Citigroup: QAOA for Portfolio Optimization with Classiq on AWS Braket

The Problem

Portfolio optimization is the process of selecting an optimal mix of assets, such as stocks, bonds, and other financial instruments, to achieve the highest possible return for a given level of risk. When the choice is which assets to hold, with limits on how many positions to take, the problem becomes combinatorial: the number of candidate portfolios grows exponentially with the number of assets, and exact optimization becomes intractable at realistic sizes.

This structure makes portfolio selection a natural candidate for quantum optimization research. Citi Innovation Labs explored the question together with Classiq, a quantum software company, and AWS, using Amazon Braket for execution.

QAOA for Portfolio Selection

The team focused on the Quantum Approximate Optimization Algorithm (QAOA), a leading near-term method for combinatorial optimization. QAOA encodes the problem as a cost function over binary variables, where each variable represents whether an asset is included in the portfolio, then uses a parameterised quantum circuit alternating cost and mixer layers to search for low-cost configurations.

Constraints such as “hold exactly k assets” are not naturally unconstrained, so they are added to the objective as penalty terms. The size of the penalty factor matters: too small and the algorithm returns infeasible portfolios, too large and it distorts the objective and slows convergence. A specific focus of the Citi, Classiq, and AWS study was how adjustments to the algorithm’s penalty factor affect its performance.

The simplified snippet below illustrates how a portfolio selection problem can be framed as a constrained binary optimization with a penalty term. It is a teaching example, not Citi’s code or a reproduction of the study’s results.

# Illustrative only: framing portfolio selection as penalised binary optimization.
# Teaching example, not Citi's code or results.
import numpy as np

n_assets, k_select = 6, 3
np.random.seed(1)

expected_return = np.random.normal(0.08, 0.04, n_assets)
cov = np.random.rand(n_assets, n_assets) * 0.01
cov = (cov + cov.T) / 2 + np.eye(n_assets) * 0.02
risk_aversion = 1.0
penalty = 2.0  # the factor whose effect the study examined

def objective(bits):
    x = np.array([int(b) for b in bits])
    ret = expected_return @ x
    risk = x @ cov @ x
    # Penalty enforces "hold exactly k_select assets"
    constraint = penalty * (x.sum() - k_select) ** 2
    return -ret + risk_aversion * risk + constraint

best = min((format(v, f"0{n_assets}b") for v in range(2 ** n_assets)),
           key=objective)
print("Best toy portfolio:", best, "value", round(objective(best), 4))

Running on Amazon Braket

The work used Amazon Braket’s on-demand simulators and quantum processing units to run and evaluate the QAOA circuits. Braket provides a single interface to classical simulators and to real quantum hardware from multiple providers, which suits exploratory studies that move between simulation and on-device runs as circuit sizes grow.

What the Study Explored

The collaboration was framed as an investigation rather than a deployment. The central questions were how tuning the penalty factor changes QAOA’s behaviour on portfolio problems, and whether QAOA would ultimately show a comparative advantage over classical optimization methods. The stated motivation was that, if successful, such methods could pave the way for improved results in portfolio optimization and other complex financial challenges.

No quantitative performance figures or claims of advantage were published on the case study page, and no production system was announced. The honest status is an early-stage exploration of an algorithm and its tuning, consistent with where near-term quantum optimization stands across the financial sector.

Why It Matters

Portfolio optimization is one of the most studied potential applications of quantum computing in finance because the problem is genuinely combinatorial and well understood. The practical reality is that near-term hardware handles only small instances, and whether QAOA can beat mature classical heuristics at useful scale is still an open research question. Studies like this one matter because they build the formulation and tooling, and surface the practical issues, such as penalty tuning, that determine whether the approach can work at all.

For students, the case study illustrates the QAOA workflow, the role of penalty terms in encoding constraints, and the difference between exploring an algorithm and demonstrating an advantage.

Learn more: Classiq Reference | Amazon Braket Reference

Sources

• Citi Explores Quantum Computing for Portfolio Optimisation (Classiq)