  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. 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.
