  The primary objective of this paper is to present an exact and general
procedure for mapping any sequence of quantum gates onto a network of
probabilistic p-bits which can take on one of two values 0 and 1. The first $n$
p-bits represent the input qubits, while the other p-bits represent the qubits
after the application of successive gating operations. We can view this
structure as a Boltzmann machine whose states each represent a Feynman path
leading from an initial configuration of qubits to a final configuration. Each
such path has a complex amplitude $\psi$ which can be associated with a complex
energy. The real part of this energy can be used to generate samples of Feynman
paths in the usual way, while the imaginary part is accounted for by treating
the samples as complex entities, unlike ordinary Boltzmann machines where
samples are positive. Quantum gates often have purely imaginary energy
functions for which all configurations have the same probability and one cannot
take advantage of sampling techniques. However, if we can use suitable
transformations to introduce a real part in the energy function then powerful
sampling algorithms like Gibbs sampling can be harnessed to get acceptable
results with far fewer samples and perhaps even escape the exponential scaling
with $nd$. This algorithmic acceleration can then be supplemented with
special-purpose hardware accelerators like Ising Machines which can obtain a
very large number of samples per second through a combination of massive
parallelism, pipelining, and clockless mixed-signal operation made possible by
codesigning circuits and architectures to match the algorithm. Our results for
mapping an arbitrary quantum circuit to a Boltzmann machine with a complex
energy function should help push the boundaries of the simulability of quantum
circuits with probabilistic resources and compare them with NISQ-era quantum
computers.
