Optimizing Circuit Reusing and its Application in Randomized Benchmarking

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 times 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 times <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math>. An optimal <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math> for various typical noise channels. We further conduct experiments on a superconducting platform, revealing a non-linear relationship between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math> and the cost, contradicting previous assumptions in the literature. Our theoretical framework successfully incorporates this non-linearity and accurately predicts the experimentally observed optimal <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math>. These findings underscore the broad applicability of our approach to experimental realizations of quantum learning protocols.