Department of Physics and Astronomy, Dartmouth College, 6127 Wilder Laboratory, Hanover, New Hampshire 03755, USA.
Phys Rev Lett. 2010 Jun 25;104(25):250501. doi: 10.1103/PhysRevLett.104.250501. Epub 2010 Jun 21.
We consider a class of random quantum circuits where at each step a gate from a universal set is applied to a random pair of qubits, and determine how quickly averages of arbitrary finite-degree polynomials in the matrix elements of the resulting unitary converge to Haar measure averages. This is accomplished by mapping the superoperator that describes t order moments on n qubits to a multilevel SU(4^{t}) Lipkin-Meshkov-Glick Hamiltonian. We show that, for arbitrary fixed t, the ground-state manifold is exactly spanned by factorized eigenstates and, under the assumption that a mean-field ansatz accurately describes the low-lying excitations, the spectral gap scales as 1/n in the thermodynamic limit. Our results imply that random quantum circuits yield an efficient implementation of ϵ approximate unitary t designs.
我们考虑一类随机量子电路，其中在每一步中，一个通用集的门被应用于一对随机量子位，并且确定任意有限度多项式的平均值在矩阵元素中的结果单位元如何快速收敛到 Haar 测量平均值。这是通过将描述 t 阶矩的超算子映射到多能级 SU(4^{t}) Lipkin-Meshkov-Glick 哈密顿量来实现的。我们表明，对于任意固定的 t，基态流形是由因子化本征态精确地张成的，并且在假设平均场假设准确地描述了低能激发的情况下，在热力学极限下，谱隙与 1/n 成比例。我们的结果表明，随机量子电路产生了ϵ近似单位元 t 设计的有效实现。
