有效的量子张量积展开式和k-设计
Efficient Quantum Tensor Product Expanders and k-designs
摘要：
量子扩张器是扩张器的量子模拟，而k-张量积扩张器是对随机化k个相关步行者的图的推广。在这里，我们给出了一个有效的构造常数度，常数间隙量子k-张量积扩展。关键成分是一个有效的经典张量积扩展和量子傅里叶变换。我们的构造在k=O（n/log n）时有效，其中n是量子位数。这个结果的一个直接推论是一个有效的近似酉k-设计，这是一个量子模拟的近似k-wise独立的功能，在n个量子比特的任何k=O（n/log n）。在此之前，k>2的有效构造是未知的，而状态设计，其中酉设计是一种推广，在[Ambainis，Emerson 2007]中被有效地构造。
英文摘要：
Quantum expanders are a quantum analogue of expanders, and k-tensor product expanders are a generalisation to graphs that randomise k correlated walkers. Here we give an efficient construction of constant-degree, constant-gap quantum k-tensor product expanders. The key ingredients are an efficient classical tensor product expander and the quantum Fourier transform. Our construction works whenever k=O(n/log n), where n is the number of qubits. An immediate corollary of this result is an efficient construction of an approximate unitary k-design, which is a quantum analogue of an approximate k-wise independent function, on n qubits for any k=O(n/log n). Previously, no efficient constructions were known for k>2, while state designs, of which unitary designs are a generalisation, were constructed efficiently in [Ambainis, Emerson 2007].
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