Fig. 1: Overview of the experiment.

a Sketch of the ion trap quantum processor. Strings of up to 16 ions are trapped in a linear chain. Any single ion or pair of ions can be individually addressed by means of steerable, tightly focused laser beams (dark red) to apply resonant operations R_j or Mølmer-Sørensen entangling gates MS_i,j. Global detection, cooling (blue), and repumping (pink) beams are used to perform a mid-circuit reset of part of the qubit register³⁵. b Implemented cluster states. Cluster states with local rotation angles \({\beta }_{i}\in \{0,\frac{\pi }{4},\ldots,\frac{7\pi }{4}\}\) up to a size of 4 × 4 qubits are created in the qubit register. Each cluster state is defined by its N stabilizers S_k which are given by rotated X operators \({\tilde{X}}_{k}={X}_{k}({\beta }_{k})\) at each site k = 1, …, N multiplied with Z operators on the respective neighboring sites. c Recycling of qubits. Using sub-register reset of qubits, we prepare cluster states that are larger than the qubit register. For example, using four ions, we prepare cluster states of size 2 × 3. d Single-instance verification. In order to verify single cluster state preparations with fixed rotation angles β, we measure it in different bases. To perform fidelity estimation we measure uniformly random elements of its stabilizer group, which is obtained by drawing a random product of the N stabilizers S_k, indexed by a length-N random bitstring indicating for each S_k whether it participates in the product. To sample from the output distribution, we measure in the X-basis. These samples are verified in small instances by the empirical total-variation distance (TVD). e Average-case verification. To assess the average quality of the cluster state preparations, we perform measurements on cluster states with random rotations. By measuring a random element of the stabilizer group of each random cluster state, we obtain an estimate of the average fidelity. From the samples from random cluster states in the X-basis, we compute the cross-entropy benchmark (XEB) by averaging the ideal probabilities p_β(x) corresponding to the samples x and the cluster with angles β.