In the era of Noisy Intermediate-Scale Quantum (NISQ) computers it is crucial to design quantum algorithms which do not require many qubits or deep circuits. Unfortunately, most of the well-known quantum algorithms are too demanding to be run on currently available quantum devices. Moreover, even the state-of-the-art algorithms developed for the NISQ era often suffer from high space complexity requirements for particular problem classes. In this paper, we show that it is possible to greatly reduce the number of qubits needed for the Travelling Salesman Problem (TSP), a paradigmatic optimization task, at the cost of having deeper variational circuits. While the focus is on this particular problem, we claim that the approach can be generalized for other problems where the standard bit-encoding is highly inefficient. Finally, we also propose encoding schemes which smoothly interpolate between the qubit-efficient and the circuit depth-efficient models. All the proposed encodings have the same volume up to polylogarithmic factors and remain efficient to implement within the Quantum Approximate Optimization Algorithm framework.

在嘈杂的中级量子 (NISQ) 计算机时代，设计不需要很多量子位或深度电路的量子算法至关重要。不幸的是，大多数众所周知的量子算法要求太高，无法在当前可用的量子设备上运行。此外，即使是为 NISQ 时代开发的最先进的算法也常常受到特定问题类别的高空间复杂度要求的困扰。在本文中，我们展示了可以大大减少旅行商问题 (TSP) 所需的量子比特数量，这是一种范式优化任务，但代价是拥有更深的变分电路。虽然重点是这个特定问题，但我们声称该方法可以推广到标准位编码效率非常低的其他问题。最后，我们还提出了在量子比特高效模型和电路深度高效模型之间平滑插值的编码方案。所有提出的编码在多对数因子上都具有相同的体积，并且在量子近似优化算法框架内实现仍然有效。