# Interpreting VQE and QAOA based solutions in portfolio optimization

I am going through portfolio optimization using Qiskit. It really looks interesting. The module has both VQE and QAOA algorithms for optimizing the portfolios.

In the VQE based solution the probability of finding the lowest energy state [1 0 0 1] (with energy -0.0149 units) is 0.8456. One can interpret this as the ground state energy has the highest probability to be found (for the optimization problem we address here).

However, in the QAOA based solution, the probability of find the lowest energy state
[1 0 0 1] (with energy -0.0149 units) is 0.1680. This probability of 0.1680 is almost same as the probability of finding other states as can be seen from the table below. This means the states ([1 0 1 0], [1 1 0 0], [0 0 1 1], [0 1 0 1], [0 1 1 0]) are in equal superposition with the optimal ground state [1 0 0 1]. I find some ambiguity in interpreting the QAOA result even though both VQE and QAOA give us the same state [1 0 0 1] as the optimal value. Can you please kindly clarify this.

The link to the Qiskit portfolio optimization is given below. https://qiskit.org/documentation/finance/tutorials/01_portfolio_optimization.html

Qiskit Portfolio Optimization QAOA result.

Optimal: selection [1. 0. 0. 1.], value -0.0149

Full result is as follows:

selection Value Probability
[1 0 0 1] -0.0149 0.1680
[1 0 1 0] -0.0140 0.1679
[1 1 0 0] -0.0130 0.1677
[0 0 1 1] -0.0010 0.1656
[0 1 0 1] 0.0002 0.1654
[0 1 1 0] 0.0008 0.1653
[1 1 1 1] 4.0656 0.0000
[0 0 0 0] 4.0795 0.0000
[0 1 0 0] 1.0208 0.0000
[1 0 1 1] 1.0049 0.0000
[1 1 1 0] 1.0069 0.0000
[1 1 0 1] 1.0060 0.0000
[0 0 1 0] 1.0197 0.0000
[0 0 0 1] 1.0191 0.0000
[1 0 0 0] 1.0059 0.0000
[0 1 1 1] 1.0199 0.0000

Thank you

• It is a matter of configuring the algorithm. Have you tried playing with the random_seed, max_iters and reps? For example, it is clear that VQE has a higher number of iterations budget. It is also possible that you need to configure the COBYLA optimizer itself. Jul 9, 2022 at 4:28
• Did you run the optimization on simulator or on a real backend? If the latter is valid, try the simulator as the results may be obscured by the decoherence. Jul 9, 2022 at 5:54