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Journal of Physics Conference Series

Manipulations with qubit states by short control pulses: the interpolation method for evolution operator and fidelity

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

Published Jan 1, 2018• 2 citations
Publisher

Abstract

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.

DOI
10.1088/1742-6596/955/1/012004

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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 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.
A numerical method of simulations for the evolution operators (propagators) of coupled qubits was developed on the basis of Magnus representation. The propagator of a multi-qubit system can be expressed by a N × N matrix whose size increases as N = 2 n with the number of qubits n . Therefore, the standard approach to the propagator calculation requires the involvement of numerically expensive procedures. We have shown that the calculations of the propagator at the current time can be reduced to the inversion of the Vandermonde matrix, which occurs when constructing interpolation polynomials. In this case, due to the recurrence relations, the number of required flops increases only as ∼ O ( N 2 ) (in opposite to ∼ O ( N 3 ) flops for the matrix of the general form). This fact allows us to speed up the calculation of the propagator for multi-qubit systems.
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.
Magnus expansion (ME) provides a general way to expand the real-time propagator of a time-dependent Hamiltonian within the exponential such that the unitarity is satisfied at any order. We use this property and explicit integration of Lagrange interpolation formulas for the time-dependent Hamiltonian within each time interval and derive approximations that preserve unitarity for the differential time evolution operators of general time-dependent Hamiltonians. The resulting second-order approximation is the same as using the average of Hamiltonians for two end points of time. We identify three fourth-order approximations involving commutators of Hamiltonians at different times and also derive a sixth-order expression. A test of these approximations along with other available expressions for a two-state time-dependent Hamiltonian with sinusoidal time dependences provides information on the relative performance of these approximations and suggests that the derived expressions can serve as useful numerical tools for time evolution in time-resolved spectroscopy, quantum control, quantum sensing, real-time ab initio quantum dynamics, and open system quantum dynamics.
Analytical solutions to the time-dependent Schrödinger equation describing a driven two-level system are invaluable to many areas of physics, but they are also extremely rare. Here, we present a simple algorithm that generates an unlimited number of exact analytical solutions. We show that a general single-axis driving term and its corresponding evolution operator are determined by a single real function which is constrained only by a certain inequality and initial conditions. Any function satisfying these constraints yields an exact analytical solution. We demonstrate this method by presenting several new exact solutions to the time-dependent Schrödinger equation. Our general method and many of the new solutions we present are particularly relevant to qubit control in quantum computing applications.
Abstract Understanding how to tailor quantum dynamics to achieve the desired evolution is a crucial problem in almost all quantum technologies. Oftentimes an otherwise ideal quantum dynamics is corrupted by unavoidable interactions, and finding ways to mitigate the unwanted effects of such interactions on the dynamics is a very active field of research. Here, we present a very general method for designing high-efficiency control sequences that are fully compatible with experimental constraints on available interactions and their tunability. Our approach relies on the Magnus expansion to find order by order the necessary corrections that result in a high-fidelity operation. In the end finding, the control fields are reduced to solve a set of linear equations. We illustrate our method by applying it to a number of physically relevant problems: the strong-driving limit of a two-level system, fast squeezing in a parametrically driven cavity, the leakage problem in transmon qubit gates, and the acceleration of SNAP gates in a qubit-cavity system.