A Hybrid Classical-Quantum Annealing Algorithm for the TSP

Abstract: Hybrid quantum-classical algorithms can help mitigating the physical limitations of current quantum devices, particularly the low qubit count and the reduced topological connectivity. In this paper, we propose a hybrid technique to solve a well-known NP-hard optimization problem: the Traveling Salesperson Problem (TSP). Our approach is based on a graph contraction technique that removes most of the dimensionality of the original problem instance, producing a sub-TSP of a size suitable to be efficiently solved by a quantum device. The performance of our approach is first demonstrated on classical quantum simulation using Path Integral Monte Carlo, and then run on a D-Wave quantum annealer.

• The paper introduces a hybrid algorithm that combines classical graph contraction with quantum annealing to reduce problem dimensionality for TSP.
• It employs PIMC simulation and iterative subQUBO extraction to refine candidate tours and manage hardware limitations in NISQ-era quantum devices.
• Experimental results on TSPLIB benchmarks demonstrate superior optimality gaps and runtime performance compared to classical methods and native D-Wave solvers.

Hybrid Classical-Quantum Annealing for the Traveling Salesperson Problem

Introduction

This paper presents a hybrid classical-quantum approach for efficiently solving the Traveling Salesperson Problem (TSP), leveraging both classical graph contraction techniques and quantum annealing. The motivation is derived from the continued limitations of current NISQ-era quantum hardware, specifically the restricted qubit count and limited connectivity, which render direct mapping of large-scale combinatorial problems intractable. The algorithm utilizes classical optimization to reduce TSP instances to manageable subproblems suitable for quantum processing, followed by the application of quantum annealing—either via Path Integral Monte Carlo (PIMC) simulation or direct execution on D-Wave hardware.

Background and Theoretical Foundation

The TSP is modeled on a complete weighted graph $G = (V, E)$, where the objective is to find the shortest Hamiltonian cycle. Existing classical algorithms, such as Branch-and-Cut techniques and heuristics like Lin-Kernighan, are effective but remain resource-intensive for very large instances. Quantum approaches, especially quantum annealing, have received attention for their potential to circumvent local minima via quantum tunneling—but are bottlenecked by hardware limitations, notably the minor embedding problem for fully connected graphs [TSP_ApplicationDriven].

Casting TSP into the QUBO (Quadratic Unconstrained Binary Optimization) form is essential for Ising-based quantum annealers. Binary variables $x_{i,j}$ indicate visitation sequences, and constraints enforce valid tours through Lagrangian multipliers. The refined QUBO formulation from [TSP_EnhancedQA2025] mitigates qubit requirements by fixing the start node, eliminating subtour constraints.

The core quantum simulation employs the PIMC method [QA_2004], mapping the quantum system onto classical replicas along an auxiliary dimension. Replica coupling emulates quantum fluctuations, enabling the system to escape local minima. The PIMC moves are restricted to valid tours via 2-opt transitions, and quantum effects are captured through imaginary time dynamics.

Recent work on subQUBO extraction [Subqubo] introduces iterative selection of the most uncertain variables across solution pools, focusing quantum resources on the unresolved subset. This concept enables partitioning of large combinatorial instances, feeding substructures to quantum annealers and iteratively refining global solutions.

Proposed Hybrid Algorithm: Graph Contraction and Quantum Annealing

Building on backbone-based TSP heuristics [jager2014backbone], the algorithm first generates a pool $\mathcal{P}$ of diverse candidate tours via classical heuristics (random permutation and refinement by simulated annealing with 2-opt moves). It then identifies high-frequency edges—those most likely part of the global optimum—across sampled solutions. Edges surpassing the confidence threshold $\tau$ are contracted into super-nodes, yielding a reduced sub-TSP instance.

The reduced instance dimensionality, $N_{sub}$, is determined by active super-nodes and the cities not fixed. The new contracted distance matrix is constructed to preserve tour feasibility post-contraction. Sampling solutions at each iteration, rather than analyzing the entire pool, intentionally injects stochasticity, supporting broader exploration.

The main optimization loop alternates between classical refinement, graph contraction, and solving the contracted instance via PIMC (or quantum hardware). Upon solution expansion, the new tour is integrated into the pool. Iteration continues until a convergence criterion ($K$ failed improvements) is reached.

Parameter selection—pool size $N_I$, sample size $N_S$, threshold $\tau$, and refinement steps $\gamma$—is critical. Larger pools improve statistical reliability; aggressive threshold values induce more substantial contraction but risk fixing suboptimal edges. Balancing classical and quantum contributions is achieved by tuning $x_{i,j}$0.

Experimental Evaluation

Extensive experiments were conducted on TSPLIB benchmark instances, evaluating parameter sensitivity, overall performance, and comparing quantum simulation to D-Wave hardware execution.

Threshold and Pool Size Sensitivity

Results demonstrate that lowering $x_{i,j}$1 notably reduces the instance dimensionality, enabling quantum hardware compatibility at the expense of solution quality. For example, reducing berlin52 with $x_{i,j}$2 nearly eliminates the graph but increases the optimality gap. The trade-off is consistent across instances, and PIMC simulation time dominates total execution time post-contraction.

Optimal pool configuration ($x_{i,j}$3, $x_{i,j}$4) yields superior solution quality. Full-pool sampling becomes less advantageous as pool size grows; moderate sampling balances exploration and exploitation. In small instances, deterministic coverage improves reliability.

Benchmarking and Comparative Analysis

For small instances (burma14, ulysses22), the algorithm consistently produces the global optimum after contraction—reduced subproblems are well within quantum hardware limits. For mid-scale problems (pr2392), mean optimality gaps remain low (3.95%), and the best gap approaches optimality (3.01%), with significant dimensionality reduction.

Larger instances, like rl11849, necessitate aggressive contraction ($x_{i,j}$5) to fit hardware constraints, shifting computational burden to classical stages as quantum subproblem size decreases.

When compared to Google OR-Tools, the algorithm provides superior optimality gaps across all sizes, matching or outperforming classical runtime (pr1002: 2.17% vs. 4.16%).

D-Wave Hardware Results

On D-Wave’s Advantage QPU, the hybrid contraction approach outperforms the native D-Wave BQM Hybrid solver in optimality gap (burma14: 1.75% vs. 2.26%; ulysses22: 2.57% vs. 16.88%). QPU time is competitive, with overhead dominated by network and embedding latency. Minor embedding quality is crucial, and further contraction would support execution of larger subproblems as hardware scales [MinorEm_spagnoli].

Implications and Future Directions

Practically, the method bridges current quantum hardware constraints, enabling scalable quantum annealing for combinatorial optimization by modular contraction. Theoretically, it demonstrates the efficacy of structural backbone extraction for dimension reduction, validating frequency-based edge selection for optimality preservation.

Future developments are anticipated in:

• Algorithmic integration: Adaptive contraction thresholds with feedback from quantum outcomes.
• Quantum hardware scaling: As QPU connectivity and qubit count improve, contraction aggressiveness can be relaxed, supporting larger native instances.
• Minor embedding optimization: Enhanced mapping strategies could complement contraction, further mitigating topological limitations.
• Generalizable hybrid frameworks: Extending contraction-annealing paradigms to other NP-hard combinatorial problems.

Conclusion

This hybrid classical-quantum framework for TSP achieves a robust compromise between solution quality and computational cost, leveraging backbone contraction to enable quantum annealing for real-world-sized combinatorial instances. Numerical results underscore its viability, especially as an intermediate solution while quantum hardware remains resource-limited. The approach presents a scalable model for future hybrid quantum-classical algorithms in combinatorial optimization (2605.09616).