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Large-Scale Combinatorial Optimization via Transcendental QAOA: Resolving the 1,000-City Traveling Salesperson Problem.

Teixeira A. C

Published Feb 21, 2026• 0 citations
Publisher

Abstract

Abstract The Traveling Salesperson Problem (TSP) is a canonical NP-hard challenge in combinatorial optimization with profound implications for global and interplanetary logistics. This paper presents a quantum-computational resolution for a 1,000-city TSP instance utilizing the ARK5Q-200K infrastructure. By mapping the distance manifold into a Quadratic Unconstrained Binary Optimization (QUBO) formulation, we employ a Quantum Approximate Optimization Algorithm (QAOA) enhanced with transcendental mixers. Our results achieve a convergence gap of $< 1\%$ relative to the theoretical Held-Karp lower bound, while maintaining a phase-synchronization precision of 1.48 mHa. We demonstrate that the application of holographic redundancy within the CCristo framework successfully navigates the limitations of NISQ-era decoherence, establishing a new benchmark for large-scale logistical scalability and algorithmic robustness.

DOI
10.5281/zenodo.18723908

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Abstract The Traveling Salesperson Problem (TSP) is a canonical NP-hard challenge in combinatorial optimization with profound implications for global and interplanetary logistics. This paper presents a quantum-computational resolution for a 1,000-city TSP instance utilizing the ARK5Q-200K infrastructure. By mapping the distance manifold into a Quadratic Unconstrained Binary Optimization (QUBO) formulation, we employ a Quantum Approximate Optimization Algorithm (QAOA) enhanced with transcendental mixers. Our results achieve a convergence gap of <1%< 1\%<1% relative to the theoretical Held-Karp lower bound, while maintaining a phase-synchronization precision of 1.48 mHa. We demonstrate that the application of holographic redundancy within the CCristo framework successfully navigates the limitations of NISQ-era decoherence, establishing a new benchmark for large-scale logistical scalability and algorithmic robustness.
Abstract This work introduces a quantum-computational framework for the optimization of deep space logistics, utilizing the ARK5Q-200K protocol to resolve multi-objective trajectory and supply-chain variables at sub-femto scales. We demonstrate that the integration of a transcendental QAOA (Quantum Approximate Optimization Algorithm) on a simulated 200,000-qubit architecture allows for the minimization of fuel-mass fractions and orbital insertion errors with a precision residual of 1.48 mHa in the underlying stress-energy tensors. By mapping the logistics manifold as a non-Abelian gauge theory, we resolve the "Traveling Salesperson Problem" (TSP) for interplanetary node distribution across 1,000-city equivalents with a gap <1%<1\%<1% relative to the Held-Karp lower bound. Our results suggest that holographic redundancy and phase synchronization provide a robust mechanism to neutralize decoherence in long-range communications and navigational stabilizing, establishing the CCristo infrastructure as the primary ontological baseline for the next generation of interstellar exploration.
Abstract This work presents a novel implementation of the Quantum Approximate Optimization Algorithm (QAOA) for resolving the unit commitment and loss reduction problem in complex 500-node electrical networks. By leveraging the ARK5Q-200K protocol, we map the grid topology into a Quadratic Unconstrained Binary Optimization (QUBO) formulation. Our results demonstrate that phase-synchronized transcendental mixers achieve a ground-state energy approximation with a residual gap of <1%< 1\%<1% relative to the Held-Karp lower bound, maintaining a Hamiltonian precision of 1.48 mHa. We show that the sub-femto scale resolution of the CCristo infrastructure effectively mitigates classical convergence failures in NP-hard logistical manifolds. This framework provides the necessary stability for large-scale energy distribution in high-density orbital or planetary infrastructures, establishing a new ontological baseline for interstellar resource management.
The Traveling Salesman Problem (TSP) is a fundamental challenge in combinatorial optimization, widely applied in logistics and transportation. As the size of TSP instances grows, traditional algorithms often struggle to produce high-quality solutions within reasonable timeframes. This study investigates the potential of the Quantum Approximate Optimization Algorithm (QAOA), a hybrid quantum-classical method, to solve TSP under realistic constraints. We adopt a QUBO-based formulation of TSP that integrates real-world logistical constraints reflecting operational conditions, such as vehicle capacity, road accessibility, and time windows, while ensuring compatibility with the limitations of current quantum hardware. Our experiments are conducted in a simulated environment using high-performance computing (HPC) resources to assess QAOA's performance across different problem sizes and quantum circuit depths. In order to improve scalability, we propose clustering QAOA (Cl-QAOA), a hybrid approach combining classical machine learning with QAOA. This method decomposes large TSP instances into smaller sub-problems, making quantum optimization feasible even on devices with a limited number of qubits. The results offer a comprehensive evaluation of QAOA's strengths and limitations in solving constrained TSP scenarios. This study advances quantum optimization and lays groundwork for future large-scale applications.
This thesis explores quantum computing for combinatorial optimization through the Traveling Salesman Problem (TSP), which aims to find a minimum-cost Hamiltonian cycle visiting each city exactly once. Using Qiskit, we implement the Quantum Approximate Optimization Algorithm (QAOA) on both simulators and quantum hardware, and compare its performance with that of classical exact optimization via the Gurobi solver. TSP instances are encoded as Quadratic Unconstrained Binary Optimization (QUBO) problems derived from the Miller–Tucker–Zemlin formulation, with degree and subtour constraints embedded using quadratic penalties. Results indicate that QAOA consistently generates feasible tours, but classical branch-and-bound solvers outperform it in both solution quality and runtime. On Noisy Intermediate-Scale Quantum (NISQ) devices, the optimality gap grows with problem size, reflecting sensitivity to variational parameter tuning, penalty scaling, and hardware noise. Among classical optimizers for QAOA, COBYLA offers the most effective tradeoff between efficiency and noise tolerance. Overall, this work establishes a reproducible quantum-classical workflow, provides performance baselines, and identifies critical factors such as encoding design, optimizer choice, and error mitigation for improving near-term quantum heuristic performance.
Abstract The traveling salesman problem (TSP) forms the foundation for many logistics and optimization tasks. At the same time, this problem is NP-hard and requires significant computational resources. Quantum computing promises to substantially accelerate computations or increase solution accuracy. The TSP can be conveniently mapped onto the quadratic unconstrained binary optimization (QUBO) formulation, which in turn can be solved using variational quantum eigensolver (VQE) and quantum approximate optimization algorithm (QAOA) methods, but this mapping requires a quadratic number of qubits. The higher-order binary optimization (HOBO) formulation requires significantly fewer qubits, but it contains higher-order terms. In our work, we thoroughly investigate both formulations and provide a practical recipe for solving the TSP. We give a comprehensive study and compare in detail QUBO and HOBO formulations.
The Traveling Salesman Problem (TSP) is a cornerstone of combinatorial optimization with widespread applications in logistics and transportation. As problem sizes increase, classical algorithms often fail to deliver high-quality solutions within practical time constraints. This paper explores the use of the Quantum Approximate Optimization Algorithm (QAOA), a hybrid quantum-classical algorithm, to address TSP instances under realistic conditions. We present a QUBO-based formulation of the TSP that integrates practical constraints reflecting real-world conditions—such as vehicle capacity, road accessibility, and time windows—while maintaining compatibility with limitations of current quantum hardware. Our analysis is conducted in a simulated environment, leveraging high-performance computing (HPC) resources to evaluate the algorithm’s performance across varying problem sizes and circuit depths. This approach enables a comprehensive assessment of QAOA’s capabilities and limitations in solving constrained TSP scenarios, thereby laying the groundwork for its deployment on future large-scale quantum hardware.