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Complexity is not Enough for Randomness
by Shiyong Guo, Martin Sasieta, Brian Swingle
Submission summary
Authors (as registered SciPost users):  Martin Sasieta 
Submission information  

Preprint Link:  scipost_202410_00007v1 (pdf) 
Date submitted:  20241006 17:57 
Submitted by:  Sasieta, Martin 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

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 timeevolution 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 nonlocal 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 nonlocal timedependent 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 finetuning is present in the ensemble of timedependent Hamiltonians. We contrast the slow generation of randomness to the growth of quantum complexity of the timeevolution 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 nonlocal systems.
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 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 multipronged followup work
 Detail a groundbreaking theoretical/experimental/computational discovery
 Present a breakthrough on a previouslyidentified and longstanding 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 kdesign 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 1design 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 kdesign.
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.
Current status:
Reports on this Submission
Report
In this work, the authors study Haar randomness in Brownian systems. They show that the time to become a kdesign 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.
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