dorsal/arxiv
View SchemaNegative weights make adversaries stronger
| Authors | Peter Hoyer, Troy Lee, Robert Spalek |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0611054 |
| URL | https://arxiv.org/abs/quant-ph/0611054 |
| DOI | 10.1145/1250790.1250867 |
| Journal | Proceedings of 39th Annual Symposium on Theory of Computing (STOC), San Diego, California, pp. 526-535, 2007 |
Abstract
The quantum adversary method is one of the most successful techniques for proving lower bounds on quantum query complexity. It gives optimal lower bounds for many problems, has application to classical complexity in formula size lower bounds, and is versatile with equivalent formulations in terms of weight schemes, eigenvalues, and Kolmogorov complexity. All these formulations rely on the principle that if an algorithm successfully computes a function then, in particular, it is able to distinguish between inputs which map to different values. We present a stronger version of the adversary method which goes beyond this principle to make explicit use of the stronger condition that the algorithm actually computes the function. This new method, which we call ADV+-, has all the advantages of the old: it is a lower bound on bounded-error quantum query complexity, its square is a lower bound on formula size, and it behaves well with respect to function composition. Moreover ADV+- is always at least as large as the adversary method ADV, and we show an example of a monotone function for which ADV+-(f)=Omega(ADV(f)^1.098). We also give examples showing that ADV+- does not face limitations of ADV like the certificate complexity barrier and the property testing barrier.
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"abstract": "The quantum adversary method is one of the most successful techniques for\nproving lower bounds on quantum query complexity. It gives optimal lower bounds\nfor many problems, has application to classical complexity in formula size\nlower bounds, and is versatile with equivalent formulations in terms of weight\nschemes, eigenvalues, and Kolmogorov complexity. All these formulations rely on\nthe principle that if an algorithm successfully computes a function then, in\nparticular, it is able to distinguish between inputs which map to different\nvalues.\n We present a stronger version of the adversary method which goes beyond this\nprinciple to make explicit use of the stronger condition that the algorithm\nactually computes the function. This new method, which we call ADV+-, has all\nthe advantages of the old: it is a lower bound on bounded-error quantum query\ncomplexity, its square is a lower bound on formula size, and it behaves well\nwith respect to function composition. Moreover ADV+- is always at least as\nlarge as the adversary method ADV, and we show an example of a monotone\nfunction for which ADV+-(f)=Omega(ADV(f)^1.098). We also give examples showing\nthat ADV+- does not face limitations of ADV like the certificate complexity\nbarrier and the property testing barrier.",
"arxiv_id": "quant-ph/0611054",
"authors": [
"Peter Hoyer",
"Troy Lee",
"Robert Spalek"
],
"categories": [
"quant-ph"
],
"doi": "10.1145/1250790.1250867",
"journal_ref": "Proceedings of 39th Annual Symposium on Theory of Computing\n (STOC), San Diego, California, pp. 526-535, 2007",
"title": "Negative weights make adversaries stronger",
"url": "https://arxiv.org/abs/quant-ph/0611054"
},
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