Spectral gap analysis for efficient tunneling in quantum adiabatic optimization
We investigate the efficiency of quantum adiabatic optimization when overcoming potential barriers to get from a local to a global minimum. Specifically we look at $n$ qubit systems with symmetric cost functions $f:{0,1}{n}\ensuremath{\rightarrow}\mathbb{R} where the ground state must tunnel through a potential barrier of width {n}\ensuremath{\alpha} and height {n}\ensuremath{\beta} By the quantum adiabatic theorem the time delay sufficient to ensure tunneling grows quadratically with the inverse spectral gap during this tunneling process. We analyze barrier sizes with $1/2\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\beta} and \ensuremath{\alpha}l1/2$ and show that the minimum gap scales polynomially as {n}{1/2\ensuremath{ \ensuremath{\alpha}\ensuremath{ \ensuremath{\beta} when $2\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\le}1$ and exponentially as {n}\ensuremath{ \ensuremath{\beta}/2}exp(\ensuremath{ }C{n}(2\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{ }1)/2} when $1l2\ensuremath{\alpha}\ensuremath{\beta} Our proof uses elementary techniques and confirms and extends an unpublished folklore result by Goldstone from 2002, which used large spin and instanton methods. Parts of our result also refine recent results by Kong and Crosson [arXiv:1511.06991] and Jiang et al. [arXiv:1603.01293] about the exponential gap scaling.