The number of steps any classical computer requires in order to find the prime factors of an l-digit integer N increases exponentially with l, at least using algorithms known at present. Factoring large integers is therefore conjectured to be intractable classically, an observation underlying the security of widely used cryptographic codes. Quantum computers, however, could factor integers in only polynomial time, using Shor's quantum factoring algorithm. Although important for the study of quantum computers, experimental demonstration of this algorithm has proved elusive. Here we report an implementation of the simplest instance of Shor's algorithm: factorization of N = 15 (whose prime factors are 3 and 5). We use seven spin-1/2 nuclei in a molecule as quantum bits, which can be manipulated with room temperature liquid-state nuclear magnetic resonance techniques. This method of using nuclei to store quantum information is in principle scalable to systems containing many quantum bits, but such scalability is not implied by the present work. The significance of our work lies in the demonstration of experimental and theoretical techniques for precise control and modelling of complex quantum computers. In particular, we present a simple, parameter-free but predictive model of decoherence effects in our system.

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We report the realization of a nuclear magnetic resonance computer with three quantum bits that simulates an adiabatic quantum optimization algorithm. Adiabatic quantum algorithms offer new insight into how quantum resources can be used to solve hard problems. This experiment uses a particularly well suited three quantum bit molecule and was made possible by introducing a technique that encodes general instances of the given optimization problem into an easily applicable Hamiltonian. Our results indicate an optimal run time of the adiabatic algorithm that agrees well with the prediction of a simple decoherence model.Since the discovery of Shor's[1] and Grover's [2] algorithms, the quest of finding new quantum algorithms proved a formidable challenge. Recently however, a novel algorithm was proposed, using adiabatic evolution [3,4]. Despite the uncertainty in its scaling behavior, this algorithm remains a remarkable discovery because it offers new insights into the potential usefulness of quantum resources for computational tasks.Experimental realizations of quantum algorithms in the past demonstrated Grover's search algorithm [5,6], the Deutsch-Jozsa algorithm [7,8,9], order-finding [10], and Shor's algorithm [11]. Recently, Hogg's algorithm was implemented using only one computational step [12], however a demonstration of an adiabatic quantum algorithm thus far has remained beyond reach.Here, we provide the first experimental implementation of an adiabatic quantum optimization algorithm using three qubits and nuclear magnetic resonance (NMR) techniques [13,14]. NMR techniques are especially attractive because several tens of qubits may be accessible, which is precisely the range that could be crucial in determining the scaling behavior of adiabatic quantum algorithms [15]. Compared to earlier implementations of search problems [5,6], this experiment is a full implementation of a true optimization problem, which does not require a black box function or ancilla bits.This experiment was made possible by overcoming two experimental challenges. First, an adiabatic evolution requires a smoothly varying Hamiltonian over time, but the terms of the available Hamiltonian in our system cannot be smoothly varied and may even have fixed values. We developed a method to approximately smoothly vary a Hamiltonian despite the given restrictions by extending NMR average Hamiltonian techniques [16]. Second, general instances of the optimization algorithm may require the application of Hamiltonians that are not easily accessible. We developed methods to implement general instances of a well known classical NP-complete optimization problem given a fixed natural system Hamiltonian.We provide a concrete procedure detailing these methods. We then apply the results to our optimization problem which is known as Maximum Cut or maxcut [17]. Our experimental results indicate there exists an optimal total running time which can be predicted using a decoherence model that is based on independent stochastic relaxation of the spins. The evolut...

We report the realization of a nuclear magnetic resonance quantum computer which combines the quantum Fourier transform with exponentiated permutations, demonstrating a quantum algorithm for order finding. This algorithm has the same structure as Shor's algorithm and its speed-up over classical algorithms scales exponentially. The implementation uses a particularly well-suited five quantum bit molecule and was made possible by a new state initialization procedure and several quantum control techniques.

We report the experimental implementation of Grover's quantum search algorithm on a quantum computer with three quantum bits. The computer consists of molecules of $^{13}$C-labeled CHFBr$_2$, in which the three weakly coupled spin-1/2 nuclei behave as the bits and are initialized, manipulated, and read out using magnetic resonance techniques. This quantum computation is made possible by the introduction of two techniques which significantly reduce the complexity of the experiment and by the surprising degree of cancellation of systematic errors which have previously limited the total possible number of quantum gates.Comment: Published in Applied Physics Letters, vol. 76, no. 5, 31 January 2000, p.646-648, after minor revisions. (revtex, mypsfig2.sty, 3 figures