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High-Performance and Reliable Probabilistic Ising Machine Based on Simulated Quantum Annealing

Eleonora Raimondo1,2, Esteban Garzón3, Yixin Shao4, Andrea Grimaldi5,2, Stefano Chiappini1, Riccardo Tomasello6, Noraica Davila-Melendez7, Jordan A. Katine7, Mario Carpentieri6 et al.

Massimo Chiappini1, Marco Lanuzza3, Pedram Khalili Amiri4, and Giovanni Finocchio2,*

  • *Contact author: gfinocchio@unime.it
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Phys. Rev. X 15, 041001 – Published 1 October, 2025

DOI: https://doi.org/10.1103/pcmz-w776

Abstract

Probabilistic computing with p-bits is emerging as a computational paradigm for machine learning and for facing combinatorial optimization problems (COPs) with the so-called probabilistic Ising machines (PIMs). From a hardware point of view, the key elements that characterize a PIM are the random number generation, the nonlinearity, the network of coupled probabilistic bits, and the energy-minimization algorithm. Regarding the energy-minimization algorithm in this work we show that PIMs using the simulated quantum annealing (SQA) schedule exhibit better performance as compared to simulated annealing and parallel tempering in solving a number of COPs, such as maximum satisfiability problems, the planted Ising problem, and the traveling salesman problem. Additionally, we design and simulate the architecture of a fully connected CMOS-based PIM that is able to run the SQA algorithm having a spin-update time of 8 ns with a power consumption of 0.22 mW. Our results also show that SQA increases the reliability and the scalability of PIMs by compensating for device variability at an algorithmic level enabling the development of their implementation combining CMOS with different technologies such as spintronics. This work shows that the characteristics of the SQA are hardware agnostic and can be applied in the codesign of any hybrid analog-digital Ising machine implementation. Our results open a promising direction for the implementation of a new generation of reliable and scalable PIMs.

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Popular Summary

Finding the lowest-energy configuration of a complex system, like arranging magnets to minimize their conflict, is an NP-hard problem in the field of combinatorial optimization that can describe many real-world situations such as route planning or scheduling. One way to tackle it is by using specialized hardware called Ising machines, which are designed to mimic the behavior of spins in magnetic materials to find the best solution. Among these, probabilistic Ising machines use “p-bits,” elements that randomly switch between two states with a probability that can be controlled. In our study, we explore a new direction to help these machines find solutions more quickly and reliably, using a method called simulated quantum annealing working with a large number of replicas.

In our implementation, simulated quantum annealing creates multiple copies of the vector state of the probabilistic machine that interact with each other through a time-dependent, transverse field. This interaction helps the system explore different configurations and gradually settle into one of the lower-energy states, and eventually the best solution. Our results show that this method not only speeds up the process (at the cost of larger hardware resources) but also makes the machine more robust against differences or imperfections in the hardware itself.

Combining probabilistic bits with simulated quantum annealing can lead to faster, more stable hardware for solving tough combinatorial optimization problems. This advance could make it easier to tackle complex challenges across fields such as transportation, finance, and logistics.

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Phys. Rev. X 15, 041001– Published 1 October, 2025
  • Received 13 June 2024
  • Revised 23 December 2024
  • Accepted 11 August 2025
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