  Portfolio optimization plays a central role in finance to obtain optimal
portfolio allocations that aim to achieve certain investment goals. Over the
years, many works have investigated different variants of portfolio
optimization. Portfolio optimization also provides a rich area to study the
application of quantum computers to obtain advantages over classical computers.
In this work, we give a sampling version of an existing classical online
portfolio optimization algorithm by Helmbold et al., for which we in turn
develop a quantum version. The quantum advantage is achieved by using
techniques such as quantum state preparation, inner product estimation and
multi-sampling. Our quantum algorithm provides a quadratic speedup in the time
complexity, in terms of $n$, where $n$ is the number of assets in the
portfolio. The transaction cost of both of our classical and quantum algorithms
is independent of $n$ which is especially useful for practical applications
with a large number of assets.
