Implementation of a quantum adiabatic algorithm for factorization on two qudits

Abstract

Implementation of an adiabatic quantum algorithm for factorization on two qudits with the number of levels d 1 and d 2 is considered. A method is proposed for obtaining a time dependent effective Hamiltonian by means of a sequence of rotation operators that are selective with respect to the transitions between neighboring levels of a qudit. A sequence of RF magnetic field pulses is obtained, and a factoriza tion of the numbers 35, 21, and 15 is numerically simulated on two quadrupole nuclei with spins 3/2 (d 1 = 4) and 1 (d 2 = 3).

Sequences of selective rotation operators to engineer interactions for quantum annealing on three qutrits

“…In this case, the MW pulse will act only on the selected resonant transition, causing coherent changes in the corresponding two states. These changes are described by operators that coincide with rotation operators of a two-level system (with effective spin S = 1/2), which are called selective rotation operators 14,16,19 n k j…”

“…For the operators 2 2 2 1 4 36 a a  in (7), the corresponding evolution operators can be obtained from 16 :…”

“…First of all, as in the work 21 , in evolution operator with DDI (3) we leave one necessary interaction and destroy two unnecessary interactions using inversion operator: (17) and (18) by setting the length of evolution intervals t. The length t is chosen 16 to be a multiple of periods 2π/ε jk .…”

“…The adiabatic quantum calculation was not considered in these papers. The simulation of the adiabatic quantum factorization algorithm on two qudits (d-level quantum systems) was performed in the work 16 , and sequences of selective RF pulses needed to create a time-varying effective Hamiltonian were found. In the present work, we simulate the solution of the factorization problem by quantum annealing on three qutrits.…”

Adiabatically implementing quantum gates

“…HðsÞ ¼ ð1 À sÞH 0 þ sH 1 þ sð1 À sÞðjaihbj þ jbihajÞ: (20) Left multiplying the above equation by haj and hbj and combining with Eq. (7), we obtain…”