Performance Comparison of QAOA Mixers for Ternary Portfolio Optimization

Abstract

The Quantum Approximate Optimization Algorithm (QAOA) is a quantum algorithm proposed for Noisy Intermediate-Scale Quantum (NISQ) devices and is regarded as a promising approach to combinatorial optimization problems, with potential applications in the financial sector. In this study, we apply QAOA to the portfolio optimization problem, which is one of the central challenges in financial engineering. A portfolio consists of a combination of multiple assets, and the portfolio optimization problem aims to determine the optimal asset allocation by balancing expected return and risk. In the context of quantum optimization, portfolio optimization is often formulated using discrete variables. Unlike conventional binary formulations, we consider a ternary portfolio optimization problem that accounts for three states-holding, not holding, and short selling-and compare its performance using different mixer operators. Specifically, we implement QAOA with the standard mixer and several XY Mixers (XY Ring, XY Parity Ring, XY Full, and QAMPA), and conducted simulations using real data based on the German stock index (DAX 30) for portfolios consisting of 5 and 8 assets. Furthermore, we introduce noise based on a depolarizing channel to investigate the behavior of the algorithm in realistic environments. The results show that while XY Mixers exhibit superiority in noiseless settings, their advantage degrades in noisy environments, and the optimal choice of mixer depends on both the number of QAOA depths and the noise strength.

Figures (12)

• Figure 1: Feasible ternary portfolios in the risk-return plane for $N=8$ under the constraint $B = 4$. Each feasible portfolio $z\in\{1,0,-1\}^N$ is shown as a circular marker, plotted to its portfolio risk $\sqrt{\sum_{i,j}\sigma_{ij} z_i z_j}$ on the horizontal axis and expected return $\sum_i \mu_i z_i$ on the vertical axis. The triangular marker denotes the portfolio that minimizes the cost function $F(z)$ with $q=1/3$.
• Figure 2: Computed energy landscapes for depth parameters $p=1,3,5$, and $7$ in a portfolio with $n=5$ assets under the investment constraint $B=2$.
• Figure 3: Average approximation ratio and ground state probability for QAOA with five different Mixers, evaluated over 20 randomly generated portfolios with $N=5$ assets and investment constraint $B=2$, as functions of the QAOA depth $p$. Error bars indicate standard deviations. The top row shows results obtained using the Statevector Simulator: (a) average approximation ratio $1-r$, and (b) ground state probability $P$. The bottom row shows the results from the Qasm Simulator: (c) $1-r$, and (d) $P$.
• Figure 4: Average approximation ratio and ground state probability for QAOA with five different Mixers, evaluated over 10 randomly generated portfolios with $N=8$ assets and investment constraint $B=4$, as functions of the QAOA depth $p$. Error bars indicate standard deviations. The top row shows results obtained using the Statevector Simulator: (a) average approximation ratio $1-r$, and (b) ground state probability $P$. The bottom row shows the results from the Qasm Simulator: (c) $1-r$, and (d) $P$.
• Figure 5: Average approximation ratio as a function of depolarizing noise strength $\eta$ for five different Mixers, evaluated for $n=5$ assets with an investment constraint $B=2$ at QAOA depths $p=1,3,5$,and $7$.