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Noise-augmented Chaotic Ising Machines for Combinatorial Optimization and Sampling

Kyle Lee, Shuvro Chowdhury, Kerem Y. Camsari

TL;DR

The paper investigates chaotic bits (c-bits) as deterministic Ising-machine elements and proposes a noise-augmented scheme via Adaptive Parallel Tempering (APT) to enable Boltzmann-like sampling and robust optimization. Through 1D quantum Ising testing, 2D Ising, and 3D spin-glass benchmarks, c-bits show Boltzmann-like statistics and critical dynamics similar to p-bits, though with nuanced differences in scaling exponents. The Adaptive Parallel Tempering framework is introduced to inject stochasticity through replica swaps, yielding a hybrid chaotic-probabilistic algorithm that matches or surpasses p-bit APT in optimization tasks, including 3D spin glass and semiprime factorization. The work includes hardware-relevant discussions and argues that a hybrid c-bit/p-bit approach can enable scalable, asynchronous probabilistic computing for combinatorial optimization and sampling, with potential realizations in oscillator-based Ising machines.

Abstract

Ising machines, hardware accelerators for combinatorial optimization and probabilistic sampling problems, have gained significant interest recently. A key element is stochasticity, which enables a wide exploration of configurations, thereby helping avoid local minima. Here, we refine the previously proposed concept of coupled chaotic bits (c-bits) that operate without explicit stochasticity. We show that augmenting chaotic bits with stochasticity enhances performance in combinatorial optimization, achieving algorithmic scaling comparable to probabilistic bits (p-bits). We first demonstrate that c-bits follow the quantum Boltzmann law in a 1D transverse field Ising model. We then show that c-bits exhibit critical dynamics similar to stochastic p-bits in 2D Ising and 3D spin glass models, with promising potential to solve challenging optimization problems. Finally, we propose a noise-augmented version of coupled c-bits via the adaptive parallel tempering algorithm (APT). Our noise-augmented c-bit algorithm outperforms fully deterministic c-bits running versions of the simulated annealing algorithm. Other analog Ising machines with coupled oscillators could draw inspiration from the proposed algorithm. Running replicas at constant temperature eliminates the need for global modulation of coupling strengths. Mixing stochasticity with deterministic c-bits creates a powerful hybrid computing scheme that can bring benefits in scaled, asynchronous, and massively parallel hardware implementations.

Noise-augmented Chaotic Ising Machines for Combinatorial Optimization and Sampling

TL;DR

The paper investigates chaotic bits (c-bits) as deterministic Ising-machine elements and proposes a noise-augmented scheme via Adaptive Parallel Tempering (APT) to enable Boltzmann-like sampling and robust optimization. Through 1D quantum Ising testing, 2D Ising, and 3D spin-glass benchmarks, c-bits show Boltzmann-like statistics and critical dynamics similar to p-bits, though with nuanced differences in scaling exponents. The Adaptive Parallel Tempering framework is introduced to inject stochasticity through replica swaps, yielding a hybrid chaotic-probabilistic algorithm that matches or surpasses p-bit APT in optimization tasks, including 3D spin glass and semiprime factorization. The work includes hardware-relevant discussions and argues that a hybrid c-bit/p-bit approach can enable scalable, asynchronous probabilistic computing for combinatorial optimization and sampling, with potential realizations in oscillator-based Ising machines.

Abstract

Ising machines, hardware accelerators for combinatorial optimization and probabilistic sampling problems, have gained significant interest recently. A key element is stochasticity, which enables a wide exploration of configurations, thereby helping avoid local minima. Here, we refine the previously proposed concept of coupled chaotic bits (c-bits) that operate without explicit stochasticity. We show that augmenting chaotic bits with stochasticity enhances performance in combinatorial optimization, achieving algorithmic scaling comparable to probabilistic bits (p-bits). We first demonstrate that c-bits follow the quantum Boltzmann law in a 1D transverse field Ising model. We then show that c-bits exhibit critical dynamics similar to stochastic p-bits in 2D Ising and 3D spin glass models, with promising potential to solve challenging optimization problems. Finally, we propose a noise-augmented version of coupled c-bits via the adaptive parallel tempering algorithm (APT). Our noise-augmented c-bit algorithm outperforms fully deterministic c-bits running versions of the simulated annealing algorithm. Other analog Ising machines with coupled oscillators could draw inspiration from the proposed algorithm. Running replicas at constant temperature eliminates the need for global modulation of coupling strengths. Mixing stochasticity with deterministic c-bits creates a powerful hybrid computing scheme that can bring benefits in scaled, asynchronous, and massively parallel hardware implementations.
Paper Structure (10 sections, 28 equations, 6 figures, 2 algorithms)

This paper contains 10 sections, 28 equations, 6 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Sampling from the 1D Transverse Field Ising Hamiltonian with c-bits
  3. Critical Scaling Dynamics: c-bits vs p-bits
  4. Adaptive Parallel Tempering
  5. Optimization of the 3D Spin Glass Problem
  6. Adaptive Parallel Tempering for Semiprime Factorization
  7. Possible Hardware Realizations
  8. Appendix 1
  9. Appendix 2
  10. Appendix 3

Figures (6)

  • Figure 1: (a) A probabilistic bit (p-bit) contains a random number generator with uniform distribution between -1 and 1. $\tanh(I_i)$ is a threshold for latching $m_i$. (b) A chaotic bit (c-bit) features deterministic billiard dynamics with a tunable slope. The billiard is periodic, given that $I_i$ is held constant. When the billiard reaches $-1$ or $+1$, $m_i$ is latched, and the billiard changes direction. (c) p-bits and c-bits form similar, asynchronous architectures, employing the same synaptic function. (d) A 20-bit factorizer conceptualized with p-bits, represented as a graph with 310 nodes and 1200 edges. (e) The 3D spin glass problem, with coupling strengths $J_{ij} \in \{-1,+1\}$. (f) The adaptive parallel tempering (APT) algorithm. Replicas of the same network run at different inverse temperature $\beta$ in parallel. Swaps attempts are made between pairs of replicas with the closest $\beta$, according to the Metropolis criterion (Eq. \ref{['eq:metropolis']}).
  • Figure 2: (a) Distribution of spin configurations for a 5 p-bit full adder, obtained analytically from the Boltzmann distribution. (b) Analytical distribution for p-bits employing parallel updates, as opposed to sequential Gibbs sampling aadit_massively_2022. (c) Numerical results for parallel p-bit updates, collected over $10^5$ Monte Carlo sweeps. (d) c-bits exhibit similar pathological behavior when their phases $x_i$ and states $m_i$ are all initialized to the same value, $+1$. Samples are collected over $10^5$ time steps.
  • Figure 3: Chaotic bits emulate a 1D ferromagnetic chain ($J_{ij} = +2$) of 8 qubits described by the quantum Transverse Ising Hamiltonian (Eq. \ref{['eq: quantum hamiltonian']}). For emulation purposes, we use Suzuki-Trotter decomposition with $R = 250$ replicas. We show average magnetization as a function of the transverse field ($\Gamma_x$) when the system is at constant inverse temperature $\beta = 10$. A symmetry breaking magnetic field of $h_i = 1$ is applied in the $+\hat{z}$ direction such that when $\Gamma_x = 0$, all spins point in the $+\hat{z}$ direction. Red dots show the results of 80 independent trials of c-bit simulations, and a single set of spin configurations is recorded at the end of $t_a = 1000$ time steps, for each trial. Blue triangles show the average numerical result. The green dashed line is an analytical solution from solving Eq. \ref{['eq: quantum boltzmann law']} as a function of $\Gamma_x$.
  • Figure 4: (a) Residual energy density at the critical point ($\rho_E^c$) as a function of annealing time for a $50 \times 50$ ferromagnetic Ising model lattice with nonperiodic boundary conditions ($J_{ij} = +1$). We perform linear regression on the 5 leftmost points of each data series to avoid finite-size effects. Error bars show $95\%$ bootstrap confidence intervals. The inset plot shows short-term exponential divergence of c-bit systems with slightly perturbed initial conditions. The slope represents a positive (maximum) Lyapunov exponent, implying chaotic behavior. (b) $\rho_E^c$ vs. annealing time for cubic spin-glass instances ($L = 11$, $J_{ij} \in \{-1,+1\}$). (c) $\rho_E^c$ as a function of inverse annealing velocity for the 2D ferromagnetic Ising model, using p-bits. Different annealing schedules are shown in a data collapse by considering the quench rate at the critical point, $\alpha$. (d) $\rho_E^c$ vs. inverse annealing velocity for the 2D ferromagnetic Ising model, using c-bits.
  • Figure 5: Residual energy density ($\rho_E^f$) as a function of total computational time ($Nt_a$) for cubic spin-glass instances with side length $L = 7$ ($J_{ij} \in \{-1,+1\}$). Curves scale as a power law: $\rho_E^f \propto t_a^{-\kappa_f}$. Error bars show 95% bootstrap confidence intervals.
  • ...and 1 more figures