Progress on Data-Driven, Multi-Objective Quantum Optimization

Here, we present two complementary approaches that advance quadratic unconstrained binary optimization (QUBO) toward practical use in data-driven materials design and other real-valued black-box optimization tasks. First, we introduce a simple yet powerful preprocessing scheme that, when applied to a machine-learned QUBO model, entirely removes system-level equality constraints by construction. This makes cumbersome soft-penalty terms obsolete, simplifies QUBO formulation, and substantially accelerates solution search. Second, we develop a multi-objective optimization strategy inspired by Tchebycheff scalarization that is compatible with non-convex objective landscapes and outperforms existing QUBO-based Pareto front methods. We demonstrate the effectiveness of both approaches using a simplified model of a multi-phase aluminum alloy design problem, highlighting significant gains in efficiency and solution quality. Together, these methods broaden the applicability of QUBO-based optimization and provide practical tools for data-driven materials discovery and beyond.

💡 Research Summary

This paper tackles two persistent challenges in QUBO‑based quantum optimization—handling system‑level equality constraints and efficiently solving multi‑objective problems—by introducing two data‑driven techniques that integrate seamlessly with the FM+QO (Factorization Machine + QUBO) framework.

Constraint‑guided Feature Mapping (CGFM).
Traditional QUBO formulations embed equality constraints (e.g., the sum of alloy component fractions must equal 100 %) as soft quadratic penalties. These penalties shrink the energy gap, increase embedding overhead, and consume a large portion of the limited analog dynamic range of current quantum annealers. CGFM eliminates the need for any penalty term. The method first normalizes continuous design variables (phase fractions) so that they automatically satisfy the sum‑to‑one condition. These normalized variables are then discretized using a one‑hot encoding, which maps each continuous fraction to a unique binary sub‑vector. Because the encoding is performed only on feasible points, the resulting binary search space contains no infeasible configurations. Consequently, the QUBO matrix is smaller, the required chain length for hardware embedding is reduced, and the precision loss caused by large penalty coefficients disappears. Importantly, CGFM works as a pre‑processing step and therefore remains effective even on noisy NISQ devices where embedding resources are scarce.

Data‑driven Tchebycheff Scalarization (DDTS).
Multi‑objective optimization is typically addressed by weighted‑sum scalarization, but this approach fails for non‑convex Pareto fronts and requires careful tuning of preference weights. DDTS adopts the Tchebycheff scalarization, which minimizes the maximum weighted deviation from a reference point, guaranteeing coverage of non‑convex regions. The novelty lies in applying the Tchebycheff transformation directly to the raw data before training the FM surrogate. Each objective is first normalized, a reference (ideal) point is defined, and the scalar objective is computed as the maximum of the weighted absolute differences. This scalar value replaces the multi‑objective vector in the FM+QO loop, allowing the quantum optimizer to search a single‑objective landscape while still preserving the multi‑objective trade‑offs.

Case Study: Multi‑phase Aluminum Alloy Design.
The authors demonstrate the methods on a synthetic design problem involving an Al matrix (fixed 80 %) and four secondary phases (eutectic Si, Mg₂Si, Al₃Ni, Al₂Cu). Five material properties—thermal conductivity (κ), Young’s modulus (E), density (ρ), coefficient of thermal expansion (α), and solidification interval (ΔT)—are estimated using simple rule‑of‑mixtures and a coarse Al–Si phase‑diagram approximation. A dataset of random alloy compositions and their predicted properties is generated, serving as the training set for the FM surrogate.

Applying CGFM, the equality constraint on phase fractions is removed; the binary encoding contains only feasible one‑hot vectors, reducing the total number of binary variables by roughly 30 % compared with a naïve soft‑penalty formulation. Embedding the resulting QUBO on a D‑Wave‑like Chimera graph requires 25 % shorter chains, freeing up coupler precision for the actual objective terms.

Using DDTS, the multi‑objective problem is scalarized via the Tchebycheff norm. The authors compare the Pareto fronts obtained with DDTS against those from a weighted‑sum approach (both solved with simulated annealing and quantum annealing). DDTS yields a front that is ~15 % larger in hyper‑volume and captures non‑convex regions that the weighted‑sum method misses. Solution times are reduced by a factor of 2–3 because the scalarized landscape is smoother and does not suffer from the ill‑conditioning introduced by large penalty weights.

Implications and Limitations.
The combined CGFM+DDTS pipeline demonstrates that data‑driven preprocessing can dramatically simplify QUBO formulations, making them more amenable to current quantum hardware while preserving feasibility. The Tchebycheff scalarization provides a robust alternative to weighted sums for non‑convex multi‑objective problems, eliminating the need for problem‑specific weight tuning. However, the study relies on synthetic data and simplified physics models; real experimental datasets with noisy measurements may affect FM training quality and the robustness of the scalarization. Moreover, CGFM is currently limited to linear equality constraints; extending it to inequality or nonlinear constraints will require additional methodological development.

Overall, the paper offers practical algorithmic advances that bridge the gap between theoretical quantum optimization and real‑world materials design, and it sets a clear agenda for future work on constraint generalization and validation on physical quantum processors.