SciPost Submission Page
Influence of Trotterization error on single-particle tunneling
by Anton V. Khvalyuk, Kostyantyn Kechedzhi, Vadim S. Smelyansky, Lev B. Ioffe
Submission summary
| Authors (as registered SciPost users): | Anton Khvalyuk |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2312.04735v2 (pdf) |
| Date submitted: | 2024-02-26 14:33 |
| Submitted by: | Khvalyuk, Anton |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
|
| Approach: | Theoretical |
Abstract
Simulation of the single-particle tunneling problem by means of the Suzuki-Trotter approximation (STA) is analyzed. Considered is a particle hopping across a chain of sites in presence of a smooth position-dependent potential profile with several local minima that arrange a tunneling problem between the localized states in different minima. The STA error is found to manifest itself in three ways: i) perturbative energy shifts, ii) nonperturbartive renormalization of the tunneling rates, and iii) perturbative leakage of the total probability to other states. Generally, the first type of error is the most essential, as detuning of the tunneling resonance has to be compared with exponentially small tunneling rates. In absence of detuning (e.g. if the resonance is protected by symmetry), STA leads to exponential enhancement of the tunneling rates. The last type of error classifies the overall defect in the wave function and delineates the region of sufficiently weak distortion of the wave function due to STA. The conducted analysis confirms the naive criteria of applicability $\max\{T,P\}\ll\delta t^{-1}$ (with $T,P$ being the typical scales of kinetic and potential terms, respectively), while also revealing the structure of error and its behavior with system parameters. Analysis of the case of large Trotter step is also performed, with the main result being the reconstruction of low-energy spectrum due to coupling between states with energy difference close to $2\pi/\delta t$. The connection of the obtained results with rigorous upper error bounds on the STA error is discussed, with particular emphasis on why these rigorous bounds are not always saturated. We also point out that the proposed problem can be directly implemented on existing quantum devices [arXiv:2012.00921]. In particular, we give a detailed description of an experimental design that demonstrates the described physics.
Current status:
Reports on this Submission
Report
The authors analyze errors in simulations of quantum dynamics on a quantum computer due to time discretization and the use of the Trotter approximation, which replaces the exact Schroedinger dynamics by a sequence of finite-time-step evolutions with simple Hamiltonians. Specifically, they focus on the quantum tunneling problem for a double-well potential or tunneling out of a well into the continuous spectrum. The authors single out several types of the influence of trotterization and analyze them quantitatively. They discuss the potential-heating problem due to periodic oscillations of the effective Floquet hamiltonian, detuning of the nominally resonant levels, modification of the tunneling exponential, and the modification of the initial state causing leakage of probability.
The results are of current interest, and the content of the publication would be useful for the general readership.
At the same time, I cannot wholeheartedly recommend publication of the present version of the manuscript. First, presentation of the results lacks a clear concise logic, with exposition of a great number of specific (though relevant and interesting) minor questions and with repetitions of certain ideas. This hinders understanding of the results (it is hard to appreciate their importance already for this reason) and even reading the paper. On top of that, it is very hard to keep in mind all the subtleties and parameter regimes, and notations brought up at numerous positions in the text. Structuring the material in a compact way may help the reader, and in my view without such "polishing" the paper is not ready for publication.
The language in the manuscript also may need careful proofreading.
A few minor remarks:
How does eq.(67) come about: eq.66 seems to imply that the rms energy difference is majorized by something like the rhs of (67), but not necessarily each level difference.
In Sec.IID, in the comment about p=\pi turning points more details are needed (or a reference).
Eq.62: should the first less-or-equal sign be reversed?
Why is the exponential in the first lines of Sec.IIIA a short-distance operator? (One can understand this statement about the operator in the exponent though.)
Recommendation
Ask for major revision
Report
The manuscript deals with the errors induced into the quantum mechanical time evolution of a single particle in different potentials by the finite step size of the Trotterized system. A semi-classical treatment of the system is derived that allows to predict the Suzuki-Trotter errors with high accuracy.
The manuscript contains a number of interesting results. In particular, the Trotter step size required to preserve the correct tunneling amplitude between potential minima is found surprisingly small. The investigations appear thorough and the mathematical (perturbative) derivations are rigorous. I can therefore recommend the publication in general (maybe in SciPost Physics Core?) once the concerns below are addressed.
The questions addressed in this manuscript are very specific and refer to a rather simple system, namely a single quantum particle. For these reasons I do not think the paper fulfils the expectations required by SciPost Physics.
My main criticism is that the manuscript is very hard to read. It is not particularly well written with a high number of typographic mistakes and missing articles. None of these mistakes are very severe, but the sheer number of them makes the reader stumble over and over again, severely slowing down the reading process. The authors would be well advised to have someone (ideally a native speaker) carefully proof read the manuscript.
Even more importantly, there is little to no continuous thread to follow throughout the manuscript. It reads like a collection of (almost independent) facts without a clear direction where any of them is leading, especially in Sections II.C,D and III.A,B,C. The Discussion is very long and redundant. Rather than briefly summarising, it repeats a lot of the details.
Overall, I would recommend to shorten the manuscript, removing redundant text and potentially moving more parts to the appendix. This should go hand in hand with with clear explanations why a given quantity is calculated at a given point.
Here a few more specific points (typos in the text excluded):
Eq. (8): Ln -> ln
Write out abbreviations like WKB once.
Eq. (36): The integral presumably starts at 0.
Explain how the exact results have been calculated. E.g. in Fig. 4 a system with L=50 sites described by the Hamiltonian in eq. (1) is too large to be exactly diagonalised.
Why is there a maximum allowed number of Trotter steps (in Fig. 6)?
In summary, the content of the manuscript is reliable, interesting and warrants publication. However, I fail to see any groundbreaking results and therefore cannot recommend publication in SciPost Physics, but rather in a lower profile journal. In addition, the presentation has to be improved significantly before I can recommend to publish.
Recommendation
Ask for major revision