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2007

Quantum Margulis expanders

Gross, D., Eisert, J.

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We present a simple way to quantize the well-known Margulis expander map.

The result is a quantum expander which acts on discrete Wigner functions in the same way the classical Margulis expander acts on probability distributions.
The quantum version shares all essential properties of the classical counterpart, e.g., it has the same degree and spectrum.
Unlike previous constructions of quantum expanders, our method does not rely on non-Abelian harmonic analysis.

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M.B. Hastings, Phys. Rev. B 76

2007

Cited alongside, same era.

It follows directly from the definition of an ( N , D , λ ) (N,D,\lambda) -quantum expander that ‖ Λ N ( ρ ) − 𝟙 / N ‖ 2 ≤ λ ‖ ρ − 𝟙 / N ‖ 2 , \|\Lambda_{N}(\rho)-\mathbbm{1}/N\|_{2}\leq\lambda\|\rho-\mathbbm{1}/N\|_{2}, (23)

Cited in the paper.

This can be shown by writing G G as G = A A T G=AA^{T} with A ∈ ℝ 2 , | 𝒮 | A\in\mathbbm{R}^{2,|{\cal S}|} with entries A 1 , T \displaystyle A_{1,T} = \displaystyle= x T | 𝒮 | 1 / 2 − ∑ T ′ x T ′ | 𝒮 | 1 / 2 , \displaystyle\frac{x_{T}}{|{\cal S}|^{1/2}}-\sum_{T^{\prime}}\frac{x_{T}^{\prime}}{|{\cal S}|^{1/2}}, (24)

Cited in the paper.

C. Dankert, R. Cleve, J. Emerson, and E. Livine, quant-ph/0606161

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