Portfolio Optimization

Source: examples/portfolio_opt/README.md

Select a portfolio of exactly B assets from N candidates to maximise expected return while penalising higher-order risk cross-interactions.

QUBO formulation

• Input: N asset returns, cubic risk interactions (i, j, k, sigma), budget B
• Model: N binary variables. x_i = 1 if asset i is selected.
• Objective: -sum(r_i * x_i) + sum(sigma_ijk * x_i * x_j * x_k) – first term maximises return (minimising its negation), second penalises correlated three-asset risk interactions.
• Constraints: budget sum(x_i) = B (EQUALITY with unit coefficients, penalty 200)

Encoding strategy

Return terms are linear: ADDLINE(i, -r_i) per asset.

Cubic risk terms (i, j, k, sigma) are degree-reduced:

1. REDUCE(i, j, P_AUX) -> w (Rosenberg enforcement for w = x_i * x_j)
2. ADDQUAD(w, k, sigma) (sigma * w * x_k = sigma * x_i * x_j * x_k)

Budget constraint builds uniform-coefficient index/coeff vecs then calls EQUALITY with target = B and penalty = 200. EQUALITY is emitted after all objective (body) actions because it lands in the constraint section.

DSL methods used

• model.reduce(var_a, var_b, p_aux) – HOBO degree reduction for cubic risk terms
• problem.vec() – allocate index/coefficient vecs for the budget constraint
• model.apply_equality(indices, coeffs, target, penalty) – budget EQUALITY

Pipeline overview

1. CP (xqcp) – generate random returns and cubic risk interactions, declare binary variables, degree-reduce risk terms via REDUCE, and add a budget EQUALITY constraint.
2. Assemble – .xqasm text to bytecode via xquad.asm
3. Encode – run encoder on chosen XQVM to produce the XQMX model
4. Sample – solver runs SA/QPU/GPU over the model
5. Verify – verifier checks the sample is binary, that the budget constraint holds, and that each REDUCE auxiliary equals the product it stands for, then computes energy
6. Decode – decoder extracts the selected assets

Usage

uv run python examples/portfolio_opt/runner.py --seed 42
uv run python examples/portfolio_opt/runner.py --n 6 --budget 3 --interpreter rust

Choosing a solver

Solver selection and install extras are the same for every example: see Using the Examples and Solving Overview. The default is dwave-cpu, and a non-default solver will not reproduce the canonical result.

Canonical output

example-smoke validates both interpreters produce valid == 1 with --seed 42 --solver dwave-cpu. The smoke test is invariant-based – it checks validity, not exact output.