Fig. 1: Monitored circuits and space–time duality mapping.

a, A random (1 + 1)-dimensional monitored quantum circuit composed of both unitary gates and measurements. b, An equivalent dual (1 + 1)-dimensional shallow circuit of size L_x × L_y and depth T with all measurements at the final time formed from a space–time duality mapping of the circuit in a. Because of the non-unitarity nature of measurements, there is freedom as to which dimensions are viewed as ‘time’ and which as ‘space’. In this example, L_y is set by the (1 + 1)D circuit depth and L_x by its spatial size, and T is set by the measurement rate. c, Classical post-processing on a computer of the measurement record (quantum trajectory), and quantum-state readout of a monitored circuit can be used to diagnose the underlying information structures in the system.