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Optimal, Qubit-Efficient Quantum Vehicle Routing via Colored-Permutations

Chinonso Onah, Kristel Michielsen

Abstract

We formulate a global-position colored-permutation encoding for the capacitated vehicle routing problem. Each of the $K$ vehicles selects a disjoint partial permutation, and the sum of these $K$ color layers forms a full $n\times n$ permutation matrix that assigns every customer to exactly one visit position. This representation uses $n^2K$ binary decision variables arranged as $K$ color layers over a common permutation structure, while vehicle capacities are enforced by weighted sums over the entries of each color class, requiring no explicit load register and hence no extra logical qubits beyond the routing variables. In contrast, many prior quantum encodings introduce an explicit capacity or load representation with additional qubits. Our construction is designed to exploit the Constraint-Enhanced QAOA framework together with its encoded-manifold analyses. Building on a requirements-based view of quantum utility in CVRP, we develop a routing optimization formulation that directly targets one of the main near-term bottlenecks, namely the additional logical-qubit cost of vehicle labels and explicit capacity constraints. Our proposal shows strong algorithmic performance in addition to qubit efficiency. On a standard benchmark suite, our end-to-end pipeline recovers the independently verified optima. The feasibility oracle may also be of independent interest as a reusable polynomial-time decoding and certification primitive for quantum and quantum-inspired routing pipelines.

Optimal, Qubit-Efficient Quantum Vehicle Routing via Colored-Permutations

Abstract

We formulate a global-position colored-permutation encoding for the capacitated vehicle routing problem. Each of the vehicles selects a disjoint partial permutation, and the sum of these color layers forms a full permutation matrix that assigns every customer to exactly one visit position. This representation uses binary decision variables arranged as color layers over a common permutation structure, while vehicle capacities are enforced by weighted sums over the entries of each color class, requiring no explicit load register and hence no extra logical qubits beyond the routing variables. In contrast, many prior quantum encodings introduce an explicit capacity or load representation with additional qubits. Our construction is designed to exploit the Constraint-Enhanced QAOA framework together with its encoded-manifold analyses. Building on a requirements-based view of quantum utility in CVRP, we develop a routing optimization formulation that directly targets one of the main near-term bottlenecks, namely the additional logical-qubit cost of vehicle labels and explicit capacity constraints. Our proposal shows strong algorithmic performance in addition to qubit efficiency. On a standard benchmark suite, our end-to-end pipeline recovers the independently verified optima. The feasibility oracle may also be of independent interest as a reusable polynomial-time decoding and certification primitive for quantum and quantum-inspired routing pipelines.

Paper Structure

This paper contains 46 sections, 10 theorems, 132 equations, 4 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. The state of the art in quantum circuit--based routing optimization algorithms
  3. Constraint–Enhanced QAOA
  4. Two routing use cases and notation.
  5. Kernelizing Vehicle Routing Problem
  6. Encoding All Valid Routes as Coloured Permutations
  7. Setup.
  8. Constraint Enhanced Quantum Vehicle Routing Optimization
  9. Objective (travel cost on the global timeline).
  10. Qubit efficiency compared to prior CVRP formulations.
  11. Counting sanity check.
  12. Optimality and performance guarantees
  13. A hybrid quantum--classical routing algorithm
  14. Constraint handling in the hybrid pipeline.
  15. Initialization.
  16. ...and 31 more sections

Key Result

Lemma 2

Suppose $X=(x_{i,j,k})$ satisfies eq:block-onehot and eq:item-once. Then $P$ in eq:sum-over-k is a permutation matrix: for each column $j$, $\sum_i p_{i,j}=1$, and for each row $i$, $\sum_j p_{i,j}=1$, with $p_{i,j}\in\{0,1\}$. $\blacktriangleleft$$\blacktriangleleft$

Figures (4)

  • Figure 1: Global-position CE-QAOA with $3$ blocks, each of size $nK=4$ wires. The diagonal entangler $e^{-i\gamma_\ell H_C}$ spans all $3\cdot 4$ wires (green), while the mixers $U_M^{(j)}(\beta_\ell)$ (orange) act independently within each block $j$. Three layers ($\ell=1,2,3$) are shown.
  • Figure 2: Comparison of qubit counts for two binary global-position encodings across representative CVRP instance sizes. The original separated encoding scales as $Q_{\mathrm{sep}} = K N \lceil \log_2 N \rceil$, whereas the reduced colored-permutation encoding scales as $Q_{\mathrm{red}} = N \lceil \log_2(KN) \rceil$. The horizontal lines indicate illustrative hardware thresholds. The reduced encoding changes the linear factor of $K$ in the number of qubits to a logarithmic dependence and shifts several small-scale industrial routing regimes into the few-hundred to $\sim 10^3$-qubit range.
  • Figure 3: Encoded anticoncentration and design baseline. Each panel shows the empirical histogram of feasible outcomes from a depth-$p=1$ CE--QAOA run on a CVRP instance, using shots $S=\Theta(n^{3})$ per grid point. The dashed line is the encoded design baseline $1/D$ with $D=(nK)^{n}$. Peaks far above $1/D$ and the successful identification of the optimum support the anticoncentration guarantees discussed in the text.
  • Figure 4: Colored decomposition: $P=P^{(1)}+P^{(2)}$ with disjoint supports.

Theorems & Definitions (24)

  • Definition 1: CE--QAOA kernel
  • Lemma 2: One-hot $+$ item-once $\Longrightarrow$ $P$ is a permutation matrix
  • proof
  • Lemma 3: Colored completion: any permutation can be lifted to $K$ layers
  • proof
  • Theorem 4: Characterization of valid routes as colored permutations
  • proof
  • Proposition 5: Capacity enforcement requires no ancilla qubits
  • proof
  • Remark 6: Quadratic surrogate in the balanced/uniform-capacity regime
  • ...and 14 more