Pasqal Challenge: Psychics

The Capacitated Vehicle Routing Problem (CVRP) is a com- binatorial optimization problem in which customers must be assigned to a fleet of capacity-constrained vehicles while min- imizing total transportation cost [1]. Classical formulations become computationally demanding as problem size increas- es, motivating the investigation of quantum optimization methods. Quadratic Unconstrained Binary Optimization (QUBO) provides a framework for representing combinatorial objectives and constraints as quadratic functions of binary variables, which can subsequently be mapped to Ising Hamiltonians [2, 3]. A central challenge in implementing constrained optimization on neutral-atom quantum processors is the limited number of available qubits and the restricted form of physically realiz- able interactions. Conventional QUBO formulations often in- troduce auxiliary slack variables to encode inequality con- straints, increasing the number of required qubits and interac- tions. Recent research has explored slack-free penalty formu- lations to reduce this overhead [4, 5]. Neutral-atom architec- tures introduce further restrictions through their distance-de- pendent Rydberg interactions and limited detuning control [6]. We investigate a reduced-variable QUBO formulation for ca- pacity-constrained clustering, corresponding to Track A of the Pasqal Quantum Hackathon. For Tier S, consisting of four customers and two vehicles, we encode each customer assign- ment using a single binary variable, requiring only four atoms. Capacity constraints are approximated through a qua- dratic load-balancing penalty rather than explicit slack vari- ables. The resulting QUBO is embedded into a neutral-atom register and evaluated through analog quantum emulation us- ing prescribed Rabi-frequency and detuning waveforms. We assess solution feasibility, optimality gaps, and embedding fi- delity against exact classical references, including the CVRP cost obtained by routing feasible clusters exactly.a

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