Quantum Theory of NMR Adiabatic Pulses and Their Applications.

Jonathan Vandermause, Chandrasekhar Ramanathan

Long Yu, Guanru Feng, Yongchao Tang +2 more

Having the advantages of inherent robustness and fault tolerance, adiabatic quantum algorithms have attracted much attention in recent years. In this paper, we report the NMR experimental realization of adiabatic algorithms for the modified Simon's problem in a NMR quantum information processor. We first realized an adiabatic quantum algorithm by Rao [Phys. Rev. A 67, 052306 (2003)] and then proposed a quantum algorithm by using the method of enhanced symmetry of the Hamiltonian given in Schaller and Schtzhold [Quantum Inf. Comput. 10, 109 (2010)]. We demonstrated this symmetry-enhanced algorithm in experiment. We studied the effects of the number of Trotter evolution steps on the validity of adiabatic evolution and decoherence. In practical applications, a trade-off between pulse imperfection and adiabatic requirement must be made in the Trotter evolution step number. An optimal evolution step number and run time were indicated by our results. The experimental demonstration also showed that the symmetry-enhanced algorithm gave a better performance.

Tetsuro Nishino

Avik Mitra, T. S. Mahesh, Anil Kumar

NMR implementation of adiabatic algorithms face severe problems in homonuclear spin systems since the qubit selective pulses are long and during this period, evolution under the Hamiltonian and decoherence cause errors. The decoherence destroys the answer as it causes the final state to evolve to mixed state and in homonuclear systems, evolution under the internal Hamiltonian causes phase errors preventing the initial state to converge to the solution state. The resolution of these issues is necessary before one can proceed to implement an adiabatic algorithm in a large system where homonuclear coupled spins will become a necessity. In the present work, we demonstrate that by using "strongly modulated pulses" (SMPs) for the creation of interpolating Hamiltonian, one can circumvent both the problems and successfully implement the adiabatic SAT algorithm in a homonuclear three qubit system. This work also demonstrates that the SMPs tremendously reduce the time taken for the implementation of the algorithm, can overcome problems associated with decoherence, and will be the modality in future implementation of quantum information processing by NMR.

Matías Chávez, Matthias Ernst

This article presents the application of continuous Floquet theory in solid-state nuclear magnetic resonance (NMR). Continuous Floquet theory extends the traditional Floquet theory to non-continuous Hamiltonians, enabling the description of observable effects not fully captured by the traditional Floquet theory due to its requirement for a periodic Hamiltonian. We present closed-form expressions for computing first- and second-order effective Hamiltonians, streamlining integration with the traditional Floquet theory and facilitating application in NMR experiments featuring multiple modulation frequencies. Subsequently, we show examples of the practical application of continuous Floquet theory by investigating several solid-state NMR experiments. These examples illustrate the importance of the duration of the pulse scheme regarding the width of the resonance conditions and the near-resonance behavior.

T. S. Mahesh, C. S. Sudheer Kumar, Udaysinh T. Bhosale