Ultimate Speed Limits to the Growth of Operator Complexity

TL;DR

A rigorous bound on the Krylov complexity growth rate is established based on the uncertainty principle and it is shown that the presence of quantum chaos is not strictly necessary to saturation of the bound.

Abstract

In an isolated system, the time evolution of a given observable in the Heisenberg picture can be efficiently represented in Krylov space. In this representation, an initial operator becomes increasingly complex as time goes by, a feature that can be quantified by the Krylov complexity. We introduce a fundamental and universal limit to the growth of the Krylov complexity by formulating a Robertson uncertainty relation, involving the Krylov complexity operator and the Liouvillian, as generator of time evolution. We further show the conditions for this bound to be saturated and illustrate its validity in paradigmatic models of quantum chaos.

Figures (3)

• Figure 1: Growth of Krylov complexity at the speed limit. Saturation of the dispersion bound occurs in three different scenarios, each of which is associated with a different complexity algebra, that is specified by the sign of $\alpha$. a Time-dependence of the Krylov complexity. b The corresponding growth of the Lanczos coefficients in the Krylov lattice. The plots are representative of the three different scenarios. The Krylov dimension in the $\textrm{SU}(2)$ case is $D=100$, and infinite in all other cases. In b we choose $\alpha=4$ and $-4$ for the $\textrm{SL}(2,\mathbb{R})$ and $\textrm{SU}(2)$ algebras, respectively, while $\alpha$ is always zero in the $\textrm{HW}$ case. Finally, the parameter $\gamma$ in b is chosen in each case such that the corresponding Lanczos coefficients share the same behavior near the origin of the Krylov lattice. Specifically, $\gamma= 202,\, 200,\, 198$ for the $\textrm{SL}(2,\mathbb{R})$, $\textrm{HW}$ and $\textrm{SU}(2)$ algebras, respectively.
• Figure 2: Growth of Krylov complexity in a generic system.a Squares of the Lanczos coefficients for a single realization (gray points) and an average over $100$ random Hamiltonian matrices (black line). b Operator growth in the Krylov lattice as displayed by the dynamics of the amplitudes $|\varphi_n (t)|^2$ for a single random matrix realization. c Krylov complexity (green solid lines) together with the deviation time (gray dashed line) for three independent random matrix realizations. d The corresponding absolute value of the growth rate of the Krylov complexity (blue solid lines), together with the dispersion bound (red dashed lines), Eq. \ref{['dispersion bound']}. In all figures the random Hamiltonian matrices are sampled from $\mathrm{GOE}(d)$ with standard deviation $\sigma=1$, maximal Krylov dimension $D = 993$ and a uniform initial observable operator $\mathcal{O}$.
• Figure 3: For the SYK model, the complexity rate (solid light blue line) saturates the dispersion bound (dashed red line) at any time. In the plot we fix the parameters appearing in Eq. \ref{['SYK-b_n']} to be $\nu=\eta=1$, that is we consider an exact linear growth of the Lanczos coefficients: $b_n=n$ for $n>1$.

Theorems & Definitions (14)

• Proposition 1
• Lemma 2
• proof
• Corollary 3
• Proposition 4
• proof
• Proposition 5
• proof
• Lemma 6
• proof