24 research outputs found
The Groverian Measure of Entanglement for Mixed States
The Groverian entanglement measure introduced earlier for pure quantum states
[O. Biham, M.A. Nielsen and T. Osborne, Phys. Rev. A 65, 062312 (2002)] is
generalized to the case of mixed states, in a way that maintains its
operational interpretation. The Groverian measure of a mixed state of n qubits
is obtained by a purification procedure into a pure state of 2n qubits,
followed by an optimization process based on Uhlmann's theorem, before the
resulting state is fed into Grover's search algorithm. The Groverian measure,
expressed in terms of the maximal success probability of the algorithm,
provides an operational measure of entanglement of both pure and mixed quantum
states of multiple qubits. These results may provide further insight into the
role of entanglement in making quantum algorithms powerful.Comment: 6 pages, 2 figure
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.Comment: 13 page
Quantum Searching via Entanglement and Partial Diffusion
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(sqrt{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(sqrt{N/M}). A performance comparison with Grover's algorithm will be
provided.Comment: 19 pages. Submitted to IJQI. Please forward comments/enquires for the
first author to [email protected]