Optimal twirling depth for classical shadows in the presence of noise.

The classical shadows protocol represents an efficient strategy for estimating properties of an unknown state ρ using a finite number of copies and measurements. In its original form, it involves twirling the state with unitaries randomly selected from a fixed ensemble and measuring the twirled state in a predetermined basis. To compute local properties of the system, it has been demonstrated that optimal sample complexity (the minimal number of required copies) is remarkably achieved when unitaries are drawn from shallow-depth circuits composed of local entangling gates, as opposed to purely local (zero-depth) or global twirling (infinite-depth) ensembles.

In this presentation, I will discuss an improvement of these ideas by considering the sample complexity as a function of the circuit's depth, in the presence of noise. Noise is bound to be present in any experimental implementation of such shallow-depth circuits, which has important implications for determining the optimal twirling ensemble. Under fairly general conditions, I will: i) show that any single-site noise can be accounted for using a depolarizing noise channel with an appropriate damping parameter, f; ii) discuss thresholds f_th at which optimal twirling reduces to local twirling for arbitrary operators; iii) conduct a similar analysis for n^th-order Renyi entropies (where n is greater than or equal to 2); and iv) provide a meaningful upper bound, t_max , on the optimal circuit depth for any finite noise strength f. This upper bound applies to all operators and entanglement entropy measurements. These thresholds strongly constrain the search for optimal strategies to implement the classical shadows protocol and can be easily tailored to the experimental system at hand.