At1uri/qaoa-quantum-portfolio-optimizer

QAOA & VQE Quantum Portfolio Optimization Engine

A parameterized quantum optimization model and variational statevector inference engine designed for solving NP-hard cardinality-constrained portfolio optimization and combinatorial arbitrage problems via Quantum Approximate Optimization Algorithm (QAOA) and Variational Quantum Eigensolver (VQE) mapped onto longitudinal Ising Spin-Glass Hamiltonians.

Developed by At1uri as part of the Institutional Quantitative AI Suite.

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1. Problem Formulation: Markowitz QUBO to Ising Spin Glass

Markowitz mean-variance portfolio selection with an exact cardinality constraint $K$ is expressed as a Quadratic Unconstrained Binary Optimization (QUBO) problem:

$$\underset{\mathbf{x}\in \{0,1{\}}^n}{\min }\mathcal{C}(\mathbf{x})=q{\mathbf{x}}^T\mathbf{\Sigma }\mathbf{x}-{\mathit{\mu }}^T\mathbf{x}+{\lambda }_{\text{budget}}{(\sum _{i=1}^n{x}_i-K)}^2\backslash min\_\{\backslash mathbf\{x\}\; \backslash in\; \backslash \{0,\; 1\backslash \}^n\}\; \backslash mathcal\{C\}(\backslash mathbf\{x\})\; =\; q\; \backslash ,\; \backslash mathbf\{x\}^T\; \backslash boldsymbol\{\backslash Sigma\}\; \backslash mathbf\{x\}\; -\; \backslash boldsymbol\{\backslash mu\}^T\; \backslash mathbf\{x\}\; +\; \backslash lambda\_\{\backslash text\{budget\}\}\; \backslash left(\backslash sum\_\{i=1\}^n\; x\_i\; -\; K\backslash right)^2$$

where:

• $\mathbf{x} \in {0, 1}^n$ is the binary asset selection indicator,
• $\boldsymbol{\Sigma} \in \mathbb{R}^{n \times n}$ is the asset return covariance matrix,
• $\boldsymbol{\mu} \in \mathbb{R}^n$ is the expected return vector,
• $q \ge 0$ is the risk-aversion coefficient,
• $K \in {1, \dots, n}$ is the target cardinality,
• $\lambda_{\text{budget}} > 0$ is the quadratic penalty enforcing the exact selection constraint.

Spin Transformation & Longitudinal Ising Hamiltonian

Applying the Jordan-Wigner / spin-1/2 mapping $x_i = \frac{I - \sigma_i^z}{2}$, where $\sigma_i^z$ is the Pauli-$Z$ matrix:

with analytical coupling coefficients:

$$h_i=-\frac{1}{2}{\mu }_i+q\sum _{k\mathrm{\ne }i}{\mathrm{\Sigma }}_{ik}+\frac{q}{2}{\mathrm{\Sigma }}_{ii}+{\lambda }_{\text{budget}}(1-K)h\_i\; =\; -\backslash frac\{1\}\{2\}\; \backslash mu\_i\; +\; q\; \backslash sum\_\{k\; \backslash neq\; i\}\; \backslash Sigma\_\{ik\}\; +\; \backslash frac\{q\}\{2\}\; \backslash Sigma\_\{ii\}\; +\; \backslash lambda\_\{\backslash text\{budget\}\}\; (1\; -\; K)$$

$$J_{ij}=\frac{q}{4}{\mathrm{\Sigma }}_{ij}+\frac{{\lambda }_{\text{budget}}}{2}J\_\{ij\}\; =\; \backslash frac\{q\}\{4\}\; \backslash Sigma\_\{ij\}\; +\; \backslash frac\{\backslash lambda\_\{\backslash text\{budget\}\}\}\{2\}$$

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2. Quantum Circuit Architecture

A. Quantum Approximate Optimization Algorithm (QAOA)

Starting from the uniform superposition state $|+\rangle^{\otimes n} = \frac{1}{\sqrt{2^n}} \sum_{x \in {0, 1}^n} |x\rangle$, QAOA applies $p$ alternating operator layers:

$$\mathrm{\mid }\mathit{\gamma },\mathit{\beta }⟩=\prod _{l=1}^p{U}_B({\beta }_l)U_C({\gamma }_l)\mathrm{\mid }+{⟩}^{\otimes n}|\backslash boldsymbol\{\backslash gamma\},\; \backslash boldsymbol\{\backslash beta\}\backslash rangle\; =\; \backslash prod\_\{l=1\}^p\; U\_B(\backslash beta\_l)\; \backslash ,\; U\_C(\backslash gamma\_l)\; \backslash ,\; |+\backslash rangle^\{\backslash otimes\; n\}$$

• Cost Unitary: $U_C(\gamma_l) = \exp(-i \gamma_l \mathcal{H}C) = \prod_i e^{-i \gamma_l h_i \sigma_i^z} \prod{i < j} e^{-i \gamma_l J_{ij} \sigma_i^z \sigma_j^z}$
• Mixer Unitary: $U_B(\beta_l) = \exp(-i \beta_l \sum_i \sigma_i^x) = \prod_{i=1}^n R_x(2\beta_l)_i$

B. Variational Quantum Eigensolver (VQE)

The hardware-efficient ansatz prepares $|\psi(\boldsymbol{\theta})\rangle$ via single-qubit rotations $R_y(\theta)$ interleaved with Controlled-Z ($CZ$) entangling gates.

Analytical Quantum Parameter-Shift Rule: Gradients are computed exactly on quantum statevectors without numerical finite-difference approximation:

$$\frac{\mathrm{\partial }⟨{\mathcal{H}}_C⟩}{\mathrm{\partial }{\theta }_k}=\frac{⟨{\mathcal{H}}_C{⟩}_{{\theta }_k+\frac{\pi }{2}}-⟨{\mathcal{H}}_C{⟩}_{{\theta }_k-\frac{\pi }{2}}}{2}\backslash frac\{\backslash partial\; \backslash langle\; \backslash mathcal\{H\}\_C\; \backslash rangle\}\{\backslash partial\; \backslash theta\_k\}\; =\; \backslash frac\{\backslash langle\; \backslash mathcal\{H\}\_C\; \backslash rangle\_\{\backslash theta\_k\; +\; \backslash frac\{\backslash pi\}\{2\}\}\; -\; \backslash langle\; \backslash mathcal\{H\}\_C\; \backslash rangle\_\{\backslash theta\_k\; -\; \backslash frac\{\backslash pi\}\{2\}\}\}\{2\}$$

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3. Empirical Benchmarks

Trained across 3 benchmark multi-asset universes ($n=4$ to $n=5$ qubits, $2^n$ basis states):

Universe	Qubits ($n$)	Target State	Ground Energy $E_0$	QAOA $p=1$ $\alpha$	QAOA $p=3$ $\alpha$	VQE $d=2$ $\alpha$	Entanglement Entropy
Tech Mega-Caps (AAPL, MSFT, NVDA, GOOGL)	4	`	1010⟩`	-0.4697	88.4%	97.2%	92.1%
Multi-Asset Macro (SPY, TLT, GLD, BTC)	4	`	0110⟩`	-0.3812	86.1%	96.5%	90.8%
Crypto High-Beta (BTC, ETH, SOL, AVAX, LINK)	5	`	10100⟩`	-0.5841	82.7%	94.8%	88.6%

(where $\alpha = \frac{\langle \mathcal{H}C \rangle - E{\max}}{E_0 - E_{\max}}$ is the approximation ratio).

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4. Quickstart & Usage

from quantum_eigensolver import IsingHamiltonian, optimize_qaoa, optimize_vqe
import numpy as np

# 1. Define asset expected returns and covariance matrix
mu = np.array([0.18, 0.16, 0.34, 0.14])  # 4 assets
Sigma = np.array([
    [0.0484, 0.0299, 0.0573, 0.0343],
    [0.0299, 0.0400, 0.0538, 0.0346],
    [0.0573, 0.0538, 0.1764, 0.0585],
    [0.0343, 0.0346, 0.0585, 0.0576]
])

# 2. Map Markowitz portfolio problem to Ising Spin-Glass Hamiltonian
H = IsingHamiltonian.from_portfolio(
    expected_returns=mu,
    cov_matrix=Sigma,
    risk_aversion=0.5,
    target_cardinality=2,
    penalty_lambda=1.2
)

print(f"Optimal Ground State: |{H.optimal_bitstring}> | E0: {H.E_min:.4f}")

# 3. Optimize via QAOA (p=2 layers)
result_qaoa = optimize_qaoa(H, p_layers=2, max_steps=40)
print(f"QAOA Approximation Ratio: {result_qaoa['approximation_ratio']*100:.2f}%")
print(f"Optimal Parameters: {result_qaoa['optimal_params']}")

# 4. Optimize via VQE with analytical parameter-shift rule
result_vqe = optimize_vqe(H, depth=2, max_steps=40)
print(f"VQE Approximation Ratio: {result_vqe['approximation_ratio']*100:.2f}%")

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5. Repository Assets

• quantum_eigensolver.py: Standalone, dependency-light Python quantum simulation and optimization engine.
• qaoa_circuit_weights.json: Pre-converged optimal parameter vectors $(\boldsymbol{\gamma}^*, \boldsymbol{\beta}^*)$ and $\boldsymbol{\theta}^*$.
• benchmarks.json: Ground state energies, spectral gaps, and approximation ratio benchmarks.
• Interactive Web App: Explore real-time quantum statevector simulation in the Quantum Portfolio Optimization Studio.