发表机构
Los Alamos National Laboratory; Quantum Science Center（洛斯阿拉莫斯国家实验室; 量子科学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究提出在一般连通图下用随机游走构造匹配门设计，深度近最优，并改进费米子路由，实现指数加速。
AI 中文摘要
理解在各种群上生成近似随机酉算子所需的资源是量子信息理论的一个自然目标。关于近似的一种概念，即设计，已知全酉群可以通过最近邻二局部门的一维电路在对数深度内被近似。另一方面，值得注意的是，具有这种连通性的电路无法在亚线性深度内形成匹配门群上的设计。这里我们表明，当使用路由数为 ${\rm rt}(\mathsf{G})$ 的一般量子比特连通图 $\mathsf{G}$ 时，这种戏剧性的减慢可以消失。确实，在这种设置中，可以在深度 $\mathcal{O}(k^2{\rm rt}(\mathsf{G})\log n\log(n/\varepsilon))$ 内获得（强）$\varepsilon$-近似相对误差匹配门 $k$-设计。对于全连接连通性，${\rm rt}(\mathsf{G})=2$。我们的构造在概念上简单，涉及匹配门群上的随机游走，且不需要辅助量子比特。作为技术副产品，我们改进了费米子路由的现有技术水平，获得了深度为 $\mathcal{O}({\rm rt}(\mathsf{G})\log n)$ 的路由器。此外，对于 $k=3$，我们在 $\mathcal{O}({\rm rt}(\mathsf{G})\log n)$ 深度内获得了精确的强匹配门设计，同样无需辅助量子比特。在全连接连通性下，我们的费米子路由器和3-设计是最优的。值得注意的是，我们的结果意味着，需要从匹配门3-设计中抽样的费米子层析成像量子算法，在全连接连通性的量子计算机上可以相对于严格一维对应物指数级加速。
英文摘要
Understanding the resources required to generate approximately random unitaries over various groups is a natural goal of quantum information theory. With respect to one notion of approximation, that of a design, it is known that the full unitary group can be approximated in logarithmic depth by one-dimensional circuits of nearest-neighbour 2-local gates. On the other hand, remarkably, circuits with this connectivity cannot form designs over the matchgate group in sublinear depth. Here we show that this dramatic slowdown can disappear when using a general qubit connectivity graph $\mathsf{G}$ of routing number ${\rm rt}(\mathsf{G})$. Indeed, in this setting one can obtain (strong) $\varepsilon$-approximate relative error matchgate $k$-designs in depth $\mathcal{O}(k^2{\rm rt}(\mathsf{G})\log n\log(n/\varepsilon))$. For all-to-all connectivity, ${\rm rt}(\mathsf{G})=2$. Our construction is conceptually simple, involving a random walk on the matchgate group, and no ancillae. As a technical byproduct, we improve upon the state of the art for fermionic routing, obtaining an $\mathcal{O}({\rm rt}(\mathsf{G})\log n)$ depth router. Additionally, for $k=3$, we obtain an exact strong matchgate design in $\mathcal{O}({\rm rt}(\mathsf{G})\log n)$ depth, again without ancillae. Under all-to-all connectivity, our fermionic router and 3-designs are optimal. Notably, our results imply that quantum algorithms for fermionic tomography which require drawing from a matchgate 3-design may be exponentially sped up on quantum computers with all-to-all connectivity, relative to their strictly one-dimensional counterparts.
