Andrew Baczewski

Here we describe an approach for simulating quantum chemistry on quantum computers with significantly lower asymptotic complexity than prior work. The approach uses a real-space first-quantised representation of the molecular Hamiltonian which we propagate using high-order product formulae. Essential for this low complexity is the use of a technique similar to the fast multipole method for computing the Coulomb operator with O(eta) complexity for a simulation with eta particles. We show how to modify this algorithm so that it can be implemented on a quantum computer. We ultimately demonstrate an approach with t(eta^{4/3} N^{1/3} + eta^{1/3} N^{2/3})(eta Nt/epsilon)^o(1) gate complexity, where N is the number of grid points, epsilon is target precision, and t is the duration of time evolution. This is roughly a speedup by O(eta) over most prior algorithms. We provide lower complexity than all prior work for N<eta^7 (the regime of practical interest), with only first-quantised interaction-picture simulations providing better performance for N>eta^7. As with the classical fast multipole method, large numbers eta>10^3 would be needed to realise this advantage.

Stopping power is the rate at which a material absorbs the kinetic energy of a charged particle passing through it – one of many properties needed over a wide range of thermodynamic conditions in modeling inertial fusion implosions. First-principles stopping calculations are classically challenging because they involve the dynamics of large electronic systems far from equilibrium, with accuracies that are particularly difficult to constrain and assess in the warm-dense conditions preceding ignition. Here, we describe a protocol for using a fault-tolerant quantum computer to calculate stopping power from a first-quantized representation of the electrons and projectile. Our approach builds upon the electronic structure block encodings of Su et al. [PRX Quantum 2, 040332 2021], adapting and optimizing those algorithms to estimate observables of interest from the non-Born-Oppenheimer dynamics of multiple particle species at finite temperature. We also work out the constant factors associated with a novel implementation of a high-order Trotter approach to simulating a grid representation of these systems. Ultimately, we report logical qubit requirements and leading-order Toffoli costs for computing the stopping power of various projectile/target combinations relevant to interpreting and designing inertial fusion experiments. We estimate that scientifically interesting and classically intractable stopping power calculations can be quantum simulated with roughly the same number of logical qubits and about one hundred times more Toffoli gates than is required for state-of-the-art quantum simulations of industrially relevant molecules such as FeMoco or P450.