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

• URL: http://arxiv.org/abs/2507.17117v1
• Date: Wed, 23 Jul 2025 01:41:39 GMT
• Title: Explicit Formulas for Estimating Trace of Reduced Density Matrix Powers via Single-Circuit Measurement Probabilities
• Authors: Rui-Qi Zhang, Xiao-Qi Liu, Jing Wang, Ming Li, Shu-Qian Shen, Shao-Ming Fei,
• Abstract summary: We propose a universal framework to simultaneously estimate the traces of the $2$nd to the $n$th powers of a reduced density matrix.<n>We develop two algorithms: a purely quantum method and a hybrid quantum-classical approach combining Newton-Girard iteration.<n>We explore various applications including the estimation of nonlinear functions and the representation of entanglement measures.
• Score: 10.43657724071918
• License: http://creativecommons.org/licenses/by/4.0/
• 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 paper, we propose a universal framework to simultaneously estimate the traces of the $2$nd to the $n$th powers of a reduced density matrix using a single quantum circuit with $n$ copies of the quantum state. Specifically, our approach leverages the controlled SWAP test and establishes explicit formulas connecting measurement probabilities to these traces. We further develop two algorithms: 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 $M=O\left(\frac{1}{\epsilon^2}\log(\frac{n}{\delta})\right)$ measurements to achieve precision $\epsilon$ with confidence $1-\delta$. Additionally, we explore various applications 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.

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