In this paper, we will define a quantum operator that performs the inversion about the mean only on a subspace of the system (Partial Diffusion Operator). This operator is used in a quantum search algorithm that runs in O( N/M ) for searching an unstructured list of size N with M matches such that 1 ≤ M ≤ N . We will show that the performance of the algorithm is more reliable than known fixed operators quantum search algorithms especially for multiple matches where we can get a solution after a single iteration with probability over 90% if the number of matches is approximately more than one-third of the search space. We will show that the algorithm will be able to handle the case where the number of matches M is unknown in advance such that 1 ≤ M ≤ N in O( N/M ). A performance comparison with Grover's algorithm will be provided.

Since its inception, the “No Free Lunch” theorem (NFL) has concerned the application of symmetry results rather than the symmetries themselves. In our view, the conflation of result and application obscures the simplicity, generality, and power of the symmetries involved. This paper separates result from application, focusing on and clarifying the nature of underlying symmetries. The result is a general set-theoretic version of NFL which speaks to symmetries when arbitrary domains and co-domains are involved. Although our framework is deterministic, we note situations where our deterministic set-theoretic results speak nevertheless to stochastic algorithms.

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