Efficient Quantum Tensor Product Expanders and k-Designs

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 approximate unitary k-designs on n qubits for any k = O(n/ logn).