  We give a detailed theoretical derivation of the quantum master equation for
the coherent Ising machine. This is a quantum computational network with
feedback, that solves NP hard combinatoric problems, including the traveling
salesman problem and various extensions and analogs. There are two types of
master equation that are obtained, either conditional on the feedback current
or unconditional. We show that both types can be accurately simulated in a
scalable way using stochastic equations in the positive-P phase-space
representation. This depends on the nonlinearity present, and we use parameter
values that are typical of current experiments. However, while the two
approaches are in excellent agreement, they are not equivalent with regard to
efficiency. We find that unconditional simulation has much greater efficiency,
and is scalable to larger sizes. This is a case where too much knowledge is a
dangerous thing. Conditioning the simulations on the feedback current is not
essential to determining the success probability, but it greatly increases the
computational complexity. To illustrate the speed improvements obtained with
the unconditional approach, we carry out full quantum simulations of the
quantum master equation with up to 1000 nodes.
