A major outstanding problem for many quantum clock synchronization protocols is the hidden assumption of the availability of synchronized clocks within the protocol. In general, quantum operations between two parties do not have consistent phase definitions of quantum states, which introduce an unknown systematic phase error. We show that despite prior arguments to the contrary, it is possible to remove this unknown phase via entanglement purification. This closes the loophole for entanglement based quantum clock synchronization protocols, which are most compatible with current photon based long-distance entanglement distribution schemes. Starting with noisy Bell pairs, we show that the scheme produces a singlet state for any combination of (i) differing basis conventions for Alice and Bob; (ii) an overall time offset in the execution of the purification algorithm; and (iii) the presence of a noisy channel. Error estimates reveal that better performance than existing classical Einstein synchronization protocols should be achievable using current technology.
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...
In recent years there has been a great deal of focus on a globe-spanning quantum network, including linked satellites for applications ranging from quantum key distribution to distributed sensors and clocks. In many of these schemes, relativistic transformations may have deleterious effects on the purity of the distributed entangled pairs. In this paper, we make a comparison of several entanglement distribution schemes in the context of special relativity. We consider three types of entangled photon states: polarization, single photon, and Laguerre-Gauss mode entangled states. All three types of entangled states suffer relativistic corrections, albeit in different ways. These relativistic effects become important in the context of applications such as quantum clock synchronization, where high fidelity entanglement distribution is required.
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