Solve portfolio optimization task with Quantum AnnealingTransparent, Reproducible Results

In this portfolio-selection task, the investment decision is encoded as a binary vector (\mathbf{x}\in\{0,1\}^N), where (x_i=1) means asset (i) is included and (x_i=0) otherwise, with a cardinality requirement (\sum_i x_i=K). The algorithm estimates the per-asset mean returns (\boldsymbol{\mu}) and the return covariance matrix (\Sigma). These statistics define a QUBO objective that trades off risk and return while enforcing the (K)-of-(N) constraint via a penalty:

The first (quadratic) term penalizes portfolio variance; the second (linear) term rewards expected return; the third expands to linear and pairwise terms that embed the constraint into the unconstrained binary objective.

Quantum annealing is a hybrid optimization method that starts from an easy "mixing" Hamiltonian (the driver, often a transverse field) and gradually shifts to the problem Hamiltonian that encodes the cost you want to minimize:

with A(t) \downarrow and B(t)\!\uparrow. If the change is slow enough relative to the minimum spectral gap, the system stays near the best (lowest-energy) solution. Intuitively, quantum tunneling helps escape narrow energy traps that stall classical heuristics. In practice, Quantum Annealing is used in hybrid workflows: classical steps do embedding and small local improvements; the annealer supplies low-energy candidates. Problems are written as Ising models, which are algebraically equivalent to QUBO.

A quadratic unconstrained binary optimization problem seeks a binary vector \mathbf{x}\in\{0,1\}^n that minimizes a quadratic objective:

It is equivalently expressed as a binary quadratic model with linear and pairwise terms. This QUBO formulation is NP-hard and subsumes many classical combinatorial problems such as max-cut, set packing, and facility location.

Expanding a penalty function F(\mathbf{x}) with x_i^2 = x_i gives:

with:

Using spins s_i = 2x_i - 1 \in \{-1,+1\}, the QUBO maps to an Ising Hamiltonian:

where:

This H_1 is the problem term used by the annealer.

The device evolves under H(t) = A(t)\cdot H_0 + B(t)\cdot H_1, where H_0 is a transverse-field driver and H_1 encodes the Ising instance above. Forward annealing decreases A(t) while increasing B(t); sufficiently slow evolution relative to the minimum spectral gap favors low-energy (good) solutions. Reverse annealing can refine a candidate by temporarily increasing quantum fluctuations before re-annealing. Because hardware connectivity is sparse, logical variables are embedded into chains of physical qubits; chain strength, spin-reversal (gauge) transforms, anneal time, and schedule shape act as practical controls that affect success rates and chain-break behavior.