Abstract
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].
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Harrow, A.W., Low, R.A. (2009). Efficient Quantum Tensor Product Expanders and k-Designs. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2009 2009. Lecture Notes in Computer Science, vol 5687. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03685-9_41
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DOI: https://doi.org/10.1007/978-3-642-03685-9_41
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