[论文解读] Quantum Circuit Depth Lower Bounds For Homological Codes

Dorit Aharonov, Yonathan Touati|arXiv (Cornell University)|Oct 9, 2018

Quantum Computing Algorithms and Architecture参考文献 20被引用 5

一句话总结

本文首次为非简并局部哈密顿量——具体为基塔耶夫球面码——的唯一基态建立了量子线路深度下界，通过一种新颖的拓扑不变量‘γ-分离’，证明了Ω(log n)的下界。该结果将可高效由常数深度线路制备的平凡态与具有拓扑序的基态区分开来，解决了量子复杂性与拓扑序领域的一个关键挑战。

ABSTRACT

We provide an $Ω(log(n))$ lower bound for the depth of any quantum circuit generating the unique groundstate of Kitaev's spherical code. No circuit-depth lower bound was known before on this code in the general case where the gates can connect qubits even if they are far away; To the best of our knowledge, this is the first time a quantum circuit-depth lower bound is given for unique ground state of a {\it gapped} local Hamiltonian. Providing a lower bound in this case seems more challenging, since such systems exhibit exponential decay of correlations and standard lower bound techniques do not apply. We prove our lower bound by introducing the new notion of $γ$-separation, and analyzing its behavior using algebraic topology arguments. We extend out methods also to a wide class of polygonal complexes beyond the sphere, and prove a circuit-depth lower bound whenever the complex does not have a small "bottle neck" (in a sense which we define). Here our lower bound on the circuit depth is only $Ω(logloglog(n))$. We conjecture that the correct lower bound is at least $Ω(log(log(n))$, but this seems harder to achieve due to the possibility of hyperbolic geometry. For general simplicial complexes the lack of geometrical restriction on the gates becomes considerably more problematic than for the sphere, and we need to thoroughly modify the original argument in order to get a meaningful bound. To the best of our knowledge, this is the first time the class of trivial quantum states is separated from the class of unique ground states of gapped local Hamiltonians; we provide a survey of the current status of this hierarchy for completeness.The tools developed here will be useful in various contexts in which quantum circuit depth lower bounds are of interest, including the study of topological order, quantum computational complexity and quantum algorithmic speedups.

研究动机与目标

• 为基塔耶夫球面码——一种非简并局部哈密顿量——的唯一基态建立线路深度下界。
• 克服在通用线路模型中证明下界时的挑战，其中门操作可能非局域，不同于以往的几何受限设置。
• 引入并形式化‘γ-分离’这一新拓扑工具，用于分析量子态的复杂度。
• 将下界技术扩展至广泛的多边形复形，识别出结构性条件（无小瓶颈）所蕴含的非平凡深度。
• 将平凡量子态类（由常数深度线路生成）与非简并局部哈密顿量的唯一基态类区分开来，澄清全局纠缠与拓扑序的层级关系。

提出的方法

• 引入γ-分离的概念，这是一种量化同调码中量子态结构复杂度的拓扑不变量。
• 利用代数拓扑分析γ-分离在局部量子操作下的行为，将拓扑结构与线路深度约束联系起来。
• 证明任何生成基塔耶夫球面码基态的量子线路，其深度至少为Ω(log n)，基于γ-分离与拓扑不变性。
• 通过定义拓扑瓶颈条件，将该方法推广至一般多边形复形；证明若不存在此类瓶颈，则深度下界为Ω(log log log n)。
• 利用非简并局部哈密顿量基态具有指数衰减关联的特性，该特性使标准下界技术失效，而通过γ-分离克服了这一障碍。
• 通过归约至扩展的CAT态，证明其与长程关联态的重叠（常数深度线路无法生成）意味着基态的非平凡性。

实验结果

研究问题

• RQ1 是否存在常数深度的量子线路，能够生成基塔耶夫球面码的唯一基态？
• RQ2 当允许非局域门操作时，制备非简并局部哈密顿量基态所需的最小线路深度是多少？
• RQ3 在缺乏几何约束的条件下，如何利用拓扑不变量推导量子线路深度的下界？
• RQ4 在何种复形的拓扑条件下，同调码的基态需要超常数深度的线路？
• RQ5 平凡量子态类（由有界深度线路生成）能否与非简并局部哈密顿量的唯一基态类相分离？

主要发现

• 本文为在允许非局域门操作的条件下，生成基塔耶夫球面码唯一基态的线路深度，建立了Ω(log n)的下界。
• 该下界在常数因子意义下是紧的，因为深度为对数级的MERA线路可制备相同态。
• 作者引入γ-分离作为新的拓扑不变量，使在非几何设置下实现深度下界成为可能。
• 对于一大类多边形复形，本文证明：当复形缺乏小拓扑瓶颈时，深度下界为Ω(log log log n)。
• 作者猜想更强的Ω(log log n)下界，但指出当前技术可能受双曲几何的阻碍而无法实现。
• 本工作首次实现了平凡量子态类（常数深度线路生成）与非简并局部哈密顿量唯一基态类的分离，推进了对拓扑序与纠缠层级的理解。