P-dits: Probabilistic d-dimensional bits for extended-variable probabilistic computing
Phys. Rev. Applied 24, 044077 – Published 24 October, 2025
DOI: https://doi.org/10.1103/4ngx-cmz7
Abstract
This paper is a contribution to the Physical Review Applied collection titled Physics-Inspired Computing.
Ising machines can solve combinatorial optimization problems by representing them as energy minimization problems. A common implementation is the probabilistic Ising machine (PIM), which uses probabilistic bits (P-bits) to represent coupled binary spins. However, many real-world problems have complex data representations that do not map naturally into a binary encoding, leading to a significant increase in hardware resources and time-to-solution. Here, we describe a generalized spin model that supports an arbitrary number of spin dimensions, each with an arbitrary real component. We define the probabilistic d-dimensional bit (P-dit) as the base unit of a P-computing implementation of this model. We further describe two restricted forms of P-dits for specific classes of common problems and implement them experimentally on an application-specific integrated circuit (ASIC): (i) isotropic P-dits, which simplify the implementation of categorical variables, resulting in ∼34× performance improvement compared with a P-bit implementation on an example three-partition problem and (ii) probabilistic integers (P-ints), which simplify the representation of numeric values and provide ∼5× improvement compared with a P-bit implementation of an example integer linear programming (ILP) problem. Additionally, we report a field-programmable gate array (FPGA) P-int-based integer quadratic programming (IQP) solver, which shows ∼64× faster time-to-solution compared with the best of a series of state-of-the-art software solvers. The generalized formulation of probabilistic variables presented here provides a path to solving large-scale optimization problems on various hardware platforms, including digital CMOS.
Physics Subject Headings (PhySH)
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This article appears in the following collection:
Collection on Physics-Inspired Computing
Physical Review Applied is pleased to present a Collection on Physics-Inspired Computing, highlighting the rapidly evolving field of energy-efficient computing techniques, from hardware technologies to algorithms, where physics inspiration serves as the crucial link. Contributions to this collection will be published throughout 2025. This Collection is being curated by Guest Editors Kerem Camsari and Supriyo Datta.
