Polynomial Equivalence of Complexity Geometries

TL;DR

This paper demonstrates that a wide class of quantum complexity geometries are polynomially equivalent, meaning they have similar computational power and can approximate each other efficiently, broadening the scope of Nielsen's geometric approach to quantum complexity.

Contribution

It establishes the polynomial equivalence of a broad class of right-invariant metrics on the unitary group, extending Nielsen's earlier results to include metrics with modest curvature.

Findings

All metrics in the equivalence class have exponential diameter.

Metrics in the operator-norm class define the BQP complexity class.

The enlarged class includes metrics with modest curvature, aiding geometric analysis.

Abstract

This paper proves the polynomial equivalence of a broad class of definitions of quantum computational complexity. We study right-invariant metrics on the unitary group -- often called `complexity geometries' following the definition of quantum complexity proposed by Nielsen -- and delineate the equivalence class of metrics that have the same computational power as quantum circuits. Within this universality class, any unitary that can be reached in one metric can be approximated in any other metric in the class with a slowdown that is at-worst polynomial in the length and number of qubits and inverse-polynomial in the permitted error. We describe the equivalence classes for two different kinds of error we might tolerate: Killing-distance error, and operator-norm error. All metrics in both equivalence classes are shown to have exponential diameter; all metrics in the operator-norm equivalence class are also shown to give an alternative definition of the quantum complexity class BQP. My results extend those of Nielsen et al., who in 2006 proved that one particular metric is polynomially equivalent to quantum circuits. The Nielsen et al. metric is incredibly highly curved. I show that the greatly enlarged equivalence class established in this paper also includes metrics that have modest curvature. I argue that the modest curvature makes these metrics more amenable to the tools of differential geometry, and therefore makes them more promising starting points for Nielsen's program of using differential geometry to prove complexity lowerbounds.