Repetitive readout and real-time control of nuclear spin qubits in $^{171}$Yb atoms

We demonstrate high fidelity repetitive projective measurements of nuclear spin qubits in an array of neutral ytterbium-171 ($^{171}$Yb) atoms. We show that the qubit state can be measured with a fidelity of 0.995(4) under a condition that leaves it in the state corresponding to the measurement outcome with a probability of 0.993(6) for a single tweezer and 0.981(4) averaged over the array. This is accomplished by near-perfect cyclicity of one of the nuclear spin qubit states with an optically excited state under a magnetic field of $B=58$ G, resulting in a bright/dark contrast of $\approx10^5$ during fluorescence readout. The performance improves further as $\sim1/B^2$. The state-averaged readout survival of 0.98(1) is limited by off-resonant scattering to dark states and can be addressed via post-selection by measuring the atom number at the end of the circuit, or during the circuit by performing a measurement of both qubit states. We combine projective measurements with high-fidelity rotations of the nuclear spin qubit via an AC magnetic field to explore several paradigmatic scenarios, including the non-commutivity of measurements in orthogonal bases, and the quantum Zeno mechanism in which measurements "freeze" coherent evolution. Finally, we employ real-time feedforward to repetitively deterministically prepare the qubit in the $+z$ or $-z$ direction after initializing it in an orthogonal basis and performing a projective measurement in the $z$-basis. These capabilities constitute an important step towards adaptive quantum circuits with atom arrays, such as in measurement-based quantum computation, fast many-body state preparation, holographic dynamics simulations, and quantum error correction.