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Operator Growth Bounds from Graph Theory

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Abstract

Let A and B be local operators in Hamiltonian quantum systems with N degrees of freedom and finite-dimensional Hilbert space. We prove that the commutator norm \(\Vert [A(t),B]\Vert \) is upper bounded by a topological combinatorial problem: counting irreducible weighted paths between two points on the Hamiltonian’s factor graph. Our bounds sharpen existing Lieb–Robinson bounds by removing extraneous growth. In quantum systems drawn from zero-mean random ensembles with few-body interactions, we prove stronger bounds on the ensemble-averaged out-of-time-ordered correlator \({\mathbb {E}}\left[ \Vert { [A(t),B]} \Vert _{\mathrm {F}}^2\right] \). In such quantum systems on Erdös–Rényi factor graphs, we prove that the scrambling time \(t_{\mathrm {s}}\), at which \(\Vert { [A(t),B]} \Vert _{\mathrm {F}}={\Theta }(1)\), is almost surely \(t_{\mathrm {s}}={\Omega }(\sqrt{\log N})\); we further prove \(t_{\mathrm {s}}={\Omega }(\log N) \) to high order in perturbation theory in 1/N. We constrain infinite temperature quantum chaos in the q-local Sachdev-Ye-Kitaev model at any order in 1/N; at leading order, our upper bound on the Lyapunov exponent is within a factor of 2 of the known result at any \(q>2\). We also speculate on the implications of our theorems for conjectured holographic descriptions of quantum gravity.

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Notes

  1. Note that \({\mathcal {L}}\) are antisymmetric, given the inner product (11). Clearly, \(\mathrm {e}^{{\mathcal {L}}t} \in \mathrm {SO}(\dim ({\mathcal {B}}))\) is an orthogonal transformation.

  2. This is because a factor graph is “bipartite” when interpreted as an ordinary graph.

  3. This notation will be used frequently in this paper to denote that j is a vertex, and lies in a certain graph – in this case, the causal tree \(T({\mathcal {M}})\). Also note that here we are explicitly thinking of the causal tree as a subgraph of the full factor graph.

  4. When thinking of \({\varGamma }\) as a line subgraph of the factor graph G, \(X^{\varGamma }_k \in {\varGamma }\) is the unique factor obeying \(d_G(i,X^{\varGamma }_k) = 2k-1\).

  5. The name is inspired by the transverse field Ising model, which is used in quantum annealing and optimization [28, 29].

  6. While there are terms in the sum on the right hand side where a coupling \(X\notin Q_{\mathrm {L,R}}\) can show up a single time, these terms are killed by the average.

  7. Note that there is an extra \(({\mathcal {L}}^\psi _{\ell _{\mathrm {R}}})^{m^\prime _{\ell _{\mathrm {R}}}}\) term on the left hand side of \({\mathbb {P}}_j\). Its forbidden factors are \(Y^\psi _{\ell _{\mathrm {R}}}\) are the same as \(({\mathcal {L}}^\psi _{\ell _{\mathrm {R}}})^{m_{\ell _{\mathrm {R}}}}\), which appears to the right of \({\mathbb {P}}_j\), because by definition \(Y^\psi _{\ell _{\mathrm {R}}}\) only depends on the relative position of factors to each other, and not on the location of the projector \({\mathbb {P}}_j\).

  8. If \(Z\sim \mathrm {Bernoulli}(p)\), \({\mathbb {P}}(Z=0) = 1-p\) and \({\mathbb {P}}(Z=1) = p\).

  9. The notion of typical being used here is that of Erdös and Rényi, as is canonical in random (hyper)graph theory [33] when a more specific ensemble is not provided.

  10. The genus 0 term in (111) is not as strong as this bound, a choice which was deliberately made to avoid cumbersome formulas in Theorem 18 and its proof.

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Acknowledgement

This work was supported by the Gordon and Betty Moore Foundation’s EPiQS Initiative through Grant GBMF4302, by a Research Fellowship from the Alfred P. Sloan Foundation through Grant FG-2020-13795, and by the Air Force Office of Scientific Research through Grant FA9550-21-1-0195.

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Authors and Affiliations

  1. Department of Physics, Stanford University, Stanford, CA, 94305, USA

    Chi-Fang Chen & Andrew Lucas

  2. Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA, USA

    Chi-Fang Chen

  3. Department of Physics, University of Colorado, Boulder, CO, 80309, USA

    Andrew Lucas

  4. Center for Theory of Quantum Matter, University of Colorado, Boulder, CO, 80309, USA

    Andrew Lucas

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Chen, CF., Lucas, A. Operator Growth Bounds from Graph Theory. Commun. Math. Phys. 385, 1273–1323 (2021). https://doi.org/10.1007/s00220-021-04151-6

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