Optimal Twirling Depth for Classical Shadows in the Presence of Noise.

The classical shadows protocol is an efficient strategy for estimating properties of an unknown state ρ using a small number of state copies and measurements. In its original form, it involves twirling the state with unitaries from some ensemble and measuring the twirled state in a fixed basis. It was recently shown that for computing local properties, optimal sample complexity (copies of the state required) is remarkably achieved for unitaries drawn from shallow depth circuits composed of local entangling gates, as opposed to purely local (zero depth) or global twirling (infinite depth) ensembles. Here, we consider the sample complexity as a function of the depth of the circuit, in the presence of noise. We find that this noise has important implications for determining the optimal twirling ensemble. Under fairly general conditions, we (i) show that any single-site noise can be accounted for using a depolarizing noise channel with an appropriate damping parameter f, (ii) compute thresholds f_{th} at which optimal twirling reduces to local twirling for Pauli operators, (iii) nth order Renyi entropies (n≥2), and (iv) provide a meaningful upper bound t_{max} on the optimal circuit depth for any finite noise strength f, which applies to observables and entanglement entropy measurements. These thresholds strongly constrain the search for optimal strategies to implement shadow tomography and are easily tailored to the experimental system at hand.