Anton Khvalyuk: Influence of Trotterization error on single-particle tunneling

Date of publication: 3. 1. 2024
Mathematical physics seminar
Thursday
11
January
Time:
15:00 - 17:00
Location:
F1, ground floor, Faculty of Mathematics and Physics, Jadranska 19.

Simulation of the single-particle tunneling problem by means of the Suzuki-Trotter approximation (STA) is analyzed. The target system describes a particle hopping across a chain of sites with position-dependent potential profile. The latter is assumed to be smooth and posses several local minima separated by a potential barrier, arranging 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 total probability to other states. In the general case, the first type of error is the most essential, as the emerging 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 is found to cause exponential enhancement of the tunneling rates. The last type of error classifies the overall defect in the wave function and thus 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 } ≪ 1/δt (with T, P being the typical scales of kinetic and potential terms, respectively), while also revealing the structure of error and its actual behavior with system parameters. Analysis of the case of large Trotter step is also performed, with the main result being the reconstruction of the low-energy spectrum due to coupling between states with energy difference close to 2π/δt. The connection of the obtained results with the rigorous upper error bounds on the STA error is discussed, with particular emphasis on the reasons for the fact that 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. The talk is based on the recent paper [arXiv:2312.04735].