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Quantum complexity phase transition in fermionic quantum circuits

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Interacting fermions shift a quantum complexity transition to p = 1/4

desk verdict Free-fermion part is clean and worth publishing; the interacting pc,k = 1/4 rests on an imported assumption and needs real support. read the letter →

arxiv 2507.22125 v1 pith:52GWY5D4 submitted 2025-07-29 quant-ph cond-mat.dis-nn

Wei Xia , Yijia Zhou , Xingze Qiu , Xiaopeng Li This is my paper · ORCID
classification quant-phcond-mat.dis-nn MSC 82B4381P68
keywords KrylovcomplexityquantumpercolationphasetransitioninteractingfermionsoperatorgrowthGriffithseffectdisorderedsystemscircuits
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 argues that Krylov complexity, a measure of how far a quantum operator spreads during time evolution, can undergo a genuine phase transition when the lattice it lives on is randomly occupied, as in a quantum percolation model. For free fermions the complexity transition occurs exactly at the classical percolation threshold, where a spanning cluster first appears. For interacting fermions in one dimension, the paper claims the transition shifts to p = 1/4, well below the classical threshold p = 1. The mechanism is an exponential competition: interactions make the Krylov dimension of a size-s cluster grow like $4^s$, which overcomes the exponentially small probability of large clusters once $4p$ exceeds 1. If correct, this establishes a setting in which quantum complexity itself, rather than entanglement or geometry, sets the location of a phase transition.

What carries the argument

The load-bearing object is the disorder average of the Krylov dimension, $\overline{D}=\sum_s [s\rho(s,p)]d(s)$, where $\rho(s,p)$ is the percolation cluster number density and $d(s)$ is the Krylov dimension of a single cluster of size $s$. In one dimension, $\rho(s,p)=(1-p)^2p^s$, so with $d(s)\propto 4^s$ the average becomes a geometric series in $4p$; its radius of convergence fixes $p_{c,k}=1/4$. The same sum with $d(s)\propto s^2$ reproduces the free-fermion result, which diverges only at $p=1$. The mechanism is a competition between exponential decay of large-cluster probability and exponential growth of single-cluster Krylov dimension.

What would settle it

Exact-diagonalize small interacting fermion clusters of sizes $s=2$ through roughly 10 on a chain, compute the Krylov dimension of a local density operator for each cluster, and check whether $d(s)$ grows as $4^s$; a fitted base smaller than 4 would shift the divergence to $p=1/c$ and invalidate the quantitative prediction.

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Extended reading notes

Core claim

The central claim is that the long-time averaged Krylov complexity of a local fermionic operator in a quantum percolation model is controlled by the disorder-averaged Krylov dimension, $\overline{D}=\sum_s [s\rho(s,p)]d(s)$, and that this average diverges at a critical probability $p_{c,k}$. For free fermions, with $d(s)\propto s^2$, the divergence coincides with the classical percolation threshold $p_c$ on the lattices studied, with exponent $\delta=(4-\tau)/\sigma$ fixed by percolation critical exponents. For interacting fermions in one dimension, the paper takes $d(s)\propto 4^s$ and obtains $p_{c,k}=1/4$, strictly below $p_c=1$; the average Krylov dimension is finite for $p<1/4$, grows like $L^2$ at the critical point, and diverges exponentially with system size for $p>1/4$. The authors interpret the intermediate regime as a Griffiths-like phase in which exponentially rare spanning clusters dominate the disorder average, decoupling the complexity transition from the geometric percolation transition.

Load-bearing premise

The argument rests on the assumption, carried over from the authors' prior work, that the Krylov dimension of an interacting fermion cluster of size s grows as $4^s$; the predicted threshold $p_{c,k}=1/4$ is exactly where the sum over $s(4p)^s$ diverges, so a smaller exponential base or polynomial growth would erase the claimed separation from the percolation transition.

Editorial extensions

If this is right

  • For any free-fermion quantum percolation model whose clusters obey the standard scaling form $\rho(s,p)=s^{-\tau}\hat F((p_c-p)s^\sigma)$, the Krylov complexity transition sits at the classical percolation threshold with exponent $\delta=(4-\tau)/\sigma$, as verified numerically on 1D, 2D, and Bethe lattices and on a Penrose tiling.
  • In one-dimensional interacting systems, long-time Krylov complexity is finite for $p<1/4$, grows as $L^2$ at $p=1/4$, and diverges exponentially with system size for $p>1/4$, so $p=1/4$ is a genuine complexity phase transition.
  • The transition is observable in principle on near-term quantum hardware by preparing time-evolved operator states through Trotterized evolution, orthogonalizing them with a quantum Gram-Schmidt circuit, and measuring overlaps to recover the Krylov complexity.
  • The threshold location is set by the balance between exponential cluster rarity and exponential complexity growth, which the authors identify as a Griffiths-type mechanism; the same balance should control other disordered many-body complexity transitions.

Reading between the lines

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

  • A direct generalization of the 1D calculation, not tested in the paper, is that any model with $d(s)\propto \lambda^s$ on a percolating 1D chain would have its Krylov threshold at $p_{c,k}=1/\lambda$, so the gap below the classical threshold sensitively probes the exponential growth base $\lambda$.
  • The divergence lives in the disorder average, so typical individual samples near $p_{c,k}$ will show much smaller complexity than the mean; an experimental test should report the full distribution, not just the average, to distinguish the Griffiths-like phase from an ordinary transition.
  • The measurement protocol uses polynomially many Krylov basis states at early times, but the saturated long-time regime for interacting systems may require exponential resources, so the protocol as proposed is most practical at small system sizes.
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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 / 5 minor

Summary. The paper studies the long-time averaged Krylov complexity of a local density operator in fermionic quantum percolation models. The authors introduce a scaling theory in which the disorder-averaged Krylov dimension is a sum over cluster sizes of the cluster size distribution times a cluster Krylov dimension d(s). For free fermions they set d(s) ~ s^2 and show that the Krylov transition coincides with the classical percolation threshold on 1D, Bethe, 2D, and Penrose lattices, with critical exponent delta = (4 - tau)/sigma. For interacting fermions in 1D they set d(s) ~ 4^s and derive p_{c,k} = 1/4, a separation from the percolation threshold p_c = 1, interpreted as a Griffiths-like amplification of rare large clusters. They also propose a quantum Gram-Schmidt based protocol for measuring Krylov complexity. The free-fermion series evaluations and their numerical checks are internally consistent; the interacting result is the main novel claim.

Significance. If the interacting prediction d(s) ~ 4^s is correct, the paper identifies a genuinely new mechanism by which a complexity phase transition can be decoupled from the geometric percolation transition, and the Griffiths analogy is conceptually appealing. The free-fermion part is a solid contribution: the 1D cluster-size averages are evaluated exactly, the scaling form delta = (4 - tau)/sigma is derived cleanly, and the numerical checks for free fermions on 1D, 2D, and Penrose lattices support the correspondence p_{c,k} = p_c. The experimental protocol, even if not fully developed, is a useful step toward measuring Krylov complexity. However, the interacting threshold is not supported within this paper because it is obtained by substituting an imported exponential growth law into the cluster average; the model-specific validity of that law is the entire content of the separation claim.

major comments (3)
  1. [Interacting fermion case; Eq. (9); S2-1] The central result p_c,k = 1/4 in Eq. (9) follows entirely from inserting the assumption d(s) ~ 4^s into Eq. (4). This exponential growth law is imported from Ref. [56] and is not derived or independently checked for the Hamiltonian in Eq. (1) on a one-dimensional chain. The threshold is exactly the value at which the geometric series sum_s (4p)^s diverges, so a smaller exponential base c would shift the threshold to p = 1/c, and a polynomial d(s) would eliminate the separation from percolation. The 1D spinless-fermion t-V model maps by Jordan-Wigner to the integrable XXZ chain, so an exponential operator-growth law established for generic graphs cannot be assumed without model-specific evidence. No interacting 1D numerical data are presented (Fig. 2 is for free fermions only), and the finite-size scaling ansatz of S2-2 is not applied to any interacting data. This missing support is load-bearing for the paper's most novel conclusion and must be fixed.
  2. [Eq. (9) and S2-2] The finite-size expressions for the interacting 1D model are mutually inconsistent. Eq. (9) gives D_int ~ L (4p)^{L+1} (1-p)/(4p-1) for p > p_c,k, while S2-2 reports a leading-order term L^2 (1-p^2) (4p)^L/(1-4p), with the sign also reversed. In addition, the finite-size scaling ansatz (S26)-(S29) is never used to collapse data, and no interacting finite-size data are shown anywhere. Since the paper claims exact critical exponents and a quantitative phase diagram, this inconsistency and the absent scaling analysis need to be resolved.
  3. [Abstract and Conclusion] The abstract states that 'for interacting systems' the KCPT develops a generic separation from the percolation transition, but the only interacting calculation in the manuscript is the 1D case of Eq. (9). The supplementary material treats Bethe, 2D, and Penrose lattices exclusively for free fermions. The word 'generic' and the corresponding claim in the Conclusion are not supported by the evidence presented; the claim should either be restricted to 1D or backed by additional interacting calculations.
minor comments (5)
  1. [S1-1, S1-2] There are typographical errors: 'thermal limit' should be 'thermodynamic limit' in S1-1, and 'Bette lattice' should be 'Bethe lattice' throughout S1-2 and S2-3.
  2. [S1-3, S2-3] The notation for the cluster number density is inconsistent: the main text and S1 use rho(s,p), while S2-3 uses n(s,p) for the same object; please unify.
  3. [Fig. 2] The figure caption does not identify which quantity (D or C) is shown on which axis, nor the units or error bars; please label both panels explicitly.
  4. [Eq. (4)] The asymptotic form poly(p)/|p-p_c,k|^delta describes the divergence as p approaches p_c,k from below; for p > p_c,k in the interacting 1D case the average D diverges with system size rather than as a power law. Please state this explicitly to avoid confusion.
  5. [Experimental protocol, S3-2] The main text claims the protocol is accessible to present experiments without qualification, while S3-2 notes that for interacting systems only short-time dynamics keeps the Krylov basis count polynomial. The main text should state this limitation prominently.

Circularity Check

2 steps flagged · score 6.0 of 10

The headline interacting threshold p_{c,k}=1/4 is the pole of an assumed exponential Krylov-dimension input d(s)~4^s imported from the authors' own Ref. [56]; the derivation in S2-1 reduces this prediction to that assumption, so the claimed separation from percolation is not independently established in this paper.

  1. ansatz smuggled in via citation [Supplementary S2-1, 'Average Krylov dimension of one-dimension quantum percolation model'; main-text Eq. (4) and Eq. (9)]
    "Building upon our previous work, we establish a reasonable assumption that the Krylov dimension of a free Hamiltonian is proportional to the square of the number of sites connected to the site where the initial operator is located. ... For interacting fermions, the Krylov dimension is proportional to the cluster sizes, represented as∼ 4s. Hence, the dimension DInt of the Krylov space for interacting fermions is determined as follows: DInt = 4p(1−p)^2/(1−4p)^2. Interestingly, the critical probability pc is 1/4."

    The only input distinguishing the interacting case is the assumption d(s)~4^s, which is imported from Ref. [56] (same authors Xia, Zou, Li) and is not derived in this paper. Substituting d(s)~4^s and the 1D cluster density rho(s,p)=(1-p)^2 p^s into Eq. (4) yields a geometric series in 4p; the denominator (1-4p)^2 vanishes at 4p=1, i.e. p=1/4. Thus the paper's headline prediction p_{c,k}=1/4 is exactly the value where the assumed exponential base 4 balances the percolation probability p; any other base c would give p_{c,k}=1/c. The numerical prediction is a restatement of the assumed growth law, not an independent result about interactions in this model, and no interacting 1D numerics or independent check of d(s) is provided.

  2. self citation load bearing [Main text, 'Complexity phase transition' paragraph, Eq. (4)]
    "For non-interacting systems, D∝ s2; for interacting systems, D∝ 4s."

    The paper's central claim that interactions generically separate the Krylov transition from the percolation transition rests entirely on the statement D∝4^s. The only support offered for this exponential scaling is the authors' own prior work [56], an arXiv preprint by the same group, and this manuscript provides no derivation, independent numerical verification, or externally checkable evidence for that scaling in the 1D t-V model under study. Consequently, the claimed dramatic lowering of the critical point from p_c=1 to p_{c,k}=1/4 is load-bearing self-citation: without [56], the interacting prediction has no support within this paper.

full rationale

The free-fermion part of the paper is self-contained and not circular: D∝s^2 follows from the dimension of the single-particle operator space, and combining it with the exact 1D cluster density rho(s,p)=(1-p)^2 p^s gives Eq. (8) with a divergence at p=1; the numerical check in Fig. 2 is consistent with that independent derivation. The general scaling argument for non-interacting lattices is also a legitimate application of standard percolation scaling theory. However, the paper's most novel claim, the interacting separation p_{c,k}=1/4 < p_c=1, is obtained by inserting d(s)~4^s into Eq. (4). The critical value 1/4 is exactly 1/base, with base=4 being the assumed exponential growth rate imported from the authors' own Ref. [56]. The derivation in S2-1 is therefore a direct algebraic consequence of that input assumption: the 'prediction' reduces by construction to the assumed growth law, rather than being established by evidence contained in this manuscript. No 1D interacting numerical data are presented to test the assumed 4^s scaling in this specific model, and the cited prior work is not machine-checked or externally falsified here. Hence the central interacting result is partially circular, supported mainly by an unverified self-cited ansatz, while the surrounding non-interacting scaling theory and experimental protocol remain independent contributions.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central 1D results rest on three inputs from prior work: C proportional to D, d(s) proportional to s^2, and d(s) proportional to 4^s. The first two are standard or easily checked; the third is not derived and controls the headline interacting threshold. No new physical entities are introduced.

free parameters (1)
  • Exponential growth base for interacting Krylov dimension = 4 (assumed, from Ref. [56])
    The threshold pc,k = 1/4 solves 4p = 1. The base 4 controls the entire interacting phase diagram and is not derived in this paper.
assumptions (5)
  • domain assumption For a noninteracting fermion cluster of s sites, the Krylov dimension is d(s) proportional to s^2.
    Invoked in the 'Free fermion case' and S2-1 to evaluate DFree. It is plausible from the dimension of bilinear fermion operator space, but the paper calls it a 'reasonable assumption' based on prior work.
  • domain assumption For an interacting fermion cluster of s sites, the Krylov dimension is d(s) proportional to 4^s.
    Invoked in the 'Interacting fermion case' and S2-1, where the Krylov dimension is said to be 'proportional to the cluster sizes, represented as about 4^s'. This is the load-bearing input that produces pc,k = 1/4; it is cited to the authors' own Ref. [56] and not derived here.
  • domain assumption Long-time averaged Krylov complexity is proportional to the Krylov dimension D.
    Used in Eq. (4) and the text ('C proportional to D'); this relies on saturation of the Krylov wavepacket and is cited to Ref. [56].
  • standard math Cluster number density scaling form rho(s,p) = s^(-tau) F((p - pc) s^sigma) applies to the relevant lattices.
    Standard percolation scaling theory, used in Eq. (5) and S1-3 to derive the divergence exponents.
  • domain assumption Disconnected clusters have commuting Hamiltonians, so the Krylov space factorizes by cluster.
    Used in the 'Complexity phase transition' paragraph; true for site percolation with nearest-neighbor hoppings and interactions on occupied sites.

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Cite this review

Pith. "Pith review of Quantum complexity phase transition in fermionic quantum circuits." pith.science (2026). https://pith.science/paper/52GWY5D4

@misc{pith2026250722125,
  author       = {Pith},
  title        = {Pith review of: Quantum complexity phase transition in fermionic quantum circuits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/52GWY5D4}},
  note         = {Machine review of arXiv:2507.22125}
}
read the original abstract

Understanding the complexity of quantum many-body systems has been attracting much attention recently for its fundamental importance in characterizing complex quantum phases beyond the scope of quantum entanglement. Here, we investigate Krylov complexity in quantum percolation models (QPM) and establish unconventional phase transitions emergent from the interplay of exponential scaling of the Krylov complexity and the number of spanning clusters in QPM. We develop a general scaling theory for Krylov complexity phase transitions (KCPT) on QPM, and obtain exact results for the critical probabilities and exponents. For non-interacting systems across diverse lattices (1D/2D/3D regular, Bethe, and quasicrystals), our scaling theory reveals that the KCPT coincides with the classical percolation transition. In contrast, for interacting systems, we find the KCPT develops a generic separation from the percolation transition due to the highly complex quantum many-body effects, which is analogous to the Griffiths effect in the critical disorder phase transition. To test our theoretical predictions, we provide a concrete protocol for measuring the Krylov complexity, which is accessible to present experiments.

Figures

Figures reproduced from arXiv: 2507.22125 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Schematic of a percolation lattice. Sites are occupied [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Krylov dimension [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Schematics of our experimental protocol for measuring the Krylov complexity. (a) Quantum Gram-Schmidt circuit: Input [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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    Average cluster size S (p) The average cluster size S is defined as S (p) =⟨s⟩ = X s=1 s sρ(s, p)P s sρ(s, p) = 1 p X s=1 s2ρ(s, p) (S5) = (1− p)2 p X s=1 s2ps (S6) = (1− p)2 p X s=1 p ∂ ∂p p ∂ ∂p ps (S7) = 1 + p 1− p (S8) Here, P∞ s=1 sρ(s, p) = (1− p)2p(1− p)−2 = p and we in...

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