Two-Qubit Implementation of QAOA for MAX-CUT on an NV-Center Quantum Processor

Abstract

We report a proof-of-principle implementation of the quantum approximate optimization algorithm (QAOA) for the smallest nontrivial MAX-CUT instance on an NV-center-based quantum processor operating at room temperature. The two-qubit register is encoded in the electron spin and the ${}^{14}\mathrm{N}$ nuclear spin of a single NV$^-$ center. Using a minimization formulation of MAX-CUT, we implement a single-layer QAOA ansatz with native entangling and single-qubit control operations. Because the optical readout of the NV$^-$ center is not projective in the computational basis, we reconstruct computational-basis populations from averaged fluorescence signals and use them to determine the experimental QAOA cost landscape by scanning the variational parameters. These results show that the core elements of QAOA can be realized on this platform and establish a baseline for future improvements in phase tracking, coherence-preserving control, and scaling to larger problem sizes.

Figures (4)

• Figure 1: QAOA circuit for the two-vertex graph with $p=1$, implemented on the electron spin ($q_{e^-}$) and nitrogen nuclear spin ($q_{{}^{14}\mathrm{N}}$) of an NV$^-$ center. The CNOT-$R_Z(\gamma)$-CNOT block realizes $RZZ(\gamma)$, and the final $R_X(2\beta)$ gates on each qubit implement the mixing unitary.
• Figure 2: Overview of the measurement protocol.(a) One measurement shot consists of optical initialization, coherent control, and optical readout. Microwave pulses address the electron spin, conditional radio-frequency pulses address the nuclear spin, and the detected fluorescence is recorded as a running average of the photon count per shot. (b) Readout circuits used for calibration and population reconstruction. The final operations $I$, $X_1$, $X_2$, and $X_1X_2$ generate four mean photon counts $\bar{n}$, which are used to determine the calibration values $\{I_s\}$ and to reconstruct the computational-basis populations according to Eq. \ref{['eq:four_measurements']}. The circuits are executed in alternating order to reduce drift.
• Figure 3: Convergence of the reconstructed computational-basis populations for the ansatz $|\psi(0.15\pi,1.5\pi)\rangle$ as a function of the accumulated number of shots. Solid lines denote the mean reconstructed populations, while the shaded regions show the corresponding standard deviation. The norm, defined as the sum of the four reconstructed populations (before avering), is shown analogously.
• Figure 4: QAOA cost landscapes for the two-vertex MAX-CUT instance. (a) The ideal QAOA cost landscape obtained from a noise-free state-vector simulation, (b) the experimentally reconstructed landscape measured on the NV-center quantum processor, and (c) the absolute difference between the two.