  We present a simplified analog quantum simulation protocol for preparing
quantum states that embed solutions of parabolic partial differential
equations, including the heat, Black-Scholes and Fokker-Planck equations. The
key idea is to approximate the heat equations by a system of hyperbolic heat
equations that involve only first-order differential operators. This scheme
requires relatively simple interaction terms in the Hamiltonian, which are the
electric and magnetic dipole moment-like interaction terms that would be
present in a Jaynes-Cummings-like model. For a d-dimensional problem, we show
that it is much more appropriate to use a single d-level quantum system - a
qudit - instead of its qubit counterpart, and d+1 qumodes. The total resource
cost is efficient in d and precision error, and has potential for realisability
for instance in cavity and circuit QED systems.
