  The Chemical Master Equation (CME) provides a highly accurate, yet extremely
resource-intensive representation of a stochastic chemical reaction network and
its kinetics due to the exponential scaling of its possible states with the
number of reacting species. In this work, we demonstrate how quantum algorithms
and hardware can be employed to model stochastic chemical kinetics as described
by the CME using the Schl\"ogl Model of a trimolecular reaction network as an
illustrative example. To ground our study of the performance of our quantum
algorithms, we first determine a range of suitable parameters for constructing
the stochastic Schl\"ogl operator in the mono- and bistable regimes of the
model using a classical computer and then discuss the appropriateness of our
parameter choices for modeling approximate kinetics on a quantum computer. We
then apply the Variational Quantum Deflation (VQD) algorithm to evaluate the
smallest-magnitude eigenvalues, $\lambda_0$ and $\lambda_1$, which describe the
transition rates of both the mono- and bi-stable systems, and the Quantum Phase
Estimation (QPE) algorithm combined with the Variational Quantum Singular Value
Decomposition (VQSVD) algorithm to estimate the zeromode (ground state) of the
bistable case. Our quantum computed results from both noisy and noiseless
quantum simulations agree within a few percent with the classically computed
eigenvalues and zeromode. Altogether, our work outlines a practical path toward
the quantum solution of exponentially complex stochastic chemical kinetics
problems and other related stochastic differential equations.
