  How to quickly solve the problem of high-order unconstrained binary
optimization (HUBO) has attracted much attention, because of its importance and
wide-range applications. Here we implement HUBO using a quantum adiabatic
algorithm and achieve algorithmic speedup by introducing gauge symmetry into
the algorithm. Gauge symmetry enforces the state to be in the instantaneous
ground state, further speeding up the computation. Specifically we map the HUBO
problem to quantum Z2 lattice gauge theory defined on the dual graph. The gauge
operators are found by using the closed-loop-search algorithm, and subsequently
the speedup scheme with gauge symmetry for HUBO problem is developed. As an
example demonstrated in the classical computers, we present the mathematical
formulation of our speedup scheme and propose the so-called gauged local
annealing (gLQA) , which is the local quantum annealing (LQA) protected by the
gauge symmetry. We then use gLQA to calculate the ground state energy of the Z2
gauge theory. gLQA reduces the computational time by one order of magnitude
from that of LQA.
