Efficient Quantum Tensor Product Expanders and k-Designs

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 /logn ), 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 /logn ). Previously, no efficient constructions were known for k > 2, while state designs, of which unitary designs are a generalisation, were constructed efficiently in [1].