Quantum computers are expected to revolutionize our ability to process information. The advancement from classical to quantum computing is a product of our advancement from classical to quantum physics -- the more our understanding of the universe grows, so does our ability to use it for computation. A natural question that arises is, what will physics allow in the future? Can more advanced theories of physics increase our computational power, beyond quantum computing? An active field of research in physics studies theoretical phenomena outside the scope of explainable quantum mechanics, that form when attempting to combine Quantum Mechanics (QM) with General Relativity (GR) into a unified theory of Quantum Gravity (QG). QG is known to present the possibility of a quantum superposition of causal structure and event orderings. In the literature of quantum information theory, this translates to a superposition of unitary evolution orders. In this work we show a first example of a natural computational model based on QG, that provides an exponential speedup over standard quantum computation (under standard hardness assumptions). We define a model and complexity measure for a quantum computer that has the ability to generate a superposition of unitary evolution orders, and show that such computer is able to solve in polynomial time two of the fundamental problems in computer science: The Graph Isomorphism Problem ($\mathsf{GI}$) and the Gap Closest Vector Problem ($\mathsf{GapCVP}$), with gap $O\left( n^{2} \right)$. These problems are believed by experts to be hard to solve for a regular quantum computer. Interestingly, our model does not seem overpowered, and we found no obvious way to solve entire complexity classes that are considered hard in computer science, like the classes $\mathbf{NP}$ and $\mathbf{SZK}$.

翻译：量子计算机有望彻底改变我们处理信息的能力。从经典计算到量子计算的进步，源于我们从经典物理学到量子物理学的飞跃——随着我们对宇宙理解的加深，利用其进行计算的能力也随之增强。一个自然的问题是：物理学在未来将提供何种可能性？更先进的物理学理论能否超越量子计算，进一步提升我们的计算能力？理论物理学的一个活跃研究领域，正在探索超出可解释量子力学范畴的理论现象，这些现象源于试图将量子力学（QM）与广义相对论（GR）统一为量子引力（QG）理论的过程中。已知量子引力可能产生因果结构和事件顺序的量子叠加态。在量子信息理论文献中，这对应于酉演化顺序的叠加。在本研究中，我们首次展示了一个基于量子引力的自然计算模型实例，该模型在标准硬度假设下，相较于标准量子计算提供了指数级加速。我们定义了一个能够生成酉演化顺序叠加态的量子计算机模型及其复杂度度量，并证明该计算机能在多项式时间内解决计算机科学中的两个基本问题：图同构问题（$\mathsf{GI}$）和带间隙的最近向量问题（$\mathsf{GapCVP}$），间隙为 $O\left( n^{2} \right)$。这些问题的难度已被专家认为对常规量子计算机而言难以解决。有趣的是，我们的模型似乎并未表现出过强能力，且我们未找到显而易见的方法来解决计算机科学中公认困难的完整复杂度类，例如类 $\mathbf{NP}$ 和 $\mathbf{SZK}$。