Improved spectral gaps for random quantum circuits: Large local dimensions and all-to-all interactions
Improved spectral gaps for random quantum circuits: Large local dimensions and all-to-all interactions
复制标题
改进随机量子电路的光谱间隙：大局部尺寸和全面相互作用
DOI：
10.1103/physreva.104.022417
复制
发表时间：
2020
影响因子：
2.9
通讯作者：
Nicholas Hunter
中科院分区：
文献类型：
--
作者：
Jonas Haferkamp;Nicholas Hunter
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强相互作用系统和黑洞中的混乱描述。随机量子电路在这些环境中的效用源于它们快速产生量子伪随机性的能力。在Brandao、Harrow和Horodecki的一篇开创性论文中，证明了在局部维数为q的量子点上的局部随机量子电路的t阶矩算子具有至少为$\Omega（n^{-1}t^{-5-3.1/\log（q）}）的谱隙，这意味着它们是近似酉设计的有效构造。作为第一个结果，我们使用无阻挫哈密顿量的谱隙的Knabe界来表明，$1D$随机量子电路的谱隙标度为$\Omega（n^{-1}）$，条件是$t$与局部维数相比很小：$t^2\leq O（q）$。这意味着在设计顺序$t$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谱隙，其下界为$\Omega（n^{-1}\log^{-1}（n）t^{-\alpha（q）}）$，对于具有全对全相互作用的随机量子电路。这改进了非局部模型的设计深度中的$n$和$t$缩放。我们证明了这一点，通过证明一个递归关系的频谱间隙涉及一个辅助随机游走。最后，我们解决了最小的非平凡的情况下，确切地和联合收割机与数值和Knabe界，以改善涉及的常数在频谱间隙小的值$t$。
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 Brandao, 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 $\Omega(n^{-1}t^{-5-3.1/\log(q)})$, 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 $\Omega(n^{-1})$, 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 $\Omega(n^{-1}\log^{-1}(n) t^{-\alpha(q)})$ 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 $t$.
影响因子：
1.3
作者：
M. Lemm;E. Mozgunov
通讯作者：
M. Lemm;E. Mozgunov
DOI：
--
发表时间：
2018-02
期刊：
arXiv: High Energy Physics - Theory
影响因子：
--
作者：
L. Susskind
通讯作者：
L. Susskind
影响因子：
5
作者：
Brown, Adam R.;Susskind, Leonard
通讯作者：
Susskind, Leonard