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Complexity is not Enough for Randomness
by Shiyong Guo, Martin Sasieta, Brian Swingle
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Martin Sasieta |
| Submission information | |
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| Preprint Link: | scipost_202410_00007v1 (pdf) |
| Date accepted: | 2024-11-07 |
| Date submitted: | 2024-10-06 17:57 |
| Submitted by: | Sasieta, Martin |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We study the dynamical generation of randomness in Brownian systems as a function of the degree of locality of the Hamiltonian. We first express the trace distance to a unitary design for these systems in terms of an effective equilibrium thermal partition function, and provide a set of conditions that guarantee a linear time to design. We relate the trace distance to design to spectral properties of the time-evolution operator. We apply these considerations to the Brownian $p$-SYK model as a function of the degree of locality $p$. We show that the time to design is linear, with a slope proportional to $1/p$. We corroborate that when $p$ is of order the system size this reproduces the behavior of a completely non-local Brownian model of random matrices. For the random matrix model, we reinterpret these results from the point of view of classical Brownian motion in the unitary manifold. Therefore, we find that the generation of randomness typically persists for exponentially long times in the system size, even for systems governed by highly non-local time-dependent Hamiltonians. We conjecture this to be a general property: there is no efficient way to generate approximate Haar random unitaries dynamically, unless a large degree of fine-tuning is present in the ensemble of time-dependent Hamiltonians. We contrast the slow generation of randomness to the growth of quantum complexity of the time-evolution operator. Using known bounds on circuit complexity for unitary designs, we obtain a lower bound determining that complexity grows at least linearly in time for Brownian systems. We argue that these bounds on circuit complexity are far from tight and that complexity grows at a much faster rate, at least for non-local systems.
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
Author comments upon resubmission
Reply to Report 1:
Indeed, if J is made to scale exponentially with the entropy then the time to Haar is O(1). However as the referee points out such an scaling is unphysical. The two possible physical scalings are to consider that J is fixed (which is what is done in the paper), or to scale J with the entropy , which is to say that the interactions are extensive in the system size. We have added clarifications of this point in the text.
We use the trace distance definition of an approximate k-design because we can compute the trace distance as a replica partition function, whereas the diamond distance is really involved. However from the bounds on appendix B it follows that a parametric linear growth given our definition implies a parametric linear growth for the diamond definition. We added a footnote to clarify this.
We thank the referee for pointing out the typo.
Reply to Report 2:
We thank the referee for the comments. We have added the corresponding clarifications throughout the text.
List of changes
In the third paragraph of page 4 we have clarified that the uniformity of the spectrum corresponds to the mean density of states.
We have added footnote 6 to clarify the physically reasonable normalizations for the couplings. We have also added footnote 10 to clarify the scaling of the time to 1-design as the scrambling time if the couplings are normalized extensively.
Before Eq. (1.4) we have added that hyperfast means that complexity saturates at an O(1) time.
We have added footnote 7 to clarify the first point raised by referee 2.
After Eq. (3.7) we have defined the fermion number operator.
We have added footnote 21 regarding the linear growth in design for the diamond definition of an approximate k-design.
We have added footnote 22 regarding Hamiltonians which contain rational relations and degeneracies in the spectrum.
We have corrected various typos, including Eqs. (2.36) and (2.37) which had typos.
We have added reference 77.
Published as SciPost Phys. 17, 151 (2024)
Reports on this Submission
Report #1 by Anonymous (Referee 2) on 2024-10-8 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202410_00007v1, delivered 2024-10-08, doi: 10.21468/SciPost.Report.9880
Report
In this work, the authors study Haar randomness in Brownian systems. They show that the time to become a k-design is linear both for a general setup satisfying certain assumptions as well as for Brownian SYK and random matrix models.
The paper meets the expectations and criteria for this journal, so I recommend it for publication.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)