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arxiv: 2606.04070 · v1 · pith:WKB6N2CGnew · submitted 2026-06-02 · 🪐 quant-ph

Quantum circuit partition as a maze: emerging percolation transition via path finding

P. Zentilini , M. Guatto , F. Preti , D. Arya , F. A. C\'ardenas-L\'opez , F. Motzoi , E. Prati This is my paper

Pith reviewed 2026-06-28 09:49 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum circuit partitioningpercolation transitionCNOT gatesmaze path findingqubit permutationssimulated annealingcircuit optimizationnetwork science
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The pith

A percolation phase transition in a CNOT maze separates partitionable quantum circuits from nonpartitionable ones.

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

The paper models the problem of partitioning a quantum circuit as finding a cutting path through a maze whose walls are the CNOT gates. Existence of such a path marks a percolation phase transition that divides circuits into a partitionable regime and a nonpartitionable regime, with the division arising from qubit permutations generated by simulated annealing. The transition appears when the number of CNOTs is nearly equal to the number of qubits, supplying a concrete numerical criterion for deciding whether a given circuit admits a partition into two CNOT clusters. The authors examine the resulting CNOT cluster structure through network-science tools and distribution analysis to support the criterion.

Core claim

Formalizing circuit partitioning as a cutting path through a maze of CNOT walls shows that the existence of the path separates quantum circuits into two classes via a percolation phase transition. The transition distinguishes a partitionable regime from a nonpartitionable one that arises from qubit permutations generated by simulated annealing. Partitioning into two CNOT clusters becomes possible when the number of CNOTs is almost equal to the number of qubits, yielding a scalable criterion for identifying whether such a partition exists.

What carries the argument

Maze representation of the circuit in which CNOT gates are walls and a cutting path is sought; percolation transition in path existence under qubit permutations generated by simulated annealing.

If this is right

  • Partitioning into two CNOT clusters is feasible precisely when the CNOT count is nearly equal to the qubit count.
  • The percolation criterion supplies a practical test for deciding whether a circuit can be split without removing CNOT gates.
  • Network-science analysis of the CNOT cluster under permutations reveals how the transition organizes the gate structure.
  • The same maze model can be used to guide algorithmic development for parallel circuit optimization across devices.

Where Pith is reading between the lines

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

  • The maze-path criterion might extend directly to deciding multi-way partitions by checking for multiple disjoint cutting paths.
  • Similar percolation signatures could appear in other quantum resource-allocation tasks that involve gate clustering.
  • Hardware experiments that attempt to partition real circuits near the reported CNOT-qubit ratio would test the criterion's utility beyond simulation.

Load-bearing premise

Representing CNOT gates as maze walls and generating permutations via simulated annealing captures the essential structure that produces a genuine percolation transition.

What would settle it

If varying the CNOT-to-qubit ratio or replacing simulated annealing with another permutation generator eliminates the sharp change in partitionability, the claimed percolation transition is falsified.

Figures

Figures reproduced from arXiv: 2606.04070 by D. Arya, E. Prati, F. A. C\'ardenas-L\'opez, F. Motzoi, F. Preti, M. Guatto, P. Zentilini.

Figure 1
Figure 1. Figure 1: Wire representation and maze representation compared to a quantum circuit addressed in [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The maze layout optimization using simulated annealing on the wire representation of the [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The evolution of the average distribution of covered cells under simulated annealing for [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Panels (a) and (d) show the degree and betweenness centrality, respectively, before the SA [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The overlap area A between the two Gaussian components of the GMM fitted to the final consensus values, shown as a function of the number of CNOT gates for circuits with Nq = 15, 20, and 25 qubits. The x-axis represents the number of CNOT gates and the y-axis the overlap area A, where small values indicate strong separation between the two partitions and larger values correspond to stronger mixing. Shaded … view at source ↗
Figure 6
Figure 6. Figure 6: Score obtained by the best performing agent among all candidates at [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The figure shows how randomly distributed CNOT gates can be thought of as links between [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: (a) An example of a multigraph with n = 14 nodes q1, ..., q14 (number of wires or qubits) and repeated pairwise connections between the vertices (number of CNOTs or other entangling gates). (b) The maze representation of the multigraph in (a) with the corresponding connections represented in each column as walls of different colors matching the colors used for the edges in (a). The vertical axis correspond… view at source ↗
Figure 9
Figure 9. Figure 9: Visualization of the evolution of the ansatz ˜p ∗ (x) – including the boundary function b(x) –with decreasing temperature T. We can see that the shape correctly describes the separation of the distribution into two clusters. The initial profile of ˜p0(x) is shown as a dotted gray line. The values of the parameters have been set to d = 0.5, c = 0.02, l = 0.1, σ = p Nq The fit model has limitations due to it… view at source ↗
Figure 10
Figure 10. Figure 10: Single-trajectory consensus dynamics of qubit values for circuits with different numbers of CNOT gates. (a) For a small number of CNOTs (10), the qubits form two well-separated clusters that remain distinct throughout the evolution. (b) For an intermediate number of CNOTs, the clusters begin to mix as interactions cross the ideal partition. (c) For a large number of CNOTs, strong mixing occurs and the fin… view at source ↗
Figure 11
Figure 11. Figure 11: Gaussian Mixture Model (GMM) analysis of the final consensus values xi(T) for 30 qubits over 500 trajectories, with increasing circuit connectivity: (a) 10, (b) 75, and (c) 150 CNOTs gates. Blue histograms represent the empirical distribution of all qubit values, while the solid curves correspond to the fitted two-component Gaussian mixture and its individual components. The shaded region denotes the over… view at source ↗
Figure 12
Figure 12. Figure 12: Conductance as a function of the number of CNOT gates for circuits with different numbers of qubits. Solid lines represent the mean conductance over multiple realizations, while shaded regions (or error bars) indicate the standard deviation. For each system size, we compare the raw circuits (before optimization) with the optimized circuits obtained via simulated annealing. The optimization consistently re… view at source ↗
Figure 13
Figure 13. Figure 13: The figure shows the score obtained by the best performing agent among all candidates at [PITH_FULL_IMAGE:figures/full_fig_p029_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Three transport regimes as a function of circuit density. Top: maze, wire, and graph [PITH_FULL_IMAGE:figures/full_fig_p030_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: The figure shows that the distance between the two peaks of the interpolated distribution [PITH_FULL_IMAGE:figures/full_fig_p032_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: The figure shows the average commutator value for [PITH_FULL_IMAGE:figures/full_fig_p032_16.png] view at source ↗
read the original abstract

In quantum circuit optimization, circuit partitioning enables the optimization process to be parallelized across multiple devices. Each device is responsible for either reducing the number of selected gates or simplifying the local circuit structure. Most existing approaches to circuit partitioning are quantum-distribution-oriented and rely on splitting CNOT gates by introducing mid-circuit measurements and qubit resets. Currently, there is no criterion to determine how a circuit can be optimally partitioned without removing the CNOT gates for circuit optimization purposes. To address this challenge, we formalize the partition problem as a cutting path through a maze, where the CNOT gates represent the walls. We show that the existence of such a path separates quantum circuits into two classes through a percolation phase transition. In particular, it distinguishes a partitionable regime from a nonpartitionable one, arising from qubit permutations. Such permutations are generated by simulated annealing. We analyze its effect on the CNOT cluster from the perspective of network science and distribution analysis. Our results show that partitioning into two CNOT clusters is possible when the number of CNOTs is almost equal to the number of qubits. Based on this observation, we provide a scalable and practical criterion for identifying whether such a partition exists. Overall, our framework provides theoretical and numerical insight into circuit partitioning within quantum circuit optimization, forming the basis for algorithmic development.

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 / 1 minor

Summary. The paper formalizes the quantum circuit partitioning problem as finding a cutting path through a maze in which CNOT gates represent walls. It claims that the existence of such a path induces a percolation phase transition that separates circuits into partitionable and non-partitionable regimes, with the transition arising from qubit permutations generated by simulated annealing. A scalable criterion is proposed: partitioning into two CNOT clusters is possible when the number of CNOTs is nearly equal to the number of qubits. The work analyzes the CNOT cluster using tools from network science and distribution analysis.

Significance. If the maze-to-partition mapping is shown to be equivalent to a valid non-destructive cut that preserves the ability to optimize sub-circuits independently, the percolation criterion could supply a practical, non-destructive test for partitionability and thereby support parallel circuit optimization. The network-science perspective on CNOT connectivity is a potentially useful angle, though its impact depends on the rigor of the claimed equivalence and the robustness of the simulated-annealing results.

major comments (2)
  1. [Formalization of the partition problem (as described in the abstract and method)] The manuscript provides no derivation establishing that connectivity of a path through the CNOT-wall maze corresponds exactly to a valid non-destructive partition that permits independent optimization of the resulting sub-circuits while preserving commutation and entanglement structure. This equivalence is load-bearing for the central claim that the path separates partitionable from non-partitionable regimes.
  2. [Analysis of simulated-annealing permutations and CNOT-cluster statistics] The percolation threshold is reported exclusively for qubit permutations obtained via simulated annealing; no comparison is given to exhaustive enumeration or to alternative search heuristics on identical circuit instances. Without such controls it is impossible to determine whether the observed transition reflects the true structural boundary or an artifact of the heuristic search.
minor comments (1)
  1. [Abstract] The abstract states the transition and the CNOT-qubit-ratio criterion but supplies neither equations nor derivation steps, making the central claim difficult to verify from the summary alone.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments on our manuscript. We address each major comment below and outline the revisions we will make.

read point-by-point responses

  1. Referee: [Formalization of the partition problem (as described in the abstract and method)] The manuscript provides no derivation establishing that connectivity of a path through the CNOT-wall maze corresponds exactly to a valid non-destructive partition that permits independent optimization of the resulting sub-circuits while preserving commutation and entanglement structure. This equivalence is load-bearing for the central claim that the path separates partitionable from non-partitionable regimes.

    Authors: We agree that an explicit derivation is necessary to establish the equivalence between a valid maze path and a non-destructive partition. In the revised manuscript we will add a dedicated subsection deriving this correspondence from the definitions of circuit partitioning, showing step-by-step how a cutting path through CNOT walls yields sub-circuits whose independent optimization preserves commutation relations and entanglement structure. revision: yes

  2. Referee: [Analysis of simulated-annealing permutations and CNOT-cluster statistics] The percolation threshold is reported exclusively for qubit permutations obtained via simulated annealing; no comparison is given to exhaustive enumeration or to alternative search heuristics on identical circuit instances. Without such controls it is impossible to determine whether the observed transition reflects the true structural boundary or an artifact of the heuristic search.

    Authors: The referee correctly notes the absence of controls. While simulated annealing enables scalability, we will add a new subsection comparing results on small circuits (up to 10 qubits) against exhaustive enumeration and alternative heuristics such as greedy search. These controls will confirm whether the percolation transition is structural. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper formalizes circuit partitioning as a maze-cutting path with CNOTs as walls and reports an observed percolation transition under simulated-annealing permutations, followed by a numerical criterion based on CNOT-to-qubit count. No equations, self-citations, or steps in the abstract reduce the claimed transition or partitionability criterion to a fitted input, self-definition, or prior author result by construction. The mapping and transition analysis are presented as independent numerical and network-science observations rather than tautological renamings or load-bearing self-references.

Axiom & Free-Parameter Ledger

1 free parameters · 0 axioms · 0 invented entities

Abstract provides insufficient detail to enumerate specific free parameters, axioms or invented entities beyond the high-level maze model itself; the observed CNOT-qubit ratio threshold may function as an empirically noted boundary.

free parameters (1)
  • CNOT-qubit ratio threshold = approximately 1
    The statement that partitioning is possible when the number of CNOTs is almost equal to the number of qubits appears to be an observed or fitted boundary from the analysis.

pith-pipeline@v0.9.1-grok · 5793 in / 1110 out tokens · 32510 ms · 2026-06-28T09:49:22.196337+00:00 · methodology

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