Microscopics of Quantum Annealing in the Disordered Dipolar Ising Ferromagnet LiHo $_{x}$ Y $_{1-x}$ F $_4$ Microscopics of Quantum Annealing in the Disordered Dipolar Ising Ferromagnet LiHo$_{x}$Y$_{1-x}$F$_4$

S. Säubert, Colin L. Sarkis, F. Ye +2 more

Quantum annealing (QA) refers to an optimization process that uses quantum fluctuations to find the global minimum of a rugged energy landscape with many local minima. Conceptually, QA is often framed in the context of the disordered transverse field Ising model, in which a magnetic field applied perpendicular to the Ising axis tunes the quantum fluctuations and enables the system to tunnel through energy barriers, and hence reach the ground state more quickly. A solid state material closely related to this model, LiHo $_{0.45}$ Y $_{0.55}$ F $_4$ , was shown to exhibit faster dynamics after a QA protocol, compared to thermal annealing (TA), but little is known about the actual process of optimization involved or the nature of the spin correlations in the state that is reached. Here, we report on the microscopics of QA in this material using diffuse magnetic neutron scattering. Comparing a QA to a TA protocol that reach the same end-point, we find very similar final diffuse scattering which consists of pinch-point scattering, largely consistent with critical scattering near a phase boundary in the dipolar Ising ferromagnetic model. However, comparing the time evolution at the end of the protocols, we find that the spin correlations evolve more significantly after TA, suggesting that QA produces a state closer to equilibrium. We also observe experimental evidence that the transverse field produces random fields, which had been previously predicted for this material and studied in other contexts. Thus, while the material does exhibit a quantum speedup under quantum annealing conditions, it is not a simple annealing problem; the energy landscape being optimized is changing as the optimization proceeds.Quantum annealing (QA) refers to an optimization process that uses quantum fluctuations to find the global minimum of a rugged energy landscape with many local minima. Conceptually, QA is often framed in the context of the disordered transverse field Ising model, in which a magnetic field applied perpendicular to the Ising axis tunes the quantum fluctuations and enables the system to tunnel through energy barriers, and hence reach the ground state more quickly. A solid state material closely related to this model, LiHo$_{0.45}$Y$_{0.55}$F$_4$, was shown to exhibit faster dynamics after a QA protocol, compared to thermal annealing (TA), but little is known about the actual process of optimization involved or the nature of the spin correlations in the state that is reached. Here, we report on the microscopics of QA in this material using diffuse magnetic neutron scattering. Comparing a QA to a TA protocol that reach the same end-point, we find very similar final diffuse scattering which consists of pinch-point scattering, largely consistent with critical scattering near a phase boundary in the dipolar Ising ferromagnetic model. However, comparing the time evolution at the end of the protocols, we find that the spin correlations evolve more significantly after TA, suggesting that QA produces a state closer to equilibrium. We also observe experimental evidence that the transverse field produces random fields, which had been previously predicted for this material and studied in other contexts. Thus, while the material does exhibit a quantum speedup under quantum annealing conditions, it is not a simple annealing problem; the energy landscape being optimized is changing as the optimization proceeds.

arXiv:2105.03408May 7, 20211 citation

Numerical and experimental studies of the dynamics of Ising spin glasses

Jan‐Olov Andersson

Jan 1, 19941 citation

Classical ergodicity and quantum eigenstate thermalization: Analysis in fully connected Ising ferromagnets

Takashi Mori

We investigate the relation between the classical ergodicity and the quantum eigenstate thermalization in the fully connected Ising ferromagnets. In the case of spin-1/2, an expectation value of an observable in a single-energy eigenstate coincides with the long-time average in the underlying classical dynamics, which is a consequence of the Wentzel-Kramers-Brillouin approximation. In the case of spin-1, the underlying classical dynamics is not necessarily ergodic. In that case, it turns out that, in the thermodynamic limit, the statistics of the expectation values of an observable in the energy eigenstates coincides with the statistics of the long-time averages in the underlying classical dynamics starting from random initial states sampled uniformly from the classical phase space. This feature seems to be a general property in semiclassical systems, and the result presented here is crucial in discussing equilibration, thermalization, and dynamical transitions of such systems.We investigate the relation between the classical ergodicity and the quantum eigenstate thermalization in the fully connected Ising ferromagnets. In the case of spin-1/2, an expectation value of an observable in a single-energy eigenstate coincides with the long-time average in the underlying classical dynamics, which is a consequence of the Wentzel-Kramers-Brillouin approximation. In the case of spin-1, the underlying classical dynamics is not necessarily ergodic. In that case, it turns out that, in the thermodynamic limit, the statistics of the expectation values of an observable in the energy eigenstates coincides with the statistics of the long-time averages in the underlying classical dynamics starting from random initial states sampled uniformly from the classical phase space. This feature seems to be a general property in semiclassical systems, and the result presented here is crucial in discussing equilibration, thermalization, and dynamical transitions of such systems.

arXiv:1609.09258Physical review. EJul 17, 201714 citations

Critical Ising Spin Dynamics on Percolation Clusters

Christopher L. Henley

Ferromagnetically interacting Ising spins are placed on a fractal network (such as a critical percolation cluster) with ${T}_{c}=0$ , and endowed with a single-spin-flip dynamics. At low temperatures the collective dynamics is determined by thermal activation over energy barriers. The barrier to overturning of the spins in a domain of size $L$ is proportional to $Z\mathrm{ln}L$ , where $Z$ is a new geometrical parameter characterizing the fractal. A new "singular" dynamic scaling is found in which the effective dynamic critical exponent diverges at criticality.Ferromagnetically interacting Ising spins are placed on a fractal network (such as a critical percolation cluster) with ${T}_{c}=0$, and endowed with a single-spin-flip dynamics. At low temperatures the collective dynamics is determined by thermal activation over energy barriers. The barrier to overturning of the spins in a domain of size $L$ is proportional to $Z\mathrm{ln}L$, where $Z$ is a new geometrical parameter characterizing the fractal. A new "singular" dynamic scaling is found in which the effective dynamic critical exponent diverges at criticality.

Physical Review LettersMay 6, 1985109 citations

Dynamics of the one dimensional randomly dilute Ising ferromagnet

H. G. E. Hentschel

The European Physical Journal BDec 1, 19805 citations

Microscopics of Quantum Annealing in the Disordered Dipolar Ising Ferromagnet LiHo

_{x}

_{1-x}

_4

_{0.45}

_{0.55}

_4

, was shown to exhibit faster dynamics after a QA protocol, compared to thermal annealing (TA), but little is known about the actual process of optimization involved or the nature of the spin correlations in the state that is reached. Here, we report on the microscopics of QA in this material using diffuse magnetic neutron scattering. Comparing a QA to a TA protocol that reach the same end-point, we find very similar final diffuse scattering which consists of pinch-point scattering, largely consistent with critical scattering near a phase boundary in the dipolar Ising ferromagnetic model. However, comparing the time evolution at the end of the protocols, we find that the spin correlations evolve more significantly after TA, suggesting that QA produces a state closer to equilibrium. We also observe experimental evidence that the transverse field produces random fields, which had been previously predicted for this material and studied in other contexts. Thus, while the material does exhibit a quantum speedup under quantum annealing conditions, it is not a simple annealing problem; the energy landscape being optimized is changing as the optimization proceeds.

Ferromagnetically interacting Ising spins are placed on a fractal network (such as a critical percolation cluster) with

{T}_{c}=0

, and endowed with a single-spin-flip dynamics. At low temperatures the collective dynamics is determined by thermal activation over energy barriers. The barrier to overturning of the spins in a domain of size

L

is proportional to

Z\mathrm{ln}L

, where

Z

is a new geometrical parameter characterizing the fractal. A new "singular" dynamic scaling is found in which the effective dynamic critical exponent diverges at criticality.