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arXiv:2203.02447 · Physical Review A

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

Simon Kiesewetter, P. D. Drummond

Published Aug 12, 2022• 14 citations

Abstract

We give a detailed theoretical derivation of the master equation for the coherent Ising machine. This is a quantum computational network with feedback, that approximately solves NP-hard combinatoric problems, including the traveling salesman problem and various extensions and analogs. There are two possible types of master equation, 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. 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 more scalable to large 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 master equation with up to 1000 nodes.

DOI: 10.1103/physreva.106.022409

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We give a detailed theoretical derivation of the master equation for the coherent Ising machine. This is a quantum computational network with feedback, that approximately solves NP-hard combinatoric problems, including the traveling salesman problem and various extensions and analogs. There are two possible types of master equation, 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. 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 more scalable to large 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 master equation with up to 1000 nodes.

A new, to the best of our knowledge, technique is demonstrated for carrying out exact positive-P phase-space simulations of the coherent Ising machine quantum computer. By suitable design of the coupling matrix, general hard optimization problems can be solved. Here, computational quantum simulations of a feedback type of photonic parametric network are carried out, which is the implementation of the coherent Ising machine. Results for success rates are obtained using this scalable phase-space algorithm for quantum simulations of quantum feedback devices.

A new technique is demonstrated for carrying out exact positive-P phase-space simulations of the coherent Ising machine quantum computer. By suitable design of the coupling matrix, general hard optimization problems can be solved. Here, quantum simulations of a feedback type of photonic parametric network are carried out, which is the implementation of the coherent Ising machine. Results for success rates are obtained using a weighted algorithm for quantum simulations of quantum feedback devices.

As a solver for non-deterministic polynomial time (NP)-hard combinatorial optimization problems, the coherent Ising machine (CIM) is in the early stages of research, and the potential of this innovative physical system will be developed. Here, we propose a speed-up coherent Ising machine with a squeezed feedback system, which we call S-CIM. We couple squeezed feedback pulses generated by the squeezed feedback system into the degenerate optical parametric oscillator (DOPO) network. Simulations indicate that quantum inseparability of the coupled DOPO network is further enhanced during the whole optimization process, and quantum fluctuations are significantly smaller around the oscillation threshold. Computation experiments are performed on MAX-CUT problems of order between 4 and 20000. Numerical results demonstrate that S-CIM increases the optimal normalized output by 2.27% and significantly reduces the optimal computation time by 75.12%.

The quantum theory of coherent Ising machines, based on degenerate optical parametric oscillators and measurement-feedback circuits, is developed using the positive $P(\ensuremath{\alpha},\ensuremath{\beta})$ representation of the density operator and the master equation. The theory is composed of the c-number stochastic differential equations for describing open dissipative quantum dynamics and the replicator dynamics equations for handling measurement-induced collapse of the density operator. We apply the present theory to simulate two simple Ising spin models and elucidate the unique features of this computing machine.

The Coherent Ising Machine (CIM) is a quantum network of optical parametric oscillators (OPOs) intended to find ground states of the Ising model. This is an NP-hard problem, related to several important minimization problems, including the max-cut graph problem. In order to enhance its potential performance, we analyze the coherent coupling strategy for the CIM in a highly quantum regime. To explore this limit, without assuming gaussianity, we employ accurate numerical simulations. Due to the inherent complexity of the system, the maximum network size is limited. While master equation methods can be used, their scalability diminishes rapidly for larger systems. Instead, we use Monte Carlo wave-function methods, which scale as the wave-function dimension, and use large numbers of samples. These simulations involve Hilbert spaces exceeding

10^{7}

dimensions. To evaluate success probabilities, we use quadrature probabilities. We demonstrate the potential for quantum computational advantage by reducing the time required to reach maximum success probability in a low-dissipation regime enabled by initial quantum superpositions and entanglement. Furthermore, we demonstrate that tailored time-dependent couplings can amplify these quantum effects. Comparisons with classical CIM models give evidence that quantum tunneling effects in this strong coupling limit can overcome trapping in false minima. This can greatly increase success rates, indicating a potential for quantum advantage. Finally, we perform a coherence analysis based on the state purity to examine the role of quantum coherence in CIM performance and to determine how state purity correlates with improved optimization outcomes.