Explicit Formulas for Estimating Trace of Reduced Density Matrix Powers via Single-Circuit Measurement Probabilities
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arXiv
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| Autores principales: | , , , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911072258097152 |
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| author | Zhang, Rui-Qi Liu, Xiao-Qi Wang, Jing Li, Ming Shen, Shu-Qian Fei, Shao-Ming |
| author_facet | Zhang, Rui-Qi Liu, Xiao-Qi Wang, Jing Li, Ming Shen, Shu-Qian Fei, Shao-Ming |
| contents | 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 $M=O\left(\frac{1}{ε^2}\log(\frac{n}δ)\right)$ measurements to achieve precision $ε$ with confidence $1-δ$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_17117 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Explicit Formulas for Estimating Trace of Reduced Density Matrix Powers via Single-Circuit Measurement Probabilities Zhang, Rui-Qi Liu, Xiao-Qi Wang, Jing Li, Ming Shen, Shu-Qian Fei, Shao-Ming Quantum Physics 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 $M=O\left(\frac{1}{ε^2}\log(\frac{n}δ)\right)$ measurements to achieve precision $ε$ with confidence $1-δ$. 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. |
| title | Explicit Formulas for Estimating Trace of Reduced Density Matrix Powers via Single-Circuit Measurement Probabilities |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2507.17117 |