SciPost Submission Page
Quantum adiabaticity in many-body systems and almost-orthogonality in complementary subspace
by Jyong-Hao Chen and Vadim Cheianov
Submission summary
| Authors (as registered SciPost users): | Jyong-Hao Chen |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202411_00033v1 (pdf) |
| Date submitted: | 2024-11-18 09:57 |
| Submitted by: | Chen, Jyong-Hao |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
|
| Approaches: | Theoretical, Computational |
Abstract
We investigate why, in quantum many-body systems, the adiabatic fidelity and the overlap between the initial state and instantaneous ground states often yield nearly identical values. Our analysis suggests that this phenomenon results from an interplay between two intrinsic limits of many-body systems: the limit of small evolution parameters and the limit of large system sizes. In the former case, conventional perturbation theory provides a straightforward explanation. In the latter case, a key insight is that pairs of vectors in the Hilbert space orthogonal to the initial state tend to become nearly orthogonal as the system size increases. We illustrate these general findings with two representative models of driven many-body systems: the driven Rice-Mele model and the driven interacting Kitaev chain 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
Current status:
Reports on this Submission
Strengths
1- pedagical presentation
2- clear writing
3- combination of analytical results and numerical examples
Weaknesses
1- unclear motivation
2- lack of discussion of relevant literature
Report
In Quantum adiabaticity in many-body systems and almost-orthogonality in complementary subspace, the authors compare adiabatic fidelity (overlap of time evolved state with the instantaneous ground state) with the overlap of the initial (ground) state with the instantaneous ground state. Bulding on previous work in Ref. [38, 39], in which the authors and collaborators noticed the effect, they set out to explain it in the present manuscript.
Overall, the manuscript takes a somewhat pedagogical approach, explaining step by step the ingredients needed for their result. I think this approach is fine, but it was difficult to me to see what in the end the precise statement is that the authors want to make. In my personal opinion, technical papers like the present one could be improved by having a "Summary and main result" Section II, in which the main assumptions are outlined and the main result is presented.
In a sense, the result is surprising: if the adiabatical fidelity and the overlaps of the instantaneous ground states are similar, then either (i) both are large, and nothing has happened (because the ground state is still essentially the same as at t=0), or (ii) both are small (the case the authors are interested in, in which case adiabatic preparation has failed, because the adiabatic fidelity is zero. Neither regime really can be called adiabatic state preparation. If adiabatic state preparation succeeds, then adiabatic fidelity should be large, and the overlap with the initial ground state small.
The difference seems to be that the authors consider a linear ramp (Eq. (7)) with a constant driving rate. Typically, when adiabatic preparation is studied, the interpolation between initial and final Hamiltonian is smooth (e.g. Gevrey class or approximations thereof) and the ramp speed is scaled down with system size, to ensure high adiabatic fidelity. There is a large body of mathematical results many-body adiabatic state preparation that the authors seem to disregard. It would be useful to understand their result in the light of what is already rigorously known about adiabatic state preparation:
[1] Sabine Jansen, Mary-Beth Ruskai, and Ruedi Seiler. “Bounds for the Adiabatic Approximation with Applications to Quantum Computation”. In: J. Math. Phys. 48.10 (Mar. 2007), p. 102111. arXiv:0603175[quant-ph].
[2] G. Nenciu. “Linear Adiabatic Theory. Exponential Estimates”. In: Commun. Math. Phys. 152.3 (1993), pp. 479–496.
[3] George A. Hagedorn and Alain Joye. “Elementary Exponential Error Estimates for the Adiabatic Approximation”. In: Journal of Mathematical Analysis and Applications 267.1 (Mar. 2002), pp. 235–246.
[4] Yimin Ge, Andr´as Moln´ar, and J. Ignacio Cirac. “Rapid Adiabatic Preparation of Injective Projected Entangled Pair States and Gibbs States”. In: Phys. Rev. Lett. 116.8 (Feb. 2016). arXiv: 1508.00570.
[5] Sven Bachmann, Wojciech De Roeck, and Martin Fraas. “Adiabatic Theorem for Quantum Spin Systems”. In: Physical Review Letters 119.6 (Aug. 11, 2017), p. 060201. arXiv: 1612.01505.
[6] Sven Bachmann, Wojciech De Roeck, and Martin Fraas. “The Adiabatic Theorem and Linear Response Theory for Extended Quantum Systems”. In: Communications in Mathematical Physics 361.3 (Aug.2018), pp. 997–1027. arXiv: 1705.02838.
[7] Sven Bachmann, Wojciech De Roeck, and Martin Fraas. The Adiabatic Theorem in a Quantum Many-Body Setting. Mar. 18, 2019. arXiv: 1808.09985.
I did not spot major omissions in the manuscript, but I am sceptical that the problem studied is really a "long-standing research stumbling block", because the studied regime is not so relevant for adiabatic state preparation and I am thus not convinced by the motivation underlying the research. As a result, I recommend publication in a more specialized venue, for example SciPost Core.
Requested changes
1- Discussion of literature
2- Clearer presentation of the main result & setting
Recommendation
Accept in alternative Journal (see Report)