We discuss the problem of filtering out abnormal states from a larger number of quantum states. For this type of problem with [Formula: see text] items to be searched, both the traditional search by enumeration and classical Grover search algorithm have the complexity about [Formula: see text]. In this letter a novel quantum search scheme with exponential speed up is proposed for abnormal states. First, a new comprehensive quantum operator is well-designed to extract the superposition state containing all abnormal states with unknown number [Formula: see text] with complexity [Formula: see text] in probability 1 via well-designed parallel phase comparison. Then, every abnormal state is achieved respectively from [Formula: see text] abnormal states via [Formula: see text] times’ measurement. Finally, a numerical example is given to show the efficiency of the proposed scheme.
We discuss the problem of filtering out abnormal states from a larger number of quantum states. For this type of problem with [Formula: see text] items to be searched, both the traditional search by enumeration and classical Grover search algorithm have the complexity about [Formula: see text]. In this letter a novel quantum search scheme with exponential speed up is proposed for abnormal states. First, a new comprehensive quantum operator is well-designed to extract the superposition state containing all abnormal states with unknown number [Formula: see text] with complexity [Formula: see text] in probability 1 via well-designed parallel phase comparison. Then, every abnormal state is achieved respectively from [Formula: see text] abnormal states via [Formula: see text] times’ measurement. Finally, a numerical example is given to show the efficiency of the proposed scheme.
We generalize Grover's unstructured quantum search algorithm to enable it to work with arbitrary starting superpositions and arbitrary unitary operators. We show that the generalized quantum search algorithm, when cast in a special orthonormal basis, can be understood as performing an exact rotation of a starting superposition into a target superposition. We derive a formula for the success probability of the generalized quantum search algorithm after n rounds of amplitude amplification. We then use this formula to determine the optimal strategy for a punctuated quantum search algorithm, i.e., one in which the amplitude amplified state is observed before the point of maximum success probability. On average, the optimal strategy is about 12% better than the naive use of Grover's algorithm. The speedup obtained is not dramatic but it illustrates that a hybrid use of quantum computing and classical computing techniques can yield a performance that is better than either alone. In addition, we show that a punctuated quantum algorithm that takes the same average computation time as Grover's standard algorithm only requires half the coherence time. We then extend the analysis to the case of a society of k quantum searches acting in parallel. We derive an analytic formula that connects the degree of parallelism with the expected computation time for k-parallel quantum search. The resulting parallel speedup scales as O(k), while the minimum number of agents needed to ensure success, k, decreases as the inverse of the square of the achievable coherence time. This result has practical significance for the design of rudimentary quantum computers that are likely to have a limited coherence time.
IntroductionFor a search problem associated with a unsorted database, the remarkable Grover's quantum algorithm (1) provides a quadratic speedup over its classical counterpart.The search problem can be described as follows: for a given function f , there exists one unknown element in the set {0, 1, ..., N -1} that satisfies f (x)=-1, say x = τ, whereas the other N -1o n e sg i v e f (x)=1.How many times of evaluations of f are required to determine the element τ for f (τ)=-1?Through a conventional algorithm, one needs O(N) trials to achieve this aim.How about the utility of quantum algorithm for the search?For the scenario in the quantum world, the search problem can be rephrased in the quantum mechanical language: for a given unitary operator I τ , that is sometimes called the oracle operator, and a set of state vectors (orthonormal basis): s = {|0 , |1 , ..., |N -1 }, I τ |x = |x for all states in the set except I τ |x = -|x for x = τ.H owma n yqueriesofI τ are required to determine |τ ?By Grover's algorithm, one needs only O( √ N) quantum mechanical steps to find the marked state |τ out.It has been shown that Grover's algorithm is optimal since it needs minimal oracle calls to perform a quantum search (2).The quantum searching process will be briefly reviewed as follows.The first step of Grover's algorithm is to prepare a superposition state of all elements with uniform probability amplitude:Then apply the Grover kernel G = -I η I τ to |s ,w h er eI η is a unitary operator and contains no bias against the marked state.For large N,a f t e ra b o u tm = π √ N/4 repetitions of G operations, the probability to observe |τ is close to one, i.e., G m |s ∼ |τ .( 2 ) Since every single G involves one query of I τ ,onlyO( √ N) searching steps are required for a quantum search task.In what follows, we will first investigate on the general SU(2) formulation for the kernel of Grover's searching operator G.The discussions of quantum searching certainty, robustness, and the analog analogue version of the Grover's algorithm will be given afterwards.
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.
Grover search algorithm is applied to the current number of objectives in quantum unsorted database, the meaning of each goal difference is not taken into account. When the target is more than a quarter of the total items, the probability of the target falls quickly identify and target increases, and when the target is more than half of the total items, the algorithm will fail. To solve these problems, first of all, based on a weighted index improvement is proposed in which each target in accordance with the right given to the importance of the weight coefficient. These different weight coefficients, we give all the target state, which may make the probability of each target equal weights of the quantum superposition. Secondly, an improved phase matching is proposed the phase rotation in both directions is the same, and the inner product of two states by the target amplitude superposition and initial state of the system is determined. When this inner product is 1/2 or more, when using the improved phase matching, the target may be 100% probability and by only one Grover iteration. Finally, the effectiveness of these measures by a simple search instance validation.
As searching targets in an unordered database with Grover's algorithm,difference in marked items is not taken into consideration.If fraction of marked items is greater than 1/4,success probability of search decreases rapidly with increase of marked items.When the fraction of marked items is greater than 1/2,the algorithm is disabled.Aiming at above problems,an improved Grover's algorithm with weighted targets is proposed in which every target is assigned a weight coefficient according to its significance.With these weight coefficients,targets are represented as quantum superpositions which make probability getting target equal to its weight coefficient.An adaptive phase matching method is proposed based on weighted targets.The directions of phase rotations are contrary,and amplitudes of the two phase rotations are determined by inner-product of target quantum superposition and initial state of the system.As the inner-product is greater than(((3-5)/8)~(1/2),) success probability is equal to 1 with two steps of Grover iteration at most.The improved quantum searching algorithm and the new phase matching are verified by an example.
Grover's algorithm, which was discovered in 1996, is faster than any classical algorithm can ever be. Quantum systems are in a superposition of states and can examine multiple data simultaneously. This work conducted a comprehensive analysis to understand how Grover's algorithm works. Grover's search provides quadratic speed in comparison to the classical one; it explores target data in the unstructured database. Grover search starts with the superposition of all Qubits, which results in the same amplitude for all data points. To achieve superposition, the Hadamard gate was applied to all Qubits. Further, a black box function named Oracle is applied to mark the desired state. In the next phase, amplitude amplification is done, which enhances the probability of the desired data point. Thus, Grover's search yields results with an O (√ N) complexity, while its counterpart classical algorithm searches the desired data point with an O (N) complexity. Experiments have been performed on the Qiskit simulator using 2 and 3 qubits for the detailed analysis of Grover's search.