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Krylov complexity in a natural basis for the Schrödinger algebra
by Dimitrios Patramanis, Watse Sybesma
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Authors (as registered SciPost users):  Watse Sybesma 
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Preprint Link:  https://arxiv.org/abs/2306.03133v2 (pdf) 
Date submitted:  20230704 11:05 
Submitted by:  Sybesma, Watse 
Submitted to:  SciPost Physics 
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Academic field:  Physics 
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Approach:  Theoretical 
Abstract
We investigate operator growth in quantum systems with twodimensional Schr\"odinger group symmetry by studying the Krylov complexity. While feasible for semisimple Lie algebras, cases such as the Schr\"odinger algebra which is characterized by a semidirect sum structure are complicated. We propose to compute Krylov complexity for this algebra in a natural orthonormal basis, which produces a pentadiagonal structure of the time evolution operator, contrasting the usual tridiagonal Lanczos algorithm outcome. The resulting complexity behaves as expected. We advocate that this approach can provide insights to other nonsemisimple algebras.
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Anonymous Report 1 on 2023822 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2306.03133v2, delivered 20230822, doi: 10.21468/SciPost.Report.7691
Report
This work suggests using a different construction for the Krylov basis for a specific case where the Liouvillian can be described as a general element in the 2dimensional Schrödinger (Lie) algebra. The authors show that the resulting "Krylov complexity" is greater than the sum of the Krylov complexity of the two subalgebras which make up the Schrödinger algebra via a semidirect sum. The Liouvillian the authors propose to use is pentadiagonal instead of tridiagonal as a result of using an algebraadapted orthonormal basis as the Krylov basis. The algebraic structure makes the solution for the wavefunctions $\phi_k(t)$ possible to perform analytically, using coherent states, and from them the associated "Kcomplexity" is computed.
I find the work interesting, but I have several comments which I believe the authors should address.
General comments:
1. The original notion of Kcomplexity as defined and studied in [1, 39] is general for any hermitian Hamiltonian and operator, and it does not seem to need a redefinition. In particular in [39, 38] it was shown to satisfy the general behaviour expected from complexity of operators at all time scales, including saturation at time scales exponential in the number of DoF.
2. It is not clear to me how this approach could be generalized to other systems where the abovementioned algebra cannot be used as a guideline. This is also related to the first point: could it be shown that such a definition follows the general behaviour of complexity, or otherwise, and what properties will it have?
3. Did the authors try probing the usual Krylov basis and complexity for the system they study? Even numerically? For example via the autocorrelation function (55).
It would be appropriate if the authors mention these points and discuss them further in their work.
More detailed comments:
1. Below equation (4) the authors say that the tridiagonal structure of the Liouvillian is a result of the properties of the innerproduct. In fact, as the authors can check, to have zeros on the diagonal the operator should also be hermitian.
2. I believe the authors should cite the work "Building Krylov complexity from circuit complexity" by Lv, Zhang and Zhou in relation to the use of coherent states to study Krylov complexity as well as in the discussion of the relation of Kcomplexity to other notions of complexity.
3. It would be good if the authors provide more details on the "Krylov" basis they construct and not only on the wavefunctions.
4. Unlike the statement the authors made about reference [47] by Rabinovici et al, [47] has indeed established a rigorous bulkboundary holographic duality between Kcomplexity and a geometric bulk quantity, in the particular setup of 2dimensional JT gravity.
Author: Watse Sybesma on 20231009 [id 4033]
(in reply to Report 1 on 20230822)We thank the reviewer for their insightful comments, which we address below individually (we've added a modified draft to reflect changes accordingly):
Comment by referee:
"The original notion of Kcomplexity as defined and studied in [1, 39] is general for any hermitian Hamiltonian and operator, and it does not seem to need a redefinition. In particular in [39, 38] it was shown to satisfy the general behaviour expected from complexity of operators at all time scales, including saturation at time scales exponential in the number of DoF. "
Response:
The aim of our work is not to redefine Krylov complexity as a general concept. Indeed it has a perfectly good definition which can, in a number of cases, capture the essential features of operator spreading at all time scales. Our aim is to provide an alternative yet closely related measure which can still be analytically computed in cases when the implementation of the Lanczos algorithm is not tractable. Given that this measure is still probing the Krylov subspace defined by the Liouvillian, we believe it is accurate to use the term Krylov complexity with the additional remark that it is computed in what we dub as the “natural basis”, as we explain both in the introduction and in the discussion section.
Comment by referee:
"It is not clear to me how this approach could be generalized to other systems where the abovementioned algebra cannot be used as a guideline. This is also related to the first point: could it be shown that such a definition follows the general behaviour of complexity, or otherwise, and what properties will it have? "
Response:
Generally more complicated symmetries are not always expressible in terms of the raising and lowering operators which we are indeed using as a guideline in our work. However, the approach itself which consists of constructing a Liouvillian (or Hamiltonian for state complexity) out of the algebra generators and considering its action on a given state can be implemented for arbitrary Lie groups. Of course one might expect that this will lead to more complicated constructions, for example more elaborate dynamics on the Krylov chain or more involved forms of coherent states, but we still believe that it can serve as a useful tool that provides us with a handle for analytic calculations in cases which this is otherwise not possible. In terms of the properties of the complexity we are computing, we have shown that in certain limits and regimes its behaviour matches what we expect from Krylov complexity. For a rigorous understanding of the relationship between the actual Krylov basis and the “natural basis”, it is necessary to have access to the formal so it is not possible to check directly, but the checks that we have performed hint that the time dependence of complexity should be the same in both cases up to a multiplicative constant and so the essential features of operator growth should be the same. However, in general our approach would apply to any nonsemisimple sum of Lie algebra’s that can be expressed in a common basis.
Comment by referee:
"Did the authors try probing the usual Krylov basis and complexity for the system they study? Even numerically? For example via the autocorrelation function (55)"
Response:
As we mention in the paper this has been the subject of reference [36] in which the authors provide an approximation scheme which numerically computes the Krylov basis and complexity. We compare our results to theirs in the discussion section. The autocorrelation function (55) can be used to compute the moments and Lanczos coefficients, which however appear to be problematic since they become complex. This is not surprising since the autocorrelation function is not Fourier transformable because it is doubly exponential and so obtaining the power spectrum which is directly related to the Lanczos coefficients would require a much more careful treatment. Potentially a more refined numerical approach could resolve the issue, but this is beyond the scope of our work.
Comment by referee: "Below equation (4) the authors say that the tridiagonal structure of the Liouvillian is a result of the properties of the innerproduct. In fact, as the authors can check, to have zeros on the diagonal the operator should also be hermitian."
Response:
We corrected the statement as suggested.
Comment by referee:
"I believe the authors should cite the work "Building Krylov complexity from circuit complexity" by Lv, Zhang and Zhou in relation to the use of coherent states to study Krylov complexity as well as in the discussion of the relation of Kcomplexity to other notions of complexity."
Response:
We added the reference as suggested.
Comment by referee:
"It would be good if the authors provide more details on the "Krylov" basis they construct and not only on the wavefunctions."
Response:
The basis that we use is simply the Fock basis on which the Liouvillian acts naturally. In other words we exploit the fact that the algebra of raising and lowering operators is associated with an automatically orthonormal number basis which we can use as a probe for the Krylov subspace. Part of the appeal of our approach is the fact that we do not have to construct the basis, but it is rather automatically provided to us by the algebra.
Comment by referee:
"Unlike the statement the authors made about reference [47] by Rabinovici et al, [47] has indeed established a rigorous bulkboundary holographic duality between Kcomplexity and a geometric bulk quantity, in the particular setup of 2dimensional JT gravity."
Response:
It is true that Rabinovici et al establish a rigorous correspondence between the length eigenstates in the bulk and the Krylov basis in the triple scaling limit of SYK. However, this does not constitute a general entry into the holographic dictionary, although one would hope that I can be generalized to setups in higher dimensions. For example, already for asymptotically AdS2 models beyond JT it would be worthwhile to check this entry into the holographic dictionary and establish this duality even further. Regardless, we have altered the statement in the paper to eliminate any ambiguity.
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