Experimental Realization of Non-Abelian Geometric Gates

TL;DR

This paper demonstrates the experimental realization of non-abelian holonomic quantum gates on a superconducting three-level system, confirming their non-commutative nature and potential for robust quantum computation.

Contribution

It provides the first full characterization and implementation of non-abelian geometric gates in a superconducting qubit system, including process tomography and demonstration of non-commutativity.

Findings

Fidelity of non-abelian gates exceeds 95%

Sequence of paths yields inequivalent transformations

Holonomic gates form a universal set for quantum computation

Abstract

The geometric aspects of quantum mechanics are underlined most prominently by the concept of geometric phases, which are acquired whenever a quantum system evolves along a closed path in Hilbert space. The geometric phase is determined only by the shape of this path and is -- in its simplest form -- a real number. However, if the system contains degenerate energy levels, matrix-valued geometric phases, termed non-abelian holonomies, can emerge. They play an important role for the creation of synthetic gauge fields in cold atomic gases and the description of non-abelian anyon statistics. Moreover, it has been proposed to exploit non-abelian holonomic gates for robust quantum computation. In contrast to abelian geometric phases, non-abelian ones have been observed only in nuclear quadrupole resonance experiments with a large number of spins and without fully characterizing the geometric process and its non-commutative nature. Here, we realize non-abelian holonomic quantum operations on a single superconducting artificial three-level atom by applying a well controlled two-tone microwave drive. Using quantum process tomography, we determine fidelities of the resulting non-commuting gates exceeding 95 %. We show that a sequence of two paths in Hilbert space traversed in different order yields inequivalent transformations, which is an evidence for the non-abelian character of the implemented holonomic quantum gates. In combination with two-qubit operations, they form a universal set of gates for holonomic quantum computation.