• Corpus ID: 118662299

The Clifford group fails gracefully to be a unitary 4-design

@article{Zhu2016TheCG,
  title={The Clifford group fails gracefully to be a unitary 4-design},
  author={Huangjun Zhu and Richard Kueng and Markus Grassl and David Gross},
  journal={arXiv: Quantum Physics},
  year={2016}
}
A unitary t-design is a set of unitaries that is "evenly distributed" in the sense that the average of any t-th order polynomial over the design equals the average over the entire unitary group. In various fields -- e.g. quantum information theory -- one frequently encounters constructions that rely on matrices drawn uniformly at random from the unitary group. Often, it suffices to sample these matrices from a unitary t-design, for sufficiently high t. This results in more explicit… 

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References

SHOWING 1-10 OF 96 REFERENCES

The Clifford group forms a unitary 3-design

  • Zak Webb
  • Mathematics
    Quantum Inf. Comput.
  • 2016
TLDR
It is proved that the Clifford group is a 3-design, showing that it is a better approximation to Haar-random unitaries than previously expected and characterizing how well random Clifford elements approximateHaar- random unitaries.

Multiqubit Clifford groups are unitary 3-designs

Unitary $t$-designs are a ubiquitous tool in many research areas, including randomized benchmarking, quantum process tomography, and scrambling. Despite the intensive efforts of many researchers,

Unitary designs and codes

TLDR
In this paper, irreducible representations of the unitary group are used to find a general lower bound on the size of a unitary t-design in U(d), for any d and t.

Evenly distributed unitaries: On the structure of unitary designs

We clarify the mathematical structure underlying unitary t-designs. These are sets of unitary matrices, evenly distributed in the sense that the average of any tth order polynomial over the design

QUANTUM DESIGNS: FOUNDATIONS OF A NONCOMMUTATIVE DESIGN THEORY

This is a one-to-one translation of a German-written Ph.D. thesis from 1999. Quantum designs are sets of orthogonal projection matrices in finite(b)-dimensional Hilbert spaces. A fundamental

Galois unitaries, mutually unbiased bases, and mub-balanced states

TLDR
It is shown that there exist transformations that cycle through all the bases in all dimensions d = pn where p is an odd prime and the exponent n is odd, and it is conjecture that this construction yields all such states in odd prime power dimension.

Representations of the multi-qubit Clifford group

The Clifford group is a fundamental structure in quantum information with a wide variety of applications. We discuss the tensor representations of the $q$-qubit Clifford group, which is defined as

Nonmalleable encryption of quantum information

We introduce the notion of nonmalleability of a quantum state encryption scheme (in dimension d): in addition to the requirement that an adversary cannot learn information about the state, here we

Low rank matrix recovery from Clifford orbits

TLDR
It is proved that low-rank matrices can be recovered efficiently from a small number of measurements that are sampled from orbits of a certain matrix group and argued that stabilizer states form an ideal model for structured measurements for phase retrieval.

Hudson's theorem for finite-dimensional quantum systems

We show that, on a Hilbert space of odd dimension, the only pure states to possess a non-negative Wigner function are stabilizer states. The Clifford group is identified as the set of unitary
...