论文标题
拓扑有序系统中的电路深度与能量
Circuit depth versus energy in topologically ordered systems
论文作者
论文摘要
我们证明，假设电路是几何局部的，则证明了在局部相互作用的量子多体系统中制备局部相互作用量子多体系统的低能状态的非平凡电路深度下限。为了准备任何具有$ε$的能量密度相对于基塔耶的复曲面hamiltonian在二维晶格上$λ$的状态，我们证明了$ω\ left的下限（\ min \ left（\ min \ left）任何$α> 0 $。我们讨论两个含义。首先，我们的界限意味着可以从大量现有的变化电路（例如，汉密尔顿变异的ANSATZ）获得的最低能量密度，通常不能随电路深度呈指数衰减。其次，如果存在远距离纠缠处的基态，即使在非零的能量密度下，这也可能导致非平凡的电路深度下限。与以前证明电路深度下限用于制备低能状态的方法不同，我们的证明技术并不依赖于基态退化。
We prove a nontrivial circuit-depth lower bound for preparing a low-energy state of a locally interacting quantum many-body system in two dimensions, assuming the circuit is geometrically local. For preparing any state which has an energy density of at most $ε$ with respect to Kitaev's toric code Hamiltonian on a two dimensional lattice $Λ$, we prove a lower bound of $Ω\left(\min\left(1/ε^{\frac{1-α}{2}}, \sqrt{|Λ|}\right)\right)$ for any $α>0$. We discuss two implications. First, our bound implies that the lowest energy density obtainable from a large class of existing variational circuits (e.g., Hamiltonian variational ansatz) cannot, in general, decay exponentially with the circuit depth. Second, if long-range entanglement is present in the ground state, this can lead to a nontrivial circuit-depth lower bound even at nonzero energy density. Unlike previous approaches to prove circuit-depth lower bounds for preparing low energy states, our proof technique does not rely on the ground state to be degenerate.
