Numerical simulations of quantum magnetism in two spatial dimensions are often constrained by the area law of entanglement entropy, which heavily limits the accessible system sizes in tensor network methods. In this work, we investigate how the choice of mapping from a two-dimensional lattice to a one-dimensional path affects the accuracy of the two-dimensional density matrix renormalization group algorithm. We systematically evaluate all mappings corresponding to a subset of the Hamiltonian paths of the $N\ifmmode\times\else\texttimes\fi{}N$ grid graphs up to $N=8$ and demonstrate that the fractal space-filling curves generally lead to faster convergence in ground state searches compared to the commonly used ``snake'' path. To explain this performance gain, we analyze various locality metrics and propose a scalable method for constructing high-performing paths on larger lattices by tiling smaller optimal paths. Our results show that such paths consistently improve simulation convergence, with the advantage increasing with system size.Numerical simulations of quantum magnetism in two spatial dimensions are often constrained by the area law of entanglement entropy, which heavily limits the accessible system sizes in tensor network methods. In this work, we investigate how the choice of mapping from a two-dimensional lattice to a one-dimensional path affects the accuracy of the two-dimensional density matrix renormalization group algorithm. We systematically evaluate all mappings corresponding to a subset of the Hamiltonian paths of the $N\ifmmode\times\else\texttimes\fi{}N$ grid graphs up to $N=8$ and demonstrate that the fractal space-filling curves generally lead to faster convergence in ground state searches compared to the commonly used ``snake'' path. To explain this performance gain, we analyze various locality metrics and propose a scalable method for constructing high-performing paths on larger lattices by tiling smaller optimal paths. Our results show that such paths consistently improve simulation convergence, with the advantage increasing with system size.

arXiv:2507.11820Physical review. B./Physical review. BDec 5, 20250 citations

Complexity, Fractal Dimension for Quantum States

Masanori Ohya

WORLD SCIENTIFIC eBooksFeb 1, 20080 citations

Measurement-Induced Perturbations of Hausdorff Dimension in Quantum Paths

You-Wei Ding, Yen Chin Ong, Hao Xu

In a seminal paper, Abbott et al. analyzed the relationship between a particle's trajectory and the resolution of position measurements performed by an observer at fixed time intervals. They predicted that quantum paths exhibit a universal Hausdorff dimension that transitions from $d=2$ to $d=1$ as the momentum of the particle increases. However, although measurements were assumed to occur at intervals of time, the calculations only involved evaluating the expectation value of operators for the free evolution of wave function within a single interval, with no actual physical measurements performed. In this work we investigate how quantum measurements alter the fractal geometry of quantum particle paths. By modelling sequential measurements using Gaussian wave packets for both the particle and the apparatus, we reveal that the dynamics of the measurement change the roughness of the path and shift the emergent Hausdorff dimension towards a lower value in nonselective evolution. For selective evolution, feedback control forces must be introduced to counteract stochastic wave function collapse, stabilising trajectories and enabling dimensionality to be tuned. When the contribution of the measurement approaches zero, our result reduces to that of Abbott et al. Our work can thus be regarded as a more realistic formulation of their approach, and it connects theoretical quantum fractality with measurement physics, quantifying how detectors reshape spacetime statistics at quantum scales.In a seminal paper, Abbott et al. analyzed the relationship between a particle's trajectory and the resolution of position measurements performed by an observer at fixed time intervals. They predicted that quantum paths exhibit a universal Hausdorff dimension that transitions from $d=2$ to $d=1$ as the momentum of the particle increases. However, although measurements were assumed to occur at intervals of time, the calculations only involved evaluating the expectation value of operators for the free evolution of wave function within a single interval, with no actual physical measurements performed. In this work we investigate how quantum measurements alter the fractal geometry of quantum particle paths. By modelling sequential measurements using Gaussian wave packets for both the particle and the apparatus, we reveal that the dynamics of the measurement change the roughness of the path and shift the emergent Hausdorff dimension towards a lower value in nonselective evolution. For selective evolution, feedback control forces must be introduced to counteract stochastic wave function collapse, stabilising trajectories and enabling dimensionality to be tuned. When the contribution of the measurement approaches zero, our result reduces to that of Abbott et al. Our work can thus be regarded as a more realistic formulation of their approach, and it connects theoretical quantum fractality with measurement physics, quantifying how detectors reshape spacetime statistics at quantum scales.

quant-pharXiv:2512.13046The European Physical Journal PlusDec 16, 20250 citations

Seeing quantum fractals

Physics TodayFeb 23, 20100 citations

N=8

and demonstrate that the fractal space-filling curves generally lead to faster convergence in ground state searches compared to the commonly used ``snake'' path. To explain this performance gain, we analyze various locality metrics and propose a scalable method for constructing high-performing paths on larger lattices by tiling smaller optimal paths. Our results show that such paths consistently improve simulation convergence, with the advantage increasing with system size.

d=2

d=1

as the momentum of the particle increases. However, although measurements were assumed to occur at intervals of time, the calculations only involved evaluating the expectation value of operators for the free evolution of wave function within a single interval, with no actual physical measurements performed. In this work we investigate how quantum measurements alter the fractal geometry of quantum particle paths. By modelling sequential measurements using Gaussian wave packets for both the particle and the apparatus, we reveal that the dynamics of the measurement change the roughness of the path and shift the emergent Hausdorff dimension towards a lower value in nonselective evolution. For selective evolution, feedback control forces must be introduced to counteract stochastic wave function collapse, stabilising trajectories and enabling dimensionality to be tuned. When the contribution of the measurement approaches zero, our result reduces to that of Abbott et al. Our work can thus be regarded as a more realistic formulation of their approach, and it connects theoretical quantum fractality with measurement physics, quantifying how detectors reshape spacetime statistics at quantum scales.