Unitary 2designs from random X and Zdiagonal unitaries
Abstract
Unitary 2designs are random unitaries simulating up to the second order statistical moments of the uniformly distributed random unitaries, often referred to as Haar random unitaries. They are used in a wide variety of theoretical and practical quantum information protocols and also have been used to model the dynamics in complex quantum manybody systems. Here, we show that unitary 2designs can be approximately implemented by alternately repeating random unitaries diagonal in the PauliZ basis and PauliX basis. We also provide a converse about the number of repetitions needed to achieve unitary 2designs. These results imply that the process after ℓ repetitions achieves a Θ (d^{ℓ}) approximate unitary 2design. Based on the construction, we further provide quantum circuits that efficiently implement approximate unitary 2designs. Although a more efficient implementation of unitary 2designs is known, our quantum circuit has its own merit that it is divided into a constant number of commuting parts, which enables us to apply all commuting gates simultaneously and leads to a possible reduction of an actual execution time. We finally interpret the result in terms of the dynamics generated by timedependent Hamiltonians and provide for the first time a random disordered timedependent Hamiltonian that generates a unitary 2design after switching interactions only a few times.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 May 2017
 DOI:
 10.1063/1.4983266
 arXiv:
 arXiv:1502.07514
 Bibcode:
 2017JMP....58e2203N
 Keywords:

 Quantum Physics
 EPrint:
 16 pages, 1 figure, v2: some minor changes and added references, v3: 21 pages, 1 figure, both results and presentations were much improved. v4: 20 pages, 1 figure, published version