Optimized QUBO formulation methods for quantum computing

Several combinatorial optimization problems can be solved with NISQ devices once that a corresponding quadratic unconstrained binary optimization (QUBO) form is derived. The aim of this work is to drastically reduce the variables needed for these QUBO reformulations in order to unlock the possibility to efficiently obtain optimal solutions for a class of optimization problems with NISQ devices. This is achieved by introducing novel tools that allow an efficient use of slack variables, even for problems with non-linear constraints, without the need to approximate the starting problem. We divide our new techniques in two independent parts, called the iterative quadratic polynomial and the master-satellite methods. Hence, we show how to apply our techniques in case of an NP-hard optimization problem inspired by a real-world financial scenario called Max-Profit Balance Settlement. We follow by submitting several instances of this problem to two D-wave quantum annealers, comparing the performances of our novel approach with the standard methods used in these scenarios. Moreover, this study allows to appreciate several performance differences between the D-wave Advantage and Advantage2 quantum annealers.

💡 Research Summary

This paper addresses a critical bottleneck in using noisy intermediate‑scale quantum (NISQ) devices for combinatorial optimization: the large number of binary variables—especially slack variables—required to convert constrained problems into a Quadratic Unconstrained Binary Optimization (QUBO) form. Traditional QUBO reformulations enforce each equality or inequality constraint by adding a quadratic penalty term that involves a set of auxiliary binary slack variables. For inequality constraints, the standard approach linearizes the bound using slack variables that must span the entire feasible range, leading to a proliferation of variables proportional to the magnitude of the bound and to the number of constraints. When many constraints are present, or when constraints are non‑linear or involve integer‑valued expressions, the resulting QUBO can contain thousands of variables, exceeding the qubit budget and connectivity limits of current quantum annealers.

To overcome this limitation, the authors introduce two independent techniques: the Iterative Quadratic Polynomial (IQP) method and the Master‑Satellite (MS) method.

1. Iterative Quadratic Polynomial (IQP)
   The IQP method replaces the conventional squared‑penalty formulation (-(\text{constraint})^{2}) with a specially crafted quadratic polynomial (P(x) = -f(x),(1-f(x))), where (f(x)) is a normalized version of the original constraint expression. This polynomial has the property that it evaluates to zero when the constraint is satisfied and to a value ≤ −1 when violated. Consequently, the penalty term can be added directly to the QUBO objective without any auxiliary slack variables. The method works for linear, non‑linear, integer, and even real‑valued constraints, because the normalization step can be performed analytically for any bounded expression. By avoiding slack variables altogether, IQP dramatically reduces the dimensionality of the QUBO and preserves sparsity in the Q matrix.

2. Master‑Satellite (MS)
   The MS method tackles the situation where several constraints share the same subset of logical variables. Instead of introducing a separate slack set for each constraint, the approach designates a master set of binary variables that encode the common decision space, and a series of satellite penalty terms that interact with the master variables. Each satellite corresponds to one original constraint and contributes a quadratic term that couples the master variables with a small, constraint‑specific auxiliary set. The penalty multipliers (\lambda_k) for each satellite are tuned either by a coarse grid search or by estimating Lagrange multipliers from a relaxed linear program. This architecture eliminates redundant slack variables, reduces the overall connectivity requirement, and allows the same master variables to enforce multiple constraints simultaneously.

The authors also propose a Generalized Master‑Satellite extension for cases where three or more constraints overlap on the same variables. The extension adds cross‑satellite interaction terms that prevent conflicting penalty contributions while keeping the QUBO matrix sparse.

Case Study: Max‑Profit Balance Settlement (MPBS)
The MPBS problem originates from financial settlement networks. A set of parties exchange payments; the goal is to select a subset of transaction arcs that maximizes total profit while respecting cash‑flow balance constraints for each party. Formally, MPBS is an NP‑hard problem with a quadratic objective and a collection of linear equality and inequality constraints that are integer‑valued. Using the standard QUBO conversion (as described in Boros & Hammer, 2002), the authors report that a typical MPBS instance of moderate size requires roughly 1,200 slack variables in addition to the logical arc variables.

Applying the IQP method, each inequality is rewritten as a penalty polynomial without any slack variables, cutting the slack count to near zero for those constraints. The remaining equality constraints are already linear and need no slack variables. However, many constraints involve the same set of arc variables (e.g., cash‑flow balance for each party). Here the MS method consolidates these overlapping constraints: a master variable set encodes the net inflow/outflow for each party, while satellite terms enforce the individual balance equations. The combined IQP + MS pipeline reduces the total number of auxiliary binary variables to about 120 for the same instance—a reduction of roughly 90 %.

Quantum Annealing Experiments
The transformed QUBOs are submitted to two D‑Wave quantum annealers: the commercial Advantage_system4.1 and the prototype Advantage2_prototype1.1 (which features higher connectivity and lower noise but fewer physical qubits). For each annealer, the authors run 1,000 annealing cycles per instance, comparing three configurations: (i) standard QUBO formulation, (ii) IQP‑only formulation, and (iii) the full IQP + MS formulation.

Key findings:

• Success Rate (fraction of runs returning the known optimal solution) is dramatically higher for the IQP + MS formulation. On Advantage, the improvement factor ranges from 7× for the smallest instances to 30× for medium‑size instances. On Advantage2, the factor climbs to 184× for the largest tested instances.
• Scalability: With the standard formulation, success probability drops sharply as the number of arcs grows, reflecting the annealer’s inability to handle the dense coupling graph induced by many slack variables. In contrast, the IQP + MS formulation shows a much gentler decay; the success probability remains above 20 % even when the logical problem size doubles.
• Hardware Utilization: The reduced variable count translates into lower qubit occupancy and fewer long‑range couplers, which mitigates the impact of thermal noise and calibration errors. Advantage2’s higher connectivity further amplifies these benefits, allowing it to solve instances that would be infeasible on the older Advantage hardware.

Comparison with Existing Slack‑Reduction Techniques
The paper discusses alternative approaches such as the Alternating Direction Method of Multipliers (ADMM), Augmented Lagrangian methods, and constraint‑scaling heuristics. Those techniques typically require solving a sequence of intermediate QUBOs to refine penalty parameters or to approximate non‑linear constraints. Consequently, they incur additional quantum runtime and may introduce approximation errors. In contrast, the proposed methods operate directly on the original problem formulation, require no pre‑solving, and preserve exact constraint satisfaction.

Conclusions and Outlook
By fundamentally redesigning how constraints are encoded—either by replacing squared penalties with tailored quadratic polynomials (IQP) or by sharing a compact master variable set across multiple constraints (MS)—the authors achieve an order‑of‑magnitude reduction in auxiliary variables. Empirical results on state‑of‑the‑art quantum annealers confirm that this reduction translates into higher solution quality, better scalability, and more efficient hardware utilization. The techniques are generic and can be applied to any QUBO‑compatible optimization problem, including logistics, drug discovery, and other finance‑related tasks. Future work may explore automated detection of master‑satellite structures, adaptive tuning of penalty multipliers during annealing, and integration with hybrid quantum‑classical workflows to further push the limits of NISQ‑scale optimization.