Approximate unitary 3-designs from transvection Markov chains

@article{Tan2022ApproximateU3,
  title={Approximate unitary 3-designs from transvection Markov chains},
  author={Xinyu Tan and Narayanan Rengaswamy and Robert Calderbank},
  journal={Designs, Codes and Cryptography},
  year={2022}
}
Unitary $t$-designs are probabilistic ensembles of unitary matrices whose first $t$ statistical moments match that of the full unitary group endowed with the Haar measure. In prior work, we showed that the automorphism group of classical $\mathbb{Z}_4$-linear Kerdock codes maps to a unitary $2$-design, which established a new classical-quantum connection via graph states. In this paper, we construct a Markov process that mixes this Kerdock $2$-design with symplectic transvections, and show that… 
Approximate 3-designs and partial decomposition of the Clifford group representation using transvections
X iv :2 11 1. 13 67 8v 1 [ qu an tph ] 2 6 N ov 2 02 1 Approximate 3-designs and partial decomposition of the Clifford group representation using transvections Tanmay Singal∗ Institute of Physics,

References

SHOWING 1-10 OF 32 REFERENCES
The Clifford group forms a unitary 3-design
  • Z. 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,
Exact and approximate unitary 2-designs and their application to fidelity estimation
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
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.
Decoupling with unitary approximate two-designs
Consider a bipartite system, of which one subsystem, A, undergoes a physical evolution separated from the other subsystem, R. One may ask under which conditions this evolution destroys all initial
Synthesis of Logical Clifford Operators via Symplectic Geometry
TLDR
A mathematical framework for synthesizing physical circuits that implement logical Clifford operators for stabilizer codes, and a proof of concept synthesis of universal Clifford gates for the well-known $k(k+1)/2) code is provided.
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.
Random Quantum Circuits are Approximate 2-designs
TLDR
It is shown that random circuits of only polynomial length will approximate the first and second moments of the Haar distribution, thus forming approximate 1- and 2-designs.
Near-linear constructions of exact unitary 2-designs
TLDR
It is shown that exact unitary 2-designs on n qubits can be implemented by quantum circuits consisting of ~O(n) elementary gates in logarithmic depth.
Exponential quantum speed-ups are generic
TLDR
It is shown that for almost any sufficiently long quantum circuit one can construct a black-box problem which is solved by the circuit with a constant number of quantum queries, but which requires exponentially many classical queries, even if the classical machine has the ability to postselect.
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