Quantum Complexity and Negative Curvature

TL;DR

The paper investigates how quantum circuit complexity in chaotic, fast-scrambling systems exhibits a universal growth pattern that mirrors the geometry behind black-hole interiors. It introduces a classical analog—a particle moving on a compact negatively curved surface—whose geodesic dynamics reproduce key features of complexity growth, including linear rise, exponential saturation, scrambling time $\tau_* = \log K$, and the switchback effect seen with precursors. By analyzing single and multiple perturbations within both the quantum and analog frameworks, the work demonstrates a deep correspondence between complexity, black-hole dynamics, and hyperbolic geometry, extending to an action-length relationship and to the holographic interpretation via Einstein-Rosen bridges. The results suggest a geometric, Nielsen-inspired understanding of complexity and provide a simple, tractable model that could guide the development of a continuum definition of complexity in quantum systems and holographic contexts.

Abstract

As time passes, once simple quantum states tend to become more complex. For strongly coupled k-local Hamiltonians, this growth of computational complexity has been conjectured to follow a distinctive and universal pattern. In this paper we show that the same pattern is exhibited by a much simpler system: classical geodesics on a compact two-dimensional geometry of uniform negative curvature. This striking parallel persists whether the system is allowed to evolve naturally or is perturbed from the outside.

Figures (18)

• Figure 1: An example of a random circuit with $K= 6$, $k=2$, and depth 4. The six qubits (black lines) are randomly grouped into three ordered pairs, and then a gate (blue box) is applied to each pair. At the next time-step, they are randomly re-paired.
• Figure 2: The evolution of the computational complexity of the operator $e^{i H \tau}$ for a generic $k$-local Hamiltonian $H$. At $t=0$ the operator $e^{iHt}$ is the identity and so has complexity zero. At early and intermediate times the complexity increases linearly, with coefficient $K/2$. After a time exponential in the number of qubits $K$, the complexity saturates at a value $\mathcal{C}_\textrm{max}$ that is exponential in $K$. It then fluctuates near that maximum value. Very very rarely---so rarely that we must wait the double exponentially long quantum recurrence time for it to be likely to have happened even once---the complexity of the system may fluctuate down to near zero, before growing again.
• Figure 3: The size of the precursor as a function of $\tau-\tau_*$ for the epidemic function Eq. \ref{['epi2']}. For $\tau - \tau_* \ll 1$ the size is growing exponentially as the epidemic spreads throughout the system, but it's growing from such a small baseline as to be visually indistinguishable from zero. When $\tau = \tau_*$ almost every site is infected and $s(\tau)$ abruptly saturates at $K$.
• Figure 4: The multiple precursor operator $W_\textrm{multi}(\tau_n, \tau_{n-1}, \ldots , \tau_1)$. When $\tau_i - \tau_{i-1}$ and $\tau_{i+1} - \tau_i$ have the same sign, as at $\tau_2$ and $\tau_6$, $W_i$ is called a 'through-going' insertion and there is generically no cancellation. When $\tau_i - \tau_{i-1}$ and $\tau_{i+1} - \tau_i$ have opposite signs, as at the other $\tau_i$, this is called a 'switchback' and there is a partial cancellation that reduces the complexity by $K \tau_*$.
• Figure 5: An example of a compactification of the hyperbolic plane. Start with a ginormous equilateral polygon centered at the origin, and then identify sides to make the compact and conical-deficit-free space ${\cal{H}}_g$.