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Coherent disorder in two-qubit gate angles drives two crossovers—loss of anticoncentration and collapse of entanglement—that push a square-lattice IQP sampling circuit toward classical simulability, with finite-size scaling suggesting both

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2026-08-01 13:48 UTC pith:H2ZPTAKI

load-bearing objection Solid finite-size result: coherent disorder drives two crossovers (anticoncentration loss near σ≈0.2, entanglement collapse near σ≈0.5) bounding the hard regime of a 2D IQP architecture; the 'continuous transition' claim is not supported by the FSS evidence presented. the 3 major comments →

arxiv 2607.18938 v1 pith:H2ZPTAKI submitted 2026-07-21 quant-ph

Coherent-disorder-driven complexity transitions in a quantum-advantage architecture

Sung-Bin B. Lee , Chae-Yeun Park , Changhun Oh , Seung-Sup B. Lee This is my paper
classification quant-ph
keywords IQP circuitsquantum random samplingcoherent disorderanticoncentrationentanglement transitionstensor networksfinite-size scalingclassical simulability
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper tries to establish that spatial disorder in two-qubit gate angles—not just decoherence—erodes the classical hardness of a square-lattice IQP random-sampling circuit. Using exact tensor-network simulations up to 576 qubits, it finds two separate crossovers as disorder strength grows: first the output distribution stops anticoncentrating at disorder strength about 0.2, then the boundary-state entanglement collapses from volume-law to logarithmic at about 0.5, turning exact contraction cost from exponential to polynomial. Finite-size scaling collapses are consistent with both crossovers becoming genuine continuous transitions in the thermodynamic limit. If true, this means realistic gate miscalibration alone can place a would-be quantum-advantage device in the classically simulable regime.

Core claim

The central claim is that coherent two-qubit disorder destroys the two conjectural pillars of IQP sampling hardness in sequence, not simultaneously. The rescaled collision probability, which measures anticoncentration, grows from the Porter–Thomas value 2 to a regime where the probability distribution is no longer spread across bitstrings, with a finite-size-scaling collapse locating the crossover at σ_Z ≈ 0.2 and giving critical exponent ν_Z ≈ 1 and a sub-leading exponent e_Z ≈ 0.2. Independently, the maximum entanglement entropy of the boundary matrix product state used in exact contraction drops from volume-law √n ln√2 to logarithmic 0.13 ln n, with collapse at σ_S ≈ 0.5 and ν_S ≈ 1.4; th

What carries the argument

The engine of the analysis is the representation of the commuting IQP circuit as a projected entangled pair state (PEPS) with bond dimension 2 (or projected entangled pair operator, PEPO, for dephased mixed states), contracted exactly with a boundary matrix product state (BMPS)—a one-dimensional matrix product state that sweeps across the square lattice and stores the entanglement that determines contraction cost. The paper uses two finite-size scaling ansätze, one for ln(Z̃/2) and one for S_max, whose data collapses locate the two disorder thresholds and yield critical exponents. The contrast between real-time evolution (with disorder destroying entanglement) and imaginary-time evolution (w

Load-bearing premise

The finite-size scaling ansatz used to extrapolate the crossovers—the claim that ln(Z̃/2) and S_max collapse onto a single curve when rescaled with the fitted exponents (σ_Z, ν_Z, e_Z) and (σ_S, ν_S)—is the load-bearing premise; if the scaling form is not asymptotically correct, the finite-system crossovers remain but the transition interpretation collapses.

What would settle it

Computing the same quantities for larger systems (e.g., n = 1024 or 2048) with the same exact BMPS contraction and checking whether the fitted parameters (σ_Z, ν_Z, e_Z, σ_S, ν_S) continue to collapse the data; a drift of the fitted exponents with n, or a growth of Z̃ slower than exp(n^0.4) for σ > σ_Z, would falsify the claim that the crossovers sharpen into continuous transitions. A second falsifier: direct computation of the entanglement spectrum at σ = 0.5 for large n showing that the singular values develop a power-law tail rather than remaining exponentially decaying, which would indicat

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • If the transitions are genuine, error-budget bounds for near-term IQP devices must include coherent disorder strength σ as a separate axis: below σ_Z anti-concentration holds but above it the standard hardness argument loses one leg.
  • For σ > σ_S, exact or high-fidelity tensor-network contraction becomes polynomial, so the architectural regime of claimed quantum advantage is limited to disorder below about 0.5.
  • Dephasing acts in the opposite direction on the collision probability: it pushes the output distribution toward uniform, so the combined effect of disorder and dephasing on hardness cannot be captured by a single noise strength.
  • Small-disorder scalings (squared relative error, squared TVD, and entanglement reduction ∝ σ^2 and ε^2; KLD ∝ ε^1.69) provide concrete predictions for experiments probing the hard regime.
  • The distinction from deep random circuits—where generic angle changes stay Haar-like—marks IQP circuits as uniquely sensitive to coherent gate-angle disorder.

Where Pith is reading between the lines

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

  • The two thresholds imply a phase diagram with a narrow intermediate window between σ_Z and σ_S where the standard Stockmeyer-based hardness argument has already failed (no anticoncentration) yet exact tensor-network contraction remains exponentially costly; hardness in that window would have to rest on other arguments. The paper does not explore this window.
  • The logarithmic-entanglement phase at large disorder resembles the area-law regime found in disordered many-body systems; connecting these transitions to many-body localization or to the stability of commuting-circuit area laws is a natural next step the paper leaves open.
  • A programmable analog Ising simulator could measure the collision probability and a proxy for half-chain entanglement while sweeping σ; the predicted crossings near 0.2 and 0.5 give concrete targets.
  • The fitted exponents (ν_Z ≈ 1, ν_S ≈ 1.4) may identify the transitions with a known universality class; the paper does not make that identification.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. Lee et al. study a square-lattice IQP architecture with random two-qubit coupling angles (disorder strength σ) and optional single-qubit dephasing ε. Using exact boundary-MPS contractions of the corresponding PEPS/PEPO tensor networks for square lattices up to 576 qubits, they compute the rescaled collision probability Z̃, statistical distances, and the maximum BMPS entanglement entropy Smax. They report two disorder-driven crossovers in the dephasing-free case: Z̃ grows and anticoncentration is lost near σ_Z≈0.2, and Smax drops from volume-law to logarithmic near σ_S≈0.5. They further report finite-size scaling (FSS) collapses under second-order-type ansätze and interpret these as evidence that the crossovers sharpen into continuous transitions in the large-system limit. In the End Matter they examine dephasing and combined disorder/dephasing scaling.

Significance. If the finite-size scaling were sound, the paper would provide a concrete mechanism by which coherent gate-angle disorder degrades both conjectural ingredients of IQP sampling hardness, and it would give useful error-budget estimates for near-term experiments. The work's strengths are the exact, fully contracted tensor networks up to n=576 (no bond truncation in the main results), the direct spectral evidence in Fig. 4, and the clean separation of the two crossovers in the raw data. The distinction between coherent disorder, which broadens the bitstring PDF and increases Z̃, and dephasing, which drives the PDF toward uniform, is a useful observation. The significance of the paper therefore rests on the finite-size numerical characterization, which is solid and relevant; the thermodynamic-limit transition interpretation is the fragile part.

major comments (3)
  1. [Output statistics, Fig. 2(b)] The FSS analysis is the only evidence for the σ_Z transition, but the collapse is performed only for σ≲0.4, as the paper itself notes later ('Meanwhile, only the data points for σ≲0.4 show a collapse onto F_Z'). The parameters (σ_Z,ν_Z,e_Z) are obtained by fitting the same data that are then shown to collapse, and no uncertainty estimates, collapse residuals, or independent estimates of σ_Z are reported. The range of linear sizes (√n up to 24) is modest for a critical-point determination. Thus the raw data do show a clean crossover, but the inference that the crossover sharpens into a continuous transition at σ_Z is not supported at the requested level. A concrete improvement would be to extract σ_Z from a crossing quantity, e.g., the intersections of statistical-distance curves for different n, and to quote fit uncertainties.
  2. [Entanglement of BMPS, Fig. 3(c)–(d)] The Smax scaling ansatz Smax=(ln n)F_S[(σ−σ_S)(√n)^{1/ν_S}] cannot describe the small-σ behavior reported in Fig. 3(c), where Smax=√n ln√2 − c n σ². Writing x=(σ−σ_S)(√n)^{1/ν_S}, the left branch of F_S would have to reproduce the explicit n-dependent prefactor √n/ln n and the nσ²/ln n term; no function of x alone can do this for fixed σ as n→∞. The statement that 'the left branch ... is not a straight line' is not a resolution; it is a symptom of the inconsistency. To claim an FSS collapse, the authors should subtract the regular volume-law contribution before scaling, or identify a well-defined order parameter whose singular part is expected to obey a scaling form. Without this, the collapse in Fig. 3(d) cannot be taken as evidence for a continuous entanglement transition.
  3. [Conclusion and abstract] The abstract and conclusion state that the FSS collapses are 'consistent with continuous transitions' or that the crossovers 'sharpen into ... transitions in the large-n limit.' Given the two issues above, this wording is too strong. The finite-size crossovers near σ≈0.2 and σ≈0.5 are interesting and well supported by the raw data, and I would regard them as the paper's central result. The revision should either provide a substantially stronger scaling analysis—with error bars, subtraction of regular parts, wider σ and n ranges, and independent critical-point estimates—or explicitly limit the conclusions to finite-size crossovers. In particular, the reduction of exact tensor-network contraction cost from exponential to polynomial is a statement about finite n unless the thermodynamic-limit transition is established; the current text blurs this distinction.
minor comments (4)
  1. [Fig. 2(b) and Fig. 3(d)] The axis labels are hard to read: the Greek symbols appear as boxed placeholders in the figure, e.g., '( Z) ( n)1/ Z' for (σ−σ_Z)(√n)^{1/ν_Z}. Please ensure all axis labels are legible and consistent with the text.
  2. [Output statistics, Fig. 2(d)] The sentence 'δ_KL ≃ 4 δ²_TVD ... resembles the bound given by the Pinsker inequality ... up to a factor of 2' is confusing. Pinsker gives δ_TVD ≤ √(δ_KL/2), so δ_KL ≃ 4δ_TVD² is a factor of 2 above that bound. Please clarify whether the factor refers to the Pinsker bound or to the observed constant.
  3. [Entanglement of BMPS, Fig. 3] Smax is defined for a single output bitstring (0), while the statistical quantities are averaged over 4096 bitstrings and 100 circuit instances. This difference should be stated in the main text near Fig. 3, together with the number of circuit instances used for Smax.
  4. [End Matter, Fig. 8] The scaling collapses for small σ and ε are shown only for n=8×8. The main-text statement that 'the squared relative error, squared TVD, and entanglement reduction scale as ε²' should explicitly mention this system-size dependence, since the exponents are not verified at larger n.

Circularity Check

0 steps flagged

No load-bearing circularity; the FSS analysis is a transparent fit-and-extrapolation, and self-citations are not load-bearing.

full rationale

The central results are exact tensor-network data for the rescaled collision probability Z~ and the BMPS entanglement Smax, and the two crossovers are direct observations from those data. The finite-size-scaling analysis fits parameters (σZ, νZ, eZ) and (σS, νS) to the same data and uses the collapse to argue consistency with continuous transitions; this is a fit plus an extrapolation, not a derivation in which the transition is identical to the input by construction. The paper explicitly states 'Assuming the scaling form persists to larger systems' and 'we expect the entanglement crossover ... to sharpen into a transition,' so it does not disguise the fitted parameters as independent predictions. The Smax ansatz's left-branch behavior and the restricted σ range are legitimate statistical/correctness risks, but they are not circularity. Self-citations (Refs. [14] and [38]) appear in a standard-hardness citation list and in a supporting remark about truncated-PEPS PDFs; neither is load-bearing for the paper's central claim. No step reaches the threshold of Eq.-equals-input or fitted-parameter-renamed-as-prediction.

Axiom & Free-Parameter Ledger

7 free parameters · 6 axioms · 0 invented entities

The central claims rest on FSS fitting parameters rather than on an external benchmark or a first-principles derivation. There are no invented physical entities. The key additional assumptions are the tensor-network representations, the FSS ansatz, and the representative-ness of the disorder distribution.

free parameters (7)
  • σ_Z = 0.2
    Fitted anticoncentration-transition point in the FSS ansatz for the rescaled collision probability Z̃(n,σ).
  • ν_Z = 1.0
    Correlation-length exponent fitted in the Z̃ FSS collapse.
  • e_Z = 0.2
    Exponent in the Z̃ FSS ansatz controlling the n-dependence of the scaling function.
  • σ_S = 0.5
    Fitted entanglement-transition point in the Smax FSS collapse.
  • ν_S = 1.4
    Correlation-length exponent fitted in the Smax FSS collapse.
  • logarithmic-plateau coefficient = 0.13
    Right-branch plateau of the Smax collapse: Smax ≈ 0.13 ln n for σ > σ_S.
  • End-Matter exponents α_X, β_X = α≈2, β≈2 (β_KL=1.69)
    Fitted exponents for δRel², δTVD², δKL, and entanglement reduction in the small-σ, small-ε scaling collapses of Fig. 8.
axioms (6)
  • standard math Sampling hardness of ideal IQP relies on average-case #P-hardness and anticoncentration conjectures, with Stockmeyer's argument implying PH collapse.
    Invoked in the Introduction to define the hard regime the paper studies; not proven in this paper.
  • domain assumption A dephasing-free IQP circuit on a 2D lattice has a PEPS representation with bond dimension 2, and with dephasing a PEPO with bond dimension 4; these contractions are exact without truncation.
    Stated in 'Setup and method'; validity depends on the known tensor-network representation and on exact BMPS contraction.
  • ad hoc to paper The FSS ansatz ln(Z̃/2)=(√n)^(-eZ)FZ[(σ−σZ)(√n)^(1/νZ)] and its Smax analogue are the correct scaling forms for second-order-type transitions.
    Introduced for this paper's data collapse (Figs. 2b, 3d); no rigorous derivation from a microscopic model.
  • domain assumption The chosen disorder distribution (Gaussian Jij=π/4(1+σηij), hi∈{π/4,3π/8}) represents realistic coherent gate miscalibration and is representative for the conclusions.
    The thresholds and transitions may depend on this distribution; the paper does not test other disorder distributions.
  • standard math Random nonzero hi avoids the Pfaffian-solvable zero-field planar Ising case, so the ideal circuit retains the intended hardness.
    Invoked in Setup and method, based on Refs 15,36,37.
  • domain assumption BMPS entanglement Smax is the correct proxy for exact tensor-network contraction cost, O(nχ^4) with χ=2^{Smax}.
    Used to translate entanglement suppression into polynomial simulation cost; standard for this method but not a rigorous complexity bound.

pith-pipeline@v1.3.0-alltime-deepseek · 3651 in / 4121 out tokens · 128725 ms · 2026-08-01T13:48:51.737427+00:00 · methodology

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read the original abstract

While decoherence is known to erode classical hardness in quantum random sampling, the impact of coherent spatial disorder remains an open question. We study a square-lattice instantaneous quantum polynomial-time (IQP) architecture subject to two-qubit gate-angle disorder and single-qubit dephasing using exact tensor-network simulations up to 576 qubits. For finite systems without dephasing, increasing disorder drives two consecutive crossovers toward classical simulability: the output distribution first loses anticoncentration, and then the tensor-network simulation cost drops from exponential to polynomial as entanglement is suppressed. The finite-size scaling collapses are consistent with continuous transitions in the large-system limit. Dephasing further reduces the complexity. We characterize the computationally hard regime through scaling laws that provide quantitative error-budget bounds for realistic near-term devices.

Figures

Figures reproduced from arXiv: 2607.18938 by Chae-Yeun Park, Changhun Oh, Seung-Sup B. Lee, Sung-Bin B. Lee.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) An instantaneous quantum polynomial-time (IQP) circuit [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) The rescaled collision probability [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a), (b) Maximum BMPS entanglement entropy [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. BMPS singular-value spectra at the bond where [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The maximum BMPS entanglement entropy [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: (a) shows that Smax decreases as either σ or ε grows. Note that here we contract square-lattice tensor networks with bond dimensions up to 4, which originate from PEPOs that describe dephased mixed states. Hence the maximal value of Smax is √ n ln 2 here, in contrast to √ n ln √ 2 from the PEPS-based simulations [cf [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. (a),(b) Scaling collapses of the squared relative error and [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

51 extracted references · 4 linked inside Pith

  1. [1]

    A. W. Harrow and A. Montanaro, Quantum computational supremacy, Nature549, 203 (2017)

    2017

  2. [2]

    physical

    Then we apply U:= Q ⟨ij⟩ Uij Q i Ui, where Uij = exp(iJijZiZj) and Ui = exp(−ihiZi) are, re- spectively, two- and single-qubit gates generated by Pauli- Z operators. U can be viewed as real-time evolution for a constant time under a nearest-neighbor Ising Hamiltonian H=− P ⟨ij⟩ JijZiZj + P i hiZi. We draw each “mag- arXiv:2607.18938v1 [quant-ph] 21 Jul 20...

    Pith/arXiv arXiv 2026

  3. [3]

    Boixo, S

    S. Boixo, S. V. Isakov, V. N. Smelyanskiy, R. Babbush, N. Ding, Z. Jiang, M. J. Bremner, J. M. Martinis, and H. Neven, Charac- terizing quantum supremacy in near-term devices, Nat. Phys.14, 595 (2018)

    2018

  4. [4]

    Bouland, B

    A. Bouland, B. Fefferman, C. Nirkhe, and U. Vazirani, On the complexityandverificationofquantumrandomcircuitsampling, Nat. Phys.15, 159 (2019)

    2019

  5. [5]

    D.HangleiterandJ.Eisert,Computationaladvantageofquantum random sampling, Rev. Mod. Phys.95, 035001 (2023)

    2023

  6. [6]

    Movassagh, The hardness of random quantum circuits, Nat

    R. Movassagh, The hardness of random quantum circuits, Nat. Phys.19, 1719 (2023)

    2023

  7. [7]

    P. W. Shor, Polynomial-time algorithms for prime factorization anddiscretelogarithmsonaquantumcomputer,SIAMJ.Comput. 26, 1484 (1997)

    1997

  8. [8]

    Shepherd and M

    D. Shepherd and M. J. Bremner, Temporally unstructured quan- tum computation, Proc. R. Soc. A465, 1413 (2009)

    2009

  9. [9]

    M. J. Bremner, A. Montanaro, and D. J. Shepherd, Average- case complexity versus approximate simulation of commuting quantum computations, Phys. Rev. Lett.117, 080501 (2016)

    2016

  10. [10]

    Bluvstein, S

    D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T. Manovitz, S. Ebadi, M. Cain, M. Kalinowski, D. Hangleiter, et al., Logical quantum processor based on reconfigurable atom arrays, Nature626, 58 (2024)

    2024

  11. [11]

    M.J.Bremner,R.Jozsa,andD.J.Shepherd,Classicalsimulation of commuting quantum computations implies collapse of the polynomial hierarchy, Proc. R. Soc. A467, 459 (2010)

    2010

  12. [12]

    M. J. Bremner, A. Montanaro, and D. J. Shepherd, Achieving quantum supremacy with sparse and noisy commuting quantum computations, Quantum1, 8 (2017)

    2017

  13. [13]

    Aaronson and A

    S. Aaronson and A. Arkhipov, The computational complexity of linear optics, inProceedings of the forty-third annual ACM symposium on Theory of computing, STOC 11 (ACM, 2011) pp. 333–342

    2011

  14. [14]

    Bermejo-Vega, D

    J. Bermejo-Vega, D. Hangleiter, M. Schwarz, R. Raussendorf, and J. Eisert, Architectures for quantum simulation showing a quantum speedup, Phys. Rev. X8, 021010 (2018)

    2018

  15. [15]

    Park and M

    C.-Y. Park and M. J. Kastoryano, Complexity phase transitions in instantaneous quantum polynomial-time circuits, Phys. Rev. Res.7, 013001 (2025)

    2025

  16. [16]

    Fujii and T

    K. Fujii and T. Morimae, Commuting quantum circuits and complexity of Ising partition functions, New J. Phys.19, 033003 (2017)

    2017

  17. [17]

    Jerrum, Two-dimensional monomer-dimer systems are com- putationally intractable, J

    M. Jerrum, Two-dimensional monomer-dimer systems are com- putationally intractable, J. Stat. Phys.48, 121 (1987)

    1987

  18. [18]

    Jaeger, D

    F. Jaeger, D. L. Vertigan, and D. J. A. Welsh, On the computa- tional complexity of the jones and tutte polynomials, Math. Proc. Cambridge Philos. Soc.108, 35 (1990)

    1990

  19. [19]

    Stockmeyer, The complexity of approximate counting, in Proceedings of the fifteenth annual ACM symposium on Theory of computing - STOC ’83, STOC ’83 (ACM Press, 1983) pp

    L. Stockmeyer, The complexity of approximate counting, in Proceedings of the fifteenth annual ACM symposium on Theory of computing - STOC ’83, STOC ’83 (ACM Press, 1983) pp. 118–126

    1983

  20. [20]

    Toda, PP is as hard as the polynomial-time hierarchy, SIAM J

    S. Toda, PP is as hard as the polynomial-time hierarchy, SIAM J. Comput.20, 865 (1991)

    1991

  21. [21]

    Y. Han, L. A. Hemaspaandra, and T. Thierauf, Threshold com- putation and cryptographic security, SIAM J. Comput.26, 59 (1997)

    1997

  22. [22]

    Aaronson, Quantum computing, postselection, and probabilis- tic polynomial-time, Proc

    S. Aaronson, Quantum computing, postselection, and probabilis- tic polynomial-time, Proc. R. Soc. A461, 3473 (2005)

    2005

  23. [23]

    Z. Liu, D. Devulapalli, D. Hangleiter, Y.-K. Liu, A. J. Kollár, A. V. Gorshkov, and A. M. Childs, Efficiently verifiable quan- tum advantage on near-term analog quantum simulators, PRX Quantum6, 010341 (2025)

    2025

  24. [24]

    Gao, S.-T

    X. Gao, S.-T. Wang, and L.-M. Duan, Quantum supremacy for simulating a translation-invariant Ising spin model, Phys. Rev. Lett.118, 040502 (2017)

    2017

  25. [25]

    Bermejo-Vega, Closing gaps of a quantum advantage with short-time Hamiltonian dynamics, Phys

    J.Haferkamp,D.Hangleiter,A.Bouland,B.Fefferman,J.Eisert, and J. Bermejo-Vega, Closing gaps of a quantum advantage with short-time Hamiltonian dynamics, Phys. Rev. Lett.125, 250501 (2020)

    2020

  26. [26]

    Rep.6, 25598 (2016)

    K.FujiiandS.Tamate,Computationalquantum-classicalbound- ary of noisy commuting quantum circuits, Sci. Rep.6, 25598 (2016)

    2016

  27. [27]

    Park and C

    S. Park and C. Oh, Matrix product state approach to lossy boson sampling and noisy IQP sampling (2025), arXiv:2510.24137 [quant-ph]

    Pith/arXiv arXiv 2025

  28. [28]

    Rajakumar, J

    J. Rajakumar, J. D. Watson, and Y.-K. Liu, Polynomial-time classical simulation of noisy iqp circuits with constant depth, inProceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)(Society for Industrial and Applied Mathematics, 2025) pp. 1037–1056

    2025

  29. [29]

    Placidi, E

    L. Placidi, E. Rinaldi, K. Fujii, and C.-Y. Liu, The impact of qubit connectivity on quantum advantage in noisy iqp circuits, (2026), arXiv:2604.12635 [quant-ph]

    Pith/arXiv arXiv 2026

  30. [30]

    Greenbaum and Z

    D. Greenbaum and Z. Dutton, Modeling coherent errors in quantum error correction, Quantum Sci. Technol.3, 015007 (2017)

    2017

  31. [31]

    Bravyi, M

    S. Bravyi, M. Englbrecht, R. König, and N. Peard, Correct- ing coherent errors with surface codes, npj Quantum Inf.4, 10.1038/s41534-018-0106-y (2018)

    doi:10.1038/s41534-018-0106-y 2018

  32. [32]

    S.Hakkaku,K.Mitarai,andK.Fujii,Sampling-basedquasiprob- ability simulation for fault-tolerant quantum error correction on the surface codes under coherent noise, Phys. Rev. Res.3, 043130 (2021)

    2021

  33. [33]

    Huang, A

    E. Huang, A. C. Doherty, and S. Flammia, Performance of quantum error correction with coherent errors, Phys. Rev. A99, 022313 (2019)

    2019

  34. [34]

    G. Feng, J. J. Wallman, B. Buonacorsi, F. H. Cho, D. K. Park, T. Xin, D. Lu, J. Baugh, and R. Laflamme, Estimating the coherence of noise in quantum control of a solid-state qubit, Phys. Rev. Lett.117, 260501 (2016)

    2016

  35. [35]

    Ahsan, S

    M. Ahsan, S. A. Z. Naqvi, and H. Anwer, Quantum circuit engineering for correcting coherent noise, Phys. Rev. A105, 022428 (2022)

    2022

  36. [36]

    Z. Cai, X. Xu, and S. C. Benjamin, Mitigating coherent noise using pauli conjugation, npj Quantum Inf.6, 10.1038/s41534- 019-0233-0 (2020)

    doi:10.1038/s41534- 2020

  37. [37]

    Kasteleyn, The statistics of dimers on a lattice, Physica27, 1209 (1961)

    P. Kasteleyn, The statistics of dimers on a lattice, Physica27, 1209 (1961)

    1961

  38. [38]

    M. E. Fisher, Statistical mechanics of dimers on a plane lattice, Phys. Rev.124, 1664 (1961)

    1961

  39. [39]

    S.-B. B. Lee, H. R. Choi, D. D. Ohm, and S.-S. B. Lee, Scalable projectedentangled-pairstaterepresentationofrandomquantum circuit states, Phys. Rev. Res.7, 033252 (2025)

    2025

  40. [40]

    Verstraete and J

    F. Verstraete and J. I. Cirac, Renormalization algorithms for quantum-many-body systems in two and higher dimensions (2004), arXiv:cond-mat/0407066

    Pith/arXiv arXiv 2004

  41. [41]

    V. Murg, F. Verstraete, and J. I. Cirac, Variational study of hard- core bosons in a two-dimensional optical lattice using projected entangled pair states, Phys. Rev. A75, 033605 (2007)

    2007

  42. [42]

    C. E. Porter and R. G. Thomas, Fluctuations of nuclear reaction widths, Phys. Rev.104, 483 (1956). 6

    1956

  43. [43]

    M. E. Fisher and M. N. Barber, Scaling theory for finite-size effects in the critical region, Phys. Rev. Lett.28, 1516 (1972)

    1972

  44. [44]

    A.M.Dalzell,N.Hunter-Jones,andF.G.S.L.Brandão,Random quantum circuits anticoncentrate in log depth, PRX Quantum3, 010333 (2022)

    2022

  45. [45]

    Barahona, On the computational complexity of Ising spin glass models, J

    F. Barahona, On the computational complexity of Ising spin glass models, J. Phys. A15, 3241 (1982)

    1982

  46. [46]

    Istrail, Statistical mechanics, three-dimensionality and NP- completeness: I

    S. Istrail, Statistical mechanics, three-dimensionality and NP- completeness: I. universality of intractability for the partition functionoftheIsingmodelacrossnon-planarlattices,inProceed- ings of the Thirty-Second Annual ACM Symposium on Theory of Computing, STOC ’00 (Association for Computing Machinery, New York, NY, USA, 2000) pp. 87–96

    2000

  47. [47]

    Grundner, P

    M. Grundner, P. Westhoff, F. B. Kugler, O. Parcollet, and U. Schollwöck, Complex time evolution in tensor networks andtime-dependentGreen’sfunctions,Phys.Rev.B109,155124 (2024)

    2024

  48. [48]

    X. Cao, Y. Lu, E. M. Stoudenmire, and O. Parcollet, Dynamical correlation functions from complex time evolution, Phys. Rev. B 109, 235110 (2024)

    2024

  49. [49]

    A. W. Harrow and R. A. Low, Random quantum circuits are approximate 2-designs, Commun. Math. Phys.291, 257 (2009)

    2009

  50. [50]

    F. G. S. L. Brandão, A. W. Harrow, and M. Horodecki, Local random quantum circuits are approximate polynomial-designs, Commun. Math. Phys.346, 397 (2016)

    2016

  51. [51]

    A. W. Harrow and S. Mehraban, Approximate unitaryt-designs by short random quantum circuits using nearest-neighbor and long-range gates, Commun. Math. Phys.401, 1531 (2023). 7 End Matter 0 10 3 10 2 10 1 100 0 2 4 6 8Smax (a) n = 12×12 nln 2 0 10 3 10 2 10 1 100 0 10 4 10 3 10 2 10 1 100 101 102 Z 1 PTD (b) n = 8×8 0.0 0.2 0.4 0.6 0.8 1.0 FIG. 6. (a) Smax...

    2023