Optimal Twirling Depth for Classical Shadows in the Presence of Noise and Improved Post-Processing

The classical shadow protocol offers a highly efficient method for learning numerous properties of unknown quantum states using minimal measurements and state copies. This protocol involves applying randomly selected unitaries from a predefined ensemble. Recently, it was demonstrated that shallow-depth quantum circuits, composed of local entangling gates, form an optimal set for learning local properties of quantum states.

In this work, we examine the sample complexity as a function of circuit depth in the presence of noise. We find that noise plays a crucial role in determining the optimal twirling ensemble. Under broad conditions, we: (i) demonstrate that any single-site noise can be effectively modeled by a depolarizing noise channel with an appropriate damping parameter, f; (ii) establish thresholds where optimal twirling simplifies to local twirling for arbitrary Pauli operators and n-th order Rényi entropies (for n≥2); (iii) derive an upper bound on the optimal circuit depth for any finite noise strength f, applicable to both Pauli operators and entanglement entropy measurements; and (iv) optimize the post-processing scheme by adapting it to the specific noise features.

These findings significantly restrict the search for optimal strategies in shadow tomography and can be readily adapted to various experimental systems. The enhanced protocol notably reduces sample complexity in the presence of strong gate-dependent noise.