Why is it said that Portfolio Optimization is a good case for quantum computing? Is it only speed? IF speed is the only benefit, why cant we build more powerful supercompuetrs?
1$\begingroup$ Why can't we build more powerful supercomputers? This depends massively on the complexity scaling of your problem and how large a problem you want to solve. If the difficulty of your problem scales too badly it will become increasingly more difficult to build the computer large enough to solve the problem. $\endgroup$– RammusJun 24, 2022 at 6:53
$\begingroup$ I edited my answer, maybe it helps. $\endgroup$– Martin VeselyJun 24, 2022 at 16:27
It depends on algorithm you use for the optimization. In certain cases, you can use HHL algorithm because the optimization is converted to solution of a linear system. In this case we can obtain up to exponential speed-up. However, this depends on structure of covariance matrix.
You can also use QUBO solvers like QAOA or VQE. In this case the speed-up is not rigorously proven but in some cases, quantum computer should be faster than classical.
See more on this in my article Application of Quantum Computers in Foreign Exchange Reserves Management.
EDIT: Concerning the speed, portfolio optimization based on Markowitz model is a quadratic programming task. In general, QP is NP-hard task. This means that computational time is increasing exponentially with increasing size of the task. In this case supecomputers do not help. You have to decrease complexity of the task. And this would be possible on quantum computers.