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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2408.00314 |
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| _version_ | 1866915471572336640 |
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| author | Shin, Myeongjin Lee, Junseo Lee, Seungwoo Jeong, Kabgyun |
| author_facet | Shin, Myeongjin Lee, Junseo Lee, Seungwoo Jeong, Kabgyun |
| contents | Estimating the trace of quantum state powers, $\text{Tr}(ρ^k)$, for $k$ identical quantum states is a fundamental task with numerous applications in quantum information processing, including nonlinear function estimation of quantum states and entanglement detection. On near-term quantum devices, reducing the required quantum circuit depth, the number of multi-qubit quantum operations, and the copies of the quantum state needed for such computations is crucial. In this work, inspired by the Newton-Girard method, we significantly improve upon existing results by introducing an algorithm that requires only $\mathcal{O}(\widetilde{r})$ qubits and $\mathcal{O}(\widetilde{r})$ multi-qubit gates, where $\widetilde{r} = \min\left\{\text{rank}(ρ), \left\lceil\ln\left({2k}/ε\right)\right\rceil\right\}$. This approach is efficient, as it employs the $\tilde{r}$-entangled copy measurement instead of the conventional $k$-entangled copy measurement, while asymptotically preserving the known sample complexity upper bound. Furthermore, we prove that estimating $\{\text{Tr}(ρ^i)\}_{i=1}^{\tilde{r}}$ is sufficient to approximate $\text{Tr}(ρ^k)$ even for large integers $k > \widetilde{r}$. This leads to a rank-dependent complexity for solving the problem, providing an efficient algorithm for low-rank quantum states while also improving existing methods when the rank is unknown or when the state is not low-rank. Building upon these advantages, we extend our algorithm to the estimation of $\text{Tr}(Mρ^k)$ for arbitrary observables and $\text{Tr}(ρ^k σ^l)$ for multiple quantum states. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_00314 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Resource-efficient algorithm for estimating the trace of quantum state powers Shin, Myeongjin Lee, Junseo Lee, Seungwoo Jeong, Kabgyun Quantum Physics Estimating the trace of quantum state powers, $\text{Tr}(ρ^k)$, for $k$ identical quantum states is a fundamental task with numerous applications in quantum information processing, including nonlinear function estimation of quantum states and entanglement detection. On near-term quantum devices, reducing the required quantum circuit depth, the number of multi-qubit quantum operations, and the copies of the quantum state needed for such computations is crucial. In this work, inspired by the Newton-Girard method, we significantly improve upon existing results by introducing an algorithm that requires only $\mathcal{O}(\widetilde{r})$ qubits and $\mathcal{O}(\widetilde{r})$ multi-qubit gates, where $\widetilde{r} = \min\left\{\text{rank}(ρ), \left\lceil\ln\left({2k}/ε\right)\right\rceil\right\}$. This approach is efficient, as it employs the $\tilde{r}$-entangled copy measurement instead of the conventional $k$-entangled copy measurement, while asymptotically preserving the known sample complexity upper bound. Furthermore, we prove that estimating $\{\text{Tr}(ρ^i)\}_{i=1}^{\tilde{r}}$ is sufficient to approximate $\text{Tr}(ρ^k)$ even for large integers $k > \widetilde{r}$. This leads to a rank-dependent complexity for solving the problem, providing an efficient algorithm for low-rank quantum states while also improving existing methods when the rank is unknown or when the state is not low-rank. Building upon these advantages, we extend our algorithm to the estimation of $\text{Tr}(Mρ^k)$ for arbitrary observables and $\text{Tr}(ρ^k σ^l)$ for multiple quantum states. |
| title | Resource-efficient algorithm for estimating the trace of quantum state powers |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2408.00314 |