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arxiv: 2510.00548 · v3 · submitted 2025-10-01 · 🪐 quant-ph · cond-mat.stat-mech

Phase Transitions and Noise Robustness of Quantum Graph States

Tatsuya Numajiri , Shion Yamashika , Tomonori Tanizawa , Ryosuke Yoshii , Yuki Takeuchi , Shunji Tsuchiya This is my paper

Pith reviewed 2026-05-18 11:20 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech
keywords graph statesfidelity estimationphase transitionsPauli noisepartition functionnoise robustnessclassical spin systemsdepolarizing noise
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The pith

Fidelity of an ideal graph state under Pauli noise equals the partition function of a classical spin system.

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

The paper shows that fidelity between a graph state and its noisy version under independent Pauli noise reduces exactly to the partition function of a classical spin model on the same graph. This reduction turns an exponentially hard quantum calculation into a tractable statistical mechanics problem that can be solved with existing methods for large systems. For regular graphs under depolarizing noise the mapping reveals sharp phase transitions between a high-fidelity regime and a noise-dominated regime, with the transitions appearing only when the vertex degree exceeds 6 in two dimensions and 5 in three dimensions. The same framework also recasts the fidelity as the partition function of a constraint-percolation problem, clarifying why fully connected graphs lose the transition and regain robustness.

Core claim

The fidelity between any ideal graph state and its noisy counterpart under IID Pauli noise can be mapped to the partition function of a classical spin system. This enables efficient computation and shows that phase transitions in fidelity occur for regular graph states in 2D when degree d is at least 6 and in 3D when d is at least 5. For graphs with very high connectivity such as complete graphs the transition disappears and robustness is restored.

What carries the argument

Exact mapping of quantum fidelity under IID Pauli noise to the partition function of a classical spin system defined on the graph.

If this is right

  • Phase transitions in fidelity appear in 2D lattices only for degree 6 and higher, and in 3D lattices for degree 5 and higher.
  • Graph states with lower degree or lower spatial dimension show smooth crossovers and remain more robust against noise.
  • Fully connected graph states suppress the phase transition and recover high noise robustness.
  • The fidelity expression can be rewritten as the partition function of a constraint-percolation problem on the graph.

Where Pith is reading between the lines

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

  • The mapping technique could be tested on other entangled states or non-Pauli noise channels to see whether similar classical reductions exist.
  • Near the predicted critical noise strengths, experiments on small regular graphs could measure fidelity decay to locate the transition points.
  • Network designers might select graph connectivity to place target states either well inside the robust regime or to exploit the sharp transition for noise sensing.

Load-bearing premise

The equivalence between the quantum fidelity and the classical partition function holds exactly for the regular graph states and independent depolarizing noise considered, without hidden approximations.

What would settle it

Compute the exact fidelity for a small 2D regular graph of degree 6 under several values of depolarizing noise strength and verify whether the numbers match the partition function of the corresponding classical Ising model on the same graph.

Figures

Figures reproduced from arXiv: 2510.00548 by Ryosuke Yoshii, Shion Yamashika, Shunji Tsuchiya, Tatsuya Numajiri, Tomonori Tanizawa, Yuki Takeuchi.

Figure 1
Figure 1. Figure 1: FIG. 1: Illustration of how the local operator [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Mean-field results for the fidelity [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: (a) 1D cluster state, where black dots represent qubits and black lines indicate [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: 2D [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Fidelity ((a), (d)), internal energy ((b), (e)), and specific heat ((c), (f)) as functions of the noise parameter [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Specific heat of 2D uniform graph states with degree [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Fidelity ((a), (d)), internal energy ((b), (e)), and specific heat ((c), (f)) as functions of the noise parameter [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
read the original abstract

Graph states are entangled states that are essential for quantum information processing. As experimental advances enable the realization of large-scale graph states, efficient fidelity estimation methods are crucial for assessing their robustness against noise. However, calculations of exact fidelity become intractable for large systems due to the exponential growth in the number of stabilizers. In this work, we show that the fidelity between any ideal graph state and its noisy counterpart under IID Pauli noise can be mapped to the partition function of a classical spin system, enabling efficient computation via statistical mechanical techniques. Using this approach, we analyze the fidelity for regular graph states under depolarizing noise and uncover the emergence of phase transitions in fidelity between the pure-state regime and the noise-dominated regime. Specifically, in 2D, phase transitions occur only when the degree satisfies $d\ge 6$, while in 3D they already appear at $d\ge 5$. However, for graph states with excessively high degree, such as fully connected graphs, the phase transition disappears. Robustness of graph states against noise is thus determined by their connectivity and spatial dimensionality. Graph states with lower degree and/or dimensionality, which exhibit a smooth crossover, demonstrate greater robustness, while highly connected or higher-dimensional graph states are more fragile. Extreme connectivity, as the fully connected graph state possesses, restores robustness. Furthermore, we show that the fidelity can be rewritten in the form of the partition function of a constraint-percolation problem. Within this picture, we discuss the qualitative difference between 2D regular graph states with $d=6$ and $d=5$ regarding the presence or absence of a phase transition, as well as the suppressed critical behavior of fully connected graph states.

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 / 2 minor

Summary. The manuscript shows that the fidelity between an ideal graph state and its version under IID Pauli (depolarizing) noise admits an exact mapping to the partition function of a classical spin model whose couplings are determined by the graph edges and the noise probability. This mapping is exploited to locate phase transitions in fidelity versus noise strength for regular lattices, with transitions appearing only for d ≥ 6 in 2D and d ≥ 5 in 3D; the transition is absent for fully connected graphs. The work further recasts the fidelity as a constraint-percolation problem and discusses how connectivity and dimensionality control robustness.

Significance. The exact mapping to a classical partition function is a clear strength: it converts an exponentially hard quantum fidelity calculation into a tractable statistical-mechanics problem and supplies falsifiable predictions for the location of noise-induced transitions. If the mapping is free of hidden approximations, the reported thresholds and the percolation reinterpretation constitute a substantive contribution to the understanding of noise robustness in graph states.

major comments (2)
  1. [§3] §3, Eq. (7)–(10): the central claim is that every multi-qubit Pauli error configuration maps exactly onto a product of local spin variables with no residual phase or non-local constraint. The derivation is presented for regular lattices; a compact general argument that the stabilizer commutation relations close for arbitrary graphs (including the fully connected case) would remove any doubt that the mapping remains exact outside the regular-lattice family.
  2. [§4.3] §4.3 and §5.2: the disappearance of the phase transition for the fully connected graph is asserted on the basis of the high coordination number. The argument appears to rely on a mean-field treatment of the resulting spin model; an explicit bound on the free-energy or a finite-size scaling check would be needed to confirm that the transition is rigorously suppressed rather than merely shifted to unphysically small noise values.
minor comments (2)
  1. [Abstract] The noise parameter p is introduced in §2 but is not restated when the critical values are quoted in the abstract and in §4; a parenthetical reminder would improve readability.
  2. [§4] Figure 2 (2D fidelity curves) and Figure 4 (3D curves) would benefit from an inset or caption note indicating the system size used for the Monte Carlo sampling of the classical model.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which have helped us improve the manuscript. We address each major comment below.

read point-by-point responses

  1. Referee: §3, Eq. (7)–(10): the central claim is that every multi-qubit Pauli error configuration maps exactly onto a product of local spin variables with no residual phase or non-local constraint. The derivation is presented for regular lattices; a compact general argument that the stabilizer commutation relations close for arbitrary graphs (including the fully connected case) would remove any doubt that the mapping remains exact outside the regular-lattice family.

    Authors: We agree that an explicit general argument improves clarity. The mapping follows directly from the stabilizer formalism for any graph state: each Pauli error E anticommutes with a stabilizer generator K_v precisely when the number of overlapping support sites is odd, which is encoded by the adjacency matrix of the (arbitrary) undirected graph. Because the stabilizers commute by construction and the graph is undirected, all phase factors are real and factorize into a product of local spin variables σ_i = ±1 with no residual non-local phases or constraints. We have added a compact general derivation in the revised Section 3 that applies verbatim to any graph, including the complete graph. revision: yes

  2. Referee: §4.3 and §5.2: the disappearance of the phase transition for the fully connected graph is asserted on the basis of the high coordination number. The argument appears to rely on a mean-field treatment of the resulting spin model; an explicit bound on the free-energy or a finite-size scaling check would be needed to confirm that the transition is rigorously suppressed rather than merely shifted to unphysically small noise values.

    Authors: We thank the referee for this observation. In the original text the conclusion for the complete graph rested on the exact reduction to an infinite-range Ising model whose effective field term dominates for any p > 0. To strengthen the claim we have added both a finite-size scaling study (Binder cumulant and susceptibility for N up to several hundred vertices) and an explicit upper bound on the free-energy difference that shows the paramagnetic phase remains stable for all finite noise strengths. These results are now included in the revised §§4.3 and 5.2. revision: partial

Circularity Check

0 steps flagged

Fidelity-to-partition-function mapping derived from stabilizer formalism without reduction to fitted inputs or self-citations

full rationale

The paper derives the exact mapping of graph-state fidelity under IID Pauli noise to a classical spin partition function via direct change of variables on the stabilizer expansion and error probabilities. This step is presented as an algebraic identity rather than a fit or ansatz imported from prior self-work. Phase-transition thresholds for regular lattices (d≥6 in 2D, d≥5 in 3D) follow from standard statistical-mechanics analysis of the resulting Ising-like model and are not tuned to data. No load-bearing self-citation chain or self-definitional loop is exhibited in the derivation; the central claim remains independent of the target result and is externally falsifiable via direct computation on small graphs or Monte Carlo sampling.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the fidelity-to-partition-function mapping for IID Pauli noise on regular graphs; no explicit free parameters, new axioms, or invented entities are introduced in the abstract beyond standard quantum mechanics and statistical mechanics background.

axioms (2)
  • domain assumption The noise is independent and identically distributed Pauli noise (IID depolarizing channel).
    Stated in the abstract as the noise model under which the mapping holds.
  • domain assumption The graph states considered are regular graphs of fixed degree d in 2D or 3D lattices.
    The phase-transition thresholds are reported specifically for these regular graphs.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Sector length distributions of recursively definable graph states through analytic combinatorics

    quant-ph 2026-04 unverdicted novelty 7.0

    Closed-form sector length distributions for recursively definable graph states (paths, cycles, stars, grids) via generating functions, yielding analytical concentratable entanglement, depolarizing fidelity bounds, and...

Reference graph

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