发表机构
University of Illinois at Urbana-Champaign; Syracuse University; Eindhoven University of Technology（伊利诺伊大学厄巴纳-香槟分校; 雪城大学; 埃因霍温理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出连续通用量子计算概念，通过向局域控制算符添加魔法哈密顿量（如伊辛哈密顿量）生成完整李代数，实现任意酉算符的指数化实现，并分析所需时间与相变。
AI 中文摘要
在传统的量子计算理论中，通用性是通过酉门集的性质来讨论的。然而，在许多实验设置中，我们转而可以访问一组参数化的哈密顿量，这些哈密顿量可以在任意所需时间内进行指数化。因此，我们提出了通用性的连续模拟，其中向一组廉价的局域控制算符添加一个昂贵的哈密顿量控制即可实现通用计算。我们找到了几个具体的1-或2-局域泡利生成元集合$S$和$n$量子比特的“魔法”哈密顿量$H$，它们共同生成完整的李代数$\frak{su}(2^n)$，特别关注伊辛哈密顿量$J$。我们提出了一种通过指数化这些生成元集合来直接实现任意酉算符的方法，并给出了需要应用魔法哈密顿量$H$的时间界限。我们发现，为了以精度$\epsilon$逼近任意酉算符所需的$H$的应用次数，随量子比特数量呈指数增长，这与标准Solovay-Kitaev定理的应用预期一致。我们还发现，当伊辛哈密顿量的参数变化时，生成的李代数的大小会出现相变。我们找到了Chow/Rashevskii技术的一种算法应用，用于实现对应于迭代交换子的酉算符。
英文摘要
In the conventional theory of quantum computation, universality is discussed in terms of properties of unitary gate sets. In many experimental setups, however, we instead have access to a parametrized set of Hamiltonians, which can be exponentiated for any desired time. Accordingly, we formulate a continuous analogue of universality, where adding one expensive Hamiltonian control to a cheap set of local control operators allows for universal computation. We find several concrete sets of 1- or 2-local Pauli generators $S$ and $n$-qubit ``magic" Hamiltonians $H$ that together generate the full Lie algebra $\mathfrak{su}(2^n)$, with a special focus on the Ising Hamiltonian $J$. We propose a method to directly implement any unitary by exponentiating these generator sets, and give bounds on the needed time of application of the magic Hamiltonian $H$. We find that the number of applications of $H$ needed to approximate any unitary up to accuracy $ε$ scales exponentially in the number of qubits as expected from the standard application of the Solovay-Kitaev theorem. We also find phase transitions in the size of the generated Lie algebras as the parameters of the Ising Hamiltonian are varied. We find an algorithmic application of the Chow/Rashevskii technique to implement unitaries corresponding to iterated commutators.
Comments41 pages
