Quantum computation of stopping power for inertial fusion target design

Abstract

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 Quant. 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 an 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.

Keywords: algorithms; fault-tolerance; fusion; quantum computing; stopping-power.

Fig. 1.

(Left) First-principles stopping power calculations involve time evolving a projectile (red) passing through a target medium (blue) while monitoring observables related to energy transfer between them. The initial velocity, $v_{\mathrm{proj}}$, is chosen to mitigate trajectory sampling and finite-size error using techniques from ref. . The coupled electron-projectile dynamics are time evolved subject to this initial condition and the work or average force on the projectile is calculated throughout the trajectory. (Right) The stopping power is related to the slope of the work that the target does on the projectile as a function of its displacement from its original position (solid). A moving average for this slope (dashed) illustrates the rate at which the stopping power estimate converges. Close collisions involve large impulses in the work that are essential to capture on average. However, if these relatively rare events are included in the sample, they can dominate the variance for sample-efficient estimates.

Fig. 2.

The Toffoli cost of estimating the projectile kinetic energy with traditional Monte Carlo sampling and the mean estimation algorithm from Kothari and O’Donnell (81) (KO). Both techniques have SE that linearly depends on the square root of the variance of the observable. The number of samples required for fixed SE in Standard Monte Carlo mean estimation scales as $\mathcal{O}(1/{ϵ}^2)$ while the KO algorithm scales as $\mathcal{O}(1/ϵ)$ but with larger constant factors originating from code (circuit) for the random variable and quantum phase estimation on the Grover like iterate used in the algorithm. More details of the algorithm’s main subroutines are provided in SI Appendix, section VI.

Fig. 3.

Subroutine costs for each component of implementing the block-encoding. The labels C$\{n\}$ correspond to the costs enumerated in protocol SI Appendix, section IV.D. CR is the reflection cost which given the additional register and augmented $\lambda$ is $n_T+2n_{\eta }+6n_n+n_M+16$ where $n_T=10+log(\lambda /ϵ)$, with $ϵ$ being the target precision of time evolution, and $n_{\eta }=\lceil log(\eta )\rceil$.

Fig. 4.

(Left) Toffoli complexity to synthesize the system propagator of the Alpha + Hydrogen system for time $t=10,20,30,40$ (in atomic units) for a range of fidelities. Using the lowest sample complexity considered $N_s\approx 50$ to 100 and $10$ total times to estimate the slope the Toffoli complexity is $\approx$200 times the Toffoli costs shown. (Right) Toffoli scaling with respect to particle number at fixed $r_s$ and a fixed number of planewaves (increasing grid resolution). Time evolution cost is expected to scale as somewhere in between $\mathcal{O}({\eta }^{4/3})$ and $\mathcal{O}({\eta }^{8/3})$ using quantum signal processing. In blue the time evolution cost for $t=1$ is shown demonstrating the expected scaling along with constant factors. Constant factors associated with QPE are shown in red for $1/t=ϵ={10}^{-3}$ which is should proportionally increase the cost. The displayed constant factor resources are in line with what is demonstrated in ref. for constant $r_s\approx 1$.