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 [1][2][3][4] 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 5 . They play an important role for the creation of synthetic gauge fields in cold atomic gases 6 and the description of non-abelian anyon statistics 7 . Moreover, it has been proposed to exploit non-abelian holonomic gates for robust quantum computation [8][9][10] . In contrast to abelian geometric phases 11 , nonabelian 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 12,13 . Here, we realize non-abelian holonomic quantum operations 14,15 on a single superconducting artificial three-level atom 16 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.A cyclic evolution of a non-degenerate quantum system is in general accompanied by a phase change of its wave function. The acquired abelian phase can be divided into two parts: The dynamical phase which is proportional to the evolution time and the energy of the system, and the geometric phase which depends only on the path of the system in Hilbert space. This characteristic feature leads to a resilience of the geometric phase to certain fluctuations during the evolution 17-19 , a property which has attracted particular attention in the field of quantum information processing 20 . However, universal quantum computation cannot be based on simple phase gates, which modify only the relative phase of a superposition state, unless they act on specific basis states 21 . Furthermore, geometric operations acting on degenerate subspaces have * abdumalikov@phys.ethz.ch † Now at Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA been proposed for holonomic quantum computation fully based on geometric concepts 8 . In this scheme, quantum bits are encoded in a doubly degenerate eigenspace of the system hamiltonian h( λ). The parameters λ are varied to induce a cyclic evolution of the system. When the system returns back to its initial state, it can acquire not only a simple geometric phase factor, but also undergoes a path-dependent unitary transformation, a non-abelian holonomy, which causes a transition bet...

Classical realism demands that system properties exist independently of whether they are measured, while noncontextuality demands that the results of measurements do not depend on what other measurements are performed in conjunction with them. The Bell–Kochen–Specker theorem states that noncontextual realism cannot reproduce the measurement statistics of a single three-level quantum system (qutrit). Noncontextual realistic models may thus be tested using a single qutrit without relying on the notion of quantum entanglement in contrast to Bell inequality tests. It is challenging to refute such models experimentally, since imperfections may introduce loopholes that enable a realist interpretation. Here we use a superconducting qutrit with deterministic, binary-outcome readouts to violate a noncontextuality inequality while addressing the detection, individual-existence and compatibility loopholes. This evidence of state-dependent contextuality also demonstrates the fitness of superconducting quantum circuits for fault-tolerant quantum computation in surface-code architectures, currently the most promising route to scalable quantum computing.