Clock Factorized Quantum Monte Carlo Method for Long-range Interacting Systems

1) Novel quantum Monte Carlo algorithm for quantum systems with long-range interaction with potentially improved performance.

1) Calculated quantity not well defined.
2) No physical results calculated.
3) Work and results are not well embedded in existing literature.

The article entitled "Clock Factorized Quantum Monte Carlo Method for Long-range Interacting Systems" by Fan, Zhang, and Deng describes an improved quantum Monte Carlo algorithm in the path-integral formalism to treat quantum systems with long-range interactions. This novel algorithm is called clock factorized quantum Monte Carlo and is adapted from classical to quantum systems by the authors. The latter are intensively studied in current research and are relevant for quantum-optical platforms and condensed matter systems. On the one hand, the described method sounds interesting (and flexible) and the article is well written. On the other hand, the algorithm is not well embedded in the current status of research and literature and the presented results are of very limited use. In fact, the present article does not contain any new physical result. In case the authors are able to resolve the issues mentioned, the article is in my opinion therefore better suited for SciPost Core.

Specifically, let me address the following major points:

1) In chapter 4, for all examples, the authors present numerical results of the new algorithm for a quantity which is called "Complexity". I think it would be important to define this quantity precisely in an equation and refer to this equation and quantity whenever possible.

2) In chapter 4, for all examples, it would be important to also include physical quantities like the ground-state energy or order parameter and compare the obtained results quantitatively with the known ones from literature. Current simulations are done at rather large temperatures. How do the simulations perform by a proper scaling of temperature and length scales of systems when extracting quantum critical properties?

3) I am surprised that the authors do not refer to stochastic series expansion quantum Monte Carlo pioneered in Sandvik, A.W. Stochastic series expansion method for quantum Ising models with arbitrary interactions. Phys. Rev. E 2003, 68, 056701, which is heavily used for quantum systems with long-range interactions. For more bibligraphy please also check the recent review Entropy 2024, 26(5), 401 on Monte-Carlo approaches to long-range interactions in quantum systems, which in particular coveres also the physics of the three examples discussed by the authors.

Minor points:

4) Lines 33-36: There are no references.
5) Line 299: "production form" - "product form"
6) Line 370: "1-p_j,rel)" -> "(1-p_j,rel)"
7) Line 480: Line too long.
8) Line 605: There are no references.
9) Line 619: Line too long.
10) Line 700: The Bose-Hubbard model is more relevant in quantum optics compared to condensed matter physics.

Accept in alternative Journal (see Report)

1) Novel quantum Monte Carlo approach to simulate quantum systems with long-range interactions
2) Flexible method which can be incorporated into existing codes
3) Detailed self-contained presentation and discussion of the method

1) Unclear what is new in the first 3 sections
2) No physics shown

In the manuscript "Clock Factorized Quantum Monte Carlo Method for Long-range Interacting Systems", the authors present a path integral quantum Monte Carlo (PIMC) approach to simulate quantum systems with long-range interactions. This is done by generalizing a previously introduced method for classical systems in [Phys. Rev. E 99 010105 (2019)]. The authors provide a detailed, clear, and self-contained discussion of their method. In particular, they further show the implementation for several quantum models, which comes with a generalization of the worm algorithm for long-range interactions.

Developing efficient and unbiased quantum Monte Carlo (QMC) methods for quantum systems with long-range interactions is of importance, and this manuscript provides a good contribution to this field. Further, the "clock-factorized method" can be incorporated into existing QMC codes. In my opinion, this is clearly interesting work and does meet the SciPost criterion "Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work".

Before recommending publishing it, there are, however, a few points I kindly ask the authors to address, which are listed in 1)-3) below.

1) In the abstract the authors mention that in addition to the PIMC generalization, this manuscript contains further technical improvements. To me, it is, however, not really clear what is new and what isn't in the first three sections. While I highly appreciate the self-contained nature of the discussion, it should be made more transparent in case this is largely a review of the approach presented in [Phys. Rev. E 99 010105 (2019)]. If there are some technical improvements, highlighting them more could also further strengthen the manuscript.

2) In terms of efficiency and complexity for the transverse field Ising model (TFIM) they compare their method to a "naive" PIMC implementation with complexity $\mathcal{O}(N^2)$. For the TFIM there already exist, however, efficient QMC methods with overall complexity $\mathcal{O}(N\log(N))$, which to my knowledge can also be extended to the XXZ model (see DOI 10.1103/PhysRevE.68.056701 and related works). In my opinion, this is a more relevant comparison for their method that should be discussed.

3) To my understanding of the method, there is essentially a tradeoff between a lower overall acceptance rate but (much) less computation time per Monte Carlo step. Further the authors argue, that the clock factorized algorithm does not change the dynamics of the algorithm and resort to only showing the complexity/runtime per update step. I think, however, it would still be beneficial to show the efficiency in terms of measuring some actual physical observable, either in terms of computation time to reach similar accuracy (for a comparison) or showing results for system sizes previously not reachable. Since the acceptance rate for the XXZ model algorithm is as low as around 5% this could further erase additional doubts about the efficiency.

Following are a few minor/optional questions and comments that the authors could consider for their revision.

4) The authors state that the overall "complexity" per step is given by $C\sim \mathcal{O}(\log(P_B)/\log(P_{\mathrm{fac}}))$, which to my understanding is the key part for the efficiency since in case where $P_B$ scales with $N$ the same way as $P_\mathrm{fac}$ a complexity of $\mathcal{O}(1)$ is achieved. Since $P_B$ depends on the choice of the hazard rates, my question is: When can the condition that $P_B$ scales with $N$ like $P_\mathrm{fac}$ be met? Is this always the case for extensive systems? A clarification in the manuscript might be beneficial and could provide more clarity for the QMC community when to use the method.

5) While it was clearly introduced, the term "complexity" used in the manuscript could still be a bit misleading as it only refers to the complexity per update step (of which $N$ are performed). Changing it to something like "complexity per step" (also in the figures) could thus be more appropriate.

6) The authors mentioned that the clock factorized method does not affect the general dynamics of the worm algorithm. Does the clock factorized approach affect the measurement of Green's functions? It might be beneficial to (shortly) comment on this in the manuscript.

7) I have to admit being a bit confused regarding the statements for frustrated systems in the discussion. Can the box technique here help reduce the QMC sign problem?


While the article is overall very well-written, here are some typos that caught my eye, which the authors may want to address. In the following "l." refers to line.

- several times "trails" is used instead of "trials"
- l.94: which "can" be considered as a generalization
- l. 377: efficently → efficiently
- l.509 Is this one sentence instead of two? (, one starts with)
- l.519 hardly occur (without s)
- l.789 allows one "to" determine

In several instances some article seems to be missing:
- l.192 "an" a priori
- l.391 "a" configuration independent
- l.706 "the" EBHM

Ask for minor revision