Fig. 2: Sketch of the circuit with qubit recycling.

a Recycling. After detection, a measured qubit is either still in the \(\left\vert 1\right\rangle\) state ("dark” outcome) or in one of the two S levels ("bright” outcome). We reset it to the \(\left\vert 0\right\rangle\) state by first applying an addressed π-pulse (1) on the \(\left\vert S^{\prime} \right\rangle \to \left\vert D^{\prime} \right\rangle\)-transition. A subsequent global 854 nm quench pulse (2) transfers population from all D-levels to the P_3/2 manifold, from which (3) spontaneous decay occurs, preferentially to the \(\left\vert 0\right\rangle\) state in the S manifold. We repeat this process twice, which is sufficient to return about 99% of the population to the \(\left\vert 0\right\rangle\) state. b Circuit. The individual qubits are prepared in a product state depending on the random angles β_i and entangled via XX interactions and some single-qubit gates (white boxes) to create a cluster state; see Supplementary Note 3. The measurement of the qubits is achieved by exciting the P ↔ S transition. In order to perform a circuit with recycling, a coherent π-pulse on the \(S\to D^{\prime}\) transition (denoted by H_S) is applied to `hide' the qubits which should not be measured in the D-manifold. After the measurement, the chain is cooled using polarization-gradient cooling. The reset makes use of local pulses on the measured qubit that transfer the remaining population of \(\left\vert S^{\prime} \right\rangle\) to the D<sub>5/2</sub>-manifold (denoted by P) and global pulses that transfer the population of that manifold back to \(\left\vert 0\right\rangle\). Prior to the reset, all unmeasured qubits are `hidden' in the S_1/2-manifold. For this, the population which was in \(\left\vert 0\right\rangle\) prior to the measurement is coherently transferred back to \(\left\vert S\right\rangle\) via a π-pulse (\({{\rm{H}}}_{S}^{-1}\)), and the population which is in \(\left\vert 1\right\rangle\) is transferred to \(\left\vert S^{\prime} \right\rangle\) via a π-pulse on the \(D\to S^{\prime}\) transition (H_D). After the reset procedure (a), a π-pulse (\({{\rm{H}}}_{D}^{-1}\)) is applied to the unmeasured qubits to transfer the population which was previously in \(\left\vert 1\right\rangle\) back from \(S^{\prime}\).