Polynomial-time quantum algorithm for the simulation of chemical dynamics

Abstract

The computational cost of exact methods for quantum simulation using classical computers grows exponentially with system size. As a consequence, these techniques can be applied only to small systems. By contrast, we demonstrate that quantum computers could exactly simulate chemical reactions in polynomial time. Our algorithm uses the split-operator approach and explicitly simulates all electron-nuclear and interelectronic interactions in quadratic time. Surprisingly, this treatment is not only more accurate than the Born-Oppenheimer approximation but faster and more efficient as well, for all reactions with more than about four atoms. This is the case even though the entire electronic wave function is propagated on a grid with appropriately short time steps. Although the preparation and measurement of arbitrary states on a quantum computer is inefficient, here we demonstrate how to prepare states of chemical interest efficiently. We also show how to efficiently obtain chemically relevant observables, such as state-to-state transition probabilities and thermal reaction rates. Quantum computers using these techniques could outperform current classical computers with 100 qubits.

Fig. 1.

The quantum simulation algorithm. The potential and kinetic energy unitaries are applied to a quantum state in turn, with the transformation between position and momentum representations being performed with the efficient quantum Fourier transform (QFT). The ancilla register is required for phase kickback and remains unchanged throughout the simulation, whereas the boxed time step is repeated t/δt times. The proposed algorithm, unlike that of Zalka (15), does not require that functions be uncomputed and is therefore twice as fast.

Fig. 2.

Resource requirements for a quantum simulation of B particles interacting through a pairwise potential. The chemical symbols correspond to the simulation of the full Coulomb dynamics of the corresponding atom (nucleus and electrons). The vertical dashed line represents the approximate current limit of numerically exact quantum simulation on classical computers on a grid (10). (A) Total qubits required. We require n qubits for each degree of freedom and m qubits for each ancilla, 4 of which are needed for the Coulomb potential. Hence, a total of n(3B − 6) + 4m qubits are needed (see SI for details). The horizontal dotted line represents a 300-qubit quantum computer, which is believed to be feasible with near-future technology (19). We assume a grid of 2³⁰ points, which corresponds to n = 10 and would suffice for the simulation of many chemical reactions or the strong-field ionization of atoms (9, 27). (B) Total elementary gates required. The 300-qubit computer is expected to achieve 1 billion elementary quantum operations. The dotted line represents the largest possible simulation of 1,000 time steps, assuming 10 bits of numerical accuracy (m = 10). Computing the Coulomb potential requires ($\frac{75}{4}$m³ + $\frac{51}{2}$m²) gates per pair of particles (see SI for details).

Fig. 3.

Estimated number of elementary quantum operations (gates) required for the simulation of chemical reactions. Standard Born–Oppenheimer potential-energy-surface calculations require time resources exponential in the size of the system (full line), whereas a fully nonadiabatic, nuclear and electronic calculation would require only polynomial time (dotted lines). The resulting cutoff indicates that for all reactions with more than four atoms (dashed line), the Born–Oppenheimer approximation is always less efficient on a quantum computer than a diabatic treatment. The complexity of the diabatic computation depends only on the atomic number Z, whereas the potential energy surfaces are modeled as polynomials of degree K along each axis. A value of K ≥ 15 is required to obtain 0.1% agreement with surfaces such as BKMP2 (28). The position of the cutoff does not significantly depend on the accuracy of the evaluated potential (m). To obtain the gate counts, we assume 20 bits of accuracy (m = 20), enough for chemical precision. The gate counts reflect the fact that an appropriate nuclear time step is approximately 1,000 times longer than an electronic time step.