Direct quantum-annealer portfolio optimization is commonly formulated as a penalty-encoded QUBO and submitted to D-Wave hardware. We show that this standard formulation fails on current devices and identify the structural reason: the cardinality penalty A(1^T x - K)^2 contributes a dense rank-one matrix A·11^T that makes the logical graph complete regardless of the underlying covariance structure, producing chain-break fractions of 83% at N=24 rising to 92% at N=49 on both Pegasus and Zephyr, and zero feasible samples. Attempting to fix this through topology-aware sparsification reveals a second problem: any sparsifier that removes off-diagonal entries also dilutes the cardinality constraint. Raw samples remain infeasible even when chains no longer break. An ablation shows that for structurally favorable cases (betting with settlement-graph priors), the classical feasibility projector alone explains the result, not the QPU. We propose dropping the penalty entirely: build the objective-only QUBO Q_obj = -diag(mu) + lambda*Sigma, sample on hardware, and enforce cardinality classically. On D-Wave Advantage and Advantage2 for equities up to N=49 (full FF49 universe) and betting up to N=48, mean chain-break fractions (per sample, averaged overreads) drop from 71-92% to at most 0.04%. The QPU returns lower-energy feasible portfolios than the greedy heuristic on betting at N in {39, 48} (an energy comparison, not a proof of optimality), and equity post-processed regret is at most 0.03% at all tested scales. These results establish that the penalty encoding, not the sparse hardware topology, is the binding constraint for direct QPU portfolio optimization at currently accessible scales.
Direct quantum-annealer portfolio optimization is commonly formulated as a penalty-encoded QUBO and submitted to D-Wave hardware. We show that this standard formulation fails on current devices and identify the structural reason: the cardinality penalty A(1^T x - K)^2 contributes a dense rank-one matrix A·11^T that makes the logical graph complete regardless of the underlying covariance structure, producing chain-break fractions of 83% at N=24 rising to 92% at N=49 on both Pegasus and Zephyr, and zero feasible samples. Attempting to fix this through topology-aware sparsification reveals a second problem: any sparsifier that removes off-diagonal entries also dilutes the cardinality constraint. Raw samples remain infeasible even when chains no longer break. An ablation shows that for structurally favorable cases (betting with settlement-graph priors), the classical feasibility projector alone explains the result, not the QPU. We propose dropping the penalty entirely: build the objective-only QUBO Q_obj = -diag(mu) + lambda*Sigma, sample on hardware, and enforce cardinality classically. On D-Wave Advantage and Advantage2 for equities up to N=49 (full FF49 universe) and betting up to N=48, mean chain-break fractions (per sample, averaged overreads) drop from 71-92% to at most 0.04%. The QPU returns lower-energy feasible portfolios than the greedy heuristic on betting at N in {39, 48} (an energy comparison, not a proof of optimality), and equity post-processed regret is at most 0.03% at all tested scales. These results establish that the penalty encoding, not the sparse hardware topology, is the binding constraint for direct QPU portfolio optimization at currently accessible scales.
Cardinality-constrained portfolio selection is routinely cast as a quadratic unconstrained binary optimization (QUBO) and submitted to a quantum processing unit (QPU) for direct annealing. We show that this standard penalty encoding is the binding constraint for direct-QPU execution on current D-Wave Pegasus and Zephyr hardware. Expanding the exact cardinality penalty contributes a dense rank-one term that makes the logical interaction graph complete regardless of the covariance, producing chain-break fractions from 83% at small universes up to 92% at the full forty-nine-industry Fama--French universe, and zero feasible raw samples at every tested scale. Topology-aware sparsification reduces chain breaks to near zero, but any sparsifier that removes off-diagonal entries also dilutes the cardinality constraint; an ablation reveals that this sparsify-and-project pipeline is dominated by the classical projector, not the QPU. We propose removing the penalty entirely: sample an objective-only QUBO built from expected returns and the risk-scaled covariance on hardware, and enforce cardinality classically through a deterministic feasibility projector. Across 4,468 saved embedding records on live Pegasus and Zephyr hardware, spanning equities up to forty-nine assets and football-betting instances up to forty-eight, this penalty-free pipeline reduces mean chain-break fractions from 71%--92% down to at most 0.04%, and post-processed regret is at most 0.03% relative to greedy classical references at every tested scale. We do not claim quantum advantage; the penalty encoding, not the sparse hardware topology, is the limiting factor for direct-QPU portfolio optimization at currently accessible scales.
We audit how much of D-Wave's hybrid quantum-classical portfolio-optimization service is actually quantum. On cardinality-constrained mean-variance-turnover instances spanning N = 10 to 640 with a Gurobi MIQP optimality anchor, the constraint-native LeapHybridCQM service matches Gurobi's proven optimum on all 54 instances where Gurobi proves optimality, but mean QPU access time is only 0.034 seconds out of a 5-second wall-clock budget, roughly 0.7% of the run. The remaining ~99% is the service's classical decomposition, sub-problem assembly, and feasibility-aware reassembly. The reported D-Wave hybrid win on this problem class is therefore a constraint-native classical pipeline with a small QPU contribution, not a quantum-sampling win. Two structural results sharpen this audit: (i) the cardinality penalty contributes a dense rank-one term that makes the penalty-encoded logical graph fully connected regardless of the original covariance density, collapsing the intended density benchmark axis for all penalty-encoded paths while leaving CQM's sparsity intact; (ii) the CQM service returns identical solutions at every tested wall-clock budget from 5 to 300 seconds and across 10 repeated calls, a determinism property of the service on this problem class. Together with two classical baselines (Gurobi MIQP and simulated annealing) and a comparison against the penalty-encoded LeapHybridBQM interface, these results extend the prior CQM-vs-BQM observation of Sakuler et al. (2025) from "CQM handles constraints natively" to the operational decomposition of where the win actually originates, a finding that reframes how D-Wave hybrid performance should be reported in quantum-finance benchmarks.
Hybrid quantum-classical solvers conceal how reported performance divides between quantum-processing-unit (QPU) access and other service time. We audit D-Wave's Leap service on cardinality-constrained mean-variance portfolio instances from N=10 to 640, comparing constraint-native CQMs, penalty-encoded BQMs, Gurobi MIQP, simulated annealing, and a matched-budget Tabu baseline, and we propose a four-metric operational audit of the exposed timing fields. LeapHybridCQM matches Gurobi's proven optimum on all 54 head-to-head instances at N<=120, yet its mean QPU access is 0.034 s: 0.68% of the nominal five-second budget; telemetry does not resolve the residual into classical computation, orchestration, or budget granularity. Across 108 fixed-seed five-second CPU runs on 36 instances, TabuSampler reaches CQM-level objectives with mean absolute delta 0.00078 (maximum 0.00798); at N in {400,640} the mean is 3.04e-5. This establishes classical objective achievability, not an internal QPU ablation. We prove that quadratic encoding of exact cardinality adds a dense rank-one term and, apart from isolated exact coefficient cancellations, yields a complete logical graph independent of covariance support, with a bounded-degree embedding lower bound; the resulting density collapse is consistent with hybrid-BQM degradation. A descriptive five-window Fama-French overlay of post-projection direct-QPU portfolios yields mean Sharpe 1.94 versus 2.22 for 1/N. These results do not establish a quantum-sampling advantage; they motivate reporting QPU wall-clock fraction, optimality gap, density amplification, service-output variance, and matched-budget classical controls, keeping operational accounting separate from causal attribution.
We present an end-to-end pipeline for large-scale portfolio selection with cardinality constraints and experimentally demonstrate it on trapped-ion quantum processors using hardware-aware decomposition. Building on RMT-based correlation-matrix denoising and community detection, we identify correlated asset groups and introduce a correlation-guided greedy splitting scheme that caps each cluster by the executable qubit budget. Each cluster defines a hardware-embeddable QUBO subproblem that we solve using bias-field digitized counterdiabatic quantum optimization (BF-DCQO), a non-variational method that avoids classical parameter-training loops. We recombine low-energy candidates into global portfolios and enforce feasibility with a two-stage post-processing routine: fast repair followed by a cardinality-preserving swap local search. We benchmark the workflow on a 250-asset universe taken from the S&P 500 and execute subproblems on a 64-qubit Barium development system similar to the forthcoming IonQ Tempo line. We observe that larger executable subproblem sizes reduce decomposition error and systematically improve final objective values and risk-return trade-offs relative to randomized baselines under identical post-processing. Overall, the results establish a hardware-tested route for scaling financial optimization problems, defined by a trade space in which executable problem size and circuit cost are balanced against the resulting solution quality.
Quadratic unconstrained binary optimization (QUBO) is the standard interface to quantum annealers, yet a single constrained task admits many QUBO encodings whose penalty choices reshape the energy landscape experienced by hardware. We study a Job Shop Scheduling instance using a two-parameter family of encodings controlled by penalty weights psum (one-hot/sum constraints) and ppair (precedence constraints). Sweeping (psum,ppair), we observe sharp transitions in feasibility and solver success across classical annealing-inspired heuristics and on a D-Wave Advantage processor. Going beyond solution probability, we treat the annealer as an open thermodynamic system and perform cyclic reverse-annealing experiments initialized from thermal samples, measuring the stochastic processor energy change. From the first two moments of this energy change we infer lower bounds on entropy production, work, and exchanged heat via thermodynamic uncertainty relations, and corroborate the observed trends with adiabatic master equation simulations. We find that the same encoding transitions that govern computational hardness also reorganize dissipation: weak penalties generate low-energy infeasible manifolds, while overly strong penalties suppress the effective problem energy scale and increase irreversibility, reducing the thermodynamic efficiency. Our results establish QUBO penalties as thermodynamic control knobs and motivate thermodynamics-aware encoding strategies for noisy intermediate-scale quantum annealers.
Several optimization solvers inspired by quantum annealing have been recently developed, either running on actual quantum hardware or simulating it on traditional digital computers. Industry and academics look at their potential in solving hard combinatorial optimization problems. Formally, they provide heuristic solutions for Ising models, which are equivalent to quadratic unconstrained binary optimization (QUBO). Constraints on solutions feasibility need to be properly encoded. We experiment on different ways of performing such an encoding. As benchmark we consider the cardinality constrained quadratic knapsack problem (CQKP), a minimal extension of QUBO with one inequality and one equality constraint. We consider different strategies of constraints penalization and variables encoding. We compare three QUBO solvers: quantum annealing on quantum hardware (D-Wave Advantage), probabilistic algorithms on digital hardware and mathematical programming solvers. We analyze their QUBO resolution quality and time, and the persistence values extracted in the quantum annealing sampling process. Our results show that a linear penalization of CQKP inequality improves current best practice. Furthermore, using such a linear penalization, persistence values produced by quantum hardware in a generic way allow to match a specific CQKP metric from literature. They are therefore suitable for general purpose variable fixing in core algorithms for combinatorial optimization.