Conditions for Equivalent Noise Sensitivity of Geometric and Dynamical Quantum Gates

TL;DR

This paper analytically compares the noise sensitivity of geometric and dynamical quantum gates, showing under certain conditions they can be equivalent in robustness, challenging the assumption that geometric gates are inherently more resilient.

Contribution

It provides a general analytical framework using invariant theory and filter functions to identify conditions where geometric and dynamical gates have identical noise sensitivities.

Findings

Geometric and dynamical gates can have equivalent noise sensitivity under certain conditions.

The framework applies to arbitrary-dimensional Hilbert spaces.

Explicit examples in single-qubit systems demonstrate the theoretical results.

Abstract

Geometric quantum gates are often expected to be more resilient than dynamical gates against certain types of error, which would make them ideal for robust quantum computing. However, this is still up for debate due to seemingly conflicting results in the literature. Here we use dynamical invariant theory in conjunction with filter functions in order to analytically characterize the noise sensitivity of an arbitrary quantum gate. For any control Hamiltonian that produces a geometric gate, we find that under certain common conditions one can construct another control Hamiltonian that produces an equivalent dynamical gate with identical noise sensitivity (as characterized by the filter function). Our result holds for a Hilbert space of arbitrary dimensions, but we illustrate our result by examining experimentally relevant single-qubit scenarios and providing explicit examples of equivalent geometric and dynamical gates.