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Journal of Experimental and Theoretical Physics

Dynamics of Qubits in the Field of Unipolar Pulses: Magnus Propagator, Generalized Area Theorem, and Motion on Groups

M. V. Denisenko, N. V. Klenov, Сатанин Аркадий Михайлович

Published Feb 1, 2019• 4 citations
Publisher

Abstract

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.

DOI
10.1134/s1063776119010096

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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.
In this article the dynamics of the qubits states based on solution of the time-dependent Schrödinger equation is investigated. Using the Magnus method we obtain an explicit interpolation representation for the propagator, which allows to find wave function at an arbitrary time. To illustrate the effectiveness of the approach, the population of the levels a single and two coupled qubits have been calculated by applying the Magnus propagator and the result have been compared with the numerical solution of the Schrödinger equation. As a measure of the approximation of the wave function, we calculate fidelity, which indicates proximity when the exact and approximate evolution operator acts on the initial state. We discuss the possibility of extending the developed methods to multi-qubits system, when high-speed calculation methods of the operators of evolution is particularly relevant.
Abstract Methods for controlling states of interacting superconducting flux qubits using power-efficient devices of fast single-quantum logic (Josephson nonlinearity cavities) are studied. One- and two-qubit quantum logical operations performed within the conventional control technique using Rabi pulses and using picosecond single unipolar magnetic field pulses are comparatively analyzed. It is shown that all main operations can be implemented with an accuracy of better than 97% due to optimization of the shape and parameters of unipolar control pulses (associated with, e.g., propagation of fluxons in transmission lines). The efficiency of the developed technique for programming a two-qubit quantum processor implementing the simplest Deutsch–Jozsa algorithm is demonstrated.
The present study is devoted to development of a technique for numerical simulation of the wave function dynamics the single Josephson qubits and arrays of noninteracting qubits controlled by ultra-short pulses. We wish to demonstrate the feasibility of a new principle of basic logical operations on the picosecond timescale. The influence of the unipolar pulse (“fluxon”) form on the evolution of the state during the execution of the quantum one-qubit operations – “NOT”, “READ” and “ NOT ” – is investigated in the presence of decoherence. In the array of non interacting qubits, the question of the influence of the spread of their energy parameters (tunnel constants) is studied. It is shown that a single unipolar pulse can control a huge array of artificial atoms with 10% spread of geometric parameters in the array.
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 conventional technique for controlling the states of individual qubits and quantum registers is based on the use of high-frequency modulated pulses whose carrier frequency is close to the transition frequency between the ground states of “artificial atoms” and “molecules” (the Rabi technique). In this paper, an alternative concept for controlling superconducting qubits is developed that is based on the idea of fast excitation of a quantum system by solitary unmodulated pulses. The formation of entangled Bell states in a two-qubit superconducting register under the action of field pulses obtained by rapid single flux quantum (RSFQ) devices is demonstrated by numerical simulation. The tomography of quantum states is performed by solving the density matrix equation. It is shown that the technique of “unipolar pulses” allows one to speed up the formation of entangled states (at picosecond times) with a fidelity of 95–98%. The execution of the basic quantum algorithms is demonstrated: the Deutsch–Jozsa algorithm with an accuracy of more than 98% and Grover’s algorithm with an accuracy of 93%.
The work develops an approach to researching the dynamics of qubits that makes it possible to monitor the behavior of the system in the process of quantum computing using computer simulation. The study deals with individual and interacting qubits based on spin ½ in a magnetic field. The goal of the work is to obtain evolutionary equations (equations of motion) for observables that adequately describe the state of the system and provide opportunities for computer modeling of processes. The dynamics of qubits are described using the formalism of the density matrix of pure states. The paper shows derivation of the equations of motion for observables being the average values of operators of physical quantities that describe the state of the system. For spin ½ in magnetic fields, the derivation of the dynamic equation for the spin polarization vector, which is a unit vector in real three-dimensional space, is shown. The shape and parameters of radiofrequency pulses that implement single-qubit operations are presented. In the case of two interacting spins ½, an expression is given for the density matrix and the Hamiltonian of the system for scalar indirect interaction of the magnetic moments of nuclei through the electron shells of atoms in the molecule. Evolution equations are derived for the average dipole moments and quadrupole moment of the system. It is shown that in the case of weak interaction the dynamics of the system can be described using a closed system of equations for the average dipole moments, which in this case coincide with the spin polarization vectors of individual spins. The presentation is accompanied by examples of numerical solutions of the resulting equations describing one-qubit and two-qubit operations. A conclusion is made about the adequacy of the created mathematical model and the possibility of its use for computer modeling of quantum computing processes, the influence of the environment on them and the stabilization of states to prevent or reduce errors in quantum computing.