Amortized Stabilizer R\'enyi Entropy of Quantum Dynamics

Unraveling the secrets of how much nonstabilizerness a quantum dynamic can generate is crucial for harnessing the power of magic states, the essential resources for achieving quantum advantage and realizing fault-tolerant quantum computation. In this work, we introduce the amortized $\alpha$-stabilizer R\'enyi entropy, a magic monotone for unitary operations that quantifies the nonstabilizerness generation capability of quantum dynamics. Amortization is key in quantifying the magic of quantum dynamics, as we reveal that nonstabilizerness generation can be enhanced by prior nonstabilizerness in input states when considering the $\alpha$-stabilizer R\'enyi entropy, while this is not the case for robustness of magic or stabilizer extent. We demonstrate the versatility of the amortized $\alpha$-stabilizer R\'enyi entropy in investigating the nonstabilizerness resources of quantum dynamics of computational and fundamental interest. In particular, we establish improved lower bounds on the $T$-count of quantum Fourier transforms and the quantum evolutions of one-dimensional Heisenberg Hamiltonians, showcasing the power of this tool in studying quantum advantages and the corresponding cost in fault-tolerant quantum computation.