multi-Stochastic Core Architecture for Scaling Probabilistic Ising Machines

Scaling Approach

• m-SCA integrates multiple PASS cores to expand problem sizes beyond the limit imposed by the number of neurons in a single core.The architecture scales machine size and computational throughput rather than scaling up one core.Source: Scaling Approach · p. 3
• Block Gibbs sampling partitions the mapped problem among four cores and samples each block conditional on neighboring blocks' states from the previous iteration.Adjacent edge neurons are clamped to set the conditional probability.Source: Scaling Approach · p. 3
• Because each block conditions on neighboring blocks' previous states, the other blocks can continue sampling while one block is sampled, enhancing throughput.Source: Scaling Approach · p. 4

Max-Cut Optimization Problem

• The benchmarks compare m-SCA with GPU simulated annealing using parallelized Gibbs sampling and CPU tabu search on an Intel Xeon Gold 6330.Source: Max-Cut Optimization Problem · p. 4
• Four PASS accelerators were tested on 100 randomly generated Max-Cut problems per graph size, spanning 4 to 784 nodes with weights {−1, 1}.The sparsely connected instances used King's move graphs and were directly mapped onto m-SCA.Source: Max-Cut Optimization Problem · p. 4
• For the 784-node configuration, the four-core cluster supports 6.2k connections and reaches 940.8 GMACs per second at 150 MHz.It reaches 14.7 GigaSamples/s at an 18.75 MHz sampling clock.Source: Max-Cut Optimization Problem · p. 4
• The study estimates ground-state solutions using the lowest-energy sampled state, rather than mode estimation, because mode estimation is affected by low-probability states in early sampling steps.Experiments were run 200 times per problem instance to measure energy distributions and calculate success probability.Source: Max-Cut Optimization Problem · pp. 4–5
• Success means a cut value greater than 95% of the best qbsolv solution; the reported median success probability is across 100 random problems per graph size.The authors state that higher success probability implies fewer attempts to reach the target and thus shorter TTS and lower energy consumption.Source: Max-Cut Optimization Problem · p. 5
• ∼1000 times acceleration is reported for m-SCA versus GPU simulated annealing and CPU qbsolv, with the acceleration trade-off visible above 196 nodes.The authors identify the scaling architecture's computing advantage for large-scale optimization problems.Source: Max-Cut Optimization Problem · p. 5

Quantum 1D Transverse Ising

• The one-dimensional quantum chain is mapped onto m-SCA to sample equilibrium statistics and model the basis-state probability distribution associated with the quantum wavefunction.The system follows a stoquastic Transverse Field Ising Hamiltonian; the approach is described as Block Gibbs sampling based.Source: Quantum 1D Transverse Ising · pp. 5–6
• With Ji,i+1 = 1 and Γz = 1, larger transverse fields Γx weaken z-direction spin order and ⟨mz⟩, while higher transverse fields and larger-chain interaction terms lower quantum energy.The reported field settings are Γx = 1.5 and 2.0.Source: Quantum 1D Transverse Ising · p. 6
• Quantum energy from m-SCA sampled states approximately matches DMRG within 6% error, while the sampled distribution includes ordered states with probability as low as 10^-8.Exponentially growing sample requirements limit the study to 24 spins with 24 replicas; the authors attribute energy error to low-probability states below 10^-8.Source: Quantum 1D Transverse Ising · p. 6
• The m-SCA sampled order parameter ⟨mz⟩ agrees well with theoretical DMRG calculations.QMC is run once during the sampling phase to approximate system properties.Source: Quantum 1D Transverse Ising · p. 6
• At least 4 orders of magnitude acceleration of QMC methods compared to state-of-the-art GPUs is reported, with approximately constant m-SCA scaling versus exponential GPU scaling.The m-SCA TTS curve changes near 14 spins as computation transitions from one core to four cores.Source: Quantum 1D Transverse Ising · p. 7

Conclusion

• m-SCA solves Max-Cut graphs up to 784 nodes with approximately 1000 times speed-up relative to GPU, and accelerates quantum Monte Carlo by at least 4 orders of magnitude compared to GPUs.For the quantum spin chain, it captures the equilibrium state distribution with probability precision of 10^-8 and determines order parameter and quantum energy with accuracy above 94%.Source: Conclusion · p. 7
• The multi-Stochastic Cores Architecture scales processing capabilities by a factor of 3.37 relative to a single-core architecture and achieves 18.67 binary-TOPS/W.The conclusion identifies Block Gibbs sampling as the core concept guiding the implementation.Source: Conclusion · p. 7
• The reported acceleration is attributed to asynchronous updates, uncorrelated samples produced at 150 MHz, and sampling at 18.75 Megasamples per neuron per sec.Average power is reported as around 64.3 µW per neuron operation, including synapse calculations.Source: Conclusion · p. 7
• The conclusion presents m-SCA as a candidate for scaling computational and sampling throughput of stochastic systems and accelerating classical and quantum problems over GPUs in a restricted power budget.The stated scope includes NP-hard classical and quantum problems.Source: Conclusion · p. 7

PASS Stochastic Core

• Each PASS core contains 256 stochastic mixed-signal neurons operating at 0.6 V, locally connected synapses in King’s move configuration, and a built-in sampler.Each neuron uses locally generated reverse-bias-diode shot noise amplified for a sigmoid comparator that produces a digitized stochastic signal.Source: PASS Stochastic Core · p. 8
• The core has no computation clock; digitized stochastic neuron signals serve as internal clocks for asynchronous updates, with Poisson-process autocorrelation decay setting a 150 MHz internal clock speed.Source: PASS Stochastic Core · p. 8
• Synapses multiply and accumulate neighboring neuron outputs and interaction weights, using digital multiplexers and a DAC to convert accumulated values into analog Vin for asynchronous neuron updates.The FPGA host samples neuron data at 300 MHz, yielding an effective 18.75 MHz sampling rate for each neuron.Source: PASS Stochastic Core · p. 8

Max-Cut Problem Setup

• Max-Cut partitions graph nodes into V+ (si = 1) and V− (si = −1) to maximize the sum of edge weights between the two sets.The formulation is given for a graph G with edges E and interaction weights Jij.Source: Max-Cut Problem Setup · p. 8
• The cut δ(V) consists of edges interconnecting V+ and V−, while E(V+) and E(V−) denote edges connecting nodes within each respective set.Source: Max-Cut Problem Setup · p. 8
• Minimizing the Ising Hamiltonian H(s) corresponds to finding the maximum cut; sampled cut values are evaluated by their Hamiltonian energy to derive success probability.Source: Max-Cut Problem Setup · p. 9

• Suzuki-Trotter decomposition converts the transverse Ising Hamiltonian of the quantum chain into its classical equivalent.Source: Transverse Ising 1D Setup · p. 9
• The classical representation is a 2D lattice with N spins and M replicas; infinite replicas reproduce the quantum chain, while finite M keeps error permissible.Source: Transverse Ising 1D Setup · p. 9
• Real-valued weights and biases are adapted to m-SCA’s 8-bit integer precision using constant weight multiplication and bias addition.Source: Transverse Ising 1D Setup · p. 9
• At Γx = 2, negligible perpendicular interaction makes the replicas effectively independent chains that can map to contiguous N-neuron blocks.Source: Transverse Ising 1D Setup · p. 9
• The finite M×N lattice forms a King’s-move subgraph that is mapped onto m-SCA and partitioned into sampling blocks.Source: Transverse Ising 1D Setup · p. 9

• Equation 11 calculates quantum energy using the states sampled from m-SCA.Source: Quantum Energy Calculation · p. 10

• A PASS core arranges 256 neurons in a King’s-move grid, with noisy sigmoid neuron circuits and synapse circuits that accumulate weighted neighboring signals.The neuron circuit includes a noise source, amplifier, sigmoid comparator, and digitization stage; synapse circuitry uses multiplexers and digital-to-analog conversion.Source: Figure S1 · p. 18
• An exponential fit to PASS-core autocorrelation defines the rate of independent stochastic samples and the 150 MHz internal spin-update clock.Source: Figure S2 · p. 19
• The Trotterization figure depicts decomposition of a one-dimensional quantum spin chain into a two-dimensional spin lattice.Source: Figure S3 · p. 19