Quantum Adiabatic Computation and the Travelling Salesman Problem

The NP-complete problem of the travelling salesman (TSP) is considered in the framework of quantum adiabatic computation (QAC). We first derive a remarkable lower bound for the computation time for adiabatic algorithms in general as a function of the energy involved in the computation. Energy, and not just time and space, must thus be considered in the evaluation of algorithm complexity, in perfect accordance with the understanding that all computation is physical. We then propose, with oracular Hamiltonians, new quantum adiabatic algorithms of which not only the lower bound in time but also the energy requirement do not increase exponentially in the size of the input. Such an improvement in both time and energy complexity, as compared to all other existing algorithms for TSP, is apparently due to quantum entanglement. We also appeal to the general theory of Diophantine equations in a speculation on physical implementation of those oracular Hamiltonians.