Enumerative combinatorics
Classical and Quantum Computation
Classical and Quantum Computation
Random Measurement Bases, Quantum State Distinction and Applications to the Hidden Subgroup Problem
CCC '06 Proceedings of the 21st Annual IEEE Conference on Computational Complexity
Quantum t-designs: t-wise Independence in the Quantum World
CCC '07 Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity
Quantum expanders from any classical Cayley graph expander
Quantum Information & Computation
Quantum Information & Computation
Exponential quantum speed-ups are generic
Quantum Information & Computation
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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].