Geometric Quantum Computing and the Diffeomorphism Field

Zach Andersen

The complexity geometry approach to quantum computing treats the space of unitary gates of an (n qubit) quantum computer as an SU(2^n)
manifold. In this approach, constructing algorithms corresponds to synthesizing a desired unitary gate from a set of one and two qubit gates. This is approached geometrically by finding geodesics in SU(2^n) under a metric that penalizes motion in the direction of three or more qubit gates. It can be shown that the length of such geodesics gives bounds on the complexity of synthesizing a given unitary gate with a chosen universal gate set. Since the geodesics that arise in the geometric quantum computing context are only physically meaningful up to reparameterization invariance, we can apply the projective equivalence arguments of Thomas-Whitehead gravity and motivate the construction of a Thomas Cone on the SU(2^n) manifold.

In this talk, I will review concepts from geometric quantum computing and find the connection components of the Thomas Cone built on the SU(2^n) manifold. Specifically, we follow the work of my advisor, Dr. Vincent Rodgers, and treat the diffeomorphism field as a dynamical variable in the theory for which we can find equations of motion from a choice of a modified projective Gauss-Bonnet action.

To participate in this event virtually via Zoom, go to https://uiowa.zoom.us/j/99570315915.