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Extended-variable probabilistic computing with p-dits

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Probabilistic p-dits beat binary p-bits by up to 34x and software solvers by 64x

desk verdict Real hardware and a useful generalization, but the p-int update's stationary distribution is unproven and may be sampling at 2β, which undercuts the integer-programming claims until fixed. read the letter →

arxiv 2506.00269 v1 pith:ZFCHOBWW submitted 2025-05-30 physics.app-ph

Christian Duffee , Jordan Athas , Andrea Grimaldi , Deborah Volpe , Giovanni Finocchio , Ermin Wei , Pedram Khalili Amiri This is my paper
classification physics.app-ph
keywords probabilisticcomputingp-ditp-bitp-intIsingmachinecombinatorialoptimizationcategoricalvariablesintegerlinearprogramming
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that the binary p-bit, the standard unit of probabilistic Ising machines, can be usefully generalized to a p-dit that holds more than two states, and that this generalization translates directly into faster probabilistic optimization on real hardware. The authors define a probabilistic d-dimensional bit and then specialize it to two practical forms: isotropic p-dits, which encode categorical variables with no invalid states, and probabilistic integers, which encode numeric variables directly on a number line. On a 130 nm CMOS ASIC, the isotropic p-dit3 solves a 14-number 3-partition problem in about 34x fewer trials than the p-bit implementation, and the p-int solves a change-making integer linear program in about 5.3x fewer trials. An FPGA p-int solver on a non-convex integer quadratic program reaches the true solution about 64x faster than the best of several state-of-the-art software solvers. If the formulation scales, p-dits give Ising machines a way around the resource and energy-barrier costs of binary and one-hot encodings.

What carries the argument

The central object is the probabilistic d-dimensional bit (p-dit), a spin that stochastically oscillates among discrete states in $|D|$-dimensional space. Its update rule is a multi-input sigmoid over the energy differences to all candidate states, so lower-energy states are selected with higher probability. Two restricted forms carry the hardware results: an isotropic p-dit, whose allowed states are the orthogonal coordinate axes and whose couplings reduce to a single $N \times N$ matrix, and a p-int, whose allowed states are consecutive integers along one axis and whose updates are limited to stepping to a neighboring value. These restrictions make extended probabilistic variables compact enough to fabricate, and the isotropic p-dit is explicitly related to the planar/clock Potts model.

What would settle it

Measure the long-run state histogram of a single p-int on the ASIC with a fixed local field $I$, no self-coupling, and no other spins, and compare it with the Boltzmann weights $\exp(\beta I m)$ over its full range; a systematic mismatch would show that the restricted update does not sample the distribution implied by the temperature parameter.

Watch

Extended reading notes

Core claim

The central claim is that probabilistic computing does not have to be built from two-state p-bits: one p-dit can represent a d-dimensional spin, and two restricted classes of p-dit remove the main encoding bottlenecks that arise when Ising machines are applied to categorical and integer optimization. An isotropic p-dit restricts the spin to lie along exactly one of the orthogonal axes in its state space, so a categorical variable is always in a valid state; this eliminates the exponentially growing set of invalid one-hot assignments and the need to tune a constraint-to-objective constant. A p-int restricts the spin to integer levels along one axis and updates by increments or decrements of one, which avoids the energy barriers that binary representations create between adjacent integer values. The authors derive Glauber-like update probabilities for both cases, argue in supplementary notes that the resulting Markov chains converge, and demonstrate the performance gains experimentally on an ASIC and an FPGA.

Load-bearing premise

The load-bearing premise is that a p-int restricted to stepping only to its current value and its two immediate neighbors converges to a stationary distribution matching the intended energy landscape, with the proof deferred to Supplementary Note 5.

Editorial extensions

If this is right

  • For problems with many categorical variables, the fraction of invalid one-hot states grows exponentially, so the advantage of isotropic p-dits should widen as problem size grows.
  • One p-int replaces roughly $\lceil \log_2(B-A) \rceil$ p-bits for an integer variable with bounds $A$ and $B$, reducing hardware footprint and the number of energy barriers a solver must cross.
  • Inequality constraints encoded with asymmetric violation variables avoid the deep energy valleys of slack variables, giving an additional ~10x improvement in trials-to-solution on a fixed-charge ILP.
  • A p-int probabilistic solver can outperform conventional software MILP/MIQCP solvers in time-to-solution on a non-convex IQP problem, suggesting a practical role for this hardware approach.
  • A single reconfigurable p-element can act as a p-bit, a p-int, or an isotropic p-dit3, so one CMOS chip can host several problem types with a two-clock-cycle update.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • We infer that the reported 34x and 5.3x ratios are likely to be lower bounds for larger instances, since both sources of overhead that p-dits remove grow with problem size and word length.
  • The violation-variable idea is not tied to p-ints: the same asymmetric constraint representation could be adapted to isotropic p-dits or other extended variables, which the paper does not explore.
  • Because isotropic p-dits coincide with the clock Potts model, annealing schedules and theoretical results from Potts-model physics could be transferred directly to tune p-dit-based solvers.
  • Graph coloring and scheduling are natural next benchmarks for isotropic p-dits, since they share the labeling-symmetry condition that makes the isotropic coupling matrix valid.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper generalizes probabilistic Ising machines from binary p-bits to p-dits, probabilistic units with d-dimensional states, and analyzes two restrictions: isotropic p-dits for categorical variables and p-ints for integer variables. The authors derive a full-state Glauber-like update that converges to the Boltzmann distribution and a restricted update for p-ints that is claimed to converge. They fabricate a 130-nm ASIC with 64 configurable p-elements and implement an FPGA-based p-int solver. Measured results include ~34x fewer trials-to-solution for a 3-partition problem using isotropic p-dit3 versus p-bits, ~5.3x for a change-making ILP using p-ints versus p-bits, and ~64x faster time-to-solution for a non-convex IQP compared with several GAMS solvers. The paper also introduces violation variables and scaled sampling as practical heuristics for inequality constraints and variable ranges.

Significance. The empirical hardware results are substantial: a fabricated ASIC implementing all three variable types, an elegant isotropic p-dit formulation that avoids invalid one-hot states, and large measured speedups on representative optimization problems. The full-state and isotropic p-dit theory is sound as far as presented, and the performance numbers are measured rather than fitted. The main significance is the demonstration that extended variables can reduce hardware resource usage and improve time-to-solution for problems with categorical or integer structure. However, the theoretical status of the restricted p-int update needs clarification before the 'probabilistic computing' framing is fully justified; the p-int results currently stand as strong heuristics rather than exact Boltzmann sampling.

major comments (3)
  1. [Probabilistic Computing with P-ints, Eq. (30)] The restricted p-int update does not yield the Boltzmann stationary distribution of the stated Hamiltonian. For a single p-int with constant local field I and J_ii=0, the transition probabilities in Eq. (30) give P_up/P_down = exp(2βI), so the birth-death chain has π_m ∝ exp(2βI m), not exp(-βE) ∝ exp(βI m). Thus p-int annealing operates at an effective inverse temperature 2β for this case, and in general the stationary distribution is not specified. The main text only states that the update 'will result in a converging probability distribution' (deferred to Supplementary Note 5). Because the paper's framing presents p-ints as probabilistic computing elements, the authors should either provide the actual stationary distribution and its relation to the Hamiltonian, or explicitly state that p-ints are a non-Boltzmann heuristic and adjust the theoretical narrative accordingly.
  2. [Violation Variables] The violation-variable construction produces an asymmetric J matrix. The Boltzmann result in Eq. (10) and the energy-difference formulas are derived under the symmetric-coupling assumption stated before Eq. (8). The paper does not specify what energy function, if any, the asymmetric update kernel is sampling, nor does it prove convergence for this case. Since the fixed-charge and IQP benchmarks rely on violation variables, the theoretical basis for these results should be stated explicitly, or the scheme should be presented as a heuristic with appropriate caveats.
  3. [Results/Integer Programming, Fig. 6] The reported 5.3x improvement in trials-to-solution is an ASIC p-int result compared with a simulated p-bit baseline ('simulated results for p-bits, and both simulated and experimental results for p-ints'). Because the ASIC supports p-bits, the comparison should either use the ASIC p-bit baseline or be explicitly labeled as hardware-versus-simulation. Similarly, the IQP comparison in Fig. 8 is based on a single problem instance, 25 hardware trials, and one trial per software solver; the generality of the ~64x claim would be strengthened by multiple instances and seeds, and these limitations should be stated.
minor comments (4)
  1. [Throughout] Typos and leftover fragments: 'dimentional' in Fig. 1, 'quadradic' in the Discussion, '3-partion' in Table 1, and the orphaned 'or53.' in §Isotropic P-dits should be corrected.
  2. [References] References [32] and [47] cite the same paper (Camsari et al., Stochastic p-bits for invertible logic) and should be consolidated.
  3. [Scaled Sampling] The scaled-sampling procedure is described only qualitatively; the formula for the sampling ratio as a function of p-int ranges should be given in the main text or Methods.
  4. [Probabilistic Computing with P-ints, Eq. (28)] The statement in Eq. (28) is consistent with Eq. (24) when I_i is evaluated at the current state; the self-coupling terms cancel in the two-step difference. The text should clarify this to avoid apparent inconsistency.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the p-dit theory and benchmarks are self-contained and measured; the main weakness is an unsupported stationary-distribution claim for the restricted p-int update, which is a correctness issue rather than a circular reduction.

full rationale

The paper's derivation chain is largely self-contained. The generalized spin Hamiltonian (Eq. 5) and the Glauber-like update rules (Eqs. 9 and 11) define the stochastic process; the claim that the full-update chain converges to the Boltzmann distribution (Eq. 10) is a standard detailed-balance result, not an input fitted from data. The isotropic p-dit and p-int restrictions are specializations of this framework, and the reported ~34x, ~5.3x, ~10x, and ~64x improvements are measured trials-to-solution or time-to-solution numbers from an ASIC and an FPGA, not quantities derived from the model, so there is no fitted-input-called-prediction. Hyperparameters (beta, C, O, cutoffs) are tuned per benchmark, which is standard practice and not circular. There are some self-citations (e.g., refs. 3, 9, 39), but they are background or roadmap references and are not load-bearing for any central claim; no uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled in via citation. The one substantive weakness is that the restricted p-int update of Eq. (30) is asserted to converge, with proof deferred to Supplementary Note 5, and the two-step energy difference in Eq. (28) appears to omit the -2J_ii term required by Eq. (24). This is a missing or invalid mathematical justification, i.e., a correctness risk, not a circular reduction: the update rule is not defined in terms of the claimed stationary distribution, and the benchmark speedups do not rest on that distribution being exactly Boltzmann. Hence the circularity score is 1.

Assumptions & free parameters 5 free parameters · 5 assumptions · 4 invented entities

The generalized spin model is a straightforward vector-spin Hamiltonian with standard Boltzmann or Glauber dynamics for the full-state update. No parameters are fitted in the theory itself. The empirical claims rest on hand-tuned hyperparameters (C, O, β, cutoffs) and on two heuristics: the restricted one-step update for p-ints and the violation-variable encoding for inequality constraints. The convergence of the restricted update to a Boltzmann distribution is asserted rather than derived and appears false in the single-spin case.

free parameters (5)
  • Inverse temperature β = e.g., 1/64 (p-int change-making), 1/128 (p-bit change-making), β0=1/256 and 1/4 for fixed-charge, β0=3.3 for IQP
    Tuned per problem and implementation to maximize success rate; not derived.
  • Constraint scaling C = e.g., 94 (3-partition), 1 (change-making, fixed-charge)
    Chosen to balance constraint enforcement and objective optimization; swept in experiments.
  • Objective scaling O = e.g., 1 (3-partition), 96 (change-making), 2 (slack fixed-charge), 1/4 (violation fixed-charge), 0.0008 (IQP)
    Tuned per problem and solver configuration.
  • IQP reset cutoff = 2^21 and 2^28 iterations
    Selected to optimize time-to-solution based on the distribution of solution times.
  • Sampling ratio for scaled sampling = proportional to p-int range
    Ad hoc method to equalize exploration across variables with different ranges.
assumptions (5)
  • standard math Boltzmann distribution and Glauber dynamics for multi-state spins
    Used in Eq. (10) and the derivation of update probabilities.
  • domain assumption Quadratic Hamiltonian captures the relevant energy landscape
    The generalized model (Eq. 5) considers only up to second-order interactions; higher-order terms are mentioned but not used.
  • ad hoc to paper Restricted state-dependent updates converge to the Boltzmann stationary distribution
    Assumed for p-ints at Eq. (30) and deferred to Supplementary Note 5; the main text equations suggest the stationary distribution is not Boltzmann.
  • ad hoc to paper Asymmetric J matrix in violation variables preserves the optimization dynamics
    Violation variables break symmetry of J to improve exploration, with no proof that the resulting dynamics converge to any desired distribution.
  • ad hoc to paper Scaled sampling improves exploration
    Proportional sampling of p-ints by range is introduced heuristically.
invented entities (4)
  • p-dit (probabilistic d-dimensional bit) independent evidence
    purpose: Base unit of the generalized spin model; oscillates among discrete states in d-dimensional space.
    Implemented as 64 multi-purpose p-elements on a fabricated ASIC and measured on partition and ILP problems.
  • Isotropic p-dit independent evidence
    purpose: Represent categorical variables with orthogonal states, avoiding invalid one-hot encodings.
    Implemented on ASIC; 3-partition and 6-partition results show solution quality improvements.
  • p-int (probabilistic integer) independent evidence
    purpose: Represent integer variables with regular steps along one dimension.
    Implemented on ASIC and FPGA; shows speedups on ILP and IQP problems.
  • Violation variables independent evidence
    purpose: Replace slack variables for inequality constraints in ILP and IQP Hamiltonians.
    Tested on fixed-charge ILP; shows about 10x improvement over slack variables with even sampling.

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Cite this review

Pith. "Pith review of Extended-variable probabilistic computing with p-dits." pith.science (2026). https://pith.science/paper/ZFCHOBWW

@misc{pith2026250600269,
  author       = {Pith},
  title        = {Pith review of: Extended-variable probabilistic computing with p-dits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZFCHOBWW}},
  note         = {Machine review of arXiv:2506.00269}
}
read the original abstract

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 (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): (A) isotropic p-dits, which simplify the implementation of categorical variables resulting in ~34x performance improvement compared to a p-bit implementation on an example 3-partition problem. (B) Probabilistic integers (p-ints), which simplify the representation of numeric values and provide ~5x improvement compared to 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 ~64x faster time-to-solution compared to 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.

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.