Geometry of quantum complexity

TL;DR

Geometry of quantum complexity investigates Nielsen's complexity metric for $n$ qubits, focusing on penalty choices that shape curvature and scrambling properties. It develops a detailed geometric framework, deriving geodesic equations, Riemannian curvature, and a closed-form state-space metric via a Riemannian submersion, and analyzes conjugate points through the Raychaudhuri equation. A key finding is that progressive penalties $q_w=\alpha^{w-1}$ produce negative scalar curvature and, in the large-$\alpha$ regime, robust indications that maximal complexity scales exponentially with $n$, aligning with expectations for holographic scrambling. The work links curvature characteristics to complexity growth and holographic interior physics, offering a principled route to understand complexity in quantum many-body systems and quantum field theories.

Abstract

Computational complexity is a new quantum information concept that may play an important role in holography and in understanding the physics of the black hole interior. We consider quantum computational complexity for $n$ qubits using Nielsen's geometrical approach. We investigate a choice of penalties which, compared to previous definitions, increases in a more progressive way with the number of qubits simultaneously entangled by a given operation. This choice turns out to be free from singularities. We also analyze the relation between operator and state complexities, framing the discussion with the language of Riemannian submersions. This provides a direct relation between geodesics and curvatures in the unitaries and the states spaces, which we also exploit to give a closed-form expression for the metric on the states in terms of the one for the operators. Finally, we study conjugate points for a large number of qubits in the unitary space and we provide a strong indication that maximal complexity scales exponentially with the number of qubits in a certain regime of the penalties space.

Figures (8)

• Figure 1: Regions of negativity of sectional curvatures in the $(P,Q)$ plane. In the white region all the sectional curvatures are positive. The blue shaded regions correspond to a negative scalar curvature.
• Figure 2: The exact value of $R/\eta$ plotted as a function of $\alpha$ in the case of progressive penalties, for $n=5,10,15,20$. The asymptotic value at $\alpha \rightarrow \infty$ is shown in black. The minimum in the picture appears for $n \geq 8$. Increasing $n$, the shape of the minimum tends to become more and more steep and it is located at a lower value of $\alpha$. Note that when $n=20,$ for values $\alpha \geq 4$ the result of the average sectional curvature at $\mathcal{O}(\alpha^0)$ is already very close to the exact result.
• Figure 3: A reproduction of a depiction of a submersion from besse.
• Figure 4: Left: Regions where each $R_{x,y,z}$ is positive. Right: Example of an exact conjugate point (the black spot) of geodesics for $P=Q=0.4$ in stereographic projection.
• Figure 5: Geodesics with length $\lambda=2.5$ for $Q=10$, $P =10$. The geodesics are plotted in different colours. The endpoints of the various curves are represented in black.