Unbalanced penalization: A new approach to encode inequality constraints of combinatorial problems for quantum optimization algorithms

TL;DR

The paper addresses the challenge of encoding inequality constraints in QUBO for quantum optimization by introducing unbalanced penalization, a slack-variable–free approach that reduces qubit and circuit resources. The method replaces the traditional penalty with -λ1 h(x) + λ2 h(x)^2, complemented by tunable parameters tuned with Nelder–Mead and demonstrated on TSP, BPP, and KP. Empirical results from simulations and hardware (D-Wave Advantage and D-Wave Hybrid) show improved solution quality and quantity over slack-variable encodings, with QAOA1 landscapes favoring the unbalanced encoding. The work introduces CoP as a metric for quantum performance and shows the approach scales better in practice, enabling larger problem instances to be tackled on current quantum devices.

Abstract

Solving combinatorial optimization problems of the kind that can be codified by quadratic unconstrained binary optimization (QUBO) is a promising application of quantum computation. Some problems of this class suitable for practical applications such as the traveling salesman problem (TSP), the bin packing problem (BPP), or the knapsack problem (KP) have inequality constraints that require a particular cost function encoding. The common approach is the use of slack variables to represent the inequality constraints in the cost function. However, the use of slack variables considerably increases the number of qubits and operations required to solve these problems using quantum devices. In this work, we present an alternative method that does not require extra slack variables and consists of using an unbalanced penalization function to represent the inequality constraints in the QUBO. This function is characterized by larger penalization when the inequality constraint is not achieved than when it is. We evaluate our approach on the TSP, BPP, and KP, successfully encoding the optimal solution of the original optimization problem near the ground state cost Hamiltonian. Additionally, we employ D-Wave Advantage and D-Wave hybrid solvers to solve the BPP, surpassing the performance of the slack variables approach by achieving solutions for up to 29 items, whereas the slack variables approach only handles up to 11 items. This new approach can be used to solve combinatorial problems with inequality constraints with a reduced number of resources compared to the slack variables approach using quantum annealing or variational quantum algorithms.

Figures (16)

• Figure 1: Number of qubits needed to solve (a) the TSP for different numbers of cities, (b) the BPP for different numbers of items, and (c) the KP for different numbers of items. The solid line represents the number of variables of the problem and the dashed line represents the variables of the problem plus the slack variables needed to represent the inequality constraints of the problem for a different number of nodes.
• Figure 2: The number of slack variables needed to solve the TSP (diamonds), the BPP (circles), and the KP (triangles) based on the number of nodes. The solid lines are guides to the eye.
• Figure 3: Comparison between $e^{-h(\mathrm{x})}$ and the unbalanced function $1 - h(\mathrm{x}) + \frac{1}{2} h(\mathrm{x})^ 2$.
• Figure 4: Schematic representation of QAOA for $p$ layers. The parameters $\gamma$ and $\beta$ for each layer are the ones to be optimized. QAOA is used to compare the slack variables method vs. the unbalanced penalization method for the encoding of the inequality constraints.
• Figure 5: Eigenvalues distribution for the TSP with 5 cities (20 qubits) for 10 randomly generated problems using the unbalanced penalization method. The inset shows the lowest 25 energy eigenvalues. The big circles are the optimal solutions for the random problem and the small circles are the different eigenvalues of the cost Hamiltonian of its QUBO representation. Note that each eigenvalue is degenerate with multiplicity two, because the problem is symmetric, e.g., one clockwise solution and another anti-clockwise.