A fundamental computational problem is to find a shortest non-zero vector in Euclidean lattices, a problem known as the Shortest Vector Problem (SVP). This problem is believed to be hard even on quantum computers and thus plays a pivotal role in post-quantum cryptography. In this work we explore how (efficiently) Noisy Intermediate Scale Quantum (NISQ) devices may be used to solve SVP. Specifically, we map the problem to that of finding the ground state of a suitable Hamiltonian. In particular, (i) we establish new bounds for lattice enumeration, this allows us to obtain new bounds (resp.~estimates) for the number of qubits required per dimension for any lattices (resp.~random q-ary lattices) to solve SVP; (ii) we exclude the zero vector from the optimization space by proposing (a) a different classical optimisation loop or alternatively (b) a new mapping to the Hamiltonian. These improvements allow us to solve SVP in dimension up to 28 in a quantum emulation, significantly more than what was previously achieved, even for special cases. Finally, we extrapolate the size of NISQ devices that is required to be able to solve instances of lattices that are hard even for the best classical algorithms and find that with approximately $10^3$ noisy qubits such instances can be tackled.

翻译：基本计算问题之一是在欧几里得格中寻找非零最短向量，即最短向量问题（SVP）。该问题被认为即使在量子计算机上也难以求解，因此在后量子密码学中具有关键作用。本研究探讨了如何（高效地）利用含噪中等规模量子（NISQ）设备求解SVP。具体而言，我们将该问题映射为寻找合适哈密顿量基态的问题。其中：（i）我们建立了格枚举的新界限，从而获得了任意格（对应随机q元格）求解SVP时每维度所需量子比特数的新界限（及估计值）；（ii）我们通过提出（a）不同的经典优化循环或（b）新的哈密顿量映射，将零向量排除在优化空间之外。这些改进使我们能够在量子模拟中求解维度高达28的SVP，远超此前特殊情况下的最高水平。最后，我们推算了要解决即便最优经典算法也难以处理的困难格实例所需的NISQ设备规模，发现约需10³个噪声量子比特即可应对此类实例。