Quantum Portfolio OptimizationThree Problems, Three Quantum Algorithms

When the constraint is a hard budget or risk-capacity cap rather than a target return, portfolio selection is a knapsack problem: choose the subset of assets that maximizes value without exceeding the cap, with optional group caps on top. The quantum approach encodes the binary include/exclude decision as constrained QAOA — a Quantum Approximate Optimization Algorithm cost function whose Quantum Walk Mixer keeps the search inside the constraint-preserving subspace, instead of penalizing violations after the fact.

Dataset and setup: on a 5-asset instance, constrained QAOA runs on 11 qubits on a simulator, with parameter schedules and mixer depth logged for reproducibility. The same instance is benchmarked against MILP and heuristic baselines.

When to pick it: choose QAOA with a quantum walk mixer when your constraint is a hard cap that must never be violated — a knapsack-style budget or risk-capacity limit, optionally with group caps — rather than a continuous risk-return trade-off. See how constrained QAOA compares against a classical solver in QAOA vs. MILP.

Mean-variance optimization looks for continuous allocation weights w that hit a target monthly return while minimizing portfolio variance, subject to a budget constraint:

Adding Lagrange multipliers turns the constrained optimization into a linear system Ax = b, where A packs the covariance matrix \Sigma and the return vector r together with the constraint rows. The HHL (Harrow–Hassidim–Lloyd) algorithm solves that system on a quantum computer by preparing a state proportional to the solution.

Dataset and setup: three assets (AAPL, GOOGL, MSFT), 1,258 rows of historical daily closing prices from 2020–2025, reduced to roughly 60 monthly observations to estimate the expected-return vector and covariance matrix. The target was at least 2.0% expected return per month (≥26.8%/year) with risk capped at 0.5% per month (≤1.8%/year) and no single asset above 40% of the portfolio. The linear system runs on 11 qubits on a simulator, with no training step.

When to pick it: choose HHL when the decision variable is continuous allocation weights rather than a discrete selection, and when you mainly need risk/return functionals — not every coordinate of the weight vector — at scale.

Cardinality-constrained selection asks for exactly K of N assets — a binary decision per asset, \mathbf{x}\in\{0,1\}^N, with \sum_i x_i = K — that balances expected return against risk. The objective folds risk, return and the K-of-N constraint into a single quadratic unconstrained binary optimization (QUBO):

The first term penalizes portfolio variance, the second rewards expected return, and the third is a penalty that enforces the exact count. QUBO maps directly onto an Ising Hamiltonian, which quantum annealing hardware is built to minimize by slowly interpolating from an easy driver Hamiltonian to the problem Hamiltonian, using quantum tunneling to escape the local minima that stall classical heuristics on rugged energy landscapes.

Dataset and setup: ten large-cap tickers (AAPL, MSFT, GOOGL, AMZN, META, NVDA, JPM, XOM, V, PG), daily prices from 2018–2024, with the cardinality constraint set to K = 6 of N = 10. The QUBO runs on 10 logical qubits, executed on a simulator in about two minutes.

When to pick it: choose quantum annealing when the constraint is an exact K-of-N count rather than a soft budget cap, and you want a single tunable QUBO formulation that can run on hybrid or fully classical samplers today and move to QPU hardware later.