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Journal of Contemporary Physics (Armenian Academy of Sciences)

Interaction of qubits in Furry–Magnus approach

Gor A. Abovyan, Gagik Yu. Kryuchkyan

Published Jul 1, 2016• 0 citations
Publisher

Abstract

The system of qubits interacting with the perturbation field of an arbitrary envelope in the Furry representation based on the Magnus expansion in quantum electrodynamics is investigated. The general expressions for quantum transitions for probabilities between the states of a qubit are derived. The generalizations of this approach to the system of two interacting qubits are presented.

DOI
10.3103/s1068337216030038

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The system of qubits interacting with the perturbation field of an arbitrary envelope in the Furry representation based on the Magnus expansion in quantum electrodynamics is investigated. The general expressions for quantum transitions for probabilities between the states of a qubit are derived. The generalizations of this approach to the system of two interacting qubits are presented.
We study qubit manipulations by a train of pulses in a systematic approach based on the Magnus expansion and Furry representation in quantum electrodynamics. Furry–Magnus expansion gives approximate exponential representation of a time-evolution operator without the Dyson time-ordering procedure for interactions with external fields of arbitrary intensity and allows us to obtain general expression for single-qubit populations as a function of time-dependent Rabi frequency. The concrete calculations are performed for tunneling quantum dynamics, for multiphoton resonance interactions of qubits, as well as for off-resonance excitations of qubits driven by Gaussian pulses beyond rotating wave approximation. In this way, the populations of qubit states are investigated for various operational regimes, including single-pulse excitation and two-pulse excitation with a phase shift between pulse envelopes being the controlling parameter, and for excitation with sequential pulses. In the last case, we demonstrate the formation of quasi-energetic states and quasi-energies of qubits driven by a train of identical pulses. In this case the transition probability of the qubit generally exhibiting aperiodic oscillations becomes periodically regular for definite values of the quasi-energy.
Abstract The problem of speeding up quantum computations through controlling the states of qubits by short large-amplitude unipolar pulses is discussed. A new method is proposed to describe the dynamics of qubits that is based on the Magnus representation for the evolution operator (propagator) of qubits, which allows one to find a solution to the quantum control problem outside the framework of perturbation theory. The evolution of the states of a multiqubit system is represented as a motion on a special unitary group SU(N), similar to the rotation of the Bloch vector on the group SU(2) in the case of a single qubit. As an example, unitary transformations of single-qubit states under different excitation modes are considered. A general formula is found that establishes a relation between the shape of a pulse and the result of its action, similar to the well-known “area theorem” for Rabi pulses. The effect of unipolar pulses on a two-qubit system is analyzed in detail. Two situations are considered: first, a symmetric configuration, when the parameters of qubits do not differ and are acted upon by the same magnetic field from a fluxon, and second, when the configuration of levels arises in which, for one of the qubits, a transition to the nearest level becomes possible under perturbation. The general case of two interacting qubits—a four-level system—is considered analytically. A direct numerical simulation of the dynamics of a multiqubit system is performed to estimate the accuracy of approximation, and fidelity is used as a criterion of accuracy of operations. It is shown that the expressions obtained for the propagator allow one to formulate conditions for the pulse parameters that are necessary to perform logical operations.
We treat the interaction of an atom with a short intense few-cycle laser pulse by the use of the Magnus expansion of the time-evolution operator. Terms of the Magnus expansion up to the third order in the pulse duration are evaluated explicitly, and expressions for the transition probability between atomic eigenstates are given. The transition operators up to this order are particularly simple, involving only momentum and position shifts arising from the laser field alone. The connection to other existing approaches is established. For a variety of generic short-pulse forms the results of the Magnus approximation are in excellent agreement with time-dependent transition probabilities obtained from accurate ab initio numerical calculations. However, the limitation of the Magnus expansion for pulses having both vanishing momentum and position shifts is demonstrated also.