantum dynamics. In particular, we focus on the derivation and application of quantum algorithms for adiabatic and non-adiabatic quantum dynamics, which include efficient approaches for the calculation of the BO potential energy surfaces (PESs). Thereafter, we study the time-evolution of a model system consisting of two coupled PESs in first and second quantization. In a first application, we discuss a recently introduced quantum algorithm for the evolution of a wavepacket in first quantization and exploit the potential quantum advantage of mapping its spatial grid representation to logarithmically many qubits. For the second demonstration, we move to the second quantization framework and review the scaling properties of two alternative time-evolution algorithms, namely, a variational quantum algorithm (VQA) (based on the McLachlan variational principle) and conventional Trotter-type evolution (based on a Lie-Trotter-Suzuki formula). Both methods clearly demonstrate the potential of quantum algorithms and their favorable scaling compared to the available classical approaches. However, a clear demonstration of quantum advantage in the context of molecular quantum dynamics may require the implementation of these algorithms in fault-tolerant quantum computers, while their application in near-term, noisy quantum devices is still unclear and deserves further investigation.
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.
In this work, we explore the implementation possibility of quantum simulation for quantum molecular dynamics, in particular for reaction dynamics, though several implementations have already reported through quantum-classical mixed simulations ({\it Acc. Chem. Res.} {\bf 54} (2021), 4229 and {\it J. Phys. Chem. Lett.} {\bf xx} (2026), XXXX). To analyze this aspect, we examine (1) the conjugacy relation between quantum simulator and the target molecular system, (2) the wave function correspondence in quantum algorithm and classical algorithm for multi-dimensional dynamics, (3) problems arisen from real-valued classical algorithms, and finally (4) geometric phase arisen from the separation among the degrees of freedom (DOFs). As is well known, the aforementioned first and second points play fundamental roles in quantum simulation of quantum many-body systems, and the third and fourth points are theoretical issues that might introduce problems in classical and quantum computing. In this work, we mainly focus on the third and fourth points by analysis of the first two points by reviewing previously reported quantum-classical mixed implementations of quantum simulation. We also consider gauge freedom in high-dimensional quantum molecular dynamics that has been introduced recently, and then discuss possibility of advantages and disadvantages of quantum simulation for molecular reaction dynamics.