The translation of Grover's search algorithm from its standard version, designed for implementation on a single quantum system amenable to projective measurements, into one suitable for an ensemble of quantum computers, whose outputs are expectation values of observables on individual qubits, is described in detail. A filtering scheme, which effectively determines expectation values on a limited portion of the quantum state, is presented and used to locate a single item for searches involving more than one marked item. A truncated version of Grover's algorithm, requiring fewer steps than the translated standard version but locating marked items just as successfully, is proposed. For quantum computational devices that only return expectation values, the truncated version is superior to its standard counterpart. This indicates that it is possible to modify quantum algorithms so as to reduce the required temporal resources by using the ensemble's spatial resources.
The translation of Grover's search algorithm from its standard version, designed for implementation on a single quantum system amenable to projective measurements, into one suitable for an ensemble of quantum computers, whose outputs are expectation values of observables on individual qubits, is described in detail. A filtering scheme, which effectively determines expectation values on a limited portion of the quantum state, is presented and used to locate a single item for searches involving more than one marked item. A truncated version of Grover's algorithm, requiring fewer steps than the translated standard version but locating marked items just as successfully, is proposed. For quantum computational devices that only return expectation values, the truncated version is superior to its standard counterpart. This indicates that it is possible to modify quantum algorithms so as to reduce the required temporal resources by using the ensemble's spatial resources.
Quantum algorithms are conventionally formulated for implementation on a single system of qubits amenable to projective measurements. However, in expectation value quantum computation, such as nuclear magnetic resonance realizations, the computer consists of an ensemble of identical qubit-systems amenable only to expectation value measurements. The prevalent strategy in such expectation value implementations of quantum algorithms has been to retain the conventional formulation's unitary operations but modify its initialization and measurement steps appropriately. This naive approach is not optimal for Grover's algorithm and a shortened version for expectation value quantum computers is presented.
In ensemble (or bulk) quantum computation, all computations are performed on an ensemble of computers rather than on a single computer. Measurements of qubits in an individual computer cannot be performed; instead, only expectation values (over the complete ensemble of computers) can be measured. As a result of this limitation on the model of computation, many algorithms cannot be processed directly on such computers, and must be modified, as the common strategy of delaying the measurements usually does not resolve this ensemble-measurement problem. Here we present several new strategies for resolving this problem. Based on these strategies we provide new versions of some of the most important quantum algorithms, versions that are suitable for implementing on ensemble quantum computers, e.g., on liquid NMR quantum computers. These algorithms are Shor's factorization algorithm, Grover's search algorithm (with several marked items), and an algorithm for quantum fault-tolerant computation. The first two algorithms are simply modified using a randomizing and a sorting strategies. For the last algorithm, we develop a classical-quantum hybrid strategy for removing measurements. We use it to present a novel quantum fault-tolerant scheme. More explicitly, we present schemes for fault-tolerant measurement-free implementation of Toffoli and σ(z)(¼) as these operations cannot be implemented "bitwise", and their standard fault-tolerant implementations require measurement.
In this paper, Grover's quantum search algorithm is analyzed using a classical computer by calculating the amplitudes and the probabilities of finding a single marked state for n=5, 10, 15, 20, 25, and 27 qubit states.The calculations show that the marked state can be found in 𝒪(√𝑁) iterations, where 𝑁 = 2 ( is the number of items.The possibility of improving Grover's search algorithm to find a single item in N search elements is discussed by calculating the amplitudes and hence the probabilities of finding a single marked state for n=5, 10, 15, 20, 25, 30, 35, 40, 45, and 50 qubit states.The calculations showed that the marked state could be found with sufficiently high probability in 𝒪(ln (𝑁)) iterations.This is quite a remarkable speed-up that can be achieved to find a single marked element in an unsorted N search element.
Using cluster state and single qubit measurement one can perform the one-way quantum computation. Here we give a detailed scheme for realizing a modified Grover search algorithm using measurements on cluster state. We give the measurement pattern for the cluster-state realization of the algorithm and estimated the number of measurement needed for its implementation. It is found that O(2 3n/2 n 2 ) number of single qubit measurements is required for its realization in a cluster-state quantum computer.
The iterative nature of Grover's quantum search algorithm may limit the feasibility of its implementation on quantum computers having relatively short coherence times. It may be possible to make a parallel version of the algorithm by extracting partial information about the solution after a limited number of iterations. Here we investigate the use of positive operator value measurements to reduce the coherence time requirements of Grover's algorithm. The utilization of single qubit measurements on multiple copies of the output after a single iteration is analyzed.
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.