Fig. 6: Logical two-copy measurement.

a, Identical scrambling circuits are performed on two copies of 12 logical qubits and then measured in the Bell basis to extract information about the state. Z-basis measurements are corrected with an [[8,3,2]] decoder (when full error detection is not applied). b, Measured entanglement entropy as a function of subsystem size, showing expected Page-curve behaviour¹⁹ for the highly scrambled state, improving with increased error detection. c, Measured and simulated magic (associated with non-Clifford operations) as a function of the number of CCZ gates applied, performed on two copies of scrambled six-logical-qubit systems. d, Pauli string measurement and zero-noise extrapolation using logical qubits. Plot shows the absolute values of all 4¹² Pauli string expectation values, which only have five discrete values for our digital circuit; Pauli strings with the same theory value are grouped. By analysing with sliding-scale error detection, we improve towards the theoretical expectation values (squares) while also improving towards a purity of 1. By extrapolating to perfect purity, we extrapolate the expectation values and better approximate the ideal values (shaded regions are statistical fit uncertainty).