Quantum search algorithm can be described as the rotation of state vectors in a Hilbert space. The state vectors uniformly rotate by iterative sequences until they hit the target position. To optimize the algorithm, it is necessary to have the precise knowledge about some parameters like the number of target positions. Here we demonstrate the implementation of optimal fixed-point quantum search (OFPQS) algorithm in a five-qubit quantum computer developed by IBM Corporation. We perform the OFPQS algorithm for one and two-iterations and confirm the accuracy of our results by state tomography process.
Quantum search algorithm can be described as the rotation of state vectors in a Hilbert space. The state vectors uniformly rotate by iterative sequences until they hit the target position. To optimize the algorithm, it is necessary to have the precise knowledge about some parameters like the number of target positions. Here we demonstrate the implementation of optimal fixed-point quantum search (OFPQS) algorithm in a five-qubit quantum computer developed by IBM Corporation. We perform the OFPQS algorithm for one and two-iterations and confirm the accuracy of our results by state tomography process.
A quantum algorithm is a set of instructions for a quantum computer, however, unlike algorithms in classical computer science their results cannot be guaranteed. Quantum search algorithm can be described as the rotation of state vectors in a Hilbert space. The state vectors uniformly rotate by iterative sequences until they hit the target position. To optimize the algorithm, it is necessary to have the precise knowledge about some parameters like the number of target positions and total number of states. Here, we demonstrate the implementation of optimal fixed-point quantum search algorithm in IBMQ simulator developed by IBM corporation. We perform the search algorithm for one-iteration for two, three, four and five qubit systems and confirm the accuracy of our results through histogram.
The optimal fixed-point quantum search (OFPQS) algorithm [Phys. Rev. Lett. 113, 210501 (2014)] achieves both the fixed-point property and quadratic speedup over classical algorithms, which gives a sufficient condition on the number of iterations to ensure the success probability is no less than a given lower bound (denoted by 1 − δ 2 ). However, this condition is approximate and not exact. In this paper, we derive the sufficient and necessary condition on the number of feasible iterations, based on which the exact least number of iterations can be obtained. For example, when δ = 0 . 8 , iterations can be saved by almost 25%. Moreover, to find a target item certainly, setting directly 1 − δ 2 = 100 % , the quadratic advantage of the OFPQS algorithm will be lost, then, applying the OFPQS algorithm with 1 − δ 2 < 100 % requires multiple executions, which leads to a natural problem of choosing the optimal parameter δ . For this, we analyze the extreme and minimum properties of the success probability and further analytically derive the optimal δ which minimizes the query complexity. Our study can be a guideline for both the theoretical and application research on the fixed-point quantum search algorithms.
The quantum search algorithm consists of an iterative sequence of selective inversions and diffusion type operations, as a result of which it is able to find a state with desired properties (target state) in an unsorted database of size N in only sqrt[N] queries. This is achieved by designing the iterative transformations in a way that each iteration results in a small rotation of the state vector in a two-dimensional Hilbert space that includes the target state; if we choose the right number of iterative steps, we will stop just at the target state. This Letter shows that by replacing the selective inversions by selective phase shifts of pi/3, the algorithm preferentially converges to the target state irrespective of the step size or number of iterations. This feature leads to robust search algorithms and also to new schemes for quantum control and error correction.
Generic quantum search algorithm searches for target entity in an unsorted database by repeatedly applying canonical Grover’s quantum rotation transform to reach near the vicinity of the target entity represented by a basis state in the Hilbert space associated with the qubits. Thus, when qubits are measured, there is a high probability of finding the target entity. However, the number of times quantum rotation transform is to be applied for reaching near the vicinity of the target is a function of the number of target entities present in the unsorted database, which is generally unknown. A wrong estimate of the number of target entities can lead to overshooting or undershooting the targets, thus reducing the success probability. Some proposals have been made to overcome this limitation. These proposals either employ quantum counting to estimate the number of solutions or fixed point schemes. This paper proposes a new scheme for stopping the application of quantum rotation transformation on reaching near the targets by measurement and subsequent processing to estimate the distance of the state vector from the target states. It ensures a success probability, which is at least greater than half for all the ratios of the number of target entities to the total number of entities in a database, which are less than half. The search problem is trivial for remaining possible ratios. The proposed scheme is simpler than quantum counting and more efficient than the known fixed-point schemes. It has same order of computational complexity as canonical Grover`s search algorithm but is slow by a factor of two and requires an additional ancilla qubit.
The standard quantum search lacks a feature, enjoyed by many classical algorithms, of having a fixed point, i.e. monotonic convergence towards the solution. Recently a fixed point quantum search algorithm has been discovered, referred to as the Phase-π/3 search algorithm, which gets around this limitation. While searching a database for a target state, this algorithm reduces the error probability from ε to ε2q+1 using q oracle queries, which has since been proved to be asymptotically optimal. A different algorithm is presented here, which has the same worst-case behavior as the Phase-π/3 search algorithm but much better average-case behavior. Furthermore the new algorithm gives ε2q+1 convergence for all integral q, whereas the Phase-π/3 search algorithm requires q to be (3n−1)/2 with n a positive integer. In the new algorithm, the operations are controlled by two ancilla qubits, and fixed point behavior is achieved by irreversible measurement operations applied to these ancillas. It is an example of how measurement can allow us to bypass some restrictions imposed by unitarity on quantum computing.
The standard quantum search lacks a feature, enjoyed by many classical algorithms, of having a fixed point, i.e. monotonic convergence towards the solution. Recently a fixed point quantum search algorithm has been discovered, referred to as the Phase-\pi/3 search algorithm, which gets around this limitation. While searching a database for a target state, this algorithm reduces the error probability from \epsilon to \epsilon^{2q+1} using q oracle queries, which has since been proved to be asymptotically optimal. A different algorithm is presented here, which has the same worst-case behavior as the Phase-\pi/3 search algorithm but much better average-case behavior. Furthermore the new algorithm gives \epsilon^{2q+1} convergence for all integral q, whereas the Phase-\pi/3 search algorithm requires q to be (3^{n}-1)/2 with n a positive integer. In the new algorithm, the operations are controlled by two ancilla qubits, and fixed point behavior is achieved by irreversible measurement operations applied to these ancillas. It is an example of how measurement can allow us to bypass some restrictions imposed by unitarity on quantum computing.