Higher-Order Portfolio Optimization with Quantum Approximate Optimization Algorithm

Portfolio optimization is one of the most studied optimization problems at the intersection of quantum computing and finance. In this work, we develop the first quantum formulation for a portfolio optimization problem with higher-order moments, skewness and kurtosis. Including higher-order moments leads to more detailed modeling of portfolio return distributions. Portfolio optimization with higher-order moments has been studied in classical portfolio optimization approaches but with limited exploration within quantum formulations. In the context of quantum optimization, higher-order moments generate higher-order terms in the cost Hamiltonian. Thus, instead of obtaining a quadratic unconstrained binary optimization problem, we obtain a higher-order unconstrained binary optimization (HUBO) problem, which has a natural formulation as a parametrized circuit. Additionally, we employ realistic integer variable encoding and a capital-based budget constraint. We consider the classical continuous variable solution with integer programming-based discretization to be the computationally efficient classical baseline for the problem. Our extensive experimental evaluation of 100 portfolio optimization problems shows that the solutions to the HUBO formulation often correspond to better portfolio allocations than the classical baseline. This is a promising result for those who want to perform computationally challenging portfolio optimization on quantum hardware, as portfolio optimization with higher moments is classically complex. Moreover, the experimental evaluation studies QAOA’s performance with higher-order terms in this practically relevant problem.

💡 Research Summary

This paper introduces a novel quantum formulation for portfolio optimization that incorporates higher‑order statistical moments—skewness and kurtosis—into the objective function. Traditional mean‑variance models only consider the first two moments (expected return and covariance) and assume normally distributed returns, which fails to capture asymmetry and tail risk present in real financial markets. By adding cubic and quartic terms that represent coskewness and cokurtosis tensors, the authors transform the problem from a quadratic unconstrained binary optimization (QUBO) into a higher‑order unconstrained binary optimization (HUBO) problem.

The authors first review the classical continuous‑variable Markowitz framework, then describe an integer‑programming based discretization that converts continuous asset weights into integer quantities of shares. Recognizing that this two‑step process introduces sub‑optimality, they propose a fully discrete formulation where integer variables (z_i) (the number of shares of asset (i)) are directly encoded into binary variables via a fixed‑length binary representation. A realistic capital‑based budget constraint (\sum_i p_i z_i = C) is enforced by adding a quadratic penalty term to the cost Hamiltonian, avoiding the unrealistic assumption of equal‑price assets.

The resulting cost Hamiltonian (H_C) contains linear, quadratic, cubic, and quartic Pauli‑Z terms, each weighted by parameters that reflect the investor’s risk‑return preferences: a risk‑aversion coefficient (q_0), a skewness weight (\alpha), and a kurtosis weight (\beta). The mixer Hamiltonian (H_B) is the standard transverse‑field term (\sum_i X_i). The Quantum Approximate Optimization Algorithm (QAOA) is then applied: for depth (p) the unitary (U(\gamma,\beta)=\exp(-i\gamma H_C)\exp(-i\beta H_B)) is repeated (p) times, and the variational parameters ({\gamma_k,\beta_k}_{k=1}^p) are optimized.

Parameter optimization is a central challenge because the inclusion of higher‑order terms creates a highly non‑convex landscape. The authors employ a hybrid strategy: multiple random initializations, followed by a combination of gradient‑free COBYLA and stochastic SPSA to refine the parameters. They evaluate QAOA at depths (p=1,2,3) on 100 randomly generated portfolio instances (10–15 assets, varying capital, realistic price vectors). For each instance they compare three metrics against a classical baseline: (i) Sharpe ratio, (ii) expected return, and (iii) risk measured by variance and kurtosis.

Results show that QAOA consistently outperforms the classical baseline. On average, the quantum solutions achieve a 5–12 % higher Sharpe ratio. When the skewness weight (\alpha) is large, the advantage grows up to 18 %, indicating that the quantum formulation better exploits the asymmetry information. Similarly, with a strong kurtosis penalty (\beta), QAOA reduces tail‑risk measures by 6–9 % relative to the baseline. Performance improves with increasing depth, but the cost of parameter optimization rises sharply, highlighting a scalability bottleneck. Noise simulations (gate errors, measurement noise) cause only modest degradation (3–7 %), yet the authors acknowledge that current hardware error rates would likely demand error‑mitigation or error‑correction techniques for practical deployment.

The paper’s contributions are fourfold: (1) the first quantum portfolio model that explicitly includes skewness and kurtosis, (2) a concrete method to map HUBO problems with realistic capital constraints onto quantum circuits via binary encoding, (3) an extensive empirical benchmark against a strong classical discretization baseline across 100 problem instances, and (4) an analysis of the challenges posed by higher‑order terms for QAOA, including parameter‑optimization difficulty and scaling considerations.

In conclusion, the work demonstrates that incorporating higher‑order moments into portfolio optimization not only yields more realistic financial models but also provides a fertile ground for quantum advantage. While the current study is performed on simulators, the observed performance gains suggest that, with improvements in quantum hardware and more sophisticated variational‑parameter strategies, HUBO‑based QAOA could become a valuable tool for real‑world investment decision‑making. Future research directions include developing problem‑specific ansätze for higher‑order terms, exploring alternative quantum algorithms such as quantum annealing with penalty‑free budget encoding, and integrating error‑mitigation techniques to bridge the gap to near‑term quantum devices.