A Feasibility-Preserved Quantum Approximate Solver for the Capacitated Vehicle Routing Problem

TL;DR

This work targets the Capacitated Vehicle Routing Problem (CVRP) by proposing a feasibility-preserving Quantum Alternating Operator Ansatz (AOA) tailored to constrained optimization. It introduces a two-part binary encoding (a permutation x and an unconstrained depot-flag y), a constraint-preserving decoding, and an ancilla-based condition encoding that yields a cost Hamiltonian without penalizing infeasible states. The approach initializes in an equal superposition of feasible states and uses constraint-preserving mixers (including Grover-type reflections) to maintain feasibility, resulting in higher probabilities of measuring optimal solutions and reduced optimality gaps compared to penalty-based QAOA encodings. Statevector simulations demonstrate orders-of-magnitude improvements in feasibility and optimality metrics on small CVRP instances, indicating the potential to extend quantum optimization to constrained combinatorial problems and related domains such as knapsack-type formulations.

Abstract

The Capacitated Vehicle Routing Problem (CVRP) is an NP-optimization problem (NPO) that arises in various fields including transportation and logistics. The CVRP extends from the Vehicle Routing Problem (VRP), aiming to determine the most efficient plan for a fleet of vehicles to deliver goods to a set of customers, subject to the limited carrying capacity of each vehicle. As the number of possible solutions skyrockets when the number of customers increases, finding the optimal solution remains a significant challenge. Recently, the Quantum Approximate Optimization Algorithm (QAOA), a quantum-classical hybrid algorithm, has exhibited enhanced performance in certain combinatorial optimization problems compared to classical heuristics. However, its ability diminishes notably in solving constrained optimization problems including the CVRP. This limitation primarily arises from the typical approach of encoding the given problems as penalty-inclusive binary optimization problems. In this case, the QAOA faces challenges in sampling solutions satisfying all constraints. Addressing this, our work presents a new binary encoding for the CVRP, with an alternative objective function of minimizing the shortest path that bypasses the vehicle capacity constraint of the CVRP. The search space is further restricted by the constraint-preserving mixing operation. We examine and discuss the effectiveness of the proposed encoding under the framework of the variant of the QAOA, Quantum Alternating Operator Ansatz (AOA), through its application to several illustrative examples. Compared to the typical QAOA approach, the proposed method not only preserves the feasibility but also achieves a significant enhancement in the probability of measuring optimal solutions.

Figures (10)

• Figure 1: Visualization of the CVRP.
• Figure 2: The energy landscapes of depth-$1$\ref{['fig:fig1a']} QAOA and \ref{['fig:fig1b']} Grover-Mixer Quantum Alternating Operator Ansatz (GM-QAOA) for solving the Traveling Salesman Problem (TSP). The red lines track the optimization process. The optimizer finds the position of the lowest energy state in the GM-QAOA, while it becomes trapped in a local minimum within the QAOA.
• Figure 3: Visualization of the proposed problem encoding. Permutation matrix $x$ exhibit a certain order of customer visits, delineated as $[3,4,5,1,2]$. The vehicle returns to the depot before visiting $V_4$ and $V_1$, as decision bit $y_2=1$ and the insufficiency of the vehicle's capacity at time step $3$ to meet the demand of $V_1$.
• Figure 4: The circuit of condition encoding operation $U_E$. The depot visit condition for each time step is stored into the ancilla register $a$, segmented by full-row barriers. Each step consists of four typical procedures separated by partial-row barriers. First, the satisfied demand of the customer is logged into register $d$. Then, register $d$ and decision qubit $y_t$ control the flip of $a_t$, where $a_t\!=\!1$ suggests a depot visit. After that, if such a depot visit occurs, the register $d$ and $c$ are recovered. Finally, the visited customer number is recorded into register $c$.
• Figure 5: The decomposition of $\left[>\! Q \right]$, where $Q\!=\!9$, can be presented in a binary format as '01001'. According to Equation (\ref{['eqn:dq']}), the $\left[>\! 9 \right]$ can be decomposed into a sequence of three multi-controlled Pauli-$X$ gates. Specifically, the control states for each gate are identified as '1', '011', and '0101', each, respectively, corresponding to control qubits $d_4$, $d_4d_3d_2$, and $d_4d_3d_2d_1$.