Quantum Lower Bounds by Polynomials
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Quantum Oracle Interrogation: Getting All Information for Almost Half the Price
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
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In the black-box model, problems constrained by a 'promise' are the only ones that admit a quantum exponential speedup over the best classical algorithm in terms of query complexity. The most prominent example of this is the Deutsch-Jozsa algorithm. More recently, Wim van Dam put forward an algorithm for unstructured problems (i.e., those without a promise). We consider the Deutsch-Jozsa algorithm with a less restrictive (or 'broken') promise and study the transition to an unstructured problem. We compare this to the success of van Dam's algorithm. These are both compared with a standard classical sampling algorithm. The Deutsch-Jozsa algorithm remains good as the problem initially becomes less structured, but the van Dam algorithm can be adapted so as to become superior to the Deutsch-Jozsa algorithm as the promise is weakened.