Efficient Quantum Tensor Product Expanders and k-designs

Details Efficient Quantum Tensor Product Expanders and k-designs Efficient Quantum Tensor Product Expanders and k-designs

2008-11-16

arxiv.org/abs/0811.2597v3

Proceedings of RANDOM 2009, LNCS, 5687:548-561, 2009

Physics

Quantum Physics

16 pages, typo in references fixed

Scientific paper

10.1007/978-3-642-03685-9_41

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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