Grover's algorithm is a quantum search algorithm that proceeds by repeated applications of the Grover operator and the Oracle until the state evolves to one of the target states. In the standard version of the algorithm, the Grover operator inverts the sign on only one state. Here we provide an exact solution to the problem of performing Grover's search where the Grover operator inverts the sign on N states. We show the underlying structure in terms of the eigenspectrum of the generalized Hamiltonian, and derive an appropriate initial state to perform the Grover evolution. This allows us to use the quantum phase estimation algorithm to solve the search problem in this generalized case, completely bypassing the Grover algorithm altogether. We obtain a time complexity of this case of sqrt[D/M^{α}], where D is the search space dimension, M is the number of target states, and α≈1, which is close to the optimal scaling.
Grover's search algorithm drives a quantum system from an initial state $|s\ensuremath{\rangle}$ to a desired final state $|t\ensuremath{\rangle}$ by using selective phase inversions of these two states. Earlier, we studied a generalization of Grover's algorithm that relaxes the assumption of the efficient implementation of
Is, the selective phase inversion of the initial state, also known as a diffusion operator. This assumption is known to become a serious handicap in cases of physical interest. Our general search algorithm works with almost any diffusion operator
Ds with the only restriction of having $|s\ensuremath{\rangle}$ as one of its eigenstates. The price that we pay for using any operator is an increase in the number of oracle queries by a factor of
O(B), where
B is a characteristic of the eigenspectrum of
Ds and can be large in some situations. Here we show that by using a quantum Fourier transform, we can regain the optimal query complexity of Grover's algorithm without losing the freedom of using any diffusion operator for quantum searching. However, the total number of operators required by the algorithm is still
O(B) times more than that of Grover's algorithm. So our algorithm offers an advantage only if the oracle operator is computationally more expensive than the diffusion operator, which is true in most search problems.
A general quantum search algorithm aims to evolve a quantum system from a known source state $|s\ensuremath{\rangle}$ to an unknown target state $|t\ensuremath{\rangle}$. It uses a diffusion operator
Ds that has the source state as one of its eigenstates and
It, where ${I}_{\ensuremath{\psi}}$ denotes the selective phase inversion of the $|\ensuremath{\psi}\ensuremath{\rangle}$ state. It evolves $|s\ensuremath{\rangle}$ to a particular state $|w\ensuremath{\rangle}$, which we will call the
w state, in $O(B/\ensuremath{\alpha})$ time steps, where $\ensuremath{\alpha}$ is $|\ensuremath{\langle}t|s\ensuremath{\rangle}|$ and
B is a characteristic of the diffusion operator. Measuring the
w state gives the target state with a success probability of
O(1/B2), and
O(B2) applications of the algorithm can boost it from
O(1/B2) to
O(1), making the total time complexity $O({B}^{3}/\ensuremath{\alpha})$. In the special case of Grover's algorithm,
Ds is
Is and
B is very close to 1. A more efficient way to boost the success probability is quantum amplitude amplification provided we can efficiently implement
Iw. Such an efficient implementation is not known so far. In this paper, we present an efficient algorithm to approximate selective phase inversions of the unknown eigenstates of an operator using a phase estimation algorithm. This algorithm is used to efficiently approximate
Iw, which reduces the time complexity of general algorithm to $O(B/\ensuremath{\alpha})$. Although $O(B/\ensuremath{\alpha})$ algorithms are known to exist, our algorithm offers physical implementation advantages.
Grover's quantum search algorithm drives a quantum computer from a prepared initial state to a desired final state by using selective transformations of these states. Here, we analyze a framework when one of the selective transformations is replaced by a more general unitary transformation. Our framework encapsulates several previous generalizations of the Grover's algorithm. We show that the general quantum search algorithm can be improved by controlling the transformations through an ancilla qubit. As a special case of this improvement, we get a faster quantum algorithm for the two-dimensional spatial search.
The search problem is to find a state satisfying certain properties out of a given set. Grover's algorithm drives a quantum computer from a prepared initial state to the target state and solves the problem quadratically faster than a classical computer. The algorithm uses selective transformations to distinguish the initial state and target state from other states. It does not succeed unless the selective transformations are very close to phase inversions. Here we show a way to go beyond this limitation. An important application lies in quantum error correction, where the errors can cause the selective transformations to deviate from phase inversions. The algorithms presented here are robust to errors as long as the errors are reproducible and reversible. This particular class of systematic errors arises often from imperfections in the apparatus setup. Hence our algorithms offer a significant flexibility in the physical implementation of quantum search.
One iterative in Grover's original quantum search algorithm consists of two Hadamard–Walsh transformations, a selective amplitude inversion and a diffusion amplitude inversion. We concentrate on the relation among the probability of success of the algorithm, the phase shifts, the number of target items and the number of iterations via replacing the two amplitude inversions by phase shifts of an arbitrary ϕ = ϕ(0 ⩽ ϕ, ϕ ⩽ 2π). Then, according to the relation we find out the optimal phase shifts when the number of iterations is given. We present a new quantum search algorithm based on the optimal phase shifts of 1.018 after 0.5π/ iterations. The new algorithm can obtain either a single target item or multiple target items in the search space with the probability of success at least 93.43%.