Random Quantum Circuits are Approximate 2-designs
Random Quantum Circuits are Approximate 2-designs
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DOI：
10.1007/s00220-009-0873-6
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发表时间：
2009-10-01
影响因子：
2.4
通讯作者：
Low, Richard A.
中科院分区：
文献类型：
--
作者：
Harrow, Aram W.;Low, Richard A.
给定两个量子比特上的通用门集合，众所周知，将集合中的随机门应用于量子比特的随机对将最终产生近似哈尔分布的幺正。然而，这需要指数时间。我们发现，只有多项式长度的随机电路将近似哈尔分布的第一和第二时刻，从而形成近似1和2设计。以前的结构需要更长的电路，只适用于特定的门组。作为我们的主要结果的推论，我们还改进了以前的Clifford群上的随机游动的收敛速度的界。
Given a universal gate set on two qubits, it is well known that applying random gates from the set to random pairs of qubits will eventually yield an approximately Haar-distributed unitary. However, this requires exponential time. We show that random circuits of only polynomial length will approximate the first and second moments of the Haar distribution, thus forming approximate 1- and 2-designs. Previous constructions required longer circuits and worked only for specific gate sets. As a corollary of our main result, we also improve previous bounds on the convergence rate of random walks on the Clifford group.