Measuring nonstabilizerness via multifractal flatness

Universal quantum computing requires nonstabilizer (magic) quantum states. Quantifying the nonstabilizerness and relating it to other quantum resources is vital for characterizing the complexity of quantum many-body systems. In this work, we prove that a quantum state is a stabilizer if and only if all states belonging to its Clifford orbit have a flat probability distribution on the computational basis. This implies, in particular, that multifractal states are nonstabilizers. We introduce multifractal flatness, a measure based on the participation entropy that quantifies the wave-function distribution flatness. We demonstrate that this quantity is analytically related to the stabilizer entropy of the state and present several examples elucidating the relationship between multifractality and nonstabilizerness. In particular, we show that the multifractal flatness provides an experimentally and computationally viable nonstabilizerness certification. Our work unravels the direct relation between the nonstabilizerness of a quantum state and its wave-function structure.