Generalized Inner Product Estimation with Limited Quantum Communication

Authors Srinivasan Arunachalam , Louis Schatzki



PDF
Thumbnail PDF

File

LIPIcs.STACS.2025.11.pdf
  • Filesize: 0.8 MB
  • 17 pages

Document Identifiers

Author Details

Srinivasan Arunachalam
  • IBM Quantum, Almaden, CA, USA
Louis Schatzki
  • Electrical and Computer Engineering, University of Illinois Urbana-Champaign, IL, USA

Cite As Get BibTex

Srinivasan Arunachalam and Louis Schatzki. Generalized Inner Product Estimation with Limited Quantum Communication. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 11:1-11:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.STACS.2025.11

Abstract

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum information theory
Keywords
  • Quantum property testing
  • Quantum Distributed Algorithms

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Pablo Andres-Martinez, Tim Forrer, Daniel Mills, Jun-Yi Wu, Luciana Henaut, Kentaro Yamamoto, Mio Murao, and Ross Duncan. Distributing circuits over heterogeneous, modular quantum computing network architectures. Quantum Science and Technology, 9(4):045021, 2024. Google Scholar
  2. Anurag Anshu and Srinivasan Arunachalam. A survey on the complexity of learning quantum states. Nature Reviews Physics, 6(1):59-69, 2024. Google Scholar
  3. Anurag Anshu, Zeph Landau, and Yunchao Liu. Distributed quantum inner product estimation. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, pages 44-51, 2022. URL: https://doi.org/10.1145/3519935.3519974.
  4. Srinivasan Arunachalam and Louis Schatzki. Distributed inner product estimation with limited quantum communication. arXiv preprint, 2024. URL: https://doi.org/10.48550/arXiv.2410.12684.
  5. Costin Bădescu, Ryan O'Donnell, and John Wright. Quantum state certification. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 503-514, 2019. URL: https://doi.org/10.1145/3313276.3316344.
  6. Adriano Barenco, Andre Berthiaume, David Deutsch, Artur Ekert, Richard Jozsa, and Chiara Macchiavello. Stabilization of quantum computations by symmetrization. SIAM Journal on Computing, 26(5):1541-1557, 1997. URL: https://doi.org/10.1137/S0097539796302452.
  7. Charles H Bennett, David P DiVincenzo, Christopher A Fuchs, Tal Mor, Eric Rains, Peter W Shor, John A Smolin, and William K Wootters. Quantum nonlocality without entanglement. Physical Review A, 59(2):1070, 1999. Google Scholar
  8. Michael Brannan. Alice and bob meet banach: The interface of asymptotic geometric analysis and quantum information theory, 2021. Google Scholar
  9. Harry Buhrman, Richard Cleve, John Watrous, and Ronald De Wolf. Quantum fingerprinting. Physical review letters, 87(16):167902, 2001. Google Scholar
  10. Kean Chen, Qisheng Wang, Peixun Long, and Mingsheng Ying. Unitarity estimation for quantum channels. IEEE Transactions on Information Theory, 69(8):5116-5134, 2023. URL: https://doi.org/10.1109/TIT.2023.3263645.
  11. Kean Chen, Qisheng Wang, and Zhicheng Zhang. Local test for unitarily invariant properties of bipartite quantum states. arXiv preprint, 2024. URL: https://doi.org/10.48550/arXiv.2404.04599.
  12. Sitan Chen, Jerry Li, Brice Huang, and Allen Liu. Tight bounds for quantum state certification with incoherent measurements. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 1205-1213. IEEE, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00118.
  13. Eric Chitambar, Debbie Leung, Laura Mančinska, Maris Ozols, and Andreas Winter. Everything you always wanted to know about locc (but were afraid to ask). Communications in Mathematical Physics, 328:303-326, 2014. Google Scholar
  14. Scott M Cohen. Understanding entanglement as resource: Locally distinguishing unextendible product bases. Physical Review A - Atomic, Molecular, and Optical Physics, 77(1):012304, 2008. Google Scholar
  15. Jacob P Covey, Harald Weinfurter, and Hannes Bernien. Quantum networks with neutral atom processing nodes. npj Quantum Information, 9(1):90, 2023. Google Scholar
  16. Andreas Elben, Benoît Vermersch, Rick van Bijnen, Christian Kokail, Tiff Brydges, Christine Maier, Manoj K Joshi, Rainer Blatt, Christian F Roos, and Peter Zoller. Cross-platform verification of intermediate scale quantum devices. Physical review letters, 124(1):010504, 2020. Google Scholar
  17. Heng Fan. Distinguishability and indistinguishability by local operations and classical communication. Physical Review Letters, 92(17):177905, 2004. Google Scholar
  18. Ranjani G Sundaram, Himanshu Gupta, and CR Ramakrishnan. Efficient distribution of quantum circuits. In 35th International Symposium on Distributed Computing (DISC 2021). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  19. Giovanni Guccione, Tom Darras, Hanna Le Jeannic, Varun B Verma, Sae Woo Nam, Adrien Cavaillès, and Julien Laurat. Connecting heterogeneous quantum networks by hybrid entanglement swapping. Science advances, 6(22):eaba4508, 2020. Google Scholar
  20. Aram W Harrow. The church of the symmetric subspace. arXiv preprint, 2013. URL: https://arxiv.org/abs/1308.6595.
  21. Marcel Hinsche, Marios Ioannou, Sofiene Jerbi, Lorenzo Leone, Jens Eisert, and Jose Carrasco. Efficient distributed inner product estimation via pauli sampling. arXiv preprint, 2024. URL: https://arxiv.org/abs/2405.06544.
  22. Christopher Monroe, Robert Raussendorf, Alex Ruthven, Kenneth R Brown, Peter Maunz, L-M Duan, and Jungsang Kim. Large-scale modular quantum-computer architecture with atomic memory and photonic interconnects. Physical Review A, 89(2):022317, 2014. Google Scholar
  23. Michael A Nielsen and Isaac L Chuang. Quantum computation and quantum information. Cambridge university press, 2010. Google Scholar
  24. A Pirker, J Wallnöfer, and W Dür. Modular architectures for quantum networks. New Journal of Physics, 20(5):053054, 2018. Google Scholar
  25. Pranab Sen. A quantum johnson-lindenstrauss lemma via unitary t-designs. arXiv preprint, 2018. URL: https://arxiv.org/abs/1807.08779.
  26. Guifré Vidal and Rolf Tarrach. Robustness of entanglement. Physical Review A, 59(1):141, 1999. Google Scholar
  27. Stephanie Wehner, David Elkouss, and Ronald Hanson. Quantum internet: A vision for the road ahead. Science, 362(6412):eaam9288, 2018. Google Scholar
  28. Jun-Yi Wu, Kosuke Matsui, Tim Forrer, Akihito Soeda, Pablo Andrés-Martínez, Daniel Mills, Luciana Henaut, and Mio Murao. Entanglement-efficient bipartite-distributed quantum computing. Quantum, 7:1196, 2023. URL: https://doi.org/10.22331/Q-2023-12-05-1196.
  29. Zhi-Chao Zhang, Fei Gao, Tian-Qing Cao, Su-Juan Qin, and Qiao-Yan Wen. Entanglement as a resource to distinguish orthogonal product states. Scientific reports, 6(1):30493, 2016. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail