{"id":"4d89d553-d6a9-46e9-805e-1088410f4bb4","arxiv_id":"2506.00269","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A new probabilistic computing primitive, the p-dit, extends binary p-bits to multi-dimensional and integer states and is demonstrated in custom ASIC and FPGA hardware with large speedups on optimization benchmarks.","lead":"This paper introduces 'p-dits', a generalization of probabilistic bits that can take more than two states, and shows they work in custom silicon and FPGA hardware. The authors report large speedups for partition, integer programming, and integer quadratic programming problems compared with standard bit-based probabilistic computers and software solvers.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Restricted p-int update is not shown to sample the stated Hamiltonian: Eq. (30) omits the self-coupling term in two-step energy differences, and even with J_ii=0 the neighbor-restricted chain converges to e^{2βIm}, not e^{βIm}.","rationale":"The paper's central claim is that generalized p-dits provide a path to efficient probabilistic optimization, with the restricted p-int update as a key component. The hardware evidence is real: a fabricated 130 nm ASIC, a working RISC-V controlled PIM, and measured improvements in trials-to-solution and time-to-solution that are consistent between simulation and hardware. Those empirical speedups stand on their own as demonstrations that the specific encoded problems can be solved faster with these extended variables. However, the theoretical framing asserts that the p-int update follows from the generalized Glauber-like dynamics of Eq. (11) and converges to a meaningful stationary distribution. The reader's weakest-assumption analysis correctly identified that the neighbor-restricted update does not Boltzmann-sample the stated Hamiltonian even in the simplest single-spin case, and my reading adds a sharper internal inconsistency: Eq. (28) contradicts Eq. (24) when J_ii is nonzero, so Eq. (30) is not actually the restriction of Eq. (11) to neighboring states. This matters because the ILP and IQP formulations generate nonzero self-couplings through the Hessian of the constraint terms, so the experiments exercise precisely the regime where the theory is least secure. I do not recommend rejection: the benchmark improvements may still be valid as heuristic optimization results, and the issue is fixable either by adding the missing self-coupling term in two-step transitions or by explicitly characterizing the stationary distribution and treating the temperature as an effective parameter. But the manuscript as written overstates the theoretical support for p-int sampling, so the conditional verdict is appropriate.","tokens_in":18361,"tokens_out":11708,"duration_ms":111670,"concrete_test":"Enumerate exactly the Markov chain induced by Eq. (30) for one p-int with range [−2,2], h=1, J_ii=1, β=0.5, and compare its stationary distribution with (a) exp(−βE) for E(m)=−hm−0.5J_ii m^2 and (b) the local Gibbs probabilities of Eq. (11) on G={m−1,m,m+1}. Repeat with J_ii=0. If the stationary distributions do not match (b) for J_ii=1, Eq. (30) is not a valid restriction of the generalized Glauber update, and the deferred convergence claim in Supplementary Note 5 cannot rescue the formulation as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing theoretical step is the claim that the restricted p-int update in Eq. (30) has a useful stationary distribution, supported only by a deferred proof in Supplementary Note 5. Two concrete problems are visible from the main text. First, for a single p-int with constant local field I and no self-coupling, Eq. (30) defines a birth-death chain with up/down ratio exp(2βI), so the stationary distribution is π_m ∝ exp(2βI m), not exp(βI m) = exp(−βE). The neighbor restriction therefore does not inherit the Boltzmann result of Eq. (10); it samples the same energy at inverse temperature 2β. Second, Eq. (28) claims ΔE_i^{m−1→m+1} = −2I_i, but Eq. (24) with k=2 gives −2I_i − 2J_ii when self-coupling is present. Since the local field in Eq. (25) includes J_ii m_i, the two-step energy difference must contain the extra self term. Eq. (30) omits it, so the transition probabilities are not even the restriction of the generalized Glauber rule Eq. (11) to G={m−1,m,m+1}; they sample a different, unstated kernel. The p-int benchmarks (5.3x ILP, 64x IQP) rely on this update, so the correctness of the optimization semantics—not just the raw speedups—depends on the missing stationary-distribution analysis.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":18799,"tokens_out":17023,"duration_ms":175227,"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":[{"comment":"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.","section":"Probabilistic Computing with P-ints, Eq. (30)"},{"comment":"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.","section":"Violation Variables"},{"comment":"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.","section":"Results/Integer Programming, Fig. 6"}],"minor_comments":[{"comment":"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.","section":"Throughout"},{"comment":"References [32] and [47] cite the same paper (Camsari et al., Stochastic p-bits for invertible logic) and should be consolidated.","section":"References"},{"comment":"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.","section":"Scaled Sampling"},{"comment":"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.","section":"Probabilistic Computing with P-ints, Eq. (28)"}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid empirical contribution with a fabricated ASIC and clear speedups. The main obstacle is the theoretical status of the p-int update: the restricted nearest-neighbor chain does not sample the Boltzmann distribution, and the paper's deferral to Supplementary Note 5 leaves this unresolved. I would ask that the supplementary notes be made available to reviewers and that the authors either prove the relevant stationary distribution or honestly reframe p-ints as a heuristic. The benchmark comparison issues (simulated p-bit baseline, single IQP instance) should also be addressed. I do not see grounds for rejection, because the isotropic p-dit contribution is sound and the p-int results are empirically valuable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read the p-dit paper. Bottom line: the framework and the hardware are real, and the isotropic p-dit part is worth taking seriously. But the p-int section has a load-bearing gap that needs fixing before I'd trust the integer-programming claims.\n\nWhat's actually new: a generalized d-dimensional spin model where each spin is a vector with arbitrary real components, plus the p-dit as its discrete probabilistic unit. The isotropic p-dit (categorical variable picking one orthogonal basis state) and the p-int (integer variable stepping by ±1) are useful restrictions, and the violation-variable encoding for inequality constraints is a nice practical trick. The paper also ships real hardware: a 130 nm ASIC with 64 reconfigurable p-elements and an FPGA IQP solver. The measured 34x on 3-partition, 5.3x on change-making, and 64x on the IQP justify reading the paper, even if they're not yet careful benchmarks.\n\nThe full-state update (Eq. 9) and the isotropic p-dit update (Eq. 22) are derived cleanly and converge to Boltzmann. The weak spot is the restricted p-int update in Eq. (30). The paper defers to Supplementary Note 5 for convergence, but from the main text the transition probabilities don't produce the stated Boltzmann distribution. For a single p-int with constant field I and no self-coupling, the up/down ratio in Eq. (30) is exp(2βI), so the stationary distribution is proportional to exp(2βI m), not exp(βI m). That's Boltzmann at an effective inverse temperature 2β, not β. The apparent missing self-coupling in Eq. (28) actually cancels when you compose the two one-step differences; the real problem is the ratio. Maybe Supp Note 5 addresses this by rescaling β or proving a different stationary distribution, but the paper doesn't say that in the main text. Since the p-int is the basis for the ILP and IQP results, the temperature semantics and the optimization claims hang on this.\n\nThe benchmark section is also a bit thin: no error bars on the speedup ratios, a single random IQP instance, and one trial per software solver. The 64x figure is suggestive, not definitive.\n\nWho is this for? Anyone working on Ising machines, p-bits, or probabilistic hardware. The p-dit/violation-variable ideas are worth discussing even if the p-int convergence gets revised. I'd send it to a referee, but with a clear request to check the p-int stationary distribution and re-run the benchmarks with proper statistics.","headline":"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.","tokens_in":19287,"tokens_out":6596,"would_cite":true,"duration_ms":60303,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Probabilistic p-dits beat binary p-bits by up to 34x and software solvers by 64x","keywords":["probabilistic computing","p-dit","p-bit","p-int","Ising machine","combinatorial optimization","categorical variables","integer linear programming"],"falsifier":"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.","tokens_in":18211,"feed_emoji":"🧮","tokens_out":10943,"duration_ms":115247,"temperature":0.7,"pith_summary":"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.","feed_headline":"New p-dit chips beat binary p-bits by up to 34x","feed_subtitle":"Categorical and integer variables encoded directly on an ASIC, beating a leading software IQP solver by 64x.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the one-hot and integer-programming Ising formulations that the p-dit encodings replace or improve.","marker":"[45]"},{"why":"Defines the p-bit and its stochastic update rule, the starting point that the generalized p-dit extends.","marker":"[47]"},{"why":"Used for the two's-complement binary encoding of integer variables in the p-bit ILP implementation.","marker":"[48]"},{"why":"Grounds the claim that binary integer encodings create energy barriers between adjacent values.","marker":"[54]"},{"why":"Establishes the planar/clock Potts model to which isotropic p-dits are related.","marker":"[28]"},{"why":"Source of the fixed-charge ILP and slack-variable formulation used as a baseline.","marker":"[55]"},{"why":"Alternative categorical/Potts implementation that requires explicit total-energy calculation, against which the energy-difference update is positioned.","marker":"[51]"},{"why":"Alternative Potts solver that decomposes the problem into Ising subproblems, which the p-dit formulation avoids.","marker":"[52]"},{"why":"Alternate k-state update with uniform state selection rather than energy-based probabilities, providing a contrast to p-dit updates.","marker":"[53]"}],"fun_headline_variants":["p-dits: beyond binary bits for faster optimization","Probabilistic d-dimensional bits speed up Ising machines 64x","Generalized p-dits solve categorical and integer problems faster","Beyond binary: p-dits give 64x speedup over software solvers"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["p-dits: beyond binary bits for faster optimization","Probabilistic d-dimensional bits speed up Ising machines 64x","Generalized p-dits solve categorical and integer problems faster","Beyond binary: p-dits give 64x speedup over software solvers"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000531,"raw_usage":{"total_tokens":2593,"prompt_tokens":1014,"completion_tokens":1579,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":630,"completion_tokens_details":{"reasoning_tokens":1505}},"tokens_in":630,"tokens_out":1579,"duration_ms":13558,"temperature":1.0,"reasoning_tokens":1505,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:10:37.512595+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Ising formulations of many np problems","cited_arxiv_id":null,"evidence_quote":"Supplies the one-hot and integer-programming Ising formulations that the p-dit encodings replace or improve."},{"cited_title":"Y., Faria, R., Sutton, B","cited_arxiv_id":null,"evidence_quote":"Defines the p-bit and its stochastic update rule, the starting point that the generalized p-dit extends."},{"cited_title":"& Bermel, P","cited_arxiv_id":null,"evidence_quote":"Used for the two's-complement binary encoding of integer variables in the p-bit ILP implementation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Grounds the claim that binary integer encodings create energy barriers between adjacent values."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the planar/clock Potts model to which isotropic p-dits are related."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Source of the fixed-charge ILP and slack-variable formulation used as a baseline."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Alternative categorical/Potts implementation that requires explicit total-energy calculation, against which the energy-difference update is positioned."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Alternative Potts solver that decomposes the problem into Ising subproblems, which the p-dit formulation avoids."}],"review_version":1}