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Efficient construction of the Feynman-Vernon influence functional as matrix product states
by Chu Guo, Ruofan Chen
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Submission summary
| Authors (as registered SciPost users): | Ruofan Chen |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2402.14350v3 (pdf) |
| Date submitted: | 2024-06-24 05:17 |
| Submitted by: | Chen, Ruofan |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Computational |
Abstract
The time-evolving matrix product operator (TEMPO) method has become a very competitive numerical method for studying the real-time dynamics of quantum impurity problems. For small impurities, the most challenging calculation in TEMPO is to construct the matrix product state representation of the Feynman-Vernon influence functional. In this work we propose an efficient method for this task, which exploits the time-translationally invariant property of the influence functional. The required number of matrix product state multiplication in our method is almost independent of the total evolution time, as compared to the method originally used in TEMPO which requires a linearly scaling number of multiplications. The accuracy and efficiency of this method are demonstrated for the Toulouse model and the single impurity Anderson model.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
List of changes
1. Added some descriptive sentences according to the referee's comments
2. Added an appendix about multiplication of GMPS
3. Added some discussion about the relation between the Prony method and linear prediction method
Current status:
Reports on this Submission
Report #3 by Anonymous (Referee 3) on 2024-7-22 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2402.14350v3, delivered 2024-07-22, doi: 10.21468/SciPost.Report.9446
Strengths
1- Succinct and well written
2- Advances the influence functional approach to calculating the dynamics of open quantum systems by means of well-known tools from tensor networks
Weaknesses
1- The overall impact of the work is somewhat obscure
2- Small grammatical mistakes throughout the paper
Report
The authors make a case for how to simplify, both computationally and conceptually, the influence functional (IF) approach to quantum impurity problems, and in doing so suggests a unified language for constructing the influence functional in problems involving quadratic bosonic and fermionic environments. This is done by the introduction of the "time-translationally invariant IF" (TTI) method, in which the IF is built from matrix product states (MPS) in which the site tensors are the same, modulo boundaries. The authors show that the TTI has lower scaling runtimes compared to the partial IF as a function of total propagation time, and tests the method by calculating the current through single Anderson impurity coupled to two leads.
The authors' approach leverages well-known ideas of the tensor network community. Here, long range interactions (in time) are approximated using sums of exponentials which allow for simple matrix product operator (MPO) representations, and exponentials of matrix product operators are computed using W(I)/W(II) approaches. In this paper MPO-MPS contractions are replaced in favor of MPS-MPS multiplications.
The overal claim that the authors make is that their TTI method is computationally superior to their partial IF method. Overall, this seems like a somewhat weak result from several perspectives:
1- While this claim on scaling may true, it relies on the number of time steps $N$ being larger than the considered bond dimension $\chi$. This is due to the overall scaling of the partial IF method being $O(N^2 \chi^3)$ versus the overall scaling of the TTI method being $O(m N \chi^4)$, where $m$ is the number of times the MPS is "squared". In the same spirit as QUAPI or TEMPO, by making use of a truncation of $\Delta^{\zeta \zeta'}_{j,k}$ such that this is set to zero for $|j-k| > N_{trunc}$, would it not be possible to reduce the scaling of the partial IF method to $O(N_{trunc} N \chi^3)$? How much would this affect the accuracy of the dynamics? If the overall effect is not too large, then would the central claim of the paper need to be weakened?
2- The authors mention that there are other recent papers that construct the IF for fermionic impurity problems though without using the Grassmann MPS formalism. At the very least it would be useful to make some comparison of efficiency against those works to at least contextualise the authors' claim.
3- IF is not the only method for solving quantum impurity problems. How does the TTI method fare against a more straightforward approach, say, with using MPS to represent wavefunctions of a discretised bath?
There is a way in which this work seems incremental, even from the term "time-translationally invariant IF". This term suggests that the IF is considered as a truly time-translationally invariant object, and so one would expect it to be infinite in extent so that it can be represented as a uniform MPS. The benefits to the scaling of the method would become immediately apparent, as there would be no dependence on $N$. This is the approach taken in a paper cited by the authors, ref. 71 by Link et al. It is rather perplexing to construct this TTI method without taking full advantage of the stated time-translational invariance. That the authors have in the subsequent month or two posted a paper fully using this time-translationally invariant property only enhances this sense of incrementality.
In summary, this paper has merits that may be of interest to those working in open quantum systems. As written, however, the overall impact of this work is weakened as it only narrowly focuses on an influence functional approach, and for that matter, only on the authors' previous method. While this is not a sin per se, it would still behoove the authors to enhance the clarity of their paper by giving more details about their calculations.
Detailed remarks:
1- It would be useful to include a plot of how the required bond dimension to converge the calculation increases as a function of time.
2- What is the temperature used for the Toulouse model calculations? It should be the case that the retarded Green's function is temperature independent, but the IF and the difficult in constructing it should depend strongly on the temperature.
3- How practical is the TTI method in absolute terms, i.e. what is the runtime? If the Toulouse model example with $\chi = 50$ is any indication, constructing the IF for the SIAM with $\chi = 160$ will require 100x more time.
4- The paper mentions the Fishman-White algorithm, which constructs the Gaussian state of the IF by applying local gates to the vacuum state (a method to construct the partial IF). In this algorithm, the number of tensor network operations also scales roughly linearly with time. Does this imply that there is a way to construct the partial IF with similar efficiency scaling as the TTI method, even without leveraging the time translational invariance of the hybridization terms?
5- For all the simulations, please specify whether you are computing the IF up to time $t_{max}$ and extracting observable values for all $t <= t_{max}$, or whether you are computing observable values at time t using the IF defined to time $t$.
6- Regarding Figure 4, can the authors comment on why the earlier times (t=4) are less accurate than longer times when compared to exact diagonalization results?
7- Fig 3b at time $Dt/2 < 5$ shows that improving the prony fitting error does not necessarily lead to similar order of magnitude improvements in the dynamics. Can the authors comment on why this is the case for the Lorentzian coupling strength function as opposed to the semi-circular one? Is the latter easier due to the finite support over $\omega$? Does the error in the retarded Green's function reflect the deviation of the Prony fit?
Minor remarks:
1- In Section II.B, is the statement "Based on the ADT, one can easily calculate any multi-time correlations of the impurity" accurate? Following the definition of the ADT from QUAPI and TEMPO, the ADT should already factor in system propagations as well. That is, the ADT is entire integrand of eq (3), as opposed to the IF which is only the $I_{\sigma}$ part of the integrand. Once the system propagation tensors $K$ are included, is it possible to freely calculate any multi-time correlations?
2- Related to the previous point, the symbol $K$ in eq (3) is not defined in any of the text surrounding the equation.
3- It would be useful to include an additional appendix with the explicit expressions of the hybridization terms, as this would make the paper much more self-contained.
4- Is there a technical reason why the authors only consider the IF up to first order error in the time discretization?
5- In the sentence below eq (9), it seems to imply that the Prony algorithm finds the optimal parameters for the exponential fit. Is this actually true?
Requested changes
See report
Recommendation
Ask for major revision
Report #1 by Anonymous (Referee 1) on 2024-7-16 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2402.14350v3, delivered 2024-07-16, doi: 10.21468/SciPost.Report.9413
Report
I thank the referees for addressing most of my concerns.
The addition of Appendix A was very helpful, indeed
(see more detailed comments on this further below).
Indeed I missed that the approach eventually concerns the impurity only, with the bath fully integrated out. As an aside then, in Eq. (3) the a's and \tilde{a}'s are only introduced as `Grassmann trajectories'. It shall also be emphasized then that these refer to the impurity only, even though indirectly implied by the notation and context.
So the Grassmann tensors are MPS/MPOs of length N=t/dt. Please specify max(N) in the paper as used for the paper, even though one may imply it from the data. Similarly, please include the response to my question on reachable time scales in the paper itself, e.g., N=600 leading to L=8N Grassmann variables. Is L the actual length of the MPS then?
With respect to Fig. 1 then, MPS multiplication of GTs considers all i=1,..,N `sites' = time steps. This appears to imply that to reach some final time t, the whole time windows [0,t] needs to be addressed simultaneously. This appears in stark contrast to wave function based approaches, where one iteratively proceeds from time t -> t+dt. It would be useful to briefly address this in the paper.
The parameters for truncation like \chi are set by hand, and convergence is checked by comparing results. However, to have a sensible estimate of the numerical cost, the following two questions need to be addressed, assuming sufficiently large \chi:
1) how does the entanglement entropy of the target MPS
scale with N=t/dt for a fixed time window [0,t]?
(i.e., making dt smaller)
2) how does the entanglement entropy of the target MPS
scale with increasing t in [0,t] for fixed dt = t/N?
The argument of linear scaling of the cost with t relies crucially on the assumption that the entanglement entropy in (1) and (2) saturates, so that the same fixed \chi may be used. However, is this justified? The answer will depend sensitively on the answer to (1) and (2) above.
Detailed remarks on App. A:
Multiplication of Grassmann tensors (GTs)
In (A3), it appears the upper indices with \xi_n^{i_n} are actual powers, not contour indices as in the main text. Please make sure this is understood unambiguously.
Typo in line after (A5): x_1 -> \xi_1
The last paragraph in App. A is rather obscure.
It may be replaced by a discussion along the following:
It appears to me that (A7) can be compactly formulated as a rank-3 fusion tensor, say X, that can be simply contracted onto a pair of matching Grassmann variables in the product of two MPS/MPOs.
Furthermore, when having an MPS representation of a GT, this likely can make use of a Z_2 parity structure to count number of particles (GVs) mod2. Then all A-tensors along the MPS are parity preserving, such that sign-strings can be efficiently incorporated in terms of parity gates on the MPS bond next to the respective site.
If the above is correct, then it appears to me that the multiplication of the GTs represented as MPS should have a simple transparent pictorial tensor network representation that includes fusion tensors X and parity gates on MPS bonds.
Corollary to the above: did the authors exploit Z_2 symmetry in their MPS/MPO simulations, or did they use full tensors throughout? This is important for interpreting numerical costs. Please clarify in the paper itself.
Requested changes
see report above.
Recommendation
Ask for minor revision
Author: Ruofan Chen on 2024-08-09 [id 4682]
(in reply to Report 1 on 2024-07-16)
We thank the referee for the refereeing and comments. Our response to the comments are shown below.
The referee writes:
Indeed I missed that the approach eventually concerns the impurity only, with the bath fully integrated out. As an aside then, in Eq. (3) the a's and \tilde{a}'s are only introduced as `Grassmann trajectories'. It shall also be emphasized then that these refer to the impurity only, even though indirectly implied by the notation and context.
Our response: We thank the referee for the suggestion, and we have added a sentence after Eq. (3) that
It should be noted that this path integral formalism only contains the impurity GVs in the temporal domain.
The referee writes:
So the Grassmann tensors are MPS/MPOs of length N=t/dt. Please specify max(N) in the paper as used for the paper, even though one may imply it from the data. Similarly, please include the response to my question on reachable time scales in the paper itself, e.g., N=600 leading to L=8N Grassmann variables. Is L the actual length of the MPS then?
Our response: We thank the referee for the suggestion. Since in a time step there are 8 GVs (two spins, two branches, and the conjugate), the actual length of the MPS is indeed L=8N for the SIAM. We have added two sentences, one after Eq. (4)
Since there are 8 GVs within each time step, the total number of GVs is 8N
and one in the first paragraph of Sec.II B
In our implementation we represent each GV as one site, therefore the MPS representations of K and I all have 8N sites for the SIAM.
to specify this point. The max(N) used in all our simulations is N=1200 corresponding to Dt=120 with delta t=0.05, which has also been specified after Eq.(18).
The referee writes:
With respect to Fig. 1 then, MPS multiplication of GTs considers all i=1,..,N `sites' = time steps. This appears to imply that to reach some final time t, the whole time windows [0,t] needs to be addressed simultaneously. This appears in stark contrast to wave function based approaches, where one iteratively proceeds from time t -> t+dt. It would be useful to briefly address this in the paper.
Our response: We thank the referee for the suggestion. We have discussed this point after Eq. (7), which is attached in the following:
Here we can also see another stark difference between the GTEMPO and the wave-function based MPS methods that in GTEMPO the whole evolution time widow [0,t] is addressed simultaneously, while in the latter the evolution proceeds from time t to t + dt iteratively. Furthermore, in GTEMPO we first build the MPS-IF for the whole time interval [0, t], which is then used for calculating any multi-time impurity correlations within this time interval.
The referee writes:
The parameters for truncation like \chi are set by hand, and convergence is checked by comparing results. However, to have a sensible estimate of the numerical cost, the following two questions need to be addressed, assuming sufficiently large \chi:
1) how does the entanglement entropy of the target MPS scale with N=t/dt for a fixed time window [0,t]? (i.e., making dt smaller)
2) how does the entanglement entropy of the target MPS scale with increasing t in [0,t] for fixed dt = t/N?
The argument of linear scaling of the cost with t relies crucially on the assumption that the entanglement entropy in (1) and (2) saturates, so that the same fixed \chi may be used. However, is this justified? The answer will depend sensitively on the answer to (1) and (2) above.
Our response: We thank the referee for the comment. We have added discussions in the last paragraph of Sec.IIIA to justify the effectiveness of the MPS representation of the IF. As suggested by the referee, we have also added Fig.4 in the revised manuscript, which shows the scaling of the operator space entanglement entropy (OSEE, which is the bipartition entanglement entropy of density operators), as a function of t for two different values of dt. We can see that the OSEE indeed approximately saturates for large t, which we think can well answer question 2) of the referee. For question 1), we have considered two different values of dt in the two panels of Fig.4, and shown that the accuracy increases for smaller dt. However, we note that there is no known theoretical guarantee that the OSEE would saturate against smaller dt. To balance numerical stability and accuracy, in both GTEMPO and the wave-function based MPS methods, one often chooses a reasonable value of dt (not too small and not too large) in practice.
The referee writes:
In (A3), it appears the upper indices with \xi_n^{i_n} are actual powers, not contour indices as in the main text. Please make sure this is understood unambiguously.
Our response: We thank the referee for the suggestion. We have added a sentence after (A4) [now (B4)] to remove the unambiguity, which is attached as follows:
where the upper indices i_k,j_k are actual powers.
The referee writes:
Typo in line after (A5): x_1 -> \xi_1
Our response: We thank the referee for pointing out the typo, it has been corrected.
The referee writes:
The last paragraph in App. A is rather obscure. It may be replaced by a discussion along the following:
It appears to me that (A7) can be compactly formulated as a rank-3 fusion tensor, say X, that can be simply contracted onto a pair of matching Grassmann variables in the product of two MPS/MPOs.
Furthermore, when having an MPS representation of a GT, this likely can make use of a Z_2 parity structure to count number of particles (GVs) mod2. Then all A-tensors along the MPS are parity preserving, such that sign-strings can be efficiently incorporated in terms of parity gates on the MPS bond next to the respective site.
If the above is correct, then it appears to me that the multiplication of the GTs represented as MPS should have a simple transparent pictorial tensor network representation that includes fusion tensors X and parity gates on MPS bonds.
Corollary to the above: did the authors exploit Z_2 symmetry in their MPS/MPO simulations, or did they use full tensors throughout? This is important for interpreting numerical costs. Please clarify in the paper itself.
Our response: We thank the referee for the comment. We reply to the comments of the referee point by point in the following:
It appears to me that (A7) can be compactly formulated as a rank-3 fusion tensor, say X, that can be simply contracted onto a pair of matching Grassmann variables in the product of two MPS/MPOs.
(A7) [now (B7)] can be implemented as rank-3 tensor, but is not the usual fusion tensor. The two input spaces are of size 2, and the output space also has size 2. In comparison a usual fusion tensor should have output space size 4. Nevertheless, if we explicitly implement this operation as the contraction with a rank-3 tensor, the code could look more elegant.
Furthermore, when having an MPS representation of a GT, this likely can make use of a Z_2 parity structure to count number of particles (GVs) mod2. Then all A-tensors along the MPS are parity preserving, such that sign-strings can be efficiently incorporated in terms of parity gates on the MPS bond next to the respective site.
Yes, this is what we do in our actual implementation, which has been thoroughly discussed in Ref.[41].
If the above is correct, then it appears to me that the multiplication of the GTs represented as MPS should have a simple transparent pictorial tensor network representation that includes fusion tensors X and parity gates on MPS bonds
Yes, this can be done. But in our current implementation we have not used the rank-3 “fusion tensor” yet.
Corollary to the above: did the authors exploit Z_2 symmetry in their MPS/MPO simulations, or did they use full tensors throughout? This is important for interpreting numerical costs. Please clarify in the paper itself.
Yes, we have used the Z_2 symmetric tensor in our GMPS simulations, which can reduce the global sign changes into local ones.
Overall, we have stressed that in our actual implementation we used the Z_2 symmetric MPS. And we have also pointed to Ref.41 for interested readers.
Author: Ruofan Chen on 2024-08-09 [id 4683]
(in reply to Report 3 on 2024-07-22)We thank the referee for the refereeing and comments. Our response to the comments are shown below.
The referee writes:
Our response: We thank the referee for the comment. The memory truncation scheme is a key technique to reduce the computational scaling in QUAPI and TEMPO method. However, such a scheme is based on the manipulation of ADT which is obtained by multiplication of K and I. The bond dimension of the ADT, \chi_{A}, is then roughly the product of those of K, denoted as \chi_{K}, and I, denoted as \chi in the paper, which can easily be very large to overwhelm any computational gain. In the spin-boson model there is no problem as there is only a single flavor and \chi_{K}=2, but the problem will be serious in the fermionic case. Therefore in GTEMPO, we never compute the ADT explicitly but to integrate the ADT only on the fly with separate K and I using the zipup algorithm. In the meantime, the zipup algorithm is incompatible with the memory truncation scheme, therefore we opt to use the zipup algorithm. We have added appendix D to thoroughly discuss the memory truncation scheme and the zipup algorithm.
The referee writes:
Our response: The major performance advantage of GTEMPO over the tensor network IF methods is that the computational cost of GTEMPO is essentially independent on the number of baths, as GTEMPO is purely dependent on the Feynman-Vernon IF in which the baths are integrated out, while the cost of the tensor network IF method reported in Ref.48 scales exponentially with number of baths. We have added a sentence in the end of the first paragraph of Sec.III B to state this difference when studying the transport problem.
The referee writes:
Our response: The IF based method has already integrated out the bath degrees of freedom, thus it is free of bath discretization error compared to the wavefunction based method. We have discussed this issue in the end of third paragraph of the introduction. And the TTI IF method described proposed in this work aims to improve the construction efficiency of the MPS-IF.
The referee writes:
Our response: We thank the referee for noticing our subsequent work [Guo and Chen, Phys. Rev. B 110, 045106 (2024)], where the infinite GTEMPO (iGTEMPO) method is represented. The iGTEMPO method employs the infinite MPS technique which takes the full advantage of time-translational invariance and its cost is essentially independent on t. In the meantime, the work by Link et al. is also a very nice work which made use of the time translational invariance but in a completely different way (in the language of our work, this latter approach found a way to explore the TTI property based on the partial IF approach). It would be interesting to make a comparison between these two different approaches to build the time translationally invariant MPS-IF in the future.
In the meantime, we donot think this work is only incremental. The contribution of this work at least includes: (1) it lays down the central techniques used for building the full time translationally invariant MPS-IF used in our later work; (2) it compares in detail the performances of the partial IF and the TTI IF methods; and (3) most importantly, the transient dynamics is an important subject in its own right, for example it is a central ingredient in non-equilibrium DMFT, and the transient dynamics will be lost if infinite MPS is used. Therefore we think that speeding up the construction of the MPS-IF for the transient real-time dynamics itself merits an individual work.
The referee writes:
Our response: We thank the referee for the comment. The first referee raised a similar comment. In this work our strategy to compress the MPS is to use a fixed bond dimension \chi. Therefore a more meaningful quantity is the operator space entanglement entropy (OSEE) which characterizes the growth of the bipartition entanglement of the MPS-IF (which is similar to a density operator). We have added Fig.4 in the revised manuscript to show the growth of OSEE against the total evolution time t, from which we can see that indeed the OSEE approximately saturates for large t.
The referee writes:
Our response: We thank the referee for the comment. The inverse beta used is 20 which is now stated after Eq.(19). In principle the temperature would affect the memory length of the hybridization function, and thus affect the bond dimension of MPS-IF.
The referee writes:
Our response: For semi-circle spectrum, the runtime to build the MPS-IF for t=60, \delta t=0.05, and \chi=50 is about 0.54 hour, while the runtime to build the MPS-IF for t=84, \delta t=0.14, \chi=160 is 49.5 hours in our implementation, which is indeed much slower (90x times slower). This is also the reason why we stop at t = 84 for the transport problem.
The referee writes:
Our response: We thank the referee for this comment. The Fishman-White algorithm is intended for building Gaussian states, which is used in the tensor network IF algorithm. In our case we work with coherent state in terms of Grassmann variables. We believe this algorithm can still be used but it is to be explored. In principle, this algorithm, if it can be applied, can be as efficient as the TTI-IF algorithm, which is something that may be explored in future studies. However, the Fishman-White algorithm breaks the TTI property, thus can not be directly generalized to work with infinite MPSs.
The referee writes:
Our response: We thank for the referee for the comment. We compute the IF up to the time t_max and extract observables for all t<=t_max. We have now added a last sentence in the end of Sec II.B to state this point.
The referee writes:
Our response: We thank for the referee for the comment. This is a fact that we have observed in all our numerical simulations for the transient dynamics including our other works (which is opposite to the wave-function based MPS methods), but we donot fully understand it yet. In principle the MPS-IF is similar to a spatially one-dimensional many-body quantum state, for which the error should spread out in the whole time window, and the errors in different locations are expected to be similar. We guess this may be due to the first-order discretization scheme we used, which may be more affected by the initial condition.
The referee writes:
Our response: We thank for the referee for the comment. As mentioned in the previous response, the error is spread out in the whole time window. Thus improving the prony fitting would reduce the total error of the whole time window. While the behavior of the distribution of the local errors, similar to our previous answer, is still not well understood.
The referee writes:
Our response: We thank for the referee for the comment. Yes, this statement is accurate. In TEMPO, K and I are tensors of system quantum numbers $s$, and when K is already included in the ADT it would be not possible to calculate the expectation of non-diagonal system operators (essentially, this is because that the ADT in TEMPO for the spin-boson model is only a simplified version of the full process tensor, see PRA 97. 012127 for definition of process tensor, due to the special form of coupling between impurity and bath). However, in GTEMPO, both $\bar{a}$ and $a$ are kept in track in the ADT (e. g., the ADT is the full process tensor), which enables us to calculate any correlations involving $\hat{a}^{\dagger}$ and $\hat{a}$, see Eq.(10) in Ref. 41 as an example.
In addition, in TEMPO we can also keep tracks of $s$ and its conjugate, where the conjugate of $s$ is just itself so it is usually merged to $s$ in the formalism. Then we can also calculate any correlations from the ADT with the cost that the ranks of K and I are doubled.
The referee writes:
Our response: We thank for the referee for the comment. We have added the definition after Eq. (3).
The referee writes:
Our response: We thank for the referee for the comment, and we have added App. D to give corresponding explicit expressions.
The referee writes:
Our response: Yes. With the same bond dimension, we found that the accuracy of the first and second order methods eventually become similar, as shown in Ref. 41. However, compared to the second order method, the first order method is preferred since it is much easier to implement.
The referee writes:
Our response: The more accurate statement is: the Prony algorithm finds the the optimal parameters to fit a given function with the sum of n exponential functions. In addition, as the lambdas can be complex, those exponential can oscillate and in principle any function can be approximated by such fit, given a large enough n.