aiqbench analysis
This paper presents a significant algorithmic advance in quantum information theory by reducing the query complexity overhead for estimating von Neumann, Tsallis, and Rényi entropies. By replacing multiplicative worst-case cost assumptions with multi-level algorithms that allocate precision locally, the authors achieve near-optimal scaling for various entropy orders and improve bounds for noninteger Rényi parameters. These efficiency gains are critical for practical quantum compression and entanglement conversion protocols where spectral estimation costs directly impact resource requirements.
Breaking the Multiplicative Overhead in Quantum Entropy Estimation
arXiv:2609.40179v1 Announce Type: new Abstract: The von Neumann, Tsallis, and R\'enyi entropies are fundamental measures of quantum information. A common estimation strategy multiplies the worst-case costs of spectral transformation and statistical readout, leaving gaps to query lower bounds. We reduce this overhead with multi-level algorithms that use local normalization and precision allocation, while variable-time estimation accounts for the probability of reaching expensive spectral tests. With controlled purified access and its inverse, we obtain bounds for additive error $\varepsilon$, rank upper bound $R$, and fixed order $\alpha$. Tsallis entropy estimation has near-optimal rank-independent query complexity $\widetilde O(\varepsilon^{-1/(\alpha-1)})$ for $1<\alpha<2$, when the allowed dimension and rank accommodate the lower-bound instances, and $\widetilde O(1/\varepsilon)$ for $\alpha\ge2$, where $\widetilde O$ suppresses logarithmic factors. The first saves a factor $1/\varepsilon$; the second extends known integer-order scaling to noninteger orders. We also improve the von Neumann entropy bound from $\widetilde O(R/\varepsilon^2)$ to $\widetilde O(R/\varepsilon)$ and obtain $\widetilde O(R/\varepsilon)$ for noninteger R\'enyi orders $1<\alpha<3$, with further bounds at other orders. Our functional-estimation theorem replaces the global product by a sum of local costs weighted by function magnitudes and spectral masses. We further estimate fixed logarithmic moments and entropy variance using $\widetilde O(R/\varepsilon)$ queries, providing efficient access to fluctuation parameters that enter finite-blocklength quantum compression and pure-state entanglement conversion. More broadly, our framework extends beyond entropy to a broad class of density-matrix functionals, offering a systematic approach toward optimal query complexity in quantum spectral estimation.
Abstract:The von Neumann, Tsallis, and Rényi entropies are fundamental measures of quantum information. A common estimation strategy multiplies the worst-case costs of spectral transformation and statistical readout, leaving gaps to query lower bounds. We reduce this overhead with multi-level algorithms that use local normalization and precision allocation, while variable-time estimation accounts for the probability of reaching expensive spectral tests. With controlled purified access and its inverse, we obtain bounds for additive error $\varepsilon$, rank upper bound $R$, and fixed order $\alpha$. Tsallis entropy estimation has near-optimal rank-independent query complexity $\widetilde O(\varepsilon^{-1/(\alpha-1)})$ for $1<\alpha<2$, when the allowed dimension and rank accommodate the lower-bound instances, and $\widetilde O(1/\varepsilon)$ for $\alpha\ge2$, where $\widetilde O$ suppresses logarithmic factors. The first saves a factor $1/\varepsilon$; the second extends known integer-order scaling to noninteger orders. We also improve the von Neumann entropy bound from $\widetilde O(R/\varepsilon^2)$ to $\widetilde O(R/\varepsilon)$ and obtain $\widetilde O(R/\varepsilon)$ for noninteger Rényi orders $1<\alpha<3$, with further bounds at other orders. Our functional-estimation theorem replaces the global product by a sum of local costs weighted by function magnitudes and spectral masses. We further estimate fixed logarithmic moments and entropy variance using $\widetilde O(R/\varepsilon)$ queries, providing efficient access to fluctuation parameters that enter finite-blocklength quantum compression and pure-state entanglement conversion. More broadly, our framework extends beyond entropy to a broad class of density-matrix functionals, offering a systematic approach toward optimal query complexity in quantum spectral estimation.
Main text: 28 pages, 2 figures, 5 tables; appendices: 38 pages, 1 table. Total: 66 pages, 2 figures, 6 tables
arXiv:2609.40179 [quant-ph]
(or arXiv:2609.40179v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2609.40179 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 343 words
