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arXiv:1903.01135

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

V. E. Zobov, I. S. Pichkovskiy

Published Mar 4, 2019• 0 citations
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Abstract

We have done simulating of factorization the number 15 on three qutrits, represented by the spins S = 1, by quantum annealing. We assume that strong one-spin interaction allow selectively operate on different transitions between levels of the each qutrit. We present a sequence of selective rotation operators to engineer from dipole-dipole interaction a time-dependent effective Hamiltonian necessary for solving the problem. Also we find dependence of fidelity versus different parameters: magnetic field, total annealing time, and duration of time step, when the continuous variation of the Hamiltonian is replaced by a discrete one.

DOI
10.48550/arxiv.1903.01135

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We have done simulating of factorization the number 15 on three qutrits, represented by the spins S = 1, by quantum annealing. We assume that strong one-spin interaction allow selectively operate on different transitions between levels of the each qutrit. We present a sequence of selective rotation operators to engineer from dipole-dipole interaction a time-dependent effective Hamiltonian necessary for solving the problem. Also we find dependence of fidelity versus different parameters: magnetic field, total annealing time, and duration of time step, when the continuous variation of the Hamiltonian is replaced by a discrete one.
We have done simulating of factorization the number 15 on three qutrits, represented by the spins S = 1, by quantum annealing. We assume that strong one-spin interaction allow selectively operate on different transitions between levels of the each qutrit. We present a sequence of selective rotation operators to engineer from dipole-dipole interaction a timedependent effective Hamiltonian necessary for solving the problem. Also we find dependence of fidelity versus different parameters: magnetic field, total annealing time, and duration of time step, when the continuous variation of the Hamiltonian is replaced by a discrete one.
Recently, it was demonstrated1 that the clustering considered in this articlec, which consists in partitioning a set of data points into subsets depending on the proximity of some properties, can be performed using quantum annealing on a system of two-level quantum elements - qubits. In previous work2, we proposed to pass from qubits to qutrits - three-level quantum elements showed the advantages of such a replacement and obtained a time-dependent effective Hamiltonian. In the present paper, we have found a sequence of selective rotation operators that allows one to realize adiabatic evolution with this effective Hamiltonian in the discrete-time approximation. On five qutrits, represented by spins S = 1, we performed a simulation of clustering a set of six points on a plane into three groups by means of quantum annealing.
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 factorization 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).
To experimentally realize quantum computations on d -level basic elements (qudits) at d > 2, it is necessary to develop schemes for the technical realization of elementary logical operators. We have found sequences of selective rotation operators that represent the operators of the quantum Fourier transform (Walsh-Hadamard matrices) for d = 3–10. For the prime numbers 3, 5, and 7, the well-known method of linear algebra is applied, whereas, for the factorable numbers 6, 9, and 10, the representation of virtual spins is used (which we previously applied for d = 4, 8). Selective rotations can be realized, for example, by means of pulses of an RF magnetic field for systems of quadrupole nuclei or laser pulses for atoms and ions in traps.
Abstract We propose a scheme of the fast forward of adiabatic spin dynamics in a triangular 3 spin cluster of quantum annealing model and reveal the structure and role of driving pair-wise interaction and three-body interactions. We settle the quasi-adiabatic spin dynamics (QASD) by adding the regularization terms to the original Hamiltonian and accelerate it with use of a large time-scaling factor which realizes QASD on shortened time scale. Assuming the candidate regularization Hamiltonian consisting of three-body interactions besides the pair-wise exchange interactions and magnetic field, we solved the regularization terms. These terms multiplied by the velocity function give rise to the state-dependent counter-diabatic terms (CDTs) for each of adiabatic states. Applying this method to quantum annealing model, we find CDTs which include pair-wise interaction and the 3-body interactions. The driving pairwise and three-body interaction in the fast-forward scheme guarantees the complete fidelity of accelerated states.