Implementation of a Walsh-Hadamard gate in a superconducting qutrit*

We present the experimental demonstration of a generalized Walsh-Hadamard gate, which is a Quantum Fourier Transform gate, for a qutrit embedded in the lowest three energy levels of a superconducting circuit. This circuit is a three Josephson junction flux qubit in which all three junctions are shunted by large coplanar capacitors. We use a decomposition of the quantum gate into two unitary operations, one implemented by an off-diagonal Hamiltonian and the other implemented by a diagonal Hamiltonian. The off-diagonal Hamiltonian is obtained by the simultaneous driving of the transitions between levels 0-1, 1-2, and 0-2, with the latter being a two-photon process. The diagonal Hamiltonian is effectively implemented by appropriately shifting the phases of the driving fields. We find that multi-level ac-Stark shifts play an important role in the dynamics, and we adjust the pulse parameters to correct for these shifts. The gate is characterized using tomography of the generated output states corresponding to a set of input states. The average fidelity exceeds 90%, in good agreement with numerical simulations that take into account the multi-level structure of the system.