Quantum Algorithm And Quantum Control Based On Nuclear Magnetic Resonance:Theoretical And Experimental Aspects

The possibility of building quantum computing machines with application of quantum mechanics has greatly changed human understanding of the nature of computation, and also attracted substantial interests in the active manipulation of the quantum world as well.In the computational aspect of view, it is generally thought that compared with classical computers (Turing machine), quantum computers are essentially more powerful. For some classically intractable problems, quantum algorithms may provide higher efficiency. Currently most quantum algorithms are developed for number theoretical (and algebraic) problems. It is one central challenge in quantum computation research field to explore the potential power of quantum algorithm. As there are still not many quantum algorithms discovered so far, which shows the urgent need for different types of quantum algorithms that will make other classes of problems accessible to efficient solutions.One major problem in building real quantum computers is to realize scaled quantum computation in practical physical systems. Along with the rapid devel-opment of quantum information science, manipulating the micro-world turns out to be more and more important and thus quantum control is attracting substan-tial interests. The goal of quantum control is to devise control methods that can perform certain desired control tasks. Scaled quantum computation requires not only the reduction of decoherence effects but also the capability of implement-ing complicated quantum networks on quantum computers with larger number of qubits. Therefore, in nuclear magnetic resonance quantum computation, the key challenges are to apply control methods to:(i) realize important quantum information processing tasks such as state engineering and high fidelity quan-tum gates, when relaxation effects are present;(ii) implement more complicated quantum circuits on samples that contain more spins.In this thesis, I introduce my works in these two aspects:(1). in the first part, a novel efficient quantum algorithm is proposed. The algo-rithm works for the square-freeness decision problem and square-free decom-position problem which, like the factorization problem, are both intractable by classical means. Our algorithm relies on properties of Gauss sums and us- es the quantum Fourier transform. It is found that the Gauss sum algorithm is not captured by the hidden subgroup problem. Compared with Shor’s algorithm, which also solves the two number theoretic problems, the Gauss sum algorithm is more efficient. Our algorithm introduces new concepts and methods that have not been used in quantum information processing so far and may be applicable to a wider class of problems.(2). in the second part, we study active control of nuclear magnetic resonance system in which relaxation effects are present. Conventionally, relaxation effects are thought to be harmful and must be reduced. However, there exist literature showing that open system control can be advantageous for some particular quantum information processing tasks, eg. algorithmic cooling and entanglement preserving. In our work, we investigate the characteristics of coherently controlled relaxation dynamics. The theory is further testified on a two qubit nuclear magnetic spin system. Under joint effects of coherent pulses and system relaxation, we are able to implement the tasks of open system polarization transfer and pseudopure state preparation with remarkably, both of them being of near optimal performance in purity. Thus our works show the great applicative potential of utilizing rather than suppressing relaxation effects in open system control protocols.(3). in appendix C, I also introduced the numerical work of pulse searching on a12qubit sample. Pulse optimization is one of the key challenges in N-MR quantum computation. Based on the codes from Raymond Laflamme’s group in the Institute of Quantum Computing in Canada, we made important improvements and finished numerical computations for labeled pesudopure state preparation experiment on the12qubit sample. The work is of practical value for larger scale quantum computation.