Exact and approximate unitary 2-designs and their application to fidelity estimation

@article{Dankert2009ExactAA,
  title={Exact and approximate unitary 2-designs and their application to fidelity estimation},
  author={Christoph Dankert and Richard Cleve and Joseph Emerson and Etera R. Livine},
  journal={Physical Review A},
  year={2009},
  volume={80},
  pages={012304}
}
We develop the concept of a unitary $t$-design as a means of expressing operationally useful subsets of the stochastic properties of the uniform (Haar) measure on the unitary group $U({2}^{n})$ on $n$ qubits. In particular, sets of unitaries forming 2-designs have wide applicability to quantum information protocols. We devise an $O(n)$-size in-place circuit construction for an approximate unitary 2-design. We then show that this can be used to construct an efficient protocol for experimentally… 

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