Polynomial-time quantum algorithm for the simulation of chemical dynamics Ivan Kassala, Stephen P. Jordanb, Peter J. Lovec, Masoud Mohsenia, and Ala´ n Aspuru-Guzika,1 aDepartment of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138; bCenter for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139; and cDepartment of Physics, Haverford College, Haverford, PA 19041 Edited by Todd J. Martinez, University of Illinois at Urbana–Champaign, Urbana, IL, and accepted by the Editorial Board October 9, 2008 (received for review August 22, 2008) The computational cost of exact methods for quantum simulation a few particles. Here, however, we show that the Born– using classical computers grows exponentially with system size. As Oppenheimer approximation would not necessarily simplify simu- a consequence, these techniques can be applied only to small lations performed on quantum computers. Indeed, except for the systems. By contrast, we demonstrate that quantum computers smallest systems, an explicit treatment of all of the particles would could exactly simulate chemical reactions in polynomial time. Our be both more accurate and more efficient, even for nearly adiabatic algorithm uses the split-operator approach and explicitly simulates chemical reactions. all electron-nuclear and interelectronic interactions in quadratic Feynman’s idea of using a quantum machine to mimic the time. Surprisingly, this treatment is not only more accurate than quantum Hamiltonian of a system of interest was one of the the Born–Oppenheimer approximation but faster and more effi- founding ideas of the field of quantum computation (13). Lloyd (14) cient as well, for all reactions with more than about four atoms. subsequently showed that quantum computers could be used to This is the case even though the entire electronic wave function is simulate systems that can be formulated in terms of local interac- propagated on a grid with appropriately short time steps. Al- tions, by using resources that scale only polynomially with system though the preparation and measurement of arbitrary states on a size. Zalka and Wiesner (15, 16) developed a quantum simulation quantum computer is inefficient, here we demonstrate how to algorithm for particles in real space and Lidar and Wang (17) prepare states of chemical interest efficiently. We also show how applied it to the calculation of the thermal rate constant of chemical to efficiently obtain chemically relevant observables, such as state- reactions. Smirnov et al. proposed an analog quantum simulator for to-state transition probabilities and thermal reaction rates. Quan- chemical reactions using quantum dots (42). We have previously tum computers using these techniques could outperform current shown (18) that quantum computers could be used to simulate the classical computers with 100 qubits. static properties of molecules, and in this work, we present a general scheme for using quantum computers for the study of dynamical electronic structure ͉ quantum computation ͉ quantum simulation chemical properties. To simulate a quantum system, we must prepare its initial quantum state, propagate it in time, and finally, extract data of CHEMISTRY ccurate simulations of quantum-mechanical processes have chemical relevance, such as rate constants. For an efficient quan- greatly expanded our understanding of the fundamentals of A tum simulation, all these tasks must be carried out by using chemical reaction dynamics. In particular, recent years have seen resources that increase polynomially with increasing system size. tremendous progress in methods development, which has enabled We present a quantum algorithm that meets these requirements. simulations of increasingly complex quantum systems. Although it We also show that for all chemical reactions with more than Ϸ4 is, strictly speaking, true that exact quantum simulation requires atoms, it is more efficient for a quantum computer to simulate the resources that scale exponentially with system size, several tech- complete nuclear and electronic time evolution rather than to use niques are available that can treat realistic chemical problems, at a the Born–Oppenheimer approximation. given accuracy, with only a polynomial cost. Certain fully quantum The polynomial scaling of these methods means they would methods—such as multiconfigurational time-dependent Hartree enable the study of systems that are, in principle, out of reach for (MCTDH) (1), matching pursuit/split-operator Fourier transform any classical computer. However, large quantum computers are far (MP/SOFT) (2), or full multiple spawning (FMS) (3)—solve the in the future, and so determining the requirements of interesting nuclear Schro¨dinger equation, including nonadiabatic effects, given calculations in absolute terms is, perhaps, of more interest than analytic expressions for the potential energy surfaces and the their scaling alone. We show that a quantum computer using these couplings between them. These techniques have permitted the techniques could outperform current classical computers using 100 simulation of large systems; as examples we can give MCTDH qubits, within the design limits of a proposed 300-qubit quantum simulations of a pentaatomic chemical reaction (4) and of a computer (19). Although we focus on chemical applications, these spin-boson model with 80 degrees of freedom (5) or an MP/SOFT techniques are generally applicable to many physical systems, from simulation of photoisomerization in rhodopsin with 25 degrees of strong-field, multielectron ionization to quantum liquids and con- freedom (6). Ab initio molecular-dynamics techniques such as ab densed-matter systems. initio multiple spawning (AIMS) (7) avoid analytic expressions for This article is organized as follows. We first review Zalka and potential energy surfaces and instead solve electronic Schro¨dinger Wiesner’s algorithm and show how the difficulty of computing a equations at every time step. This allows one to gain insight into wave function’s time evolution depends only on the complexity dynamical problems such as isomerizations through conical inter- sections (8). However, there are also chemical processes that are best treated Author contributions: I.K., S.P.J., P.J.L., M.M., and A.A.-G. designed research, performed by completely avoiding the Born–Oppenheimer approximation. As research, and wrote the paper. examples, we can cite strong-field electronic dynamics in atoms and The authors declare no conflict of interest. multielectron ionization (9, 10) or atomic and molecular fragmen- This article is a PNAS Direct Submission. T.J.M. is a guest editor invited by the Editorial Board. tation caused by collisions with energetic electrons or photons (11, 1To whom correspondence should be addressed. E-mail: [email protected]. 12). Systems that resist the application of the Born–Oppenheimer This article contains supporting information online at www.pnas.org/cgi/content/full/ approximation require very general techniques, and the consequent 0808245105/DCSupplemental. unfavorable scaling has restricted such simulations to systems with © 2008 by The National Academy of Sciences of the USA www.pnas.org͞cgi͞doi͞10.1073͞pnas.0808245105 PNAS ͉ December 2, 2008 ͉ vol. 105 ͉ no. 48 ͉ 18681–18686 Downloaded by guest on September 30, 2021 of evaluating the interaction potential. We then consider 3 approaches to the calculation of the interaction potential, in- cluding a fully nonadiabatic treatment of chemical reactions. We consider the problem of state preparation for all of the schemes and, finally, address the problem of measurement. We present 3 readout schemes for reaction dynamics—reaction probabilities, Fig. 1. The quantum simulation algorithm. The potential and kinetic energy thermal rate constants, and state-to-state probabilities—that unitaries are applied to a quantum state in turn, with the transformation be- would allow for efficient evaluation of many parameters acces- tween position and momentum representations being performed with the effi- sible to experiment. cient quantum Fourier transform (QFT). The ancilla register is required for phase kickback and remains unchanged throughout the simulation, whereas the boxed ␦ Quantum Dynamics time step is repeated t/ t times. The proposed algorithm, unlike that of Zalka (15), does not require that functions be uncomputed and is therefore twice as fast. The problem of simulating quantum dynamics is that of determin- ing the properties of the wave function ␺(t)͘ of a system at time t, given the initial wave function ␺ (0)͘ and the Hamiltonian Hˆ of the gate sequence V, which computes the potential V,actssothat system. If the final state can be prepared by propagating the initial Vx, y͘ϭx, y Q V(x)͘, where y is an m-bit integer labeling a basis state, any observable of interest may be computed. state of the ancilla register, and Q denotes addition modulo 2m. We employ an improved version of the real-space quantum We apply the diagonal unitary by phase kickback. The computer simulation technique developed by Zalka and Wiesner in which a is initialized in the state ␺͘ R 1͘m, where 1͘m in the ancilla discrete variable representation of the wave function is used (15, register represents the state 0…001͘ in m qubits. Applying the 16). In the 1-dimensional case, the domain of the wave function is inverse QFT to the ancilla register, followed by V, produces divided into a discrete position basis of n ϭ 2n equidistant points. Ϫ Ϫ The wave function is represented as: M 1 2␲iy/M M 1 2␲iy/M e ˆ e V ͉ͩ␺͘ ͸ ͉yͪ͘ ϭ eϪiV␦t͉␺͘ ͸ ͉y͘ 2nϪ1 ͱM ͱM ͉␺͑ ͒͘ ϭ ͸ ͑ ͉͒ ͘ ϭ ͉ ͘ ϩ ϩ ͉ ͘ yϭ0 yϭ0 t ax t x a0Ç0...00 ... a2nϪ1 1...11. xϭ0 n qubits 2␲ where M ϭ 2m, and we choose ␦t ϭ . The equality obtains The spatial wave function is stored in the Hilbert space of the M because the ancilla state is an eigenstate of addition (with qubits, and so the spatial resolution grows exponentially with the Ϫ2␲iq/M number of qubits. For a system with d dimensions, d registers of eigenvalue e corresponding to the addition of q) (23).