Geometry of quantum complexity

TL;DR

This paper explores the geometric structure of quantum complexity for many qubits, proposing a penalty scheme that avoids singularities and reveals exponential scaling of maximal complexity with qubit number.

Contribution

It introduces a new penalty scheme for quantum complexity that is free from singularities and relates operator and state complexities through Riemannian geometry.

Findings

Penalty scheme avoids singularities

Maximal complexity likely scales exponentially with qubits

Derived a closed-form metric for states from operators

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.