In this work, we consider the fundamental task of distributed inner product estimation when allowed limited communication. Suppose Alice and Bob are given k copies of an unknown n-qubit quantum state |ψ⟩,|ϕ⟩ respectively, are allowed to send q qubits to one another, and the task is to estimate |⟨ψ|ϕ⟩|² up to constant additive error. We show that k = Θ(√{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 also consider the task when the goal of the players is to estimate |⟨ψ|M|ϕ⟩|², for arbitrary Hermitian M. For this task we show that certain norms on M determine the sample complexity of estimating |⟨ψ|M|ϕ⟩|² when using only classical communication.
@InProceedings{arunachalam_et_al:LIPIcs.STACS.2025.11, author = {Arunachalam, Srinivasan and Schatzki, Louis}, title = {{Generalized Inner Product Estimation with Limited Quantum Communication}}, booktitle = {42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)}, pages = {11:1--11:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-365-2}, ISSN = {1868-8969}, year = {2025}, volume = {327}, editor = {Beyersdorff, Olaf and Pilipczuk, Micha{\l} and Pimentel, Elaine and Thắng, Nguy\~{ê}n Kim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2025.11}, URN = {urn:nbn:de:0030-drops-228366}, doi = {10.4230/LIPIcs.STACS.2025.11}, annote = {Keywords: Quantum property testing, Quantum Distributed Algorithms} }
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