Approximate unitary 3-designs from transvection Markov chains

Tan, Xinyu; Rengaswamy, Narayanan; Calderbank, Robert

doi:10.1007/s10623-021-01000-4

Cited by 1 publication

(3 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the proof of Lemma 21, we establish that W CS ∈⊂ suppP S . Also W CS ⊂ supp id + W (23) using the form of |w tr in equation (B116). Thus, in fact P CS |w tr = |w tr , using the form of P CS in equation (B10).…”

Section: Proofs Of Lemmas 6 Andmentioning

confidence: 99%

“…Arguably the simplest (non-Pauli) Clifford unitaries are known as transvections Clifford unitaries [26][27][28]. While transvections have been studied earlier [21,22], more interest has been shown to them recently [23][24][25][26][27][28]. These applications are based on two properties of transvections: (i) they generate the Clifford group, and (ii) they are a conjugacy class of non-Pauli Cliffords with minimal Pauli support.…”

Section: Introductionmentioning

confidence: 99%

“…A recent paper [23] gives a scheme for an asymptotic unitary 3-design: it implements a random Pauli, followed by a choice of a unitary 2-design, and then interpolates the gap between the two-design and three-design using random transvections Cliffords. The size of the set being sampled from reduces from O 2 m 2 (for the exact unitary 3-design by using the Clifford group) to O (2 m ), although the need to sample exactly from the uniform ensemble of the SL(2, 2 m )-subgroup requires sampling from another set of unitaries of size O (2 m ).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Approximate 3-designs and partial decomposition of the Clifford group representation using transvections

Singal¹,

Hsieh²

2021

Preprint

View full text Add to dashboard Cite

We study a scheme to implement an asymptotic unitary 3-design. The scheme implements a random Pauli once followed by the implementation of a random transvection Clifford by using state twirling. Thus the scheme is implemented in the form of a quantum channel. We show that when this scheme is implemented k times, then, in the k → ∞ limit, the overall scheme implements a unitary 3-design. This is proved by studying the eigendecomposition of the scheme: the +1 eigenspace of the scheme coincides with that of an exact unitary 3-design, and the remaining eigenvalues are bounded by a constant. Using this we prove that the scheme has to be implemented approximately O(m + log 1/ǫ) times to obtain an ǫ-approximate unitary 3-design, where m is the number of qubits, and ǫ is the diamond-norm distance of the exact unitary 3-design. Also, the scheme implements an asymptotic unitary 2-design with the following convergence rate: it has to be 1 sampled O(log 1/ǫ) times to be an ǫ-approximate unitary 2-design. Since transvection Cliffords are a conjugacy class of the Clifford group, the eigenspaces of the scheme's quantum channel coincide with the irreducible invariant subspaces of the adjoint representation of the Clifford group. Some of the subrepresentations we obtain are the same as were obtained in [36], whereas the remaining are new invariant subspaces. Thus we obtain a partial decomposition of the adjoint representation for 3 copies for the Clifford group. Thus, aside from providing a scheme for the implementation of unitary 3-design, this work is of interest for studying representation theory of the Clifford group, and the potential applications of this topic. The paper ends with open questions regarding the scheme and representation theory of the Clifford group.

show abstract

Section: Proofs Of Lemmas 6 Andmentioning

confidence: 99%