Approximate unitary 3-designs from transvection Markov chains

  title={Approximate unitary 3-designs from transvection Markov chains},
  author={Xinyu Tan and Narayanan Rengaswamy and Robert Calderbank},
  journal={Designs, Codes and Cryptography},
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 2 [ qu an tph ] 1 9 Ju n 20 22 Approximate 3-designs and partial decomposition of the Clifford group representation using transvections Tanmay Singal∗ Institute of Physics,


The Clifford group forms a unitary 3-design
  • Zak Webb
  • Mathematics
    Quantum Inf. Comput.
  • 2016
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
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
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
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
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
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
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.