论文标题
基质浓度不平等和随机通用量子门的效率
Matrix concentration inequalities and efficiency of random universal sets of quantum gates
论文作者
论文摘要
对于一个随机集$ \ MATHCAL {S} \ subset U（d）$的量子门，我们提供了$ \ MATHCAL {S} $形式的概率的界限，形式为a $δ$ - approximate $ t $ -design。特别是我们发现，从精确的$ t $ -t $ -DESIGN中绘制的$ \ Mathcal {s} $，形成$δ$ -2 $ -Approximate $ t $ -design的可能性满足不平等的$ \ mathbb {p} \ frac {e^{ - | \ Mathcal {s} | X \，\ Mathrm {artcanh}（x）}}} {（1-x^2）^{| \ Mathcal {s} |/2}} = o \ left（2d_t \ left（\ frac {e^{e^{ - x^2}}}} \ right）^{| \ Mathcal {s} |} \ right）$，其中$ d_t $是在$ \ u \ u \ mapsto u^{\ otimes t} \ otimes t} \ otimes \ otimes \ bar bar {U}^u}^u}^{u}^{\ otimes t} $的分解中出现的唯一唯一不可减至的表示的总和。我们使用结果表明，要获得$δ$ -TAPPROXIMATION $ t $ -DESIGN带有概率$ P $ ONE需要$ O（δ^{ - 2}（t \ log log（d） - \ log（1-p）））$许多随机门。我们还分析了$δ$如何集中其预期值$ \ mathbb {e}δ$，用于随机$ \ nathcal {s} $。我们的结果对于对称和非对称门的有效。
For a random set $\mathcal{S} \subset U(d)$ of quantum gates we provide bounds on the probability that $\mathcal{S}$ forms a $δ$-approximate $t$-design. In particular we have found that for $\mathcal{S}$ drawn from an exact $t$-design the probability that it forms a $δ$-approximate $t$-design satisfies the inequality $\mathbb{P}\left(δ\geq x \right)\leq 2D_t \, \frac{e^{-|\mathcal{S}| x \, \mathrm{arctanh}(x)}}{(1-x^2)^{|\mathcal{S}|/2}} = O\left( 2D_t \left( \frac{e^{-x^2}}{\sqrt{1-x^2}} \right)^{|\mathcal{S}|} \right)$, where $D_t$ is a sum over dimensions of unique irreducible representations appearing in the decomposition of $U \mapsto U^{\otimes t}\otimes \bar{U}^{\otimes t}$. We use our results to show that to obtain a $δ$-approximate $t$-design with probability $P$ one needs $O( δ^{-2}(t\log(d)-\log(1-P)))$ many random gates. We also analyze how $δ$ concentrates around its expected value $\mathbb{E}δ$ for random $\mathcal{S}$. Our results are valid for both symmetric and non-symmetric sets of gates.
