Operator complexity: a journey to the edge of Krylov space

Abstract

Heisenberg time evolution under a chaotic many-body Hamiltonian H transforms an initially simple operator into an increasingly complex one, as it spreads over Hilbert space. Krylov complexity, or ‘K-complexity’, quantifies this growth with respect to a special basis, generated by H by successive nested commutators with the operator. In this work we study the evolution of K-complexity in finite-entropy systems for time scales greater than the scrambling time ts> log(S). We prove rigorous bounds on K-complexity as well as the associated Lanczos sequence and, using refined parallelized algorithms, we undertake a detailed numerical study of these quantities in the SYK4 model, which is maximally chaotic, and compare the results with the SYK2 model, which is integrable. While the former saturates the bound, the latter stays exponentially below it. We discuss to what extent this is a generic feature distinguishing between chaotic vs. integrable systems.

Random matrix theory for complexity growth and black hole interiors

“…3. These universal features imply that the K-complexity of O(t), measured by its average position on the K-chain, grows exponentially for a scrambling time, before transitioning to a linear increase which is only disturbed by non-perturbative O(e −S ) effects that eventually lead to its saturation at O(e 2S ) values [15].…”

“…Our starting point is a notion of operator complexity discussed in a series of recent interesting works [13][14][15] called Krylov complexity (K-complexity) which is predicated on the simple idea reviewed in section 2: starting with a "simple" operator O and a state ρ, we build an increasing-complexity operator ladder. This is achieved by applying nested commutators with the Hamiltonian H and orthonormalizing the operators produced at every step, using as our inner product correlators in the state ρ.…”

“…In recent works [4,[13][14][15][16], a different definition of operator complexity was proposed, one attuned to the study of complexity generated by time evolution, that depends only on the system's Hamiltonian and a reference quantum state. The idea capitalizes on the intuition that acting with the Hamiltonian on an initially simple operator O via commutators generally help us ascend the complexity ladder.…”

“…A refinement of the Krylov formalism. The approach outlined above has been used in a number of recent interesting studies [4,[13][14][15][16]. For systems with a finite number of states, this formalism elegantly highlights robust signatures of quantum chaos [13,17], as we review below.…”

Operator growth in 2d CFT

“…The linear growth of b n ∼ an translates to exponential growth of the Kcomplexity, with characteristic Lyapunov exponent λ L = 2a. This operator growth hypothesis has been further studied and verified numerically in various examples [3,20,22,[34][35][36][37][38][39][40][41][42].…”