Perturbation Theory (qiskit_dynamics.perturbation)

This module contains tools for numerically computing and utilizing perturbation theory terms. Perturbation theory is an advanced topic; a brief review of the concepts and notation required to understand the contents in this module are given in the Time-dependent perturbation theory and multi-variable series expansions review discussion.

Time-dependent perturbation theory

The function solve_lmde_perturbation() computes Dyson series [1] and Magnus expansion [2][3] terms in a multi-variable setting via algorithms in [4]. It can also be used to compute Dyson-like integrals using the algorithm in [5]. Results are returned in either a PowerSeriesData or DysonLikeData class, which are data classes with functionality for indexing and accessing specific perturbation terms. See the function documentation for further details.

Truncated power-series representation and multisets

The class ArrayPolynomial represents an array-valued multivariable polynomial (i.e. a truncated power series), and provides functionality for both evaluating and transforming array-valued polynomials.

This module makes use of the multiset package for indexing multi-variable power series. See the multiset and power series notation section of the perturbation review for an explanation of this convention.

Perturbation module functions

solve_lmde_perturbation(perturbations, ...)

Compute time-dependent perturbation theory terms for an LMDE.

Perturbation module classes

ArrayPolynomial([constant_term, ...])	A polynomial with array-valued coefficients.
PowerSeriesData(data, labels[, metadata])	Storage container for power series data.
DysonLikeData(data, labels[, metadata])	Storage container for DysonLike series data.

References

[1]

F. J. Dyson. The Radiation Theories of Tomonaga, Schwinger, and Feynman. Physical Review, 75(3):486–502, 1949. doi:10.1103/PhysRev.75.486.

[2]

W. Magnus. On the exponential solution of differential equations for a linear operator. Communications on Pure and Applied Mathematics, 7(4):649–673, 1954. doi:10.1002/cpa.3160070404.

[3]

S. Blanes, F. Casas, J. A. Oteo, and J. Ros. The Magnus expansion and some of its applications. Physics Reports, 470(5–6):151–238, 2009. doi:10.1016/j.physrep.2008.11.001.

[4]

D. Puzzuoli, S. F. Lin, M. Malekakhlagh, E. Pritchett, B. Rosand, and C. J. Wood. Algorithms for perturbative analysis and simulation of quantum dynamics. arXiv:2210.11595, 2022. doi:10.48550/arXiv.2210.11595.

[5]

H. Haas, D. Puzzuoli, F. Zhang, and D. G. Cory. Engineering effective Hamiltonians. New Journal of Physics, 21(10):103011, 2019. doi:10.1088/1367-2630/ab4525.