A Quantum Pigeonhole Principle and Two Semidefinite Relaxations of Communication Complexity
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914941136535552 |
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| author | Dvořák, Pavel Loff, Bruno Sherif, Suhail |
| author_facet | Dvořák, Pavel Loff, Bruno Sherif, Suhail |
| contents | We study semidefinite relaxations of $Π_1$ combinatorial statements. By relaxing the pigeonhole principle, we obtain a new "quantum" pigeonhole principle which is a stronger statement. By relaxing statements of the form "the communication complexity of $f$ is $> k$", we obtain new communication models, which we call "$γ_2$ communication" and "quantum-lab protocols". We prove, via an argument from proof complexity, that any natural model obtained by such a relaxation must solve all Karchmer--Wigderson games efficiently. However, the argument is not constructive, so we work to explicitly construct such protocols in these two models. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_04592 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Quantum Pigeonhole Principle and Two Semidefinite Relaxations of Communication Complexity Dvořák, Pavel Loff, Bruno Sherif, Suhail Computational Complexity We study semidefinite relaxations of $Π_1$ combinatorial statements. By relaxing the pigeonhole principle, we obtain a new "quantum" pigeonhole principle which is a stronger statement. By relaxing statements of the form "the communication complexity of $f$ is $> k$", we obtain new communication models, which we call "$γ_2$ communication" and "quantum-lab protocols". We prove, via an argument from proof complexity, that any natural model obtained by such a relaxation must solve all Karchmer--Wigderson games efficiently. However, the argument is not constructive, so we work to explicitly construct such protocols in these two models. |
| title | A Quantum Pigeonhole Principle and Two Semidefinite Relaxations of Communication Complexity |
| topic | Computational Complexity |
| url | https://arxiv.org/abs/2409.04592 |