Parallel transport in rotating frames and projective holonomic quantum computation

TL;DR

This paper develops a framework for nonadiabatic holonomic quantum computation in rotating frames, addressing global phase issues and establishing bounds on gate efficiency and minimum execution time.

Contribution

It introduces a modified parallel transport condition for rotating frames and extends the isoholonomic inequality to projective gates, advancing quantum gate implementation methods.

Findings

Adjusted parallel transport condition for rotating reference frames

Extended isoholonomic inequality to projective quantum gates

Determined minimum execution time for projective holonomic gates

Abstract

Nonadiabatic holonomic quantum computation is a promising approach for implementing quantum gates that offers both efficiency and robustness against certain types of errors. A key element of this approach is a geometric constraint known as the parallel transport condition. According to the principle of covariance, this condition must be appropriately modified when changing reference frames. In this paper, we detail how to adjust the parallel transport condition when transitioning from the laboratory frame to a rotating reference frame. Furthermore, building on gauge invariance considerations, we develop a framework for nonadiabatic holonomic quantum computation with projective gates. The parallel transport condition of this framework effectively addresses the problem of global dynamical phases inherent in conventional nonadiabatic holonomic quantum computation. We extend the isoholonomic inequality, which provides a fundamental bound on the efficiency of protocols used to implement holonomic quantum gates, to encompass projective quantum gates. We also determine a minimum execution time for projective holonomic quantum gates and show that this time can be attained when the codimension of the computational space is sufficiently large.