  We present a quantum algorithm for portfolio optimization. We discuss the
market data input, the processing of such data via quantum operations, and the
output of financially relevant results. Given quantum access to the historical
record of returns, the algorithm determines the optimal risk-return tradeoff
curve and allows one to sample from the optimal portfolio. The algorithm can in
principle attain a run time of ${\rm poly}(\log(N))$, where $N$ is the size of
the historical return dataset. Direct classical algorithms for determining the
risk-return curve and other properties of the optimal portfolio take time ${\rm
poly}(N)$ and we discuss potential quantum speedups in light of the recent
works on efficient classical sampling approaches.
