  Quantum learning tasks often leverage randomly sampled quantum circuits to
characterize unknown systems. An efficient approach known as "circuit reusing,"
where each circuit is executed multiple times, reduces the cost compared to
implementing new circuits. This work investigates the optimal reusing parameter
that minimizes the variance of measurement outcomes for a given experimental
cost. We establish a theoretical framework connecting the variance of
experimental estimators with the reusing parameter R. An optimal R is derived
when the implemented circuits and their noise characteristics are known.
Additionally, we introduce a near-optimal reusing strategy that is applicable
even without prior knowledge of circuits or noise, achieving variances close to
the theoretical minimum. To validate our framework, we apply it to randomized
benchmarking and analyze the optimal R for various typical noise channels. We
further conduct experiments on a superconducting platform, revealing a
non-linear relationship between R and the cost, contradicting previous
assumptions in the literature. Our theoretical framework successfully
incorporates this non-linearity and accurately predicts the experimentally
observed optimal R. These findings underscore the broad applicability of our
approach to experimental realizations of quantum learning protocols.
