Gate Optimization via Efficient Two-Qubit Benchmarking for NV Centers in Diamond

Abstract

High-fidelity gate implementation requires sophisticated control pulses that steer the quantum system to undergo the desired transformation. Quantum Optimal Control allows to derive these control pulses in an open-loop fashion based on numerical simulations. However, their precision can be limited by incomplete knowledge of the system. Closed-loop optimization overcomes this limitation by incorporating feedback from measurements, provided a suitable and efficient measure of the gate performance can be defined. In this article, we present an efficient method to evaluate the performance of a two-qubit gate by preparation and measurement of only two quantum states, enabling experimental closed-loop optimization with a metric previously believed to be limited to open-loop control. We tailor the approach to nitrogen-vacancy centers in diamond and, through numerical simulations, demonstrate how the method can optimize a two-qubit gate while reducing the number of required measurements by two orders of magnitude compared to standard process tomography under realistic experimental settings.

Figures (5)

• Figure 1: System level scheme. An NV center in the diamond lattice is coupled to a nearby ${}^{13}$C spin, with the magnetic field aligned along the defect’s symmetry axis (structure in the inset). The two Zeeman levels $m_s=0$ and $m_s=-1$ on the electron and the two nuclear spin states $m_I=1/2$ and $m_I=-1/2$ form the computational basis states of the two-qubit system. Microwave (blue) and radio-frequency (yellow) pulses can drive the electronic transitions $\omega_{00\leftrightarrow10}$ and $\omega_{01\leftrightarrow11}$, and nuclear transitions $\omega_{00\leftrightarrow01}$ and $\omega_{10\leftrightarrow 11}$, respectively.
• Figure 2: Full pulse sequences for (a) state preparation and (b) optical readout. We illustrate the full pulse sequences to experimentally compute ${F}_\text{J}$. Green pulses represent the initialization/readout action induced by a laser, while blue pulses are required to prepare correctly the probe states and measure the trial control action on them. The QOC gate itself is represented by the time evolution map $\Phi$ (yellow). Finally, the two-step readout routine (pink), further illustrated in panel (b), allows to access all the diagonal terms of the final states as described in Eq. \ref{['eqn: OptRead conversion']}. Full details of the pulse sequence can be found in Sec. \ref{['sec: state prep and readout']}.
• Figure 3: Optimized pulse shapes. The dashed red line denotes the best microwave pulse shape obtained from the open-loop optimization, while the other twenty lines denote the rescaled pulses (both in amplitude and time) found by closed-loop optimization of the sample systems. Their performance is summarized in the first column of Fig. \ref{['fig:AllResultsSummary']}.
• Figure 4: Closed-loop optimization scores for different parameter choices. Reference infidelity achieved when adapting the original open-loop solution to twenty sample systems: for each of them we optimize over all four scaling parameters and over every combination of three of them (subscripts are removed for convenience). The background describes the reference infidelity obtained by plugging directly the open-loop solution in the sample systems: top and bottom line mark respectively the highest and lowest infidelity encountered, while the central, darker region indicates the mean value and its stochastic error.
• Figure 5: Example of closed-loop optimization. The main plot shows the optimization process, comparing the optimized infidelity (solid blue line with SPAM errors; dot-dashed black line without SPAM errors) and the reference infidelity (dashed orange line) over iterations. The inset shows the final rescaled closed-loop pulse (solid blue line), compared with the original open-loop solution (dashed red line).