Frustration by Design: Rugged Portfolio Selection on a Quantum Annealer

Quip Network Research · July 2026 · draft

Abstract

We study a cardinality-constrained portfolio problem, where the task is to hold exactly K of n assets, and we deliberately roughen its selection landscape with an explicit anti-correlation term β. We then race a D-Wave Advantage quantum annealer against simulated annealing (SA). Both solvers receive the identical selection QUBO and the identical classical finish (a greedy projection to exactly-K followed by a convex QP for the weights), so any objective difference between them measures selection-search quality and nothing else. In controlled tests on real market data, D-Wave matches the exact Gurobi MIQP optimum at every β we test, while SA lands between 1.2 and 4.4 percent above it at 28 assets and fails to improve even when granted a hundredfold more compute. In a live three-day deployment spanning 2,088 comparable races, SA never once produced a strictly better portfolio than the QPU, a result that holds at any tolerance. The finding is specific: the advantage appears only on rugged (β > 0) landscapes, it is measured against simulated annealing (with tabu search shown equally trapped at one configuration), and parallel tempering remains untested.

1. The problem

Every optimization in our system solves the same problem. From a watchlist of n assets, the solver must hold exactly K of them and weight those holdings to balance expected return against risk.

    minimize     (γ/2) · wᵀ Σ w   −   μᵀ w
       w
   subject to    Σ wᵢ  =  1                      (fully invested)
                 exactly K of the wᵢ are > 0      (hold exactly K names)
                 w_min ≤ wᵢ ≤ w_max   on held names
                 wᵢ = 0               on the rest

w      portfolio weights           (n-vector, the unknown)
μ      expected returns            (estimated from a trailing window)
Σ      covariance of returns       (estimated from a longer trailing window)
γ      risk aversion               (how much variance costs, in units of return)
K      number of names to hold
w_min, w_max   floor and cap on each held name's weight

The objective is the classic mean–variance form. The −μᵀw term pulls toward expected return, the (γ/2)·wᵀΣw term charges for variance, and γ sets the exchange rate between the two. A lower objective means a better portfolio.

The middle constraint is the source of the difficulty. “Exactly K of n” is a combinatorial choice across C(n,K) subsets layered on top of a continuous weighting problem, which makes the whole a mixed-integer quadratic program (MIQP); cardinality-constrained mean–variance optimization is NP-hard in general. Its structure, however, is particular. An outer combinatorial selection (which K names to hold, the part carrying all C(n,K) local choices) wraps around an inner convex QP (the weights on a fixed set of K names, a single bowl with one global minimum, solvable in milliseconds). The hard part and the easy part are different kinds of hard, and everything that follows comes from taking that distinction seriously.

2. The objective, term by term

The two Markowitz terms are standard, so the term that concerns this paper is the third. Writing y_i for the selection indicator (y_i = 1 when asset i is held, 0 otherwise), we add:

+ β · Σ_{i<j} ρ_ij · y_i · y_j        ρ_ij = Σ_ij / √(Σ_ii · Σ_jj)

Here ρ is the correlation matrix, so the term reads pair by pair:

ρ_ij > 0   assets move together    →  co-holding them adds +β·ρ_ij   (penalized)
ρ_ij < 0   assets move oppositely  →  co-holding them adds −β·|ρ_ij| (rewarded)
ρ_ij ≈ 0   unrelated               →  no effect

Mechanically the term rewards anti-correlation; economically it expresses a diversification preference, nudging the selection toward a decorrelated basket. That is a legitimate portfolio objective rather than a device to flatter the quantum hardware. Adding β changes the problem. A β > 0 optimum sacrifices some risk-adjusted return in exchange for decorrelation, so every comparison in this paper is made against the exact optimum of the β-modified objective, computed by an exact MIQP solver on the same instance.

The variance term (γ/2)·wᵀΣw already prices correlation, but the two terms operate at different layers:

• The variance term prices correlation only as a weight-aggregated total. The inner QP can hedge a correlated pair after the fact by shrinking both weights, so the selection stage never feels the penalty.
• The β term acts directly on the support, pair by pair, before any weights exist. It encodes a preference about which names enter the portfolio, independent of how much of each is held.

The β term competes head-on with return-seeking at the selection layer: an asset that looks excellent for the return term may be penalized for correlating with another name already in the basket. Competing pulls that cannot all be satisfied at once are the textbook ingredients of a rugged optimization landscape, which is precisely the point and the subject of the next section.

3. Frustration: why β makes the problem hard