SciPost Submission Page

Numerical evaluation of two-time correlation functions in open quantum systems with matrix product state methods: a comparison

by Stefan Wolff, Ameneh Sheikhan, Corinna Kollath

Submission summary

As Contributors: Ameneh Sheikhan
Arxiv Link: (pdf)
Date submitted: 2020-04-03
Submitted by: Sheikhan, Ameneh
Submitted to: SciPost Physics Core
Discipline: Physics
Subject area: Condensed Matter Physics - Computational
Approach: Computational


We compare the efficiency of different matrix product state (MPS) based methods for the calculation of two-time correlation functions in open quantum systems. The methods are the purification approach [1] and two approaches [2,3] based on the Monte-Carlo wave function (MCWF) sampling of stochastic quantum trajectories using MPS techniques. We consider a XXZ spin chain either exposed to dephasing noise or to a dissipative local spin flip. We find that the preference for one of the approaches in terms of numerical efficiency depends strongly on the specific form of dissipation.

Current status:
Editor-in-charge assigned

Reports on this Submission

Report 2 by Matteo Rizzi on 2020-5-14 Invited Report


1- The manuscript presents a complete, self-contained guide about three different methods to tackle an highly-relevant task, namely the study of open quantum systems dynamics.

2- The three methods are carefully compared in terms of accuracy and computational performances, in order to convey the message that the method of choice strongly depends on the model investigated. Guiding criteria to generalise the choice beyond the specific example are provided.

3- The written text is nicely complemented by informative plots and tables of high readability.


1- Some important references appear to be missing here and there (see Report): nothing too fatal but it should be amended.

2- There are some technical aspects that to be clarified, mainly about the exploitation of symmetries and the performed benchmarks: e.g., changing two relevant aspects at the same time, Lindblad operator and initial state, leaves some questions open to my eyes.

3- The large distance between the model description (Sec.2, pages 3-4) and the discussion of the method comparison (Sec.5, pages 17 -on) seems suboptimal to me. It is probably matter of taste, but I would have found more reasonable to have the general descriptions of the methods (Sec. 3-4) earlier, since they are actually almost independent of the chosen model.
Some parts could have as well be moved into appendices.


I have no doubt about the overall quality of the performed studies and the aim of the manuscript, as visible from the list of “Strengths” above. The manuscript could be publishable almost in its present form, except for a couple of extra references to be integrated into it.
Nonetheless, I would invite the Authors to consider the “Remarks” listed here below (and hinted to in the “Weaknesses”) as primarily aimed at further improving the impact of the manuscript as a (maybe "the") reference guide for people working in this relevant field.


(important) Alongside with Ref. [30], I would have expected to see the equally founding R. Dum, P. Zoller, and H. Ritsch, Phys. Rev. A 45, 4879 (1992). By the way, why is [30] not cited together with [3]?

(others) Here some references about tensor-network methods for open quantum systems, which I was quite surprised not to find cited in the overview section of the present manuscript — they are not necessarily the only ones, and should be intended as a suggestion for a more exhaustive collection of references:
T. Prosen, and M. Znidaric, Phys. Rev. E 75 015202(R) (2007);
M.J. Hartmann, et al. Phys. Rev. Lett. 102 057202 (2009);
A.H. Werner, et al., Phys. Rev. Lett. 116, 237201 (2016).

(minor) In the context of MonteCarlo-MPS methods, I would have said that Ref. 57 by A. Daley would have deserved a more prominent role, not? Maybe also some specific papers by him and co-authors (around 2010 in Zoller’s group), though a direct citation to the latters might probably be circumvented by a “… and references therein” in Ref. 57.

Technical Remarks

a) About methods for computing efficiently the time-evolution of purified states (Sec. 4.1), I wonder whether the methods introduced for thermal ensembles in the following works might be of any help here for Lindbladian dynamics, too:
C. Karrasch, J. H. Bardarson, and J. E. Moore, Phys. Rev. Lett. 108, 227206 (2012) & New J Phys 15, 083031 (2013);
T. Barthel, New J. Phys. 15 073010 (2013) & related works;
I. Pizorn, et al., New J Phys 16, 073007 (2014) & related works.

b) About purification in presence of conserved quantities (Sec. 3.2.4), I wonder whether the following works might be of relevance for the Authors’ sakes here (though they also mainly deal with thermal canonical ensembles only, not open dynamics with jumps…):
A. Nocera, G. Alvarez, Phys. Rev. B 93, 045137 (2016)
T. Barthel, Phys. Rev. B 94, 115157 (2016)

c) Incidentally, why do the Authors not introduce directly the convenient trick of Eq.(25) already at the level of representing the initial state (Sec. 3.2.2), since they claim that it brings advantages also in that task? This would help to stream-line the reading for the non-insider reader, similarly to what is nicely achieved in almost the whole rest of the manuscript.

d) Moreover, does the same trick not allow for the conservation of the total magnetization in the transformed purified state, $| \rho \rangle\rangle$? I mean that the new $\mathbb{D}_2^{\prime} \equiv \mathbb{U}^\dagger \mathbb{D}_2 \mathbb{U}$ should then read $\mathbb{D}_2^{\prime} = \gamma \left(S_c^- \otimes S_c^+ - \frac{1}{2} \left(\frac{\mathbb{I}}{2} + S^z\right) \otimes \mathbb{I} - \frac{1}{2} \mathbb{I} \otimes \left(\frac{\mathbb{I}}{2} - S^z\right) \right)$, if I am not terribly mistaken, in which case I apologize. Does it not bring at least some computational advantage, though not as much as a separate conservation of $S^z$ in both bra and ket spaces? If this has been already exploited, could the Authors convey the information in a more accessible way?

e) The benchmark for $\mathcal{D}_1$ are performed starting from a classical product state, while the ones for $\mathcal{D}_2$ from the actual, very entangled, quantum ground-state in a gapless regime.
I would be curious to know what the performances of the two (three) approaches would be for $D_1$ starting from the latter initial state. Is there a line of reasoning to spare the actual calculation and already draw conclusions from the present data?
Sorry in advance, if the questions sounds too naive to insiders.
Incidentally, among the MonteCarlo variants, is the Breuer et al. method guessed to be always better than the Mølmer et al. method, in whatever situation? Or can one at least speculate a situation (no need to prove it here and now) in which the latter would be more convenient?

f) The comparison is limited to two-point observables of a very special kind, namely $\langle S^z_{c}(t_1) S^z_{c+d}(t_2) \rangle$, where both operators are Hermitian and commute with each other. Equally interesting would be to assess performances for the computation of $\langle S^+_{c}(t_1) S^-_{c+d}(t_2) \rangle$ (and maybe their fermionic counterpart with Jordan-Wigner string in the middle).
It is fully clear to me that this would cause the loss of some important simplifications here and there, and probably result into a more expensive set of runs: nonetheless, I really think that this would considerably enlarge the content, and therefore the relevance, of this manuscript for people working (or wishing to enter) in this relevant research field.
If a clear statement on these observables could be already provided based on present data only, instead, this should be given a prominent role in the discussion section (without need for burning further computational time).

Other minor comments

A) Punctuation should be double checked again.
E.g., at page 4, "This section is structured such, that we start by giving a ..." would need no comma, in my humble opinion. Several similar examples are disseminated in the text.

B) In describing the properties of $\mathcal{D}_1$, it is stated that the jump operators being Hermitian implies that the infinite temperature state is the unique steady state of the model.
Am I right in saying that, more generally, one simply needs that the jump operators are normal (i.e., $[L,L^\dagger]=0$) to ensure the $T=\infty$ state to be a steady state, $D(\mathbb{I}) = 0$?
What about the unicity? Can the Authors sketch a proof for it?

C) While describing the general formalism of MPS and their (in principle, exact) construction via iterative SVDs, would the resulting Eq. (7) not be a MPS in the canonical $(\Gamma, \lambda)$ form? Since the aim of the explanation is clearly a didactical one for newcomers, I would not skip logical steps :-)

D) The placement of some figures (e.g., 2-3, but also others) is a bit unfortunate with respect to where they are referenced to in the main text. If possible, this should be amended.

E) Should the compression of the purified state (around Eq. 18) be performed while sweeping and keeping the canonical form with the target site as the orthogonal centre of the gauge? A brief comment would be of help for the reader.

F) Actually, another family of approaches exist on the market, namely the one based on matrix-product density-operators (MPDO), e.g., Ref. 27 & 56 (but not only): a comment about the advantage/disadvantage of the purification approach with respect to a direct MPDO approach would be nice to have here.

G) In Fig. 5, did something went wrong with the formatting? It is difficult to believe that the statistical error bar is not visible on this scale, not? Or am I dumbly mislead?

H) All comparisons are performed for extremely short distances in the correlation functions (d=0 or 1): it could be interesting to know, at least speculatively, whether and how larger d's would affect the statements drawn from the benchmarks.

I) In Fig.. 13, why are some data bouncing in a non-monotonic direction at large bond-dimensions? Simply lack of statistics (larger samples $R$ seem to be less affected), or is there something more subtle?

L) In the error estimate of Eq.(49) I would have naively thought to some quadrature sum, instead of a max(...). This would certainly not change the comparison outcome for $\mathcal{D}_1$, but what about $\mathcal{D}_2$ (let's forget for a moment about run-time)?

M) In Fig. 17, it could be useful for newcomers to plot the same error estimate for the pure-state representation alone (errors should be considerably smaller for the given $L$ and $D$'s): this would give immediate reason for the chosen D=500 for the MCWF method as a reference...

N) In Table 1 it is mentioned that "the MCWF simulation was executed in parallel on 10 cores": does it mean that the actual CPU-time would be 10 times larger, making this comparable to the purification one (845 vs 830 hours, if so)? I.e., please indicate whether the quoted run-time is CPU or wall-clock :-)

O) In Sec. 4.2.2 it is strange to read "An alternative strategy, established by Mølmer et al. [3], uses the quantum regression theorem [31]", since it sounds like an article of 1992 using a theorem of 2003...

  • validity: high
  • significance: good
  • originality: good
  • clarity: high
  • formatting: good
  • grammar: good

Anonymous Report 1 on 2020-5-7 Invited Report


1. Comparison of three different approaches to calculate the two-time correlation function for open systems are discussed.
2. Two different forms of dissipation are investigated.
3. A self containing introduction into the used approaches is given.
4. A method to purify a given MPS is given.


1. Not all approaches are investigated for both forms of dissipation.
2. Notation is inconsistent and several objects are not explicitly defined.
3. Figures and captions are sometimes not fully syncronized.


In "Numerical evaluation of two-time correlation functions in open quantum systems with matrix product state methods: a comparison" the authors compare three approaches (one based on the purification approach and two based on Monte-Carlo wave functions (MCWF)) to calculate two-time correlation functions in open quantum systems. Some approaches are applied to two different dissipation setups.

The manuscript delivers a good, self contained, in-depth overview of the approaches in question and the goal of having a comprehensive comparison is certainly interesting for everyone who wants to analyze open quantum systems with MPS.

Nevertheless, there are several open questions, some recommendations, and some requested changes that need to be answered prior to a possible publication. The question and the recommendation are listed directly below and the requested changes are listed in the separated part. Every list is ordered by the appearance within the manuscript, not by relevance.


1. The upper half of page 10 seems to be already discussed in Ref. 47. While it is certainly good practice to repeat important technical details, it is maybe helpful to stress - in this case - that it is done in order to explain the transformation of the Lindbladian.

2. In Fig. 18 the authors show the deviation of several purification approach calculations from the MCWF data and indicate the standard deviation of the Monte-Carlo average as a guidance for the error of the MCWF. In Fig. 13 the authors showed that the statistical error is much smaller than the truncation error for the bond dimension used in Fig. 18. Despite the fact that different correlation functions (equal-time vs. two-time) were measured, it is not clear why the truncation error seems to be of no interest in this comparison.

3. A closer look at Table 1 shows that the purification approach mainly takes longer because of the lack of symmetries and/or the (principled) lack of parallelism. If compared "fairer" (both approaches with a single core) the purification might even be faster. Especially because it might even be converged with a bond dimension of 600. Is that assessment correct and if so, why was this representation chosen instead?


1. It would be helpful to explain why it is fairer to compare a special case instead of the more general one.

2. As the approach by Mølmer et. al. builds up entanglement faster than the approach by Breuer et. al. it might be interesting to see how both approaches handle non exact MPS.

3. It would be interesting how the approach by Mølmer et al. compares to the approach by Breuer et al. in the case of the second dissipation setup. The authors assumably have done such calculations and it would be good to see something like "The approach by Breuer et al. shows in this setup similar behavior as in the previous setup...", if that turns out to be the case. Otherwise a more detailed explanation for not including this approach in the comparison might be necessary.

Requested changes

1. Please carefully check that references actually contain the claimed content. In particular reference 1 is referenced multiple times, but the promised informations seem not to be within the referenced paper (e.g., purification approach; two-time correlation functions (for spin-1/2 chain with dissipator $D_1$); a prove that two-time correlation functions are essential to uncover interesting dynamical regimes that display physical phenomena such as aging or hierarchical dynamics; the required time step to obtain good accuracy.).

2. Specify $t_1$ in the two-time correlation functions and if somehow possible include (maybe as additional material) some absolute data in order to establish reproducibility.

3. Check all lists of indices and alike whether or not commas are necessary/wanted. In the last line of Eq. 14 a mixture is used. Otherwise explain the chosen notation.

4. The sentence right above Eq. 5 seems to be incomplete.

5. Consider adding "Time-evolution methods for matrix-product states" ( to the list of citations concerning time-dependent matrix product states [47-50].

6. The $\hbar$ on the left hand site are missing in Eq. 12. Also the system size is usually not a part of the Trotter error, it should be removed or its origin and importance should be explained.

7. Is Fig. 2/Eq. 14 a figure or an equation? It should probably not end with a full stop.

8. Consider adding some original thermofield papers (pre MPS) to the list of citations concerning the doubled Hilbert space [28,47,53].

9. Please define the new $L$ and $D$ in Eq. 16 explicitly.

10. Please explain the choice for the black dashed and dotted lines in the caption of Fig. 4. What did you use as maximal accepted probability in the following examples?

11. The error bars in Fig. 5 are not resolvable and hence not visibly time dependent. Please consider either enhancing the scale of the error bars or removing them and state that the statistical errors are within the line width.

12. In the last sentence of point two on page 15 it should state "Here the norm of the new state is the same as the initial state".

13. In the last equation of point 4 on page 15 a ")" is missing on the right hand site.

14. The dashed lines in Fig. 9 are barely recognizable as dashed. Please use a different type of dashing or another way to seperate the maximum values from the averages. And mention them and their relevance in the caption.

15. Please elaborate what exactly can be learned from Fig. 19. As there is (apparently) no way for a comparison, can the results be discussed qualitatively?

16. Check reference 25 "[...] M. u. u. u. u. Guţă [...]"

  • validity: high
  • significance: good
  • originality: ok
  • clarity: good
  • formatting: reasonable
  • grammar: good

Login to report or comment