On the sample complexity of purity and inner product estimation

• URL: http://arxiv.org/abs/2410.12712v1
• Date: Wed, 16 Oct 2024 16:17:21 GMT
• Title: On the sample complexity of purity and inner product estimation
• Authors: Weiyuan Gong, Jonas Haferkamp, Qi Ye, Zhihan Zhang,
• Abstract summary: We study the sample complexity of the tasks quantum purity estimation and quantum inner product estimation. In purity estimation, we are to estimate $tr(rho2)$ of an unknown quantum state $rho$ to additive error $epsilon$. For quantum inner product estimation, Alice and Bob are to estimate $tr(rhosigma)$ to additive error $epsilon$ given copies of unknown quantum state $rho$ and $sigma$.
• Score: 8.94496959777308
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• Abstract: We study the sample complexity of the prototypical tasks quantum purity estimation and quantum inner product estimation. In purity estimation, we are to estimate $tr(\rho^2)$ of an unknown quantum state $\rho$ to additive error $\epsilon$. Meanwhile, for quantum inner product estimation, Alice and Bob are to estimate $tr(\rho\sigma)$ to additive error $\epsilon$ given copies of unknown quantum state $\rho$ and $\sigma$ using classical communication and restricted quantum communication. In this paper, we show a strong connection between the sample complexity of purity estimation with bounded quantum memory and inner product estimation with bounded quantum communication and unentangled measurements. We propose a protocol that solves quantum inner product estimation with $k$-qubit one-way quantum communication and unentangled local measurements using $O(median\{1/\epsilon^2,2^{n/2}/\epsilon,2^{n-k}/\epsilon^2\})$ copies of $\rho$ and $\sigma$. Our protocol can be modified to estimate the purity of an unknown quantum state $\rho$ using $k$-qubit quantum memory with the same complexity. We prove that arbitrary protocols with $k$-qubit quantum memory that estimate purity to error $\epsilon$ require $\Omega(median\{1/\epsilon^2,2^{n/2}/\sqrt{\epsilon},2^{n-k}/\epsilon^2\})$ copies of $\rho$. This indicates the same lower bound for quantum inner product estimation with one-way $k$-qubit quantum communication and classical communication, and unentangled local measurements. For purity estimation, we further improve the lower bound to $\Omega(\max\{1/\epsilon^2,2^{n/2}/\epsilon\})$ for any protocols using an identical single-copy projection-valued measurement. Additionally, we investigate a decisional variant of quantum distributed inner product estimation without quantum communication for mixed state and provide a lower bound on the sample complexity.

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