Theory

Theory

This document describes the mathematical formulation and quantum encodings used in the vqe_portfolio package. It focuses on how classical portfolio optimization problems are mapped to quantum Hamiltonians suitable for hybrid quantum–classical algorithms such as VQE and QAOA.

Table of Contents

1. Classical Mean–Variance Portfolio Optimization

Let:

  • \( \mu \in \mathbb{R}^n \) be expected returns

  • \( \Sigma \in \mathbb{R}^{n \times n} \) be the covariance matrix

  • \( w \in \mathbb{R}^n \) be portfolio weights

The classical mean–variance objective is:

\[ \min_w ; -\mu^\top w + \lambda, w^\top \Sigma w \]

where:

  • \( \lambda > 0 \) controls risk aversion

Typical constraints include:

Long-only constraint

\[ w_i \ge 0 \]

Budget constraint

\[ \sum_i w_i = 1 \]

Cardinality constraint

\[ |w|_0 = K \]

Cardinality constraints introduce combinatorial structure, motivating discrete encodings.

2. Variational Quantum Eigensolvers (VQE)

VQE solves optimization problems of the form:

\[ \min_\theta ; \langle \psi(\theta) | H | \psi(\theta) \rangle \]

where:

  • \( H \) is a problem Hamiltonian

  • \( |\psi(\theta)\rangle \) is a parameterized quantum state

  • \( \theta \) are circuit parameters optimized classically

Algorithm structure:

  1. prepare ansatz state

  2. measure expectation value

  3. update parameters using classical optimizer

  4. repeat until convergence

VQE is particularly useful when:

  • objective functions can be expressed as expectation values

  • constraints can be encoded into Hamiltonians

  • gradients can be estimated efficiently

In this project, the portfolio objective is encoded directly into \( H \).

3. Quantum Approximate Optimization Algorithm (QAOA)

QAOA is a gate-based algorithm designed for combinatorial optimization problems.

It alternates between:

  • cost evolution

  • mixing evolution

The QAOA state is:

\[ |\psi(\boldsymbol{\gamma},\boldsymbol{\beta})\rangle = \prod_{\ell=1}^p e^{-i\beta_\ell H_M} e^{-i\gamma_\ell H_C} |+\rangle^{\otimes n} \]

where:

  • \( H_C \) is the cost Hamiltonian

  • \( H_M \) is the mixer Hamiltonian

  • \( p \) is the circuit depth

Parameters:

  • \( \boldsymbol{\gamma} = (\gamma_1,\dots,\gamma_p) \)

  • \( \boldsymbol{\beta} = (\beta_1,\dots,\beta_p) \)

are optimized using classical optimization.

3.1 Cost Hamiltonian

Portfolio optimization produces a QUBO objective:

\[ C(x) = \lambda x^\top \Sigma x - \mu^\top x + \alpha(\mathbf{1}^\top x - K)^2 \]

which maps to an Ising Hamiltonian:

\[ H_C = \sum_i h_i Z_i + \sum_{i<j} J_{ij} Z_i Z_j + \text{const} \]

This Hamiltonian is shared with Binary VQE.

3.2 Mixer Hamiltonians

Two mixers are implemented.

X mixer

\[ H_M^{(X)} = \sum_i X_i \]

Promotes exploration across the full computational basis.

XY mixer

\[ H_M^{(XY)} = \sum_{i<j} (X_i X_j + Y_i Y_j) \]

Preserves approximate Hamming-weight structure, making it useful for:

  • cardinality-constrained problems

  • constrained combinatorial search spaces

3.3 Sampling interpretation

QAOA produces a probability distribution over bitstrings:

\[ p(x) = |\langle x|\psi\rangle|^2 \]

From this distribution we compute:

Marginal probabilities

\[ p_i = \mathbb{E}[x_i] \]

Top-K projection

Select the \( K \) largest marginals.

Mode bitstring

Most frequently sampled bitstring.

Best feasible candidate

Lowest-cost bitstring satisfying the constraint.

QAOA therefore provides:

  • probabilistic solutions

  • candidate discrete portfolios

  • insight into landscape structure

4. Binary Portfolio Optimization via QUBO

4.1 Binary decision variables

Let:

\[ x_i \in {0,1} \]

indicate asset inclusion.

Objective:

\[ \min_{x \in {0,1}^n} \lambda x^\top \Sigma x - \mu^\top x + \alpha(\mathbf{1}^\top x - K)^2 \]

This is a Quadratic Unconstrained Binary Optimization (QUBO) problem.

4.2 Ising mapping

Binary variables are mapped to Pauli-Z operators:

\[ x_i = \frac{1 - Z_i}{2} \]

Substitution yields:

\[ \begin{align}\begin{aligned} H =\\\sum_i h_i Z_i + \sum_{i<j} J_{ij} Z_i Z_j + \text{const} \end{aligned}\end{align} \]

This Hamiltonian is minimized via:

  • VQE

  • QAOA

4.3 Expectation vs bitstrings

Variational algorithms optimize expectation values:

\[ \min_\theta \langle H \rangle \]

but practical portfolios require discrete solutions.

Post-processing extracts:

  • deterministic selections

  • feasible bitstrings

  • empirical distributions

5. Fractional Portfolio Optimization via Simplex Encoding

5.1 Continuous parameterization

Each qubit prepares:

\[ |\psi_i\rangle = RY(\theta_i)|0\rangle \]

Measurement produces:

\[ \tilde{w}_i = \frac{1 - \langle Z_i \rangle}{2} \]

5.2 Simplex normalization

Weights:

\[ w_i = \frac{\tilde{w}_i}{\sum_j \tilde{w}_j} \]

Properties:

  • \( w_i \ge 0 \)

  • \( \sum_i w_i = 1 \)

Constraints are satisfied by construction.

5.3 Optimization objective

\[ \min_\theta - \mu^\top w(\theta) + \lambda w(\theta)^\top \Sigma w(\theta) \]

Advantages:

  • smooth landscape

  • no penalty tuning

  • deterministic feasible solutions

6. Binary vs Fractional Encodings

aspect

binary (VQE / QAOA)

fractional VQE

decision space

discrete

continuous

constraint handling

penalty

structural

objective landscape

non-convex

smooth

output

bitstrings

weights

sampling required

yes

no

post-processing

required

minimal

suitable for

asset selection

allocation

Summary

This project demonstrates three complementary quantum approaches:

Binary VQE

  • Hamiltonian expectation minimization

  • flexible ansatz design

  • probabilistic bitstring outputs

QAOA

  • structured alternating operators

  • natural fit for QUBO problems

  • interpretable circuit depth parameter

Fractional VQE

  • continuous parameterization

  • exact feasibility

  • efficient frontier construction

Together they provide a consistent framework for studying how portfolio optimization behaves under quantum-native representations.

References

  1. Markowitz, H. Portfolio Selection, Journal of Finance (1952)

  2. Farhi et al. A Quantum Approximate Optimization Algorithm, arXiv:1411.4028

  3. McClean et al. The theory of variational hybrid quantum-classical algorithms, New Journal of Physics 18 (2016)

  4. Lucas, A. Ising formulations of many NP problems, Frontiers in Physics (2014)

  5. PennyLane documentation https://docs.pennylane.ai

Author

Sid Richards

LinkedIn https://www.linkedin.com/in/sid-richards-21374b30b/

GitHub https://github.com/SidRichardsQuantum

License

MIT License — see LICENSE