Dynamical Decoupling Schemes Derived From Hamilton Cycles

We address the problem of decoupling the interactions in a spin network governed by a pair-interaction Hamiltonian. Combinatorial schemes for decoupling and for manipulating the couplings of Hamiltonians have been developed which use selective pulses. In this paper we consider an additional requirement on these pulse sequences: as few different control operations as possible should be used. This requirement is motivated by the fact that to find an optimal implementation of each individual selective pulse will be expensive since it requires to solve a pulse shaping problem. Hence, it is desirable to use as few different selective pulses as possible. As a first result we show that for d-dimensional systems, where d >= 2, the ability to implement only two control operations is sufficient to turn off the time evolution. Next, we address the case of a bipartite system with local control and show that four different control operations are sufficient. Finally, turning to networks consisting of several d-dimensional nodes, we show that decoupling can be achieved if one is able to control a number of different control operations which is logarithmic in the number of nodes. We give an explicit family of efficient decoupling schemes with logarithmic number of different pulses based on the classic Hamming codes. We also provide a table of the best known decoupling schemes for small networks of qubits.