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

The coherent Ising machine with quantum feedback: the total and conditional master equation methods