  We solve a multi-period portfolio optimization problem using D-Wave Systems'
quantum annealer. We derive a formulation of the problem, discuss several
possible integer encoding schemes, and present numerical examples that show
high success rates. The formulation incorporates transaction costs (including
permanent and temporary market impact), and, significantly, the solution does
not require the inversion of a covariance matrix. The discrete multi-period
portfolio optimization problem we solve is significantly harder than the
continuous variable problem. We present insight into how results may be
improved using suitable software enhancements, and why current quantum
annealing technology limits the size of problem that can be successfully solved
today. The formulation presented is specifically designed to be scalable, with
the expectation that as quantum annealing technology improves, larger problems
will be solvable using the same techniques.
