Efficient implementation of coupled logic gates for quantum computation

Most natural Hamiltonians do not couple specific pairs of quantum bits and spurious couplings occur along with the intended one. We present an efficient scheme that couples any designated pair of spins in heteronuclear spin systems. The scheme is based on the existence of Hadamard matrices. For a system of n spins with pairwise coupling, the scheme concatenates $\mathrm{cn}$ intervals of system evolution and uses at most ${\mathrm{cn}}^2$ pulses where $c\approx 1.$ Our results demonstrate that, in many systems, selective recoupling is possible with linear overhead, contrary to common speculation that exponential effort is always required.