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A graph-theoretic approach to chaos and complexity in quantum systems
by Maxwell West, Neil Dowling, Angus Southwell, Martin Sevior, Muhammad Usman, Kavan Modi, Thomas Quella
Submission summary
| Authors (as registered SciPost users): | Max West |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2502.16404v1 (pdf) |
| Date submitted: | March 13, 2025, 4:41 a.m. |
| Submitted by: | West, Max |
| Submitted to: | SciPost Physics Core |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
There has recently been considerable interest in studying quantum systems via dynamical Lie algebras (DLAs) -- Lie algebras generated by the terms which appear in the Hamiltonian of the system. However, there are some important properties that are revealed only at a finer level of granularity than the DLA. In this work we explore, via the commutator graph, average notions of scrambling, chaos and complexity over ensembles of systems with DLAs that possess a basis consisting of Pauli strings. Unlike DLAs, commutator graphs are sensitive to short-time dynamics, and therefore constitute a finer probe to various characteristics of the corresponding ensemble. We link graph-theoretic properties of the commutator graph to the out-of-time-order correlator (OTOC), the frame potential, the frustration graph of the Hamiltonian of the system, and the Krylov complexity of operators evolving under the dynamics. For example, we reduce the calculation of average OTOCs to a counting problem on the graph; separately, we connect the Krylov complexity of an operator to the module structure of the adjoint action of the DLA on the space of operators in which it resides, and prove that its average over the ensemble is lower bounded by the average shortest path length between the initial operator and the other operators in the commutator graph.
Current status:
Reports on this Submission
Strengths
1- Similar to ref [6], this work tries to beyond the coarse picture of the Dynamical Lie algebra (DLA), and makes statements about the scrambling dynamics of Pauli algebras. The main result is a statement about how averages over Paulis can be calculated I believe that this is a much needed contribution to the field since the DLA view can be quite limiting in making statements about short time dynamics/circuits.
2- The main text requires a good understanding representation theory to understand, but is well written and concise.
Weaknesses
1- I think the results with regards to the Graph complexity could maybe be elaborated on more. For example, I have little intuition for what Lemma 10 is telling me (Similarly for Fig. 6).
Report
I recommend this paper for publication. With regards to the main text, I have some minor questions:
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With regards to figure 2, U~\mu_G, the Haar measure on G. In the caption however, it says that U is generated as exp(i sum_j c_j H_j) with c_j~Uniform() (I assume ?). This should not give Haar distributed unitaries on G, but some other measure. Could you elaborate on this choice?
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Could the authors elaborate more on the usage of Lemma 10? What does it mean for the graph complexity to grow quadratically?
-I Am curious if the graph complexity results can be used to make statements about the complexity of Pauli propagation algorithms, whose efficiency relies on truncating the (weighted) commutator graph that is generated by applying layers of a circuit (c.f. [Phys. Rev. A 99, 062337] and arxiv.2306.05400). I feel like there must be direct connections with both the Krylov and Graph complexity. Could the authors comment on this potential connection? If not could the authors comment on if their results could potentially be used to understand the classical simulatibility of quantum dynamics? There is short comment in the conclusion but I am curious if they could elaborate more.
With regards to the proofs, I have the following questions:
1- The proof of Proposition 1 is clear to me, although the last step in A(21) took me a moment to understand. As I understand it, we get a contribution of 1/|C_B1| for each B1, so you get \sum_k \sum_S\inC_k 1/|C_k| = \sum_k where the sum over kappa is again the number of components. Maybe clarify this last step with a sentence below the proof.
2- A(41), the sum over anticommuting elements should probably be in the numerator
3- Proof Proposition 7, The Hilbert Schmidt product is already used in the proof of prop 1 and denoted as <>{HS} instead of <>
Requested changes
1- Page 3, last paragraph, I think "...the same connected component as p (see Fig. 1(c))" should refer to Fig1(a)?
2- Table 1, Ref[9], 2-local Hamiltonians on undirected graphs have also been classified: [arxiv.2409.19797]
3- There is work on "g-sim" from the early 2000s: [arxiv.org.0512209] and [Phys. Rev. Lett. 97, 190501]
4- Fig 3 is not referred to anywhere.
Recommendation
Ask for minor revision