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 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. PACS numbers: 03.75. Gg, 03.75.Mn, 42.50.Gy, 03.67.Hk Grover's algorithm [1] is one of the central algorithms in the field of quantum computing that shows a speedup in comparison to classical computing. For an unsorted search space with D elements, classical algorithms take ∝ D steps to find a solution, in comparison to Grover's algorithm taking ∝ √ D steps. While the speedup is only quadratic in comparison to other quantum algorithms such as Shor's algorithm with an exponential speedup, it is of fundamental interest as it can be applied to very wide variety of problems. Many variants and applications of Grover's algorithm have been investigated in the past. The concept of searching can be generalized to abstract solution spaces rather than literal databases, making it applicable in principle to any NP problem [2,3]. Furthermore Grover search finds many uses as a primitive in diverse applications such as cryptography [4,5], matrix and graph problems [6,7], quantum control tasks [8], optimization [9, 10], element distinctness [11], collision problems [12], and quantum machine learning [13].The standard version of Grover's algorithm proceeds by first preparing the register in a equal superposition of all states |+ = 1 √ D D−1 n=0 |n . One then repetitively applies the Oracle operator O = I − 2 n∈T |n n| where T is the set of target (i.e. solution) states, and the Grover operator G 0 = I − 2|0 0|, interspersed with Hadamard operations. The Hadamard operations can be combined with the G 0 by defining G = I − 2|+ +| such that for π 4 D M applications of GO gives with high probability a target state [14]. There is an obvious asymmetry between the operators G and O, as the Oracle inverts the phase of multiple target states, while the Grover operator only inverts the sign of one state. The generalization where both G and O inverts the phase on multiple states was previously studied by Sadhukhan and Tulsi [15]. In their work an analytic solution was found for N = 2 and M = 2, where N is the number of states that the Grover operator inverts and M is the number of target states. However, for larger N, M only numerica...
Development partners and donors have encouraged and incentivized governments in developing countries to explore ways of working with third-party service suppliers to reduce costs and increase service delivery capacity. The distribution of vaccines and medicines has for a long time shown demand for outsourcing but public health systems have struggled to develop the expertise and capital assets necessary to manage such ventures. Existing transport and logistics capacity within public health systems, in particular, is well documented as being insufficient to support existing, let alone future immunization needs. Today, a number of countries are contracting party logistics providers (3PLs) to supplement the in-house distribution operations of public health systems. This commentary reflects on recent, leading examples of outsourcing initiatives to address critical gaps in transport and logistics.
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