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arXiv:2507.09273

Defects and their Time Scales in Quantum and Classical Annealing of the Two-Dimensional Ising Model

Phillip Weinberg, Na Xu, Anders W. Sandvik

Published Jul 12, 2025• 0 citations
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Abstract

We investigate defects in the two-dimensional transverse-field Ising ferromagnet on periodic $L\times L$ lattices after quantum annealing from high to vanishing field. With exact numerical solutions for $L \le 6$, we observe the expected critical Kibble-Zurek (KZ) time scale $\propto L^{z+1/ν}$ (with $z=1$ and $1/ν\approx 1.59$) at the quantum phase transition. We also observe KZ scaling of the ground-state fidelity at the end of the process. The excitations evolve by coarsening dynamics of confined defects, with a time scale $\propto L^2$, and interface fluctuations of system-spanning defects, with life time $\propto L^3$. We build on analogies with classical simulated annealing, where we characterize system-spanning defects in detail and find differences in the dynamic scales of domain walls with winding numbers $W=(1,0)/(0,1)$ (horizontal/vertical) and $W=(1,1)$ (diagonal). They decay on time scales $\propto L^3$ (which applies also to system-spanning domains in systems with open boundaries) and $\propto L^{3.4}$, respectively, when imposed in the ordered phase. As a consequence of $L^{3.4}$ exceeding the classical KZ scale $L^{z+1/ν}=L^{3.17}$ the probability of $W=(1,1)$ domains in SA scales with the KZ exponent even in the final $T=0$ state. In QA, also the $W=(1,0)/(0,1)$ domains are controlled by the KZ time scale $L^{2.59}$. The $L^3$ scale can nevertheless be detected in the excited states, using a method that we develop that should also be applicable in QA experiments.

DOI
10.48550/arxiv.2507.09273

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We investigate defects in the two-dimensional transverse-field Ising ferromagnet on periodic L×LL\times LL×L lattices after quantum annealing from high to vanishing field. With exact numerical solutions for L≤6L \le 6L≤6, we observe the expected critical Kibble-Zurek (KZ) time scale ∝Lz+1/ν\propto L^{z+1/ν}∝Lz+1/ν (with z=1z=1z=1 and 1/ν≈1.591/ν\approx 1.591/ν≈1.59) at the quantum phase transition. We also observe KZ scaling of the ground-state fidelity at the end of the process. The excitations evolve by coarsening dynamics of confined defects, with a time scale ∝L2\propto L^2∝L2, and interface fluctuations of system-spanning defects, with life time ∝L3\propto L^3∝L3. We build on analogies with classical simulated annealing, where we characterize system-spanning defects in detail and find differences in the dynamic scales of domain walls with winding numbers W=(1,0)/(0,1)W=(1,0)/(0,1)W=(1,0)/(0,1) (horizontal/vertical) and W=(1,1)W=(1,1)W=(1,1) (diagonal). They decay on time scales ∝L3\propto L^3∝L3 (which applies also to system-spanning domains in systems with open boundaries) and ∝L3.4\propto L^{3.4}∝L3.4, respectively, when imposed in the ordered phase. As a consequence of L3.4L^{3.4}L3.4 exceeding the classical KZ scale Lz+1/ν=L3.17L^{z+1/ν}=L^{3.17}Lz+1/ν=L3.17 the probability of W=(1,1)W=(1,1)W=(1,1) domains in SA scales with the KZ exponent even in the final T=0T=0T=0 state. In QA, also the W=(1,0)/(0,1)W=(1,0)/(0,1)W=(1,0)/(0,1) domains are controlled by the KZ time scale L2.59L^{2.59}L2.59. The L3L^3L3 scale can nevertheless be detected in the excited states, using a method that we develop that should also be applicable in QA experiments.
We revisit the quantum annealing problem in the non-Hermitian transverse-field Ising model. We determine, both analytically and numerically, the intrinsic transition probabilities and the resulting defect density. Our results reveal that, unlike the Hermitian case where defect production is dominated by modes near the gap-closing point, the non-Hermitian dynamics involve significant contributions from broad momentum sectors. We find that, depending on the dissipation strength, the defect density exhibits standard Kibble-Zurek scaling, anti-Kibble-Zurek behavior, and a suppression faster than the Kibble-Zurek prediction. We demonstrate that these deviations from the standard Kibble-Zurek scaling can be understood in terms of the underlying excitation probabilities. Specifically, the fast decay of the defect density originates from a vanishing excitation probability spanning a range of annealing times across all allowed modes, even at the gap-closing points. In contrast, the anti-Kibble-Zurek behavior arises from supplementary excitations facilitated by dissipation over a broad range of allowed modes, particularly those situated away from the gap-closing region.
Geometric frustration in two-dimensional Ising models allows for a wealth of exotic universal behavior, both Ising and non-Ising, in the presence of quantum fluctuations. In particular, the triangular antiferromagnet and Villain model in a transverse field can be understood through distinct XY pseudospins, but have qualitatively similar phase diagrams including a quantum phase transition in the (2+1)-dimensional XY universality class. While the quantum dynamics of modestly-sized systems can be simulated classically using tensor-based methods, these methods become infeasible for larger lattices. Here we perform both classical and quantum simulations of these dynamics, where our quantum simulator is a superconducting quantum annealer. Our observations on the triangular lattice suggest that the dominant quench dynamics are not described by the quantum Kibble-Zurek scaling of the quantum phase transition, but rather a faster coarsening dynamics in an effective two-dimensional XY model in the ordered phase. Similarly, on the Villain model, the scaling exponent does not match the Kibble-Zurek expectation. These results demonstrate the ability of quantum annealers to perform coherent quantum dynamics simulations that are hard to classically scale beyond small systems, and open the avenue to predictive simulations of the dynamics of Ising magnetic materials on quantum simulators.
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Simulated quantum annealing (SQA) is a classical computational strategy that emulates a quantum annealing (QA) dynamics through a path-integral Monte Carlo whose parameters are changed during the simulation. Here we apply SQA to the one-dimensional transverse field Ising chain, where previous works have shown that, in the presence of disorder, a coherent QA provides a quadratic speedup with respect to classical simulated annealing, with a density of Kibble-Zurek defects decaying as ${\ensuremath{\rho}}_{\mathrm{KZ}}^{\mathrm{QA}}\ensuremath{\sim}{({log}_{10}\ensuremath{\tau})}^{\ensuremath{-}2}$ as opposed to ${\ensuremath{\rho}}_{\mathrm{KZ}}^{\mathrm{SA}}\ensuremath{\sim}{({log}_{10}\ensuremath{\tau})}^{\ensuremath{-}1}, \ensuremath{\tau}$ being the total annealing time, while for the ordered case both give the same power law ${\ensuremath{\rho}}_{\mathrm{KZ}}^{\mathrm{QA}}\ensuremath{\approx}{\ensuremath{\rho}}_{\mathrm{KZ}}^{\mathrm{SA}}\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}1/2}$. We show that the dynamics of SQA, while correctly capturing the Kibble-Zurek scaling ${\ensuremath{\tau}}^{\ensuremath{-}1/2}$ for the ordered case, is unable to reproduce the QA dynamics in the disordered case at intermediate $\ensuremath{\tau}$. We analyze and discuss several issues related to the choice of the Monte Carlo moves (local or global in space), the time-continuum limit needed to eliminate the Trotter-discretization error, and the long autocorrelation times shown by a local-in-space Monte Carlo dynamics for large disordered samples.
Based on the strong-disorder renormalization group method, a microscopic mechanism of defect formation in the quantum annealing of the random transverse-field Ising chain is proposed, which represents the annealing process as a gradual aggregation of strongly coupled spin clusters. The ferromagnetic ground state of clusters is either preserved or get excited in pairwise fusions of clusters, depending on the effective annealing rate of the fusion, the latter events being responsible for the appearance of defects in the final state. A consequence of the theory is that, although the Griffiths-McCoy phases surrounding the critical point are gapless, they are still effectively gapped from the point of view of quantum annealing. Thereby we provide an explanation of the finiteness of gap outside of the critical point, which was implicit in an early approach to the problem by Kibble-Zurek scaling [Dziarmaga, Phys. Rev. B {\bf 74}, 064416 (2006)]. Furthermore, by identifying the accessible excitations, we refine the functional form of the vanishing of the gap at the critical point. The defect density in the final state is found to decrease with the annealing time τττ, as n(τ)∼ln⁡−2(τln⁡2τ)n(τ)\sim \ln^{-2}\left(\fracτ{\ln^2τ}\right)n(τ)∼ln−2(ln2ττ​) for large τττ. In addition to this, our approach gives access also to the density of defects at intermediate times of the annealing process.