论文标题
改善随机量子电路的光谱差距：较大的局部尺寸和全面相互作用
Improved spectral gaps for random quantum circuits: large local dimensions and all-to-all interactions
论文作者
论文摘要
随机量子电路是量子信息理论中的一个核心概念，其应用范围从量子计算优势的演示到对牢固相互交互的系统和黑洞中争夺的描述。这些设置中随机量子电路的效用源于它们快速产生量子伪随机性的能力。 In a seminal paper by Brandão, Harrow, and Horodecki, it was proven that the $t$-th moment operator of local random quantum circuits on $n$ qudits with local dimension $q$ has a spectral gap of at least $Ω(n^{-1}t^{-5-3.1/\log(q)})$, which implies that they are efficient constructions of approximate unitary designs.首先，我们使用尾界对无挫败感的汉密尔顿人的频谱差距，以表明$ 1D $随机量子电路的光谱间隙缩放为$ω（n^{ - 1}）$，前提是$ t $与本地尺寸相比很小：$ t^2 \ leq oq oq o（q）$。这意味着设计顺序$ t $中电路深度的（几乎）线性缩放。我们的第二个结果是一个无条件的光谱差距，下面是$ω（n^{ - 1} \ log^{ - 1}（n）t^{ - α（q）}）$，用于随机量子电路，具有全部相互作用。这可以改善非本地模型的$ n $和$ t $缩放设计深度。我们通过证明涉及辅助随机步行的光谱间隙的递归关系来证明这一点。最后，我们准确地解决了最小的非平凡案例，并与数字和膝关节结合，以改善光谱间隙中涉及的常数，以$ t $的小值。
Random quantum circuits are a central concept in quantum information theory with applications ranging from demonstrations of quantum computational advantage to descriptions of scrambling in strongly-interacting systems and black holes. The utility of random quantum circuits in these settings stems from their ability to rapidly generate quantum pseudo-randomness. In a seminal paper by Brandão, Harrow, and Horodecki, it was proven that the $t$-th moment operator of local random quantum circuits on $n$ qudits with local dimension $q$ has a spectral gap of at least $Ω(n^{-1}t^{-5-3.1/\log(q)})$, which implies that they are efficient constructions of approximate unitary designs. As a first result, we use Knabe bounds for the spectral gaps of frustration-free Hamiltonians to show that $1D$ random quantum circuits have a spectral gap scaling as $Ω(n^{-1})$, provided that $t$ is small compared to the local dimension: $t^2\leq O(q)$. This implies a (nearly) linear scaling of the circuit depth in the design order $t$. Our second result is an unconditional spectral gap bounded below by $Ω(n^{-1}\log^{-1}(n) t^{-α(q)})$ for random quantum circuits with all-to-all interactions. This improves both the $n$ and $t$ scaling in design depth for the non-local model. We show this by proving a recursion relation for the spectral gaps involving an auxiliary random walk. Lastly, we solve the smallest non-trivial case exactly and combine with numerics and Knabe bounds to improve the constants involved in the spectral gap for small values of $t$.
