Implementation of quantum search algorithm using classical Fourier optics

We report on an experiment on Grover's quantum search algorithm showing that classical waves can search a N-item database as efficiently as quantum mechanics can. The transverse beam profile of a short laser pulse is processed iteratively as the pulse bounces back and forth between two mirrors. We directly observe the sought item being found in approximately square root[N] iterations, in the form of a growing intensity peak on this profile. Although the lack of quantum entanglement limits the size of our database, our results show that entanglement is neither necessary for the algorithm itself, nor for its efficiency.     

Design of quantum VQ iteration and quantum VQ encoding algorithm taking O（√N） steps for data compression

Vector quantization (VQ) is an important data compression method. The key of the encoding of VQ is to find the closest vector among N vectors for a feature vector. Many classical linear search algorithms take $O(N)$ steps of distance computing between two vectors. The quantum VQ iteration and corresponding quantum VQ encoding algorithm that takes $O(\sqrt N )$ steps are presented in this paper. The unitary operation of distance computing can be performed on a number of vectors simultaneously because the quantum state exists in a superposition of states. The quantum VQ iteration comprises three oracles, by contrast many quantum algorithms have only one oracle, such as Shor's factorization algorithm and Grover's algorithm. Entanglement state is generated and used, by contrast the state in Grover's algorithm is not an entanglement state. The quantum VQ iteration is a rotation over subspace, by contrast the Grover iteration is a rotation over global space. The quantum VQ iteration extends the Grover iteration to the more complex search that requires more oracles. The method of the quantum VQ iteration is universal.     

Arbitrary Phase Rotation of the Marked State Cannot Be Used for Grover's Quantum Search Algorithm

A misunderstanding that an arbitrary phase rotation of the marked state together with the inversion about average operation can be used to construct a (less efficient) quantum search algorithm is cleared. The π rotation of the phase of the marked state is not only the choice for efficiency, but also vital in Grover's quantum search algorithm. The results also show that Grover's quantum search algorithm is robust.     

量子相关词汇搜索

本文讨论了基于量子并行计算和叠加态原理的量子搜索算法,并结合概率论,给出了从无结构的海量数据(库)中搜索相关词汇(组)的方法,并说明该方法远远优越于经典搜索算法。     

Novel strategy for database searching in spin liouville space by NMR ensemble computing

Quantum computing by nuclear magnetic resonance using pseudopure spin states is bound by the maximal speed of quantum computing algorithms operating on pure states. In contrast to these quantum computing algorithms, a novel algorithm for searching an unsorted database is presented here that operates on truly mixed states in spin Liouville space. It provides an exponential speedup over Grover's quantum search algorithm with the sensitivity scaling exponentially with the number of spins, as for pseudopure state implementations. The minimal decoherence time required is exponentially shorter than that for Grover's algorithm.     

Implementation of conditional phase-shift gate for quantum information processing by NMR,using transition-selective pulses

Experimental realization of quantum information processing in the field of nuclear magnetic resonance (NMR) has been well established. Implementation of conditional phase-shift gate has been a significant step, which has lead to realization of important algorithms such as Grover's search algorithm and quantum Fourier transform. This gate has so far been implemented in NMR by using coupling evolution method. We demonstrate here the implementation of the conditional phase-shift gate using transition selective pulses. As an application of the gate, we demonstrate Grover's search algorithm and quantum Fourier transform by simulations and experiments using transition selective pulses.     

REALIZATION OF QUANTUM ALGORITHMS ON SIMULATOR AND SPECTROMETER OF NMR

报道了利用NMR谱仪和NMR模拟机实现量子算法.以天然苯为样品,我们分别用500M谱仪和NMR模拟机实现了量子D-J算法,Grover搜寻算法及受控非门(C-NOT).通过比较实验谱和模拟谱发现二者能很好符合.利用NMR模拟机实现量子算法比用NMR谱仪更为方便、清晰.     

First experimental demonstration of an exact quantum search algorithm in nuclear magnetic resonance system

The success probability of searching an objective item from an unsorted database using standard Grover's algorithm is usually not exactly 1. It is exactly 1 only when it is used to find the target state from a database with four items. Exact search is always important in theoretical and practical applications. The failure rate of Grover's algorithm becomes big when the database is small, and this hinders the use of the commonly used divide-and-verify strategy. Even for large database, the failure rate becomes considerably large when there are many marked items. This has put a serious limitation on the usability of the Grover's algorithm. An important improved version of the Grover's algorithm, also known as the improved Grover algorithm, solves this problem. The improved Grover algorithm searches arbitrary number of target states from an unsorted database with full success rate. Here, we give the first experimental realization of the improved Grover algorithm, which finds a marked state with certainty, in a nuclear magnetic resonance system. The optimal control theory is used to obtain an optimized control sequence. The experimental results agree well with the theoretical predictions.     

利用NMR谱仪及模拟机实现量子算法

报道了利用NMR谱仪和NMR模拟机实现量子算法.以天然苯为样品,我们分别用500 M谱仪和NMR模拟机实现了量子D-J算法,Grover搜寻算法及受控非门(C NOT).通过比较实验谱和模拟谱发现二者能很好符合.利用NMR模拟机实现量子算法比用NMR谱仪更为方便、清晰.     

Quantum computation through entangling single photons in multipath interferometers

Single-photon interferometry has been used to simulate quantum computations. Its use has been limited to studying few-bit applications due to rapid growth in physical size with numbers of bits. We propose a hybrid approach that employs n photons, each having L degrees of freedom yielding L(n) basis states. The photons are entangled via a quantum nondemolition measurement. This approach introduces the essential element of quantum computing, that is, entanglement into the interferometry. Using these techniques, we demonstrate a controlled-NOT gate and a Grover's search circuit. These ideas are also applicable to the study of nonlocal correlations in many dimensions.     

Efficient State Preparation via Ion Trap Quantum Computing and Quantum Searching Algorithm

We present a scheme to-prepare a quantum state in an ion trap with probability approaching to one by means of ion trap quantum computing and Grover's quantum search algorithm acting on trapped ions.     

Experimental implementation of local adiabatic evolution algorithms by an NMR quantum information processor

Quantum adiabatic algorithm is a method of solving computational problems by evolving the ground state of a slowly varying Hamiltonian. The technique uses evolution of the ground state of a slowly varying Hamiltonian to reach the required output state. In some cases, such as the adiabatic versions of Grover's search algorithm and Deutsch-Jozsa algorithm, applying the global adiabatic evolution yields a complexity similar to their classical algorithms. However, using the local adiabatic evolution, the algorithms given by J. Roland and N.J. Cerf for Grover's search [J. Roland, N.J. Cerf, Quantum search by local adiabatic evolution, Phys. Rev. A 65 (2002) 042308] and by Saurya Das, Randy Kobes, and Gabor Kunstatter for the Deutsch-Jozsa algorithm [S. Das, R. Kobes, G. Kunstatter, Adiabatic quantum computation and Deutsh's algorithm, Phys. Rev. A 65 (2002) 062301], yield a complexity of order N (where N=2(n) and n is the number of qubits). In this paper, we report the experimental implementation of these local adiabatic evolution algorithms on a 2-qubit quantum information processor, by Nuclear Magnetic Resonance.     

Computational leakage: Grover's algorithm with imperfections

We study the effects of dissipation or leakage on the time evolution of Grover's algorithm for a quantum computer. We introduce an effective two-level model with dissipation and randomness (imperfections), which is based upon the idea that ideal Grover's algorithm operates in a 2-dimensional Hilbert space. The simulation results of this model and Grover's algorithm with imperfections are compared, and it is found that they are in good agreement for appropriately tuned parameters. It turns out that the main features of Grover's algorithm with imperfections can be understood in terms of two basic mechanisms, namely, a diffusion of probability density into the full Hilbert space and a stochastic rotation within the original 2-dimensional Hilbert space. Received 12 August 2002 / Received in final form 14 October 2002 Published online 4 February 2003     

Magnetic resonance realization of decoherence-free quantum computation

We report the realization, using nuclear magnetic resonance techniques, of the first quantum computer that reliably executes a complete algorithm in the presence of strong decoherence. The computer is based on a quantum error avoidance code that protects against a class of multiple-qubit errors. The code stores two decoherence-free logical qubits in four noisy physical qubits. The computer successfully executes Grover's search algorithm in the presence of arbitrarily strong engineered decoherence. A control computer with no decoherence protection consistently fails under the same conditions.     

Quantum Query Complexity for Searching Multiple Marked States from an Unsorted Database

An important and usual sort of search problems is to find all marked states from an unsorted database with a large number of states. Grover＇s original quantum search algorithm is for finding single marked state with uncertainty, and it has been generalized to the case of multiple marked states, as well as been modified to find single marked state with certainty. However, the query complexity for finding all multiple marked states has not been addressed. We use a generalized Long＇s algorithm with high precision to solve such a problem. We calculate the approximate query complexity, which increases with the number of marked states and with the precision that we demand. In the end we introduce an algorithm for the problem on a ＂duality computer＂ and show its advantage over other algorithms.     

Quantum Query Complexity for Searching Multiple Marked States from an Unsorted Database

An important and usual sort of search problems is to find all marked states from an unsorted database with a large number of states. Grover's original quantum search algorithm is for finding single marked state with uncertainty, and it has been generalized to the case of multiple marked states, as well as been modified to find single marked state with certainty. However, the query complexity for finding all multiple marked states has not been addressed. We use a generalized Long's algorithm with high precision to solve such a problem. We calculate the approximate query complexity, which increases with the number of marked states and with the precision that we demand. In the end we introduce an algorithm for the problem on a "duality computer" and show its advantage over other algorithms.     

Programmable networks for quantum algorithms

The implementation of a quantum computer requires the realization of a large number of N-qubit unitary operations which represent the possible oracles or which are part of the quantum algorithm. Until now there have been no standard ways to uniformly generate whole classes of N-qubit gates. We develop a method to generate arbitrary controlled phase-shift operations with a single network of one-qubit and two-qubit operations. This kind of network can be adapted to various physical implementations of quantum computing and is suitable to realize the Deutsch-Jozsa algorithm as well as Grover's search algorithm.     

A Four-Phase Improvement of Grover's Algorithm

When applying Grover's algorithm to an unordered database, the probability of obtaining correct results usually decreases as the quantity of target increases. A four-phase improvement of Grover's algorithm is proposed to fix the deficiency, and the unitary and the phase-matching condition are also proposed. With this improved scheme,when the proportion of target is over 1/3, the probability of obtaining correct results is greater than 97.82%with only one iteration using two phases. When the computational complexity is O((?)), the algorithm can succeed with a probability no less than 99.63%.     

Experimental realization of one-way quantum computing with two-photon four-qubit cluster states

We report an experimental realization of one-way quantum computing on a two-photon four-qubit cluster state. This is accomplished by developing a two-photon cluster state source entangled both in polarization and spatial modes. With this special source, we implemented a highly efficient Grover's search algorithm and high-fidelity two-qubit quantum gates. Our experiment demonstrates that such cluster states could serve as an ideal source and a building block for rapid and precise optical quantum computation.