Algorithms on ensemble quantum computers.

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

NMR experiment factors numbers with Gauss sums

We factor the number 157573 using an NMR implementation of Gauss sums.Comment: 4 pages 5 figure

Factorizing Numbers with the Gauss Sum Technique: NMR Implementations

Several physics-based algorithms for factorizing large number were recently published. A notable recent one by Schleich et al. uses Gauss sums for distinguishing between factors and non-factors. We demonstrate two NMR techniques that evaluate Gauss sums and thus implement their algorithm. The first one is based on differential excitation of a single spin magnetization by a cascade of RF pulses. The second method is based on spatial averaging and selective refocusing of magnetization for Gauss sums corresponding to factors. All factors of 16637 and 52882363 are successfully obtained.Comment: 4 pages, 4 figures; Abstract and Conclusion are slightly modified. References added and formatted with Bibte

Quantum information processing by NMR using a 5-qubit system formed by dipolar coupled spins in an oriented molecule

Quantum Information processing by NMR with small number of qubits is well established. Scaling to higher number of qubits is hindered by two major requirements (i) mutual coupling among qubits and (ii) qubit addressability. It has been demonstrated that mutual coupling can be increased by using residual dipolar couplings among spins by orienting the spin system in a liquid crystalline matrix. In such a case, the heteronuclear spins are weakly coupled but the homonuclear spins become strongly coupled. In such circumstances, the strongly coupled spins can no longer be treated as qubits. However, it has been demonstrated elsewhere, that the $2^N$ energy levels of a strongly coupled N spin-1/2 system can be treated as an N-qubit system. For this purpose the various transitions have to be identified to well defined energy levels. This paper consists of two parts. In the first part, the energy level diagram of a heteronuclear 5-spin system is obtained by using a newly developed heteronuclear z-cosy (HET-Z-COSY) experiment. In the second part, implementation of logic gates, preparation of pseudopure states, creation of entanglement and entanglement transfer is demonstrated, validating the use of such systems for quantum information processing.Comment: 23 pages, 8 figure

Algorithmic Cooling and Scalable NMR Quantum Computers

We present here algorithmic cooling (via polarization-heat-bath)- a powerful method for obtaining a large number of highly polarized spins in liquid nuclear-spin systems at finite temperature. Given that spin-half states represent (quantum) bits, algorithmic cooling cleans dirty bits beyond the Shannon's bound on data compression, by employing a set of rapidly thermal-relaxing bits. Such auxiliary bits could be implemented using spins that rapidly get into thermal equilibrium with the environment, e.g., electron spins. Cooling spins to a very low temperature without cooling the environment could lead to a breakthrough in nuclear magnetic resonance experiments, and our ``spin-refrigerating'' method suggests that this is possible. The scaling of NMR ensemble computers is probably the main obstacle to building useful quantum computing devices, and our spin-refrigerating method suggests that this problem can be resolved.Comment: 21 pages, 3 figure