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Erschienen in:
2009 | OriginalPaper | Buchkapitel
Efficient Quantum Tensor Product Expanders and k-Designs
verfasst von : Aram W. Harrow, Richard A. Low
Erschienen in: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Verlag: Springer Berlin Heidelberg
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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
/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 [1].