论文标题
随机矩阵模型，用于随机大约$ t $ -Designs
A random matrix model for random approximate $t$-designs
论文作者
论文摘要
对于HAAR随机集$ \ MATHCAL {S} \ subset U（d）$的量子门$，我们考虑均匀度量$ν_\ MATHCAL {S} $，其支持由$ \ Mathcal {S s} $给出。度量$ν_\ MATHCAL {S} $可以被视为$δ（ν_\ Mathcal {s}，t）$ - 近似$ t $ -Design，$ t \ in \ in \ in \ mathbb {z} _+$。我们提出了一个随机矩阵模型，旨在描述任何$ t $的$δ（ν_\ Mathcal {s}，t）$的概率分布。我们的模型由一个块对角线矩阵给出，其块是独立的，由高斯或吉尼布尔合奏给出，其数量，大小和类型由$ t $确定。我们证明，该矩阵的运算符规范，$δ（{t}）$，是随机变量，$ \ sqrt {| \ Mathcal {s} |}δ（ν_\ Mathcal {s}，s s}，t）$收敛于分布中的分布中的$ \ nathcal niniper in Infinity的分布时。此外，对于任何$ε> 0 $，我们表征了模型，在尾部概率上给出明确的界限$ \ mathbb {p}（δ（t）> 2+ε）$。我们还表明，我们的模型满足了所谓的光谱间隙猜想，即，我们证明，使用概率$ 1 $，就有$ t \ in \ mathbb {z} _+$ in \ sup_ {k {k \ in \ mathbb {z}} _ {+}} _ {+}}} $ sup_ {k \ in \ sup_ {k \ in数值模拟给出了令人信服的证据，表明所提出的模型实际上对于$ \ Mathcal {s} $的任何基数几乎是精确的。我们提供的这种现象的启发式解释使我们猜想tail概率$ \ mathbb {p}（\ sqrt {\ sqrt {\ mathcal {s}}Δ（ν_v _ \ \ \ \ m nathcal {s}，s}，t），T）我们随机矩阵模型的$ \ mathbb {p}（δ（t）> 2+ε）$。特别是我们的猜想意味着HAAR随机集$ \ MATHCAL {s} \ subset U（d）$都以概率$ 1 $满足光谱间隙的猜想。
For a Haar random set $\mathcal{S}\subset U(d)$ of quantum gates we consider the uniform measure $ν_\mathcal{S}$ whose support is given by $\mathcal{S}$. The measure $ν_\mathcal{S}$ can be regarded as a $δ(ν_\mathcal{S},t)$-approximate $t$-design, $t\in\mathbb{Z}_+$. We propose a random matrix model that aims to describe the probability distribution of $δ(ν_\mathcal{S},t)$ for any $t$. Our model is given by a block diagonal matrix whose blocks are independent, given by Gaussian or Ginibre ensembles, and their number, size and type is determined by $t$. We prove that, the operator norm of this matrix, $δ({t})$, is the random variable to which $\sqrt{|\mathcal{S}|}δ(ν_\mathcal{S},t)$ converges in distribution when the number of elements in $\mathcal{S}$ grows to infinity. Moreover, we characterize our model giving explicit bounds on the tail probabilities $\mathbb{P}(δ(t)>2+ε)$, for any $ε>0$. We also show that our model satisfies the so-called spectral gap conjecture, i.e. we prove that with the probability $1$ there is $t\in\mathbb{Z}_+$ such that $\sup_{k\in\mathbb{Z}_{+}}δ(k)=δ(t)$. Numerical simulations give convincing evidence that the proposed model is actually almost exact for any cardinality of $\mathcal{S}$. The heuristic explanation of this phenomenon, that we provide, leads us to conjecture that the tail probabilities $\mathbb{P}(\sqrt{\mathcal{S}}δ(ν_\mathcal{S},t)>2+ε)$ are bounded from above by the tail probabilities $\mathbb{P}(δ(t)>2+ε)$ of our random matrix model. In particular our conjecture implies that a Haar random set $\mathcal{S}\subset U(d)$ satisfies the spectral gap conjecture with the probability $1$.
