Operator growth bounds from graph theory

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 $\mathbb{E}\left[\Vert [A(t),B]{\Vert }_F^2\right]$. In such quantum systems on Erdös-Rényi factor graphs, we prove that the scrambling time $t_{\mathrm{s}}$, at which $|[A(t),B]{\Vert }_F=\mathrm{\Theta }(1)$, is almost surely $t_{\mathrm{s}}=\mathrm{\Omega }(\sqrt{\log N})$; we further prove $t_{\mathrm{s}}=\mathrm{\Omega }(\log N)$ 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.

49 pages, 14 figures. v2: published version with errors fixed

References in corpus (14)

• Black holes as mirrors: quantum information in random subsystems
• A Universal Operator Growth Hypothesis
• Uncolored Random Tensors, Melon Diagrams, and the SYK Models
• The complete $1/N$ expansion of a SYK--like tensor model
• Lieb-Robinson Bounds for Harmonic and Anharmonic Lattice Systems
• Finite speed of quantum scrambling with long range interactions
• Strictly linear light cones in long-range interacting systems of arbitrary dimensions
• Euclidean operator growth and quantum chaos
• Hierarchy of linear light cones with long-range interactions
• Absence of fast scrambling in thermodynamically stable long-range interacting systems
• Tightening the Lieb-Robinson Bound in Locally-Interacting Systems
• Many-body quantum dynamics slows down at low density
• Non-perturbative dynamics of the operator size distribution in the Sachdev-Ye-Kitaev model
• Operator growth bounds in a cartoon matrix model

Cited by in corpus (20)

• Krylov Complexity in Open Quantum Systems
• The Lieb-Robinson light cone for power-law interactions
• Operator growth in 2d CFT
• Concentration for random product formulas
• Speed limits and locality in many-body quantum dynamics
• Finite speed of quantum information in models of interacting bosons at finite density
• Dynamical Transition of Operator Size Growth in Quantum Systems Embedded in an Environment
• Optimal Frobenius light cone in spin chains with power-law interactions
• Non-perturbative dynamics of the operator size distribution in the Sachdev-Ye-Kitaev model
• Prethermalization and the local robustness of gapped systems
• Operator Size Distribution in Large $N$ Quantum Mechanics of Majorana Fermions
• Riemannian quantum circuit optimization for Hamiltonian simulation
• Disordered Lieb-Robinson bounds in one dimension
• The dynamical $\alpha$-Rényi entropies of local Hamiltonians grow at most linearly in time
• Bounding the finite-size error of quantum many-body dynamics simulations
• Scrambling Transition in Free Fermion Systems Induced by a Single Impurity
• Limits of Short-Time Evolution of Local Hamiltonians
• Lieb-Robinson correlation function for the quantum transverse field Ising model
• Lieb-Robinson bounds with exponential-in-volume tails
• Dynamical quantum phase transitions on random networks