  We propose a hybrid quantum-classical algorithm for solving the
time-independent Schr\"odinger equation for atomic and molecular collisions.
The algorithm is based on the $S$-matrix version of the Kohn variational
principle, which computes the fundamental scattering $S$-matrix by inverting
the Hamiltonian matrix expressed in the basis of square-integrable functions.
The computational bottleneck of the classical algorithm -- symmetric matrix
inversion -- is addressed here using the variational quantum linear solver
(VQLS), a recently developed noisy intermediate-scale quantum (NISQ) algorithm
for solving systems of linear equations. We apply our algorithm to single and
multichannel quantum scattering problems, obtaining accurate vibrational
relaxation probabilities in collinear atom-molecule collisions. We also show
how the algorithm could be scaled up to simulate collisions of large polyatomic
molecules. Our results demonstrate that it is possible to calculate scattering
cross sections and rates for complex molecular collisions on NISQ quantum
processors, opening up the possibility of scalable digital quantum computation
of gas-phase bimolecular collisions and reactions of relevance to
astrochemistry and ultracold chemistry.
