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.

It also describes a custom variational quantum allocator whose portfolio cost is a classical function of measured expectations, rather than a Hamiltonian expectation.

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:

prepare ansatz state
measure expectation value
update parameters using classical optimizer
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¶

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

Here E is the configured ring or complete exchange graph. The implemented mixer applies the ordered edge gates

\[ U_M(\beta)=\mathcal{P}_{(i,j)\in E} \left[e^{-i\beta X_iX_j}e^{-i\beta Y_iY_j}\right]. \]

The XX and YY terms on a single edge commute, but overlapping edges generally do not. This ordered exchange circuit is an alternating-operator variant; it is generally different from the exact exponential of the summed mixer Hamiltonian. The cost Hamiltonian contains commuting Z terms, so its single product-formula layer is exact up to numerical coefficient truncation and floating-point error.

Starting from an exact-K basis or Dicke state, the ideal exchange circuit preserves Hamming weight exactly. Hardware noise and readout errors can still produce infeasible measured selections. This structure is 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. Variational Quantum Allocation via Simplex Encoding¶

This method was historically called “Fractional VQE”. It trains a quantum circuit to parameterise allocation weights and evaluates the mean-variance objective classically. It does not minimise the expectation of a fixed portfolio Hamiltonian and is not conventional Hamiltonian VQE. The default product-RY circuit is readily classically simulable; optional entangling ansatzes do not by themselves establish a computational advantage.

5.1 Continuous parameterization¶

In the default product ansatz, 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¶

Finite physical measurement expectations lie in [-1, 1]. The implementation clips numerical overshoots within 1e-8 and rejects nonfinite or out-of-range inputs. Let epsilon = 1e-12. The corrected normalisation is:

\[ w_i = \frac{\tilde{w}_i+\epsilon/n}{\sum_j \tilde{w}_j+\epsilon} \]

Properties:

\( w_i \ge 0 \)
\( \sum_i w_i = 1 \)

These properties hold mathematically and are independently checked within numerical tolerance. When every mass is zero, the allocation is uniform. Adding epsilon only to the denominator, as in historical versions, did not normalise zero or near-zero mass correctly. Smoothing changes the map slightly even away from zero, so earlier optimisation trajectories and results must retain their original source provenance.

5.3 Optimization objective¶

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

Advantages:

differentiable normalisation on the physical interior, with boundary clipping
no penalty tuning
simplex allocations within numerical tolerance, including zero measured mass

Finite-shot estimates make training and returned allocations stochastic. Differentiability does not imply a convex landscape in circuit parameters or guaranteed convergence to the best classical allocation.

6. Binary vs Fractional Encodings¶

aspect	binary (VQE / QAOA)	variational quantum allocator
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

Variational quantum allocator (legacy name: Fractional VQE)¶

continuous parameterization
smoothed simplex normalisation with independent numerical feasibility checks
efficient frontier construction

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

References¶

Markowitz, H. Portfolio Selection, Journal of Finance (1952)
Farhi et al. A Quantum Approximate Optimization Algorithm, arXiv:1411.4028
McClean et al. The theory of variational hybrid quantum-classical algorithms, New Journal of Physics 18 (2016)
Lucas, A. Ising formulations of many NP problems, Frontiers in Physics (2014)
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