Solving larger Travelling Salesman Problem networks with a penalty-free Variational Quantum Algorithm

TL;DR

This work tackles solving larger TSP networks with a penalty-free, circuit-model Variational Quantum Algorithm (VQA) that scales with qubits as $O(n\log_2(n))$, enabling simulations up to 12 locations with 29 qubits in noise-free settings. It systematically compares factorial and non-factorial encodings, Gray coding, and caching, while evaluating gradient estimators (SPSA vs parameter-shift) and warm-start strategies. A classical ML model is developed as a strong baseline, and a Monte Carlo benchmark provides a fair context for performance. Across networks up to 12 locations, the VQA achieves near-optimal solutions in noiseless simulations and remains competitive against Monte Carlo baselines, with substantial runtime gains from caching and SPSA, outlining a viable pathway toward solving larger TSP instances on quantum hardware.

Abstract

The Travelling Salesman Problem (TSP) is a well-known NP-Hard combinatorial optimisation problem, with industrial use cases such as last-mile delivery. Although TSP has been studied extensively on quantum computers, it is rare to find quantum solutions of TSP network with more than a dozen locations. In this paper, we present high quality solutions in noise-free Qiskit simulations of networks with up to twelve locations using a hybrid penalty-free, circuit-model, Variational Quantum Algorithm (VQA). Noisy qubits are also simulated. To our knowledge, this is the first successful VQA simulation of a twelve-location TSP on circuit-model devices. Multiple encoding strategies, including factorial, non-factorial, and Gray encoding are evaluated. Our formulation scales as $\mathcal{O}(nlog_2(n))$ qubits, requiring only 29 qubits for twelve locations, compared with over 100 qubits for conventional approaches scaling as $\mathcal{O}(n^2)$. Computational time is further reduced by almost two orders of magnitude through the use of Simultaneous Perturbation Stochastic Approximation (SPSA) gradient estimation and cost-function caching. We also introduce a novel machine-learning model, and benchmark both quantum and classical approaches against a Monte Carlo baseline. The VQA outperforms the classical machine-learning approach, and performs similarly to Monte Carlo for the small networks simulated. Additionally, the results indicate a trend toward improved performance with problem size, outlining a pathway to solving larger TSP instances on quantum devices.

Figures (11)

• Figure 1: An overview of a Variational Algorithm, showing how either a classical or quantum device sample bit strings, which are mapped to cycles, and an average distance evaluated. A classical optimiser changes the device parameters in a feedback loop.
• Figure 2: Algorithm 1: Diagram showing how the bit string $10_2 1_2$ is used to construct a cycle by using the bit string to point to the next item of the ordered list to be moved
• Figure 3: A classical machine learning circuit with two fully connected layers, Sine activation, binarisation, and objective function evaluation
• Figure 4: Solution quality by locations comparing our VQA and ML models with their Monte Carlo benchmarks, and a Greedy classical.
• Figure 5: Binary strings sampled by the VQA model and distinct cycles by location. As the size of the networks increases the number of binary strings required by VQA approaches an upper limit, whereas the number of distinct cycles grows rapidly.