Exponential Quantum Advantage for Simulating Open Classical Systems

A recent promising arena for quantum advantage is simulating exponentially large classical systems. Here, we show how this advantage can be used to calculate the dynamics of open classical systems experiencing dissipation, including the effects of non-Markovian baths. This is a particularly interesting class of systems since dissipation plays a key role in contexts ranging from fluid dynamics to thermalization. We adopt the Caldeira-Leggett Hamiltonian, a generic model for dissipation in which the system is coupled to a bath of harmonic oscillators with a large number of degrees of freedom. To date, the most efficient classical algorithms for simulating such systems have a polynomial dependence on the size of the bath. In this work, we give a quantum algorithm with an exponential speedup, capable of simulating $d$ system degrees of freedom coupled to $N = 2^n\gg d$ bath degrees of freedom, to within error $\varepsilon$, using $O({\rm poly}(d, n, t, \varepsilon^{-1}))$ quantum gates.