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From Cables to Qubits: A Decomposed Variational Quantum Optimization Pipeline

Paul-Niklas Ken Kandora, Adrian Asmund Fessler, Robert Fabian Lindermann, Phil Arnold, Andreas Hempel, Steffen Rebennack

TL;DR

This work targets the Cable Routing Optimization Problem (CROP) by casting it as a cable-wise separable, block-diagonal QUBO, enabling independent subproblems per cable. It introduces a decomposed gate-based quantum optimization pipeline where each cable subproblem is mapped to a small QUBO block $Q_c$ and solved via Variational Quantum Eigensolver (VQE) on $d+p$ qubits, with results merged to form a global routing. Key contributions include exact-penalty bounds for start/end and flow constraints, a per-cable penalty-augmented QUBO construction, and a practical, reproducible pipeline tested with Qiskit's SamplingVQE on room-scale instances; the experiments show feasible and near-optimal layouts under varying penalty scales. The results demonstrate the promise and current limitations of gate-based quantum optimization for multi-flow, undirected routing tasks, highlighting an actionable path toward industrial-scale deployment and providing datasets and tooling for reproducibility.

Abstract

The Cable Routing Optimization Problem (CROP) is a multi-flow routing task central to industrial layouts and smart manufacturing installations. We formulate CROP as a cable-wise separable, block-diagonal Quadratic Unconstrained Binary Optimization Problem (QUBO) and derive conservative penalty bounds that preserve feasibility. Exploiting this structure, we introduce a decomposition pipeline that builds one QUBO per cable, transforms each QUBO into a Hamiltonian and solves the subproblems with the Variational Quantum Eigensolver (VQE). Finally, the solutions per cable are merged into a global routing assignment. This procedure reduces the per-run qubits from the full problem size to those of a single-cable subproblem. We test our performance on different cable routing optimization problems varying in size using Qiskit's SamplingVQE. Our findings indicate that a decomposed VQE approach attains feasible and optimal layouts across a range of cable-routing problems.

From Cables to Qubits: A Decomposed Variational Quantum Optimization Pipeline

TL;DR

This work targets the Cable Routing Optimization Problem (CROP) by casting it as a cable-wise separable, block-diagonal QUBO, enabling independent subproblems per cable. It introduces a decomposed gate-based quantum optimization pipeline where each cable subproblem is mapped to a small QUBO block and solved via Variational Quantum Eigensolver (VQE) on qubits, with results merged to form a global routing. Key contributions include exact-penalty bounds for start/end and flow constraints, a per-cable penalty-augmented QUBO construction, and a practical, reproducible pipeline tested with Qiskit's SamplingVQE on room-scale instances; the experiments show feasible and near-optimal layouts under varying penalty scales. The results demonstrate the promise and current limitations of gate-based quantum optimization for multi-flow, undirected routing tasks, highlighting an actionable path toward industrial-scale deployment and providing datasets and tooling for reproducibility.

Abstract

The Cable Routing Optimization Problem (CROP) is a multi-flow routing task central to industrial layouts and smart manufacturing installations. We formulate CROP as a cable-wise separable, block-diagonal Quadratic Unconstrained Binary Optimization Problem (QUBO) and derive conservative penalty bounds that preserve feasibility. Exploiting this structure, we introduce a decomposition pipeline that builds one QUBO per cable, transforms each QUBO into a Hamiltonian and solves the subproblems with the Variational Quantum Eigensolver (VQE). Finally, the solutions per cable are merged into a global routing assignment. This procedure reduces the per-run qubits from the full problem size to those of a single-cable subproblem. We test our performance on different cable routing optimization problems varying in size using Qiskit's SamplingVQE. Our findings indicate that a decomposed VQE approach attains feasible and optimal layouts across a range of cable-routing problems.
Paper Structure (23 sections, 1 theorem, 31 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 23 sections, 1 theorem, 31 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Motivation
  3. Current Practices and Challenges in Cable Routing Optimization
  4. Quantum Computing and Routing Optimization
  5. Contributions
  6. CROP
  7. Mathematical model
  8. Quantum QUBO formulation
  9. Objective function
  10. Start/end segments per cable
  11. Cable flow continuity at internal nodes
  12. Selection consistency at internal nodes
  13. Putting everyhting together, we obtain the QUBO
  14. Penalty terms
  15. Cable-wise decomposition
  16. ...and 8 more sections

Key Result

Theorem 2.1

For CROP-QUBO problem formulation obj:cable - con:cable_7, the global objective decomposes as Consequently, and any collection of cable-wise minimizers $\{\textbf{z}_c^\star\}_{c\in C}$ concatenated into $\textbf{z}^\star=(\textbf{z}_1^\star,\dots,\textbf{z}_{d}^\star)$ is a global minimizer of obj:cable - con:cable_7.

Figures (3)

  • Figure 1: VQE loop. Quantum circuit $U(\boldsymbol{\theta})$ prepares a state; we measure and compute $E(\boldsymbol{\theta})$. The classical optimizer updates $\boldsymbol{\theta}$, and the loop repeats until convergence.
  • Figure 2: $\operatorname{EmpProb}^\kappa_c$ and $\operatorname{OptGap}_\text{rel}^{c,\kappa}$ for an $11$ and $16$ qubit system with different penalty scalings $\kappa\in\{ 1/4,1/2,1,2,4 \}$
  • Figure 3: Boxplots of $\operatorname{OptGap}_\text{rel}^{c,\kappa,r}$ for $c = 1,\dots, 8$, random seeds $r\in\mathcal{R}$ and $\kappa\in\{1,2,4\}$

Theorems & Definitions (1)

  • Theorem 2.1: Cable-wise separability of the CROP-QUBO