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arxiv: 2605.06037 · v1 · submitted 2026-05-07 · 💻 cs.AR

Recognition: unknown

A virtually connected probabilistic computer as a solver for higher-order, densely connected, or reconfigurable combinatorial optimisation problems

Amy J. Searle , Harry Youel , Fredrik Hasselgren , Annika M\"oslein , Ramy Aboushelbaya , Marko von der Leyen
Authors on Pith no claims yet

Pith reviewed 2026-05-08 04:14 UTC · model grok-4.3

classification 💻 cs.AR
keywords probabilistic computingphotonic hardwarecombinatorial optimizationspin glassesvirtual connectionsheuristic solversErdos-Renyi graphsdigital annealing
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The pith

Virtual connections in a photonic probabilistic computer allow direct solving of dense and higher-order optimization problems without the quality loss from transformations.

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

The paper shows how virtual hardware connections in photonic probabilistic computers based on quantum random number generators remove the need to reshape problems through embedding, sparsification, or quadratisation. This preserves native problem size and solution quality for instances with dense, higher-order, or reconfigurable geometry that physical topologies cannot handle directly. Simulations emulating the architecture predict that the approach finds ground-state approximations of Erdos-Renyi spin-glasses orders of magnitude faster than recently reported digital annealing units. Greedy graph colouring is applied to enable hardware parallelisation with favourable scaling for target solution qualities.

Core claim

The architecture uses virtual connections to let probabilistic bits interact without physical rewiring, so heuristic solvers run on the original problem graph. This avoids the size blow-up that quadratisation or sparsification would impose on dense graphs. Simulations of this setup forecast substantially shorter times to solution than digital annealing hardware achieves on the same Erdos-Renyi spin-glass ground-state task.

What carries the argument

Virtual hardware connections that emulate arbitrary pairwise or higher-order interactions between probabilistic bits in a photonic random-number-generator architecture.

If this is right

  • Native problem size is retained, avoiding the large expansions that quadratisation or sparsification would require for dense graphs.
  • Heuristic solvers become usable on higher-order or reconfigurable problems that would otherwise be incompatible with fixed hardware topologies.
  • Graph-colouring parallelisation produces scaling that improves with desired solution quality.
  • Time-to-solution gains of orders of magnitude are expected versus digital annealing on spin-glass instances.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same virtual-connection idea could be tested on other probabilistic or analogue hardware platforms that currently face topology limits.
  • If the performance holds, the method may open new application areas in logistics or machine learning where interaction graphs are naturally dense.
  • Hardware prototypes would need to verify that the virtual-connection layer scales without introducing new error sources not captured in simulation.

Load-bearing premise

Virtual connections can be realised in real photonic hardware without adding latency, noise, or overhead that erases the simulated speed advantage.

What would settle it

Fabricate the photonic hardware, run the same Erdos-Renyi instances, and measure whether the observed time to solution matches the simulation or is degraded by implementation overhead.

Figures

Figures reproduced from arXiv: 2605.06037 by Amy J. Searle, Annika M\"oslein, Fredrik Hasselgren, Harry Youel, Marko von der Leyen, Ramy Aboushelbaya.

Figure 1
Figure 1. Figure 1: (a) The Chimera architecture of D-Wave Inc., a quantum annealer which is often employed to demonstrate quantum annealing performance. In order to represent higher-order energy functions as a Chimera graph, or other imposed architectures, an expensive quadratisation process must be performed. When converting a densely connected problem, embedding or sparsification processes must also be employed. (b) A rand… view at source ↗
Figure 2
Figure 2. Figure 2: (a) A hypergraph over twelve vertices with five hyperedges. The minimal hitting set is highlighted in white. (b) An alternative tabular representation of the set cover problem. The set cover problem asks, given a set S X, and a selection {Vi}i∈I of subsets of X, such that these subsets cover X, or i∈I Vi = X, what is the minimal subset R of the sets Vi which covers X? It can be visualised in tabular form, … view at source ↗
Figure 3
Figure 3. Figure 3: (a) The number of iterations required to reach certain solution qualities for hypergraph dimensions k = 5 and k = 10, where now a single iteration updates a group of independent p-bits in parallel. (b) The growth in problem size for randomly generated instances for hypergraph dimensions k = 5 and k = 10, when quadratising the HUBO into a QUBO. (c) The performance of the PC-SA algorithm, as measured by solu… view at source ↗
Figure 4
Figure 4. Figure 4: (a) Illustration of a single KMC process on an arbitrary 10 city TSP. (b) The mask, M0, generated from solving the K1 problem, showing how KMC has reduced the number of p-bits required from 100 → 29 [35]. The existence of M can be justified by assuming that the optimal tour for the Km−1 problem will not comprise moves between different clusters until all constituent cities have been visited in a given clus… view at source ↗
Figure 5
Figure 5. Figure 5: Sparsification results for the burma14 instance. (a) The number of p-bits required to reach an architecture with a specified number of neighbours, with a low neighbour count requiring more p-bits. (b) The relationship between the ratios rN and rS representing the increase in problem size and the sparsification of the graph, respectively (for k = 2). Results are included for both KMC masked and unmasked p-b… view at source ↗
Figure 6
Figure 6. Figure 6: Spin-glass topologies of (a) Edwards–Anderson model on a 2D lattice (sparse). (b) Erdos–Rényi random ˝ graph (dense). (c) Sherrington–Kirkpatrick all-to-all model (dense & maximally connected). 0.2 0.4 0.6 0.8 1.0 Erd s-Rényi p 10 1 10 1 10 2 10 2 10 3 10 3 Iterations q = 0.5 q = 0.8 PC-SA PC-PT N = 100 N = 500 N = 750 N = 1024 (a) Iteration scaling with density at various sizes 200 400 600 800 1000 Graph … view at source ↗
Figure 7
Figure 7. Figure 7: The number of iterations required to attain solution qualities of q = 0.5, 0.8 over (a) the Erdos–Rényi graph ˝ density p and (b) the graph size N at a fixed p = 1. Note we redacted the N = 250 data in (a) for better visual clarity. it operates in polynomial time and ensures the use of no more than ∆ + 1 colors, where ∆ denotes the maximum degree of the graph. This makes it a practical and efficient fallba… view at source ↗
Figure 8
Figure 8. Figure 8: , we estimate the TTS for q = 0.8 as ∼ 10−5 seconds for the largest graph size of 1024, outperforming the digital annealing algorithm considered in Ref. [6] by several orders of magnitude. 200 400 600 800 1000 Graph size N 10 6 10 6 10 5 10 5 10 4 10 4 10 3 10 3 10 2 10 2 10 1 10 1 10 0 10 0 Time to solution (s) q = 0.5 q = 0.8 PC-SA DA (Ref.[6]) view at source ↗
Figure 9
Figure 9. Figure 9: The sparsification results for an Erdos–Rényi spin-glass instance with ˝ N = 100. (a) The number of p-bits required for a specified maximum number of neighbours for the sparsified graph (b) shows the relationship between the ratios rN and rS representing the increase in problem size and the sparsification of the graph (for k = 2). 15 view at source ↗
Figure 10
Figure 10. Figure 10: Temperature plots showing the variation in the number of colour groups |G| with the hypergraph dimension k and the number of hyperedges m, for a fixed number of vertices N. The chosen values of N were (a) N = 500, (b) N = 1000, and (c) N = 5000. F Iterations against solution quality for the TSP problem In view at source ↗
Figure 11
Figure 11. Figure 11: The average solution quality q against iterations I obtained by the PC-SA algorithm for varying hypergraph dimensions. Dimensions k = 10, 20, 30 and 40 were tested. Other parameters, such as the number of vertices and edges of the hypergraph, were kept constant. Each point was averaged over the best obtained over 20 repeats for 20 hypergraphs. We also show how the distribution of tours obtained changes wi… view at source ↗
Figure 12
Figure 12. Figure 12: The best solution qualities at various numbers of total iterations I of the SA and PT algorithms without (a) and with (b) KMC. The error bars shows the range of solution qualities obtained for 100 runs on the burma14 instance. 23 view at source ↗
Figure 13
Figure 13. Figure 13 view at source ↗
read the original abstract

Recently, there has been growing interest in unconventional computing as an approach for solving NP-hard problems, by developing dedicated hardware to find solutions more efficiently than conventional CPUs. In many of these approaches, however, certain problem geometries must be transformed into forms that are more amenable to the available hardware topology through techniques such as embedding, sparsification, and quadratisation, leading to a deterioration in solution quality. A probabilistic computing architecture based on high speed photonic quantum random number generators was recently proposed which utilises virtual hardware connections (Aboushelbaya et al., 2025), circumventing the necessity for such procedures. Here, we discuss the applicability of virtually connected hardware for running heuristic solving methods to solve a selection of problems, which due to their geometry, would suffer from topological hardware restrictions. We also employ greedy graph colouring algorithms for hardware parallelisation, allowing favourable scaling for desirable solution qualities. To emphasise the difficulty in solving these problems on physically connected hardware, we demonstrate the increase in problem size that would occur with quadratisation or sparsification. Using simulations to emulate hardware, we predict that a photonic probabilistic computer would outperform the time to solution recently reported for digital annealing units, on the ground state approximation of Erdos-Renyi graph spin-glasses, by orders of magnitude.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript proposes a virtually connected photonic probabilistic computer, building on prior work (Aboushelbaya et al., 2025), as a solver for higher-order, densely connected, or reconfigurable combinatorial optimization problems. It argues that virtual hardware connections eliminate the need for embedding, sparsification, or quadratisation that degrade solution quality on physical topologies. The paper discusses applicability to selected problems, employs greedy graph colouring for hardware parallelisation, demonstrates size increases from quadratisation/sparsification on physical hardware, and uses simulations to predict that the photonic architecture outperforms recently reported digital annealing units by orders of magnitude on ground-state approximation of Erdős–Rényi graph spin-glasses.

Significance. If the simulation results hold under realistic hardware conditions, the work could enable more efficient heuristic solvers for NP-hard problems with dense or higher-order interactions by bypassing topological restrictions, offering a promising direction in probabilistic and photonic computing hardware.

major comments (2)
  1. [Abstract / performance prediction] Abstract and performance-prediction section: the central claim of orders-of-magnitude improvement in time-to-solution rests on simulations emulating the virtually connected photonic hardware, yet no quantitative details on simulation parameters, error bars, scaling assumptions, noise models, or validation against physical photonic implementations are supplied. This directly undermines assessment of whether virtual-connection overhead remains sub-dominant for dense Erdős–Rényi graphs.
  2. [Architecture description] Virtual-connection mechanism (referenced from Aboushelbaya et al., 2025): the manuscript supplies no explicit bounds or analysis showing that latency, noise, or control overhead from virtual edges stays negligible as graph density approaches the dense limit used in the benchmarks. This assumption is load-bearing for the outperformance prediction over digital annealing units.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive review and positive assessment of the work's potential significance. The major comments identify areas where greater transparency on simulation details and overhead analysis is warranted. We address each point below and will incorporate revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract / performance prediction] Abstract and performance-prediction section: the central claim of orders-of-magnitude improvement in time-to-solution rests on simulations emulating the virtually connected photonic hardware, yet no quantitative details on simulation parameters, error bars, scaling assumptions, noise models, or validation against physical photonic implementations are supplied. This directly undermines assessment of whether virtual-connection overhead remains sub-dominant for dense Erdős–Rényi graphs.

    Authors: We agree that additional quantitative details are needed to support the performance claims. In the revised manuscript we will expand the performance-prediction section with a dedicated paragraph specifying the simulation parameters employed (Monte Carlo step counts, temperature schedules, and graph-generation seeds), report error bars obtained from repeated independent runs, state the scaling assumptions used for virtual-connection emulation, and include a short discussion of how the emulation aligns with the photonic hardware characteristics reported in the referenced prior work. These additions will allow readers to assess the sub-dominance of virtual-connection overhead for the dense graphs examined. revision: yes

  2. Referee: [Architecture description] Virtual-connection mechanism (referenced from Aboushelbaya et al., 2025): the manuscript supplies no explicit bounds or analysis showing that latency, noise, or control overhead from virtual edges stays negligible as graph density approaches the dense limit used in the benchmarks. This assumption is load-bearing for the outperformance prediction over digital annealing units.

    Authors: We acknowledge that explicit bounds on virtual-connection overhead are not supplied in the current text. While the mechanism itself is described in the cited prior work, we will add a concise analytical paragraph to the architecture section. This paragraph will derive first-order bounds on latency and control overhead as functions of edge density, showing that the incremental cost per virtual edge remains constant and constitutes only a small fraction of total update time for the Erdős–Rényi instances considered. The revised text will thereby substantiate that the overhead does not alter the reported orders-of-magnitude advantage. revision: yes

Circularity Check

1 steps flagged

Performance prediction rests on self-cited virtual-connection architecture and unvalidated simulation assumptions

specific steps
  1. self citation load bearing [Abstract]
    "A probabilistic computing architecture based on high speed photonic quantum random number generators was recently proposed which utilises virtual hardware connections (Aboushelbaya et al., 2025), circumventing the necessity for such procedures. ... Using simulations to emulate hardware, we predict that a photonic probabilistic computer would outperform the time to solution recently reported for digital annealing units, on the ground state approximation of Erdos-Renyi graph spin-glasses, by orders of magnitude."

    The outperformance prediction is obtained by simulating the virtual-connection mechanism whose correctness and overhead scaling are taken directly from the overlapping-authors 2025 citation. Because the manuscript supplies no independent derivation, noise model, or external benchmark for the assumption that virtual edges remain sub-dominant at the dense limit, the headline numerical claim reduces to the unverified premises of the self-cited prior work.

full rationale

The manuscript's central claim is a simulation-based prediction of orders-of-magnitude speedup on Erdős–Rényi spin glasses. This prediction is obtained by emulating the virtually-connected photonic architecture introduced in the cited Aboushelbaya et al. (2025) paper by overlapping authors. The abstract explicitly states that the architecture 'utilises virtual hardware connections (Aboushelbaya et al., 2025), circumventing the necessity for such procedures' and then 'using simulations to emulate hardware, we predict' the outperformance. No new quantitative bounds, noise model, or hardware validation of virtual-edge overhead appear in the provided text; the scaling assumptions and negligible-latency premise are therefore inherited from the self-citation. This constitutes partial circularity under the self-citation-load-bearing pattern, but the paper still supplies new problem mappings and graph-colouring parallelisation steps that are independent of the prior work, preventing a higher score.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract provides no explicit free parameters, axioms, or invented entities; the work inherits the probabilistic computing model and virtual-connection mechanism from prior literature without stating new postulates here.

pith-pipeline@v0.9.0 · 5559 in / 1114 out tokens · 26471 ms · 2026-05-08T04:14:02.124768+00:00 · methodology

discussion (0)

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