A comprehensive framework to simulate real-time chemical dynamics on a fault-tolerant quantum computer

We present a comprehensive end-to-end framework for simulating the real-time dynamics of chemical systems on a fault-tolerant quantum computer, incorporating both electronic and nuclear quantum degrees of freedom. An all-particle simulation is nominally efficient on a quantum computer, but practically infeasible. Hence, central to our approach is the construction of a first-quantized plane-wave algorithm making use of pseudoions. The latter consolidate chemically inactive electrons and the nucleus into a single effective dynamical ionic entity, extending the well-established concept of pseudopotentials in quantum chemistry to a two-body interaction. We explicitly describe efficient quantum circuits for initial state preparation across all degrees of freedom, as well as for block-encoding the Hamiltonian describing interacting pseudoions and chemically active electrons, by leveraging recent advances in quantum rejection sampling to optimize the implementations. To extract useful chemical information, we first design molecular fingerprints by combining density-functional calculations with machine learning techniques, and subsequently validate them through surrogate classical molecular dynamics simulations. These fingerprints are then coherently encoded on a quantum computer for efficient molecular identification via amplitude estimation. We provide an extensive analysis of the cost of running the algorithm on a fault-tolerant quantum computer for several chemically interesting systems. As an illustration, simulating the interaction between $\mathrm{NH_3}$ and $\mathrm{BF_3}$ (a 40-particle system) requires 808 logical qubits to encode the problem, and approximately $10^{11}$ Toffoli gates per femtosecond of time evolution. Our results establish a foundation for further quantum algorithm development targeting chemical and material dynamics.