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| Format: | Preprint |
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2022
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| Online-Zugang: | https://arxiv.org/abs/2212.08559 |
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| _version_ | 1866910704612671488 |
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| author | Briët, Jop Gutiérrez, Francisco Escudero Gribling, Sander |
| author_facet | Briët, Jop Gutiérrez, Francisco Escudero Gribling, Sander |
| contents | A surprising 'converse to the polynomial method' of Aaronson et al. (CCC'16) shows that any bounded quadratic polynomial can be computed exactly in expectation by a 1-query algorithm up to a universal multiplicative factor related to the famous Grothendieck constant. Here we show that such a result does not generalize to quartic polynomials and 2-query algorithms, even when we allow for additive approximations. We also show that the additive approximation implied by their result is tight for bounded bilinear forms, which gives a new characterization of the Grothendieck constant in terms of 1-query quantum algorithms. Along the way we provide reformulations of the completely bounded norm of a form, and its dual norm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_08559 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Grothendieck inequalities characterize converses to the polynomial method Briët, Jop Gutiérrez, Francisco Escudero Gribling, Sander Quantum Physics Computational Complexity A surprising 'converse to the polynomial method' of Aaronson et al. (CCC'16) shows that any bounded quadratic polynomial can be computed exactly in expectation by a 1-query algorithm up to a universal multiplicative factor related to the famous Grothendieck constant. Here we show that such a result does not generalize to quartic polynomials and 2-query algorithms, even when we allow for additive approximations. We also show that the additive approximation implied by their result is tight for bounded bilinear forms, which gives a new characterization of the Grothendieck constant in terms of 1-query quantum algorithms. Along the way we provide reformulations of the completely bounded norm of a form, and its dual norm. |
| title | Grothendieck inequalities characterize converses to the polynomial method |
| topic | Quantum Physics Computational Complexity |
| url | https://arxiv.org/abs/2212.08559 |