Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2409.10656

Eigenoperator approach to Schrieffer-Wolff perturbation theory and dispersive interactions

Gabriel Teixeira Landi

Published Sep 16, 2024• 1 citation
arXivHTMLPDF

Abstract

Modern quantum physics is very modular: we first understand basic building blocks (``XXZ Hamiltonian'' ``Jaynes-Cummings'' etc.) and then combine them to explore novel effects. A typical example is placing known systems inside an optical cavity. The Schrieffer-Wolff perturbation method is particularly suited for dealing with these problems, since it casts the perturbation expansion in terms of operator corrections to a Hamiltonian, which is more intuitive than energy level corrections, as in traditional time-independent perturbation theory. However, the method lacks a systematic approach.% and has largely remained a niche topic. In these notes we discuss how \emph{eigenoperator decompositions}, a concept largely used in open quantum systems, can be employed to construct an intuitive and systematic formulation of Schrieffer-Wolff perturbation theory. To illustrate this we revisit various papers in the literature, old and new, and show how they can instead be solved using eigenoperators. Particular emphasis is given to perturbations that couple two systems with very different transition frequencies (highly off-resonance), leading to the so-called dispersive interactions.

DOI
10.48550/arxiv.2409.10656

Related papers

A systematic method for Schrieffer-Wolff transformation and its generalizations

Rukhsan Ul Haq, Keshav Kumar Singh

Schrieffer-Wolff transformation is extensively used in quantum many-body physics to calculate the low energy effective Hamiltonian. It offers a perturbative method to understand the renormalization effects in the strong coupling regime of the quantum many-body models. The generator for Schrieffer-Wolff transformation is calculated using heuristic methods. It is highly desirable to have a rigorous, systematic and simple method for the calculation of generator for this highly important transformation. In this paper, we put forward such a method. We demonstrate our method by applying it to few well-known models from various fields like strongly correlated electron systems, quantum optics and cavity quantum electrodynamics. Our method can be applied to a broad class of quantum many-body Hamiltonians.

arXiv:2004.06534•Apr 13, 2020•1 citation

Stabilizer Perturbation Theory: A Systematic Construction via Schrieffer-Wolff Transformation

Ying, Xuzhe, Kangle Li, Hoi Chun Po

Perturbation theories provide valuable insights on quantum many-body systems. Systems of interacting particles, like electrons, are often treated perturbatively around exactly solvable Gaussian points. Systems of interacting qubits have gained increasing prominence as another class of models for quantum systems thanks to the recent advances in experimentally realizing mesoscopic quantum devices. Stabilizer states, innately defined on systems of qudits, have correspondingly emerged as another class of classically simulatable starting point for the study of quantum error-correcting codes and topological phases of matter in such devices. As a step towards analyzing more general quantum many-body problems on these platforms, we develop a systematic stabilizer perturbation theory in qubit systems. Our approach relies on the local Schrieffer-Wolff transformation, which we show can be efficiently performed through the binary encoding the Pauli algebra. As demonstrations, we first benchmark the stabilizer perturbation theory on the transverse field Ising chain in one dimension. The method is then further applied to $\mathbb{Z}_2$ toric code on square lattice and kagome lattice to probe the tendency toward confinement for anyonic excitations.

arXiv:2509.12621•Sep 16, 2025•0 citations

Quantum algorithms for Schrieffer-Wolff transformation

Zongkang Zhang, Yongdan Yang, Xiaosi Xu +1 more

The Schrieffer-Wolff transformation aims to solve degenerate perturbation problems and give an effective Hamiltonian that describes the low-energy dynamics of the exact Hamiltonian in the low-energy subspace of an unperturbed Hamiltonian. This unitary transformation decoupling the low-energy and high-energy subspaces for the transformed Hamiltonian can be realized by quantum circuits. We give a fully quantum algorithm for realizing the SW transformation. We also propose a hybrid quantum-classical algorithm for finding the effective Hamiltonian on NISQ hardware, where a general cost function is used to indicate the decoupling degree. Numerical simulations without or with noise and experiments on quantum computer ibmq_manila are implemented for a Heisenberg chain model. Our results verify the algorithm and show its validity on near-term quantum computers.

arXiv:2201.13304•Physical Review Research•Oct 12, 2022•6 citations

Tutorial: Perturbation Approximations to Asymptotic Quantum Transition Rates

Steven Kenneth Kauffmann

The orders of the perturbation approximation to the Schroedinger-picture time evolution operator in powers of the presumed-small perturbation part of the Hamiltonian operator are developed by the iteration of an identity; the possibly confusing switch to the interaction picture isn't needed. Those time-evolution operator approximations are sandwiched between two orthogonal, normalized eigenstates of the unperturbed part of the Hamiltonian operator to produce the transition amplitude approximations, whose calculation is reduced to quadrature when every occurrence of the perturbation part of the Hamiltonian operator in them is expanded in the unperturbed basis. That expansion also reveals the time-dependent parts of those approximations to be multiply nested integrals which in the long-time limit approach simple products of non-singular inverses (principle value plus delta function) of differences of unperturbed energy eigenvalues. Closely related to the long-time limits of transition amplitudes are the long-time averages of transition rates. When the unperturbed basis is that of free-particle states, sums over relevant final states of transition rates from an initial state, divided by the initial state's particle flux, produce cross sections. Here the perturbation approximations to generic quantum transition rates are parlayed to the corresponding approximations to differential cross sections for nonrelativistic-particle potential scattering.

•Aug 1, 2019•0 citations

Floquet Schrieffer-Wolff transform based on Sylvester equations

Xiao Wang, F. P. M. Méndez-Córdoba, Dieter Jaksch +1 more

We present a Floquet Schrieffer-Wolff transform (FSWT) to obtain effective Floquet Hamiltonians and micromotion operators of periodically driven many-body systems for any nonresonant driving frequency. The FSWT perturbatively eliminates the oscillatory components in the driven Hamiltonian by solving operator-valued Sylvester equations with systematic approximations. It goes beyond various high-frequency expansion methods commonly used in Floquet theory, as we demonstrate with the example of the driven Fermi-Hubbard model. In the limit of high driving frequencies, the FSWT Hamiltonian reduces to the widely used Floquet-Magnus result. We anticipate this method will be useful for designing Rydberg multiqubit gates, controlling correlated hopping in quantum simulations in optical lattices, and describing multiorbital and long-range interacting systems driven in-gap. Published by the American Physical Society 2024

arXiv:2407.08405•Physical review. B./Physical review. B•Dec 3, 2024•1 citation

Stationary State Perturbation Theory

Kurt Gottfried

The vast majority of problems in quantum mechanics cannot be solved exactly. As in all theories of physics approximation methods are therefore of great importance. Here we shall deal with time-independent problems, which means that we seek the eigenvalues and eigenstates of H . In perturbation theory we split H into two pieces, H = H 0 + H 1 , where we know the spectrum and eigenstates of H 0 , and H 1 is, in some sense, small. Needless to say, a split of this type is often not possible, or at least unknown. In that case one frequently replaces the system by a simpler model for which perturbation theory works, and which, it is hoped, captures some of the essential features of the actual problem at hand. This art of finding models is something that can only be taken up within the context of a particular subject (e.g., solid state or nuclear physics). Here we shall treat systems where good zero-order solutions are rather obvious.

CRC Press eBooks•Mar 9, 2018•0 citations
Modern quantum physics is very modular: we first understand basic building blocks (XXZ Hamiltonian'' Jaynes-Cummings'' etc.) and then combine them to explore novel effects. A typical example is placing known systems inside an optical cavity. The Schrieffer-Wolff perturbation method is particularly suited for dealing with these problems, since it casts the perturbation expansion in terms of operator corrections to a Hamiltonian, which is more intuitive than energy level corrections, as in traditional time-independent perturbation theory. However, the method lacks a systematic approach.% and has largely remained a niche topic. In these notes we discuss how \emph{eigenoperator decompositions}, a concept largely used in open quantum systems, can be employed to construct an intuitive and systematic formulation of Schrieffer-Wolff perturbation theory. To illustrate this we revisit various papers in the literature, old and new, and show how they can instead be solved using eigenoperators. Particular emphasis is given to perturbations that couple two systems with very different transition frequencies (highly off-resonance), leading to the so-called dispersive interactions.
Schrieffer-Wolff transformation is extensively used in quantum many-body physics to calculate the low energy effective Hamiltonian. It offers a perturbative method to understand the renormalization effects in the strong coupling regime of the quantum many-body models. The generator for Schrieffer-Wolff transformation is calculated using heuristic methods. It is highly desirable to have a rigorous, systematic and simple method for the calculation of generator for this highly important transformation. In this paper, we put forward such a method. We demonstrate our method by applying it to few well-known models from various fields like strongly correlated electron systems, quantum optics and cavity quantum electrodynamics. Our method can be applied to a broad class of quantum many-body Hamiltonians.
Perturbation theories provide valuable insights on quantum many-body systems. Systems of interacting particles, like electrons, are often treated perturbatively around exactly solvable Gaussian points. Systems of interacting qubits have gained increasing prominence as another class of models for quantum systems thanks to the recent advances in experimentally realizing mesoscopic quantum devices. Stabilizer states, innately defined on systems of qudits, have correspondingly emerged as another class of classically simulatable starting point for the study of quantum error-correcting codes and topological phases of matter in such devices. As a step towards analyzing more general quantum many-body problems on these platforms, we develop a systematic stabilizer perturbation theory in qubit systems. Our approach relies on the local Schrieffer-Wolff transformation, which we show can be efficiently performed through the binary encoding the Pauli algebra. As demonstrations, we first benchmark the stabilizer perturbation theory on the transverse field Ising chain in one dimension. The method is then further applied to Z2\mathbb{Z}_2Z2​ toric code on square lattice and kagome lattice to probe the tendency toward confinement for anyonic excitations.
The Schrieffer-Wolff transformation aims to solve degenerate perturbation problems and give an effective Hamiltonian that describes the low-energy dynamics of the exact Hamiltonian in the low-energy subspace of an unperturbed Hamiltonian. This unitary transformation decoupling the low-energy and high-energy subspaces for the transformed Hamiltonian can be realized by quantum circuits. We give a fully quantum algorithm for realizing the SW transformation. We also propose a hybrid quantum-classical algorithm for finding the effective Hamiltonian on NISQ hardware, where a general cost function is used to indicate the decoupling degree. Numerical simulations without or with noise and experiments on quantum computer ibmq_manila are implemented for a Heisenberg chain model. Our results verify the algorithm and show its validity on near-term quantum computers.
The orders of the perturbation approximation to the Schroedinger-picture time evolution operator in powers of the presumed-small perturbation part of the Hamiltonian operator are developed by the iteration of an identity; the possibly confusing switch to the interaction picture isn't needed. Those time-evolution operator approximations are sandwiched between two orthogonal, normalized eigenstates of the unperturbed part of the Hamiltonian operator to produce the transition amplitude approximations, whose calculation is reduced to quadrature when every occurrence of the perturbation part of the Hamiltonian operator in them is expanded in the unperturbed basis. That expansion also reveals the time-dependent parts of those approximations to be multiply nested integrals which in the long-time limit approach simple products of non-singular inverses (principle value plus delta function) of differences of unperturbed energy eigenvalues. Closely related to the long-time limits of transition amplitudes are the long-time averages of transition rates. When the unperturbed basis is that of free-particle states, sums over relevant final states of transition rates from an initial state, divided by the initial state's particle flux, produce cross sections. Here the perturbation approximations to generic quantum transition rates are parlayed to the corresponding approximations to differential cross sections for nonrelativistic-particle potential scattering.
We present a Floquet Schrieffer-Wolff transform (FSWT) to obtain effective Floquet Hamiltonians and micromotion operators of periodically driven many-body systems for any nonresonant driving frequency. The FSWT perturbatively eliminates the oscillatory components in the driven Hamiltonian by solving operator-valued Sylvester equations with systematic approximations. It goes beyond various high-frequency expansion methods commonly used in Floquet theory, as we demonstrate with the example of the driven Fermi-Hubbard model. In the limit of high driving frequencies, the FSWT Hamiltonian reduces to the widely used Floquet-Magnus result. We anticipate this method will be useful for designing Rydberg multiqubit gates, controlling correlated hopping in quantum simulations in optical lattices, and describing multiorbital and long-range interacting systems driven in-gap. Published by the American Physical Society 2024
The vast majority of problems in quantum mechanics cannot be solved exactly. As in all theories of physics approximation methods are therefore of great importance. Here we shall deal with time-independent problems, which means that we seek the eigenvalues and eigenstates of H . In perturbation theory we split H into two pieces, H = H 0 + H 1 , where we know the spectrum and eigenstates of H 0 , and H 1 is, in some sense, small. Needless to say, a split of this type is often not possible, or at least unknown. In that case one frequently replaces the system by a simpler model for which perturbation theory works, and which, it is hoped, captures some of the essential features of the actual problem at hand. This art of finding models is something that can only be taken up within the context of a particular subject (e.g., solid state or nuclear physics). Here we shall treat systems where good zero-order solutions are rather obvious.