Path Independence in Adiabatic Quantum Computing for Hadamard Gate

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The computation time in adiabatic quantum computing (AQC) is determined by the time limit of the adiabatic evolution, which in turn depends on the evolution path. In this research we have used the variational method to find an optimized path. For the simplest case involving a single qubit and for the most general path involving one or more independent interpolating functions, the result is path independent. This result does not change when there is an extra Hamiltonian term. We have also applied these two scenarios in AQC to a Hadamard gate. Adding an extra Hamiltonian gives a non-trivial result compared to the normal AQC, however it does not result in a speed-up. Moreover, we show that in these two scenarios we can choose an arbitrary path provided that it satisfies the boundary conditions.