Implementation of a Walsh-Hadamard Gate in a Superconducting Qutrit

We have implemented a Walsh-Hadamard gate, which performs a quantum Fourier transform, in a superconducting qutrit. The qutrit is encoded in the lowest three energy levels of a capacitively shunted flux device, operated at the optimal flux-symmetry point. We use an efficient decomposition of the Walsh-Hadamard gate into two unitaries, generated by off-diagonal and diagonal Hamiltonians, respectively. The gate implementation utilizes simultaneous driving of all three transitions between the three pairs of energy levels of the qutrit, one of which is implemented with a two-photon process. The gate has a duration of 35 ns and an average fidelity over a representative set of states, including preparation and tomography errors, of 99.2%, characterized with quantum-state tomography. Compensation of ac-Stark and Bloch-Siegert shifts is essential for reaching high gate fidelities.

Figure 1

(a) Schematic and microscope images of the device used in the experiment. The device is coupled to a coplanar waveguide resonator with characteristic impedance $Z_0$ for readout and a drive pad for control with pulses generated by an arbitrary waveform generator. The inset shows the magnified view of the device junctions and the loop. (b) Readout homodyne voltage $V_h$ versus the frequency of the spectroscopy tone applied to the qutrit at the flux-symmetry point, showing the 0–1, two-photon 0–2, and 1–2 transitions, respectively. The solid lines represent Lorentzian fits. (c) Peak positions for the 0–1 (blue squares) and 1–2 (red dots) transitions versus applied magnetic flux. The solid lines represent the fits with the circuit model based on the calculated qutrit energies.

Figure 2

Two-photon driving experiments. (a) Frequency of the Rabi oscillations versus the detuning between the driving frequency and the two-photon resonance frequency $({\omega }_{01}+{\omega }_{12})/2$ for a given drive amplitude. The line is a quadratic fit. (b) Two-photon resonant-driving Rabi frequency versus drive amplitude. The line is a quadratic fit. (c),(d) Results of the tomography experiments. Panel (c) shows the population of states 0 (blue squares), 1 (red disks), and 2 (green triangles), versus Rabi pulse duration, with an oscillation frequency of $2\pi \times 7.2\mathrm{MHz}$. Panel (d) shows the phase of the 0–2 component of the density matrix for two values of the phase of the driving field: 0 (triangles) and $\pi /4$ (rhombuses).

Figure 3

(a) Pulse sequence used in the experiments. (b) Real and (c) imaginary parts of the reconstructed density matrix after the gate is applied on the thermalized state shows a 99.8% gate fidelity. The differences between the reconstructed and the expected density matrices are shown for the real (d) and imaginary (e) parts.

Figure 4

Experimental (dots) and simulated (dashed curve) fidelity of the state obtained by applying the control pulse to the thermal state versus the relative phase of the 0–2 two-photon drive.