Qubit-Efficient QUBO Formulation for Constrained Optimization Problems

TL;DR

This work tackles the qubit scarcity challenge in quantum optimization by introducing a generalized exponential penalization framework for encoding inequality constraints directly into QUBO/Ising form without slack variables. It defines a flexible family of penalties $F_k(x)=f(x)-\sum_i P_k(g_i(x))$ with three forms $P_k^{(1)}$, $P_k^{(2)}$, and $P_k^{(3)}$, enabling tunable growth to enforce feasibility. Through experiments on Bin Packing and Travelling Salesman Problems, the approach achieves up to $57\%$ and $83\%$ qubit reductions respectively while maintaining competitive solution quality, with approximation probabilities reaching up to $6\%$ (BPP) and $21\%$ (TSP) under QAOA on a Qiskit Aer simulator. The results demonstrate the method's potential to scale constrained quantum optimization on NISQ devices and highlight the importance of hyperparameter tuning in practical performance. The framework offers a robust pathway toward more qubit-efficient quantum encodings for a broad class of combinatorial problems.

Abstract

Quantum computing has emerged as a promising alternative for solving combinatorial optimization problems. The standard approach for encoding optimization problems on quantum processing units (QPUs) involves transforming them into their Quadratic Unconstrained Binary Optimization (QUBO) representation. However, encoding constraints of optimization problems, particularly inequality constraints, into QUBO requires additional variables, which results in more qubits. Considering the limited availability of qubits in NISQ machines, existing encoding methods fail to scale due to their reliance on large numbers of qubits. We propose a generalized exponential penalty framework for QUBO inequality constraints inspired by a class of exponential functions, which we call exponential penalization. This paper presents an encoding strategy for inequality constraints in combinatorial optimization problems, inspired by a class of exponential functions, which we call exponential penalization. The initial idea of using exponential penalties for QUBO formulation was introduced by Montanez-Barrera et al. by applying a specific exponential function to reduce qubit requirements. In this work, we extend that approach by conducting a comprehensive study on a broader class of exponential functions, analyzing their theoretical properties and empirical performance. Our experimental results demonstrate that an exponential penalization achieves 57%, 83% qubit number reduction for Bin Packing Problem (BPP) and Traveling Salesman Problem (TSP), respectively. And we demonstrate comparable solution quality to classical with a probability of 6% and 21% accuracy for BPP with 8 and TSP with 12 qubits, respectively.

Figures (4)

• Figure 1: Overview of the optimization problem solution pipeline on classical-quantum computers, featuring the QUBO formulation derived using exponential penalty functions.
• Figure 2: Number of Qubits w.r.t. Slack and Decision Variables Required to Encode problems.
• Figure 3: QAOA objective function (energy values) for (a) BPP and (b) TSP with respect to varying $\beta, \gamma$ values on noiseless simulation.
• Figure 4: The mean convergence of QAOA with exponential encoding for single problem samples executed on noiseless simulators with hyperparameters $r=10, s=5, p=2$.