Efficient Variational Quantum Algorithms for the Generalized Assignment Problem

TL;DR

This work tackles solving the Generalized Assignment Problem (GAP) on near-term quantum devices by introducing VQGAP, a variational quantum approach that decouples the ansatz qubits from the GAP's binary decision variables to reduce quantum resource needs. It extends VQE by computing residual budgets classically and using penalty-based objective functions, enabling a smaller qubit footprint with $Q = T*(A+1)$, and further develops VQGAPe, which encodes task-agent assignments on $Q = T\lceil\log_2(A+1)\rceil$ qubits. Preliminary simulations (noiseless and IBM Brisbane noise model) show VQGAP achieves performance similar to VQE while reducing circuit width and depth, and VQGAPe can substantially shrink quantum resources while sometimes delivering superior solution quality. Overall, the proposed methods advance practical quantum optimization for GAP on NISQ hardware and offer scalable paths toward solving larger combinatorial problems with limited qubits.

Abstract

Quantum algorithms offer a compelling new avenue for addressing difficult NP-complete optimization problems, such as the Generalized Assignment Problem (GAP). Given the operational constraints of contemporary Noisy Intermediate-Scale Quantum (NISQ) devices, hybrid quantum-classical approaches, specifically Variational Quantum Algorithms (VQAs) like the Variational Quantum Eigensolver (VQE), promises to be effective approaches to solve real-world optimization problems. This paper proposes an approach, named VQGAP, designed to efficiently solve the GAP by optimizing quantum resources and reducing the required parametrized quantum circuit width with respect to standard VQE. The main idea driving our proposal is to decouple the qubits of ansatz circuits from the binary variables of the General Assignment Problem, by providing encoding/decoding functions transforming the solutions generated by ansatze in the limited quantum space in feasible solutions in the problem variables space, by exploiting the constraints of the problem. Preliminary results, obtained through both noiseless and noisy simulations, indicate that VQGAP exhibits performance and behavior very similar to VQE, while effectively reducing the number of qubits and circuit depth.

Figures (15)

• Figure 1: Sketch of the VQE algorithm, where the execution of the Quantum circuit, structured according to the Ansatz, is followed by a series of measurements, dictated by the form of the QUBO problem. The Average of the Hamiltonian (i.e., our Cost Function) is then evaluated through the measurements, and its value serves as input to the Optimizer in order to change the parameters of the variational circuit.
• Figure 2: Reference ansatz for VQE. The state of the slack qubits is determined by the values of the assignment qubits. First, slack qubits are set to the capacities of respective agents (here $B_0=3$, $B_1=2$, $B_2=1$), then the task weights ($w_1$, $w_2$, and $w_3$) are subtracted if the related task is assigned to the agents.
• Figure 3: Reference ansatz for VQGAP
• Figure 4: Ansatz VAQGAPe-RXL, consisting of a single layer of X-rotation on each qubit.
• Figure 5: Ansatz VQGAPe-ESU2, an hardware efficient SU two-local parameterized circuit.