Fig. 3: 1D entanglement phases obtained from 2D shallow quantum circuits.

a, Schematic of the 2D grid of qubits. At each cycle (blue boxes) of the circuit, random single-qubit and two-qubit iSWAP-like gates are applied to each qubit in the cycle sequence shown. The random single-qubit gate (SQ, grey) is chosen randomly from the set \(\{\sqrt{{X}^{\pm 1}},\sqrt{{Y}^{\pm 1}},\sqrt{{W}^{\pm 1}},\sqrt{{V}^{\pm 1}}\}\), where \(W=(X+Y)/\sqrt{2}\) and \(V=(X-Y)/\sqrt{2}\). At the end of the circuit, the lower M = 12 qubits are measured and post-selected on the most probable bitstring. b, Second Renyi entropy of contiguous subsystems A of the L = 7 edge qubits at various depths. The measurement is noise mitigated in the same way as in Fig. 2. c, Second Renyi mutual information \({{\mathcal{I}}}_{AB}^{(2)}\) between two-qubit subsystems A and B against depth T and distance x (the number of qubits between A and B). d, \({{\mathcal{I}}}_{AB}^{(2)}\) as a function of T for two-qubit subsystems A and B at maximum separation. e, \({{\mathcal{I}}}_{AB}^{(2)}\) versus x for T = 3 and T = 6 for different volumes of A and B.