# 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}
}
• Published 6 July 2009
• Mathematics
• Physical Review A
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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## References

SHOWING 1-10 OF 22 REFERENCES
Scalable noise estimation with random unitary operators
• Computer Science
• 2005
While the scalability of the stochastic protocol makes it most relevant in large Hilbert spaces (when quantum process tomography is infeasible), the method should be immediately useful for evaluating the degree of control that is achievable in any prototype quantum processing device.
Convergence conditions for random quantum circuits
• Computer Science, Mathematics
• 2005
It is proved that the measure over random circuits converges exponentially (with increasing circuit length) to the uniform (Haar) measure on the unitary group, though the rate for uniform convergence must decrease exponentially with the number of qubits.
Randomizing Quantum States: Constructions and Applications
• Computer Science
• 2004
It is shown that there exists a set of roughly d’log d unitary operators whose average effect on every input pure state is almost perfectly randomizing, as compared to the d2 operators required to randomize perfectly.
Standard forms of noisy quantum operations via depolarization
• Physics
• 2005
We consider completely positive maps that describe noisy, multiparticle unitary operations. We show that by random single-particle operations the completely positive maps can be depolarized to a
Generic entanglement can be generated efficiently.
• Physics, Computer Science
Physical review letters
• 2007
We find that generic entanglement is physical, in the sense that it can be generated in polynomial time from two-qubit gates picked at random. We prove as the main result that such a process
Design of strongly modulating pulses to implement precise effective Hamiltonians for quantum information processing
• Physics
• 2002
We describe a method for improving coherent control through the use of detailed knowledge of the system’s Hamiltonian. Precise unitary transformations were obtained by strongly modulating the
Remote preparation of quantum states
• Computer Science
IEEE Transactions on Information Theory
• 2005
The paper includes an extensive discussion of the results, including the impact of the choice of model on the resources, the topic of obliviousness, and an application to private quantum channels and quantum data hiding.
Pseudo-Random Unitary Operators for Quantum Information Processing
• Computer Science, Mathematics
Science
• 2003
This work uses a nuclear magnetic resonance quantum processor to realize pseudorandom unitary operators that reproduce the expected random distribution of matrix elements and enables the practical application of random unitary operator in quantum communication and information processing protocols.
Symmetrized Characterization of Noisy Quantum Processes
• Physics
Science
• 2007
This work introduces a technique based on symmetrization that enables direct experimental measurement of some key properties of the decoherence affecting a quantum system and reduces the number of experiments required from exponential to polynomial in thenumber of subsystems.
Symmetric informationally complete quantum measurements
• Mathematics
• 2003
It is conjecture that a particular kind of group-covariant SIC–POVM exists in arbitrary dimensions, providing numerical results up to dimension 45 to bolster this claim.