We present a map from the travelling salesman problem (TSP), a prototypical NP-complete combinatorial optimisation task, to the ground state associated with a system of many-qudits. Conventionally, the TSP is cast into a quadratic unconstrained binary optimisation (QUBO) problem, that can be solved on an Ising machine. The size of the corresponding physical system's Hilbert space is $2^{N^2}$, where $N$ is the number of cities considered in the TSP. Our proposal provides a many-qudit system with a Hilbert space of dimension $2^{N\log_2N}$, which is considerably smaller than the dimension of the Hilbert space of the system resulting from the usual QUBO map. This reduction can yield a significant speedup in quantum and classical computers. We simulate and validate our proposal using variational Monte Carlo with a neural quantum state, solving the TSP in a linear layout for up to almost 100 cities.

中文翻译：

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旅行商问题优化的众说纷representation表示

我们提供了一个地图，从旅行商问题（TSP）（一种典型的NP完全组合优化任务）到与多个Qudits系统相关联的基态。常规上，TSP被转换为二次无约束二进制优化（QUBO）问题，可以在Ising机器上解决。相应物理系统的希尔伯特空间的大小为$ 2 ^ {N ^ 2} $，其中$ N $是TSP中考虑的城市数。我们的建议提供了一个具有$ 2 ^ {N \ log_2N} $维度的希尔伯特空间的多Qudit系统，该空间大大小于通常的QUBO映射所得出的系统希尔伯特空间的维度。这种减少可以大大提高量子计算机和经典计算机的速度。我们使用具有神经量子态的变分蒙特卡罗模拟和验证了我们的建议，