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A numerical approach for calculating exact non-adiabatic terms in quantum dynamics
by Ewen D C Lawrence, Sebastian F J Schmid, Ieva Čepaitė, Peter Kirton, Callum W Duncan
Submission summary
| Authors (as registered SciPost users): | Peter Kirton · Ewen Lawrence |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2401.10985v1 (pdf) |
| Date submitted: | 2024-01-30 17:36 |
| Submitted by: | Lawrence, Ewen |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
Understanding how non-adiabatic terms affect quantum dynamics is fundamental to improving various protocols for quantum technologies. We present a novel approach to computing the Adiabatic Gauge Potential (AGP), which gives information on the non-adiabatic terms that arise from time dependence in the Hamiltonian. Our approach uses commutators of the Hamiltonian to build up an appropriate basis of the AGP, which can be easily truncated to give an approximate form when the exact result is intractable. We use this approach to study the AGP obtained for the transverse field Ising model on a variety of graphs, showing how the different underlying graph structures can give rise to very different scaling for the number of terms required in the AGP.
Current status:
Reports on this Submission
Strengths
1. The authors provide sufficient details of their approach.
2. The authors apply their method to the interesting system of Ising model on graphs.
Weaknesses
1. The authors do not properly review existing methods that are very closely related to the method they introduce.
2. The application of their method to more general systems is not discussed.
Report
The manuscript is appropriately prepared and the methods are explored properly.
The main advantage of their method over other methods of a similar nature are unclear. The manuscript will be suitable for the Journal once the authors have considered the changes requested.
Requested changes
I request the authors to address the following question in their manuscript:
Q 1: It is known that the choice of any orthonormal basis (Controlling and exploring quantum systems by algebraic expression of adiabatic gauge potential, PhysRevA.103.012220) will minimize the action. Depending on the problem, some bases are analytically tractable while others are not. The authors employ the Pauli basis for their computation. What are the advantages (in general) in using that basis as compared to say, the Krylov basis?
For context, the Krylov basis approach was developed for the AGP in
1. A Lanczos approach to the Adiabatic Gauge Potential (arXiv:2302.07228)
2. Shortcuts to adiabaticity in krylov space (arXiv:2302.05460)
and applied to a large class of problems. It is also known that the Krylov basis in general has lesser number of elements than the Pauli basis. Therefore it seems as if expressing the AGP in the Krylov basis is advantageous over expressing the same in the Pauli basis.
Q 2: The authors say a few words about the presence of a natural truncation point in one of their examples (due to their method). It was shown (arXiv:2302.07228) that the truncation of the Krylov chain connects to the chaoticity of the system.
Can the authors comment on the generic nature of the truncation point in their method vis-à-vis the Krylov method?
Strengths
The manuscript is clearly organized and well written.
Providing a numerical algorithm helps people who want to use counterdiabatic driving.
Weaknesses
Simple examples (the ferromagnetic Ising model with a transverse field on some graphs) are only studied.
Source code is not available.
Report
In this manuscript, the authors provide a numerical algorithm for calculating the adiabatic gauge potential. The adiabatic gauge potential is a key idea of nonadiabatic transitions and it can also be used in assisted adiabatic passage, or in other words, shortcuts to adiabaticity by counterdiabatic driving. Their algorithm is based on some previous results, i.e., the variational approach (Ref.25), the algebraic approach (Ref.40), and an idea of nested commutators (Ref.48). They use the above results in an algorithmic way. I believe that the present manuscript is interesting, clearly organized, and well written, and thus it deserves to be published in SciPost Physics.
Requested changes
Optional comments:
1. n->N in Eq.24.
2. The model in Sec.4.1 is the transverse Ising chain with the periodic boundary condition, and the adiabatic gauge potential of it is well studied in the literature. The authors should cite some reference there.
3. The absence of a peak in the adiabatic gauge potential of the Lipkin-Meshkov-Glick model sounds very interesting because we have believed that it shows a significant peak at the critical point. I recommend emphasizing this finding.
4. The present paper is useful for people who want to use counterdiabatic driving. It becomes more useful if the source code (or package code) is available.