Distributed inner product estimation with limited quantum communication
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917805118455808 |
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| author | Arunachalam, Srinivasan Schatzki, Louis |
| author_facet | Arunachalam, Srinivasan Schatzki, Louis |
| contents | We consider the task of distributed inner product estimation when allowed limited quantum communication. Here, Alice and Bob are given $k$ copies of an unknown $n$-qubit quantum states $\vert ψ\rangle,\vert ϕ\rangle$ respectively. They are allowed to communicate $q$ qubits and unlimited classical communication, and their goal is to estimate $|\langle ψ|ϕ\rangle|^2$ up to constant accuracy. We show that $k=Θ(\sqrt{2^{n-q}})$ copies are essentially necessary and sufficient for this task (extending the work of Anshu, Landau and Liu (STOC'22) who considered the case when $q=0$). Additionally, we consider estimating $|\langle ψ|M|ϕ\rangle|^2$, for arbitrary Hermitian $M$. For this task we show that certain norms on $M$ characterize the sample complexity of estimating $|\langle ψ|M|ϕ\rangle|^2$ when using only classical~communication. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_12684 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Distributed inner product estimation with limited quantum communication Arunachalam, Srinivasan Schatzki, Louis Quantum Physics Computational Complexity We consider the task of distributed inner product estimation when allowed limited quantum communication. Here, Alice and Bob are given $k$ copies of an unknown $n$-qubit quantum states $\vert ψ\rangle,\vert ϕ\rangle$ respectively. They are allowed to communicate $q$ qubits and unlimited classical communication, and their goal is to estimate $|\langle ψ|ϕ\rangle|^2$ up to constant accuracy. We show that $k=Θ(\sqrt{2^{n-q}})$ copies are essentially necessary and sufficient for this task (extending the work of Anshu, Landau and Liu (STOC'22) who considered the case when $q=0$). Additionally, we consider estimating $|\langle ψ|M|ϕ\rangle|^2$, for arbitrary Hermitian $M$. For this task we show that certain norms on $M$ characterize the sample complexity of estimating $|\langle ψ|M|ϕ\rangle|^2$ when using only classical~communication. |
| title | Distributed inner product estimation with limited quantum communication |
| topic | Quantum Physics Computational Complexity |
| url | https://arxiv.org/abs/2410.12684 |