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Importance Sampling Scheme for the Stochastic Simulation of Quantum Spin Dynamics

by Stefano De Nicola

Submission summary

As Contributors: Stefano De Nicola
Arxiv Link: (pdf)
Date submitted: 2021-04-08 12:01
Submitted by: De Nicola, Stefano
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Quantum Physics


The numerical simulation of dynamical phenomena in interacting quantum systems is a notoriously hard problem. Although a number of promising numerical methods exist, they often have limited applicability due to the growth of entanglement or the presence of the so-called sign problem. In this work, we develop an importance sampling scheme for the simulation of quantum spin dynamics, building on a recent approach mapping quantum spin systems to classical stochastic processes. The importance sampling scheme is based on identifying the classical trajectory that yields the largest contribution to a given quantum observable. An exact transformation is then carried out to preferentially sample trajectories that are close to the dominant one. We demonstrate that this approach is capable of reducing the temporal growth of fluctuations in the stochastic quantities, thus extending the range of accessible times and system sizes compared to direct sampling. We discuss advantages and limitations of the proposed approach, outlining directions for further developments.

Current status:
Editor-in-charge assigned

Submission & Refereeing History

Reports on this Submission

Anonymous Report 2 on 2021-5-24 (Invited Report)


1. New approach to studying dynamics of quantum spin systems.
2. Relevant to both theorists and experimentalists.
3. Approach seems to be powerful and versatile.
4. Paper nicely written


1. Lack of adequate benchmarking against known results.
2. Incompletediscussion on the applicability of the method - when is it applicable, what observables are measurable, etc.


The author has developed an importance sampling method to study real-time dynamics of quantum spin systems based on a disentanglement approach. This is a generalization of a similar scheme introduced earlier for imaginary time dynamics where unitary quantum dynamics is mapped to an ensemble of classical stochastic processes. The author shows that an importance sampling of trajectories close to the saddle point trajectory can significantly reduce the fluctuations in the measured expectation values of physical observables, compared to direct sampling approach.

This is an interesting work. The study of dynamics in quantum many body systems is a challenging problem. At the same time it is crucial in gaining deeper insight into quantum phases and phenomena as well as understanding several experiments. In this context, the current work is timely, relevant and will be of interest to many researchers in the field. There is no reason to doubt the correctness of the results. However, this referee feels that the manuscript can be significantly improved if the author addresses the following:

1. Can the author calculate the dynamic structure factor (DSF) using his newly developed approach? This will provide an important benchmark for the new method as the DSF for several key models are known from alternative approaches. It will also open up the new method to model inelastic neutron scattering experiments.

2. Can the author discuss whether the new method can be used to calculate observables such as the Berry curvature or fidelity?

3. The author states that the divergence in the Loschmidt ratio (LR) is related to the presence of a QCP in the ground state of the Ising model in a transverse field. Is it possible to establish a formal correspondence for the divergence of LR in any generic model and a corresponding QCP in the corresponding ground state phase diagram? Can critical exponents be extracted from the LR?

4. Is the current approach limited to weak interactions where a mean field / classical solution can be written down in terms of a single particle basis? Or is it applicable generically? A short comment on the range of applicability will be useful.

In addition to the above there are a few minor comments:

1. How are \phi(t) and \varphi(t) related?
2. Has the author studied the time evolution of a coherent state?
3. As the transverse field is reduced, the Hamiltonian approaches an exactly solvable limit. Is that connected to the accompanying suppression of fluctuations?
4. Why does the author choose to use only odd system sizes? Does he use periodic or open boundary conditions?
5. It would be better to mention the values of the Hamiltonian parameters used when discussing the results presented in a particular figure (right now, the complete set of parameters are given in the figure captions.
6. It would be more consistent to use either imaginary / real time or Euclidean / Lorentzian time – the former being possibly more readily understood by the broader community.

The referee feels that the manuscript contains important results that deserve to be published. Addressing the above comments will make it more impactful.

Requested changes

Please refer to the previous section.

  • validity: high
  • significance: high
  • originality: high
  • clarity: top
  • formatting: perfect
  • grammar: perfect

Anonymous Report 1 on 2021-5-12 (Invited Report)


This work deals with an algorithmic development for the stochastic simulation of the real-time dynamics of quantum spin systems. In recent years a numerical approach has been introduced (involving also the present author) where the dynamics generated by Schrödinger's equation is mapped onto a classical stochastic process. While this mapping is asymptotically exact upon taking into account sufficiently many classical noise trajectories, it has turned out that the initially introduced noise sampling scheme is facing a severe limitation in that the realization-to-realization fluctuations have appeared to grow exponentially both with system size and simulation time. In the end this has naturally led to strong constraints on this numerical approach.

In the present work the author introduces an importance sampling scheme which significantly reduces the resulting fluctuations by orders of magnitude. These advances are certainly important for the approach and significantly extend its regime of applicability. This is also demonstrated in the present manuscript where the short-time dynamics of Ising models on 2D lattices up to a size of 15x15=225 spins has been computed, which I consider a convincing achievement. As the author shows, the fluctuations still grow exponentially in size and time. Importantly, however, the importance sampling scheme allows to reduce the prefactor of the growth by several orders of magnitude. Although the approach therefore still faces its limitation due to an exponential growth of the required computational resources, this is in line with many other state-of-the-art numerical approaches to solve Schrödinger's equation, especially for 2D quantum matter. Consequently, even being able to access transient dynamics in this regime is an achievement, which certainly warrants publication in SciPost in my opinion.

The paper is nicely written. It provides a very good introduction to the field and the method. First the method is applied to a 2D Ising model by computing the magnetization dynamics and comparing the different sampling schemes. In a second step the author also considers the Loschmidt amplitude and identifies a dynamical quantum phase transition, signaled by temporal singular behavior in that quantity. In this context the emergence of this temporal critical behavior is set into context with an abrupt switching of the dominant contributing trajectories. This highlights another strength of the manuscript which at various instances tries to interpret the numerical solutions in terms of physical consequences.

In view of the current efforts towards accessing the real-time dynamics of 2D quantum matter both theoretically and experimentally, I consider the contributions of the present manuscript important. As a consequence I am convinced that the work deserves publishing in SciPost and I recommend publication in its present form.

I have just one final question for my own curiosity: is it possible to interpret the exponential growth of fluctuations as a kind of Lyapunov exponent of the underlying nonlinear classical differential equations?

  • validity: top
  • significance: high
  • originality: high
  • clarity: high
  • formatting: perfect
  • grammar: excellent

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