QAOA for Portfolio Optimization

Overview

Portfolio optimization is a fundamental problem in finance: select assets to maximize expected return while minimizing risk. This circuit uses QAOA to find optimal asset allocations.

The Problem

Given:

n assets with expected returns r₁, r₂, ..., rₙ
Covariance matrix Σ (measuring correlations)
Risk aversion parameter q

Maximize: Return - q × Risk

CODE
Objective = Σᵢ rᵢxᵢ - q × Σᵢⱼ σᵢⱼxᵢxⱼ

where xᵢ ∈ {0,1} indicates asset selection.

Encoding

Each qubit represents one asset:

|1⟩ = Include in portfolio
|0⟩ = Exclude from portfolio

Qubit	Asset	Return	Risk
q₀	Bond Fund	5%	Low
q₁	Stock Index	8%	Medium
q₂	Tech Stocks	12%	High
q₃	Crypto	15%	Very High

The Circuit

CODE
┌───┐┌──────────┐┌──────────┐┌─────────┐
q_0: ┤ H ├┤ RZ(-2γr₀)├┤ RZZ(γσ) ├┤ RX(2β) ├─...
     ├───┤├──────────┤├──────────┤├─────────┤
q_1: ┤ H ├┤ RZ(-2γr₁)├┤ RZZ(γσ) ├┤ RX(2β) ├─...
     ...

Running the Circuit

PYTHON
from circuit import run_circuit
import numpy as np

# Custom returns and covariance
returns = np.array([0.05, 0.10, 0.15])
cov = np.array([
    [0.01, 0.002, 0.001],
    [0.002, 0.03, 0.01],
    [0.001, 0.01, 0.05]
])

result = run_circuit(returns=returns, covariance=cov, risk_factor=0.3)
print(f"Best portfolio: {result['best_portfolio']}")

Expected Output

Metric	Example Value
Selected assets	[0, 2]
Expected return	17%
Portfolio risk	0.042
Sharpe ratio	0.83

Risk-Return Tradeoff

risk_factor	Strategy	Result
0.0	Pure return	Highest risk assets
0.5	Balanced	Mix of assets
1.0	Risk-averse	Lower risk assets

Real-World Applications

Retirement planning: Balance growth and stability
Hedge funds: Optimize risk-adjusted returns
Index tracking: Match benchmark with fewer assets

Quantum Advantage

Combinatorial explosion: 2ⁿ possible portfolios
Correlation handling: Naturally encodes asset interactions
Constraint flexibility: Budget, sector limits, etc.