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On Quantum Versions of the Yao Principle

Mart de Graaf·Ronald de Wolf
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Abstract

The classical Yao principle states that the complexity R_epsilon(f) of an optimal randomized algorithm for a function f with success probability 1-epsilon equals the complexity max_mu D_epsilon^mu(f) of an optimal deterministic algorithm for f that is correct on a fraction 1-epsilon of the inputs, weighed according to the hardest distribution mu over the inputs. In this paper we investigate to what extent such a principle holds for quantum algorithms. We propose two natural candidate quantum Yao principles, a ``weak'' and a ``strong'' one. For both principles, we prove that the quantum bounded-error complexity is a lower bound on the quantum analogues of max mu D_epsilon^mu(f). We then prove that equality cannot be obtained for the ``strong'' version, by exhibiting an exponential gap. On the other hand, as a positive result we prove that the ``weak'' version holds up to a constant factor for the query complexity of all symmetric Boolean functions

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Mart de Graaf, Ronald de Wolf. 2001-09-14. On Quantum Versions of the Yao Principle. https://arxiv.org/abs/quant-ph/0109070

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