Y. Shimoni | Searcharxiv

arXiv2005

Algebraic analysis of quantum search with pure and mixed states

An algebraic analysis of Grover's quantum search algorithm is presented for the case in which the initial state is an arbitrary pure quantum state of n qubits. This approach reveals the geometrical structure of the quantum search process, which turns out to be confined to a four-dimensional subspace of the Hilbert space. This work unifies and generalizes earlier results on the time evolution of the amplitudes during the quantum search, the optimal number of iterations and the success probability. Furthermore, it enables a direct generalization to the case in which the initial state is a mixed state, providing an exact formula for the success probability.

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arXiv2003

Analysis of Grover's quantum search algorithm as a dynamical system

Grover's quantum search algorithm is analyzed for the case in which the initial state is an arbitrary pure quantum state $|ϕ>$ of $n$ qubits. It is shown that the optimal time to perform the measurement is independent of $| ϕ>$, namely, it is identical to the optimal time in the original algorithm in which $| ϕ> = | 0>$, with the same number of marked states, $r$. The probability of success $P_{\rm s}$ is obtained, in terms of the amplitudes of the state $| ϕ>$, and is shown to be independent of $r$. A class of states, which includes fixed points and cycles of the Grover iteration operator is identified. The relevance of these results in the context of using the success probability as an entanglement measure is discussed. In particular, the Groverian entanglement measure, previously limited to a single marked state, is generalized to the case of several marked states.

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