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arXiv:1809.02124 · Physical review. B./Physical review. B

Dynamics of simulated quantum annealing in random Ising chains

Glen Bigan Mbeng, Lorenzo Privitera, Luca Arceci, Giuseppe E. Santoro

Published Feb 1, 2019• 11 citations
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Abstract

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.

DOI
10.1103/physrevb.99.064201

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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 \ensuremathρKZQA\ensuremath∼(log10\ensuremathτ)\ensuremath−2{\ensuremath{\rho}}_{\mathrm{KZ}}^{\mathrm{QA}}\ensuremath{\sim}{({log}_{10}\ensuremath{\tau})}^{\ensuremath{-}2}\ensuremathρKZQA​\ensuremath∼(log10​\ensuremathτ)\ensuremath−2 as opposed to \ensuremathρKZSA\ensuremath∼(log10\ensuremathτ)\ensuremath−1,\ensuremathτ{\ensuremath{\rho}}_{\mathrm{KZ}}^{\mathrm{SA}}\ensuremath{\sim}{({log}_{10}\ensuremath{\tau})}^{\ensuremath{-}1}, \ensuremath{\tau}\ensuremathρKZSA​\ensuremath∼(log10​\ensuremathτ)\ensuremath−1,\ensuremathτ being the total annealing time, while for the ordered case both give the same power law \ensuremathρKZQA\ensuremath≈\ensuremathρKZSA\ensuremath∼\ensuremathτ\ensuremath−1/2{\ensuremath{\rho}}_{\mathrm{KZ}}^{\mathrm{QA}}\ensuremath{\approx}{\ensuremath{\rho}}_{\mathrm{KZ}}^{\mathrm{SA}}\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}1/2}\ensuremathρKZQA​\ensuremath≈\ensuremathρKZSA​\ensuremath∼\ensuremathτ\ensuremath−1/2. We show that the dynamics of SQA, while correctly capturing the Kibble-Zurek scaling \ensuremathτ\ensuremath−1/2{\ensuremath{\tau}}^{\ensuremath{-}1/2}\ensuremathτ\ensuremath−1/2 for the ordered case, is unable to reproduce the QA dynamics in the disordered case at intermediate \ensuremathτ\ensuremath{\tau}\ensuremathτ. 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.
Simulated quantum annealing based on the path-integral Monte Carlo is one of the most common tools to simulate quantum annealing on classical hardware. Nevertheless, it is in principle highly nontrivial whether or not this classical algorithm can correctly reproduce the quantum dynamics of quantum annealing, particularly in the diabatic regime. We study this problem numerically through the generalized Kibble-Zurek mechanism of defect distribution in the simplest ferromagnetic one-dimensional transverse-field Ising model with and without coupling to the environment. We find that, in the absence of coupling to the environment, simulated quantum annealing correctly describes the annealing-time dependence of the average number of defects, but a detailed analysis of the defect distribution shows clear deviations from the theoretical prediction. When the system is open (coupled to the environment), the average number of defects does not follow the theoretical prediction but is qualitatively compatible with the numerical result by the infinite-time-evolving block decimation combined with the quasiadiabatic propagator path integral, which is valid in a very-short-time region. The distribution of defects in the open system turns out to be not far from the theoretical prediction. It is surprising that the classical stochastic dynamics of simulated quantum annealing ostensibly reproduces some aspects of the quantum dynamics. However, a serious problem is that it is hard to predict for which physical quantities in which system it is reliable. Those results suggest the necessity to exert a good amount of caution in using simulated quantum annealing to study the detailed quantitative aspects of the dynamics of quantum annealing.
We study errors of quantum annealing and simulated annealing to highlight better performance of quantum annealing over simulated annealing. Quantum annealing and simulated annealing perform optimization through a quantum adiabatic evolution and a quasi-static evolution respectively. In both methods, dynamics across a phase transition plays a crucial role. The Kibble-Zurek mechanism is known as an underlying physics of defect formation during a time-evolution across a phase transition. We apply an argument for the Kibble-Zurek mechanism to the error generation of quantum annealing and simulated annealing. We show that, for the disordered Ising chain, the kink density and residual energy per spin of quantum annealing decay as (ln τ ) −2 and (ln τ ) −4 respectively, whereas those of simulated annealing decay as (ln τ ) −1 and (ln τ ) −2 with the annealing rate −1/ τ . These results imply better performance of quantum annealing. We also develop our theory for a two-dimensional spin-glass model.
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 present here our study of the adiabatic quantum dynamics of a random Ising chain across its quantum critical point. The model investigated is an Ising chain in a transverse field with disorder present both in the exchange coupling and in the transverse field. The transverse field term is proportional to a function $\ensuremath{\Gamma}(t)$ which, as in the Kibble-Zurek mechanism, is linearly reduced to zero in time with a rate ${\ensuremath{\tau}}^{\ensuremath{-}1}$, $\ensuremath{\Gamma}(t)=\ensuremath{-}t∕\ensuremath{\tau}$, starting at $t=\ensuremath{-}\ensuremath{\infty}$ from the quantum disordered phase $(\ensuremath{\Gamma}=\ensuremath{\infty})$ and ending at t=0t=0t=0 in the classical ferromagnetic phase $(\ensuremath{\Gamma}=0)$. We first analyze the distribution of the gaps, occurring at the critical point ${\ensuremath{\Gamma}}_{c}=1$, which are relevant for breaking the adiabaticity of the dynamics. We then present extensive numerical simulations for the residual energy Eres{E}_{\mathrm{res}}Eres​ and density of defects ${\ensuremath{\rho}}_{k}$ at the end of the annealing, as a function of the annealing inverse rate $\ensuremath{\tau}$. Both the average ${E}_{\mathrm{res}}(\ensuremath{\tau})$ and ${\ensuremath{\rho}}_{k}(\ensuremath{\tau})$ are found to behave logarithmically for large $\ensuremath{\tau}$, but with different exponents, $[{E}_{\mathrm{res}}(\ensuremath{\tau})∕L{]}_{\mathrm{av}}\ensuremath{\sim}1∕{\mathrm{ln}}^{\ensuremath{\zeta}}(\ensuremath{\tau})$ with $\ensuremath{\zeta}\ensuremath{\approx}3.4$, and $[{\ensuremath{\rho}}_{k}(\ensuremath{\tau}){]}_{\mathrm{av}}\ensuremath{\sim}1∕{\mathrm{ln}}^{2}(\ensuremath{\tau})$. We propose a mechanism for $1∕{\mathrm{ln}}^{2}\ensuremath{\tau}$ behavior of $[{\ensuremath{\rho}}_{k}{]}_{\mathrm{av}}$ based on the Landau-Zener tunneling theory and on a Fisher's-type real-space renormalization group analysis of the relevant gaps. The model proposed shows therefore a paradigmatic example of how an adiabatic quantum computation can become very slow when disorder is at play, even in the absence of any source of frustration.
We show clear evidence of a quadratic speedup of a quantum annealing (QA) Schr\"odinger dynamics over a Glauber master equation simulated annealing (SA) for a random Ising model in one dimension, via an equal-footing exact deterministic dynamics of the Jordan-Wigner fermionized problems. This is remarkable, in view of the arguments of H. G. Katzgraber et al. [Phys. Rev. X 4, 021008 (2014)], since SA does not encounter any phase transition, while QA does. We also find a second remarkable result: that a ``quantum-inspired'' imaginary-time Schr\"odinger QA provides a further exponential speedup, i.e., an asymptotic residual error decreasing as a power law ${\ensuremath{\tau}}^{\ensuremath{-}\ensuremath{\mu}}$ of the annealing time $\ensuremath{\tau}$.
The quantum dynamics of many-qubit systems is an outstanding problem that has recently driven significant advances in both numerical methods and programmable quantum processing units. In this work, we employ a comprehensive toolbox of state-of-the-art numerical approaches to classically simulate the dynamics of the two-dimensional transverse field Ising model. Our methods include three different tensor network techniques -- matrix product states, tree-tensor networks, and two-dimensional tensor-networks under the belief propagation approximation -- as well as time-dependent variational Monte Carlo with Neural Quantum States. We focus on two paradigmatic dynamical protocols: (i) quantum annealing through a critical point and (ii) post-quench dynamics. Our extensive results show the quantitative predictions of various state-of-the-art numerical methods providing a benchmark for future numerical investigations and experimental studies with the aim to push the limitations on classical and QPUs. In particular, our work connects classical simulability to different regimes associated with quantum dynamics in Rydberg arrays - namely, quasi-adiabatic dynamics, the Kibble-Zurek mechanism, and quantum quenches.