Self-contained relaxation-based dynamical Ising machines
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909416172814336 |
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| author | Erementchouk, Mikhail Shukla, Aditya Mazumder, Pinaki |
| author_facet | Erementchouk, Mikhail Shukla, Aditya Mazumder, Pinaki |
| contents | Dynamical Ising machines are based on continuous dynamical systems evolving from a generic initial state to a state strongly related to the ground state of the classical Ising model on a graph. Reaching the ground state is equivalent to finding the maximum (weighted) cut of the graph, which presents the Ising machines as an alternative way to solving and investigating NP-complete problems. Among the dynamical models, relaxation-based models are distinguished by their relations with guarantees of performance achieved in time scaling polynomially with the problem size. However, the terminal states of such machines are essentially non-binary, necessitating special post-processing relying on disparate computing. We show that an Ising machine implementing a special continuous dynamical system (called the V${}_2$ model) solves the rounding problem dynamically. We prove that the V${}_2$ model, starting from an arbitrary non-binary state, terminates in a state that trivially rounds to a binary state with the cut at least as big as obtained by optimal rounding of the initial state. Besides showing that relaxation-based dynamical Ising machines can be made self-contained, this result presents a non-Boolean realization of solving a non-trivial information processing task on Ising machines. Moreover, we prove that if the initial state of the V${}_2$-machine is a random limited amplitude perturbation of a binary state, the machine progresses to a state with at least as high cut as that of the initial binary state. Since the probability of improving the cut is finite, this shows that the V${}_2$-machine with random agitations converges to a maximum cut state almost surely. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_06414 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Self-contained relaxation-based dynamical Ising machines Erementchouk, Mikhail Shukla, Aditya Mazumder, Pinaki Emerging Technologies Dynamical Systems 90C27, 37B15, 82C20 Dynamical Ising machines are based on continuous dynamical systems evolving from a generic initial state to a state strongly related to the ground state of the classical Ising model on a graph. Reaching the ground state is equivalent to finding the maximum (weighted) cut of the graph, which presents the Ising machines as an alternative way to solving and investigating NP-complete problems. Among the dynamical models, relaxation-based models are distinguished by their relations with guarantees of performance achieved in time scaling polynomially with the problem size. However, the terminal states of such machines are essentially non-binary, necessitating special post-processing relying on disparate computing. We show that an Ising machine implementing a special continuous dynamical system (called the V${}_2$ model) solves the rounding problem dynamically. We prove that the V${}_2$ model, starting from an arbitrary non-binary state, terminates in a state that trivially rounds to a binary state with the cut at least as big as obtained by optimal rounding of the initial state. Besides showing that relaxation-based dynamical Ising machines can be made self-contained, this result presents a non-Boolean realization of solving a non-trivial information processing task on Ising machines. Moreover, we prove that if the initial state of the V${}_2$-machine is a random limited amplitude perturbation of a binary state, the machine progresses to a state with at least as high cut as that of the initial binary state. Since the probability of improving the cut is finite, this shows that the V${}_2$-machine with random agitations converges to a maximum cut state almost surely. |
| title | Self-contained relaxation-based dynamical Ising machines |
| topic | Emerging Technologies Dynamical Systems 90C27, 37B15, 82C20 |
| url | https://arxiv.org/abs/2305.06414 |