论文标题
Epsilon网络，统一设计和随机量子电路
Epsilon-nets, unitary designs and random quantum circuits
论文作者
论文摘要
epsilon-net和近似统一$ t $ - 设计是自然概念，可捕获与量子信息和量子计算中许多应用相关的单一操作的属性。前者构成了统一通道的子集，这些渠道是epsilon关闭到钻石规范中任何统一通道的子集。后者是（大约）（大约）在单一渠道的条目中恢复多项式的HAAR平均值的一部分，直到订购$ t $。 在这项工作中，我们建立了这两个概念之间的定量联系。具体而言，我们证明，对于希尔伯特空间的固定尺寸$ d $，构成$δ$ -2 $ -T $ t $ t $ - expanders for $ t $ε$ -Nets的一职级，用于$ t \ simeq \ simeq \ frac {d^{5/2}}}ε$ $ $ $ $δ= \ left（\ frac {ε^{3/2}}} {d} \ right）^{d^2} $。我们还表明，$ε$ -NET可用于构建$δ$ - 适合$ t $ t $ -designs，$δ=εt$。最后，我们证明，获得$ε$ -NET所需的确切统一$ t $ -DESIGN的程度至少必须快速增长为$ \frac1ε$（对于固定$ d $），而不慢于$ d^2 $（对于固定$ε$）。这显示了我们连接$ t $ -Designs和$ε$ -NET的结果的最佳性。 我们在量子计算的背景下应用我们的发现。首先，我们表明，近似的t设计可以是由由平行和顺序局部架构中的一组通用的两极门形成的浅随机电路生成的。我们的栅极组不必是对称的（即包含门与它们的对流）或由具有代数条目的门组成。我们还展示了通用门集的Solovay-Kitaev定理的非构造版本。我们的主要技术贡献是在量子通道空间中对Dirac Delta的有效多项式近似的新结构，这可能是独立的。
Epsilon-nets and approximate unitary $t$-designs are natural notions that capture properties of unitary operations relevant for numerous applications in quantum information and quantum computing. The former constitute subsets of unitary channels that are epsilon-close to any unitary channel in the diamond norm. The latter are ensembles of unitaries that (approximately) recover Haar averages of polynomials in entries of unitary channels up to order $t$. In this work we establish quantitative connections between these two notions. Specifically, we prove that, for a fixed dimension $d$ of the Hilbert space, unitaries constituting $δ$-approximate $t$-expanders form $ε$-nets for $t\simeq\frac{d^{5/2}}ε$ and $δ=\left(\frac{ε^{3/2}}{d}\right)^{d^2}$. We also show that $ε$-nets can be used to construct $δ$-approximate unitary $t$-designs for $δ= εt$. Finally, we prove that the degree of an exact unitary $t$-design necessary to obtain an $ε$-net must grow at least fast as $\frac1ε$ (for fixed $d$) and not slower than $d^2$ (for fixed $ε$). This shows near optimality of our result connecting $t$-designs and $ε$-nets. We apply our findings in the context of quantum computing. First, we show that that approximate t-designs can be generated by shallow random circuits formed from a set of universal two-qudit gates in the parallel and sequential local architectures. Our gate sets need not to be symmetric (i.e. contain gates together with their inverses) or consist of gates with algebraic entries. We also show a non-constructive version of the Solovay-Kitaev theorem for general universal gate sets. Our main technical contribution is a new construction of efficient polynomial approximations to the Dirac delta in the space of quantum channels, which can be of independent interest.
