Explicit Formulas for Estimating Trace of Reduced Density Matrix Powers via Single‐Circuit Measurement Probabilities

Abstract In the fields of quantum mechanics and quantum information science, the traces of reduced density matrix powers play a crucial role in the study of quantum systems and have numerous important applications. In this study, a universal framework is proposed to simultaneously estimate the traces of the 2nd to the th powers of a reduced density matrix using a single quantum circuit with copies of the quantum state. Specifically, this approach leverages the controlled SWAP test and establishes explicit formulas connecting measurement probabilities to these traces. Further, two algorithms are developed: a purely quantum method and a hybrid quantum‐classical approach combining Newton–Girard iteration. Rigorous analysis via Hoeffding inequality demonstrates the method's efficiency, requiring only measurements to achieve precision with confidence . Additionally, various applications are explored including the estimation of nonlinear functions and the representation of entanglement measures. Numerical simulations are conducted for two maximally entangled states, the GHZ state and the W state, to validate the proposed method.