Deriving effective Hamiltonian models plays an essential role in quantum theory, with particular emphasis in recent years on control and engineering problems. In this work, we present two symbolic methods for computing effective Hamiltonian models: the Non-perturbative Analytical Diagonalization (NPAD) and the Recursive Schrieffer-Wolff Transformation (RSWT). NPAD makes use of the Jacobi iteration and works without the assumptions of perturbation theory while retaining convergence, allowing to treat a very wide range of models. In the perturbation regime, it reduces to RSWT, which takes advantage of an in-built recursive structure where remarkably the number of terms increases only linearly with perturbation order, exponentially decreasing the number of terms compared to the ubiquitous Schrieffer-Wolff method. Both methods consist of elementary algebra and can be easily automated to obtain closed-form expressions. To demonstrate the application of the methods, we study the ZZ interaction of superconducting qubits systems in two different parameter regimes. In the near-resonant regime, the method produces an analytical expression for the effective ZZ interaction strength, with an improvement of at least one order of magnitude compared to neglecting the comparably further detuned state. In the dispersive regime, we calculate perturbative corrections up to the 8th order and accurately identify a regime where ZZ interaction is suppressed in a simple model with only resonator-mediated coupling.
The Schrieffer-Wolff transformation (SWT) is a foundational perturbative method for deriving effective Hamiltonians in quantum systems by systematically eliminating couplings between pairs of energy distant subspaces. Despite recent advancements, the implementation of SWTs for sufficiently complex systems remains computationally challenging and often requires extensive calculations that are prone to errors. In this work, we introduce an analytical software tool, SymPT (Symbolic Perturbation Theory), designed to automate the SWT and its extensions. Building on a universal framework developed in recent research, SymPT provides a systematic and generalizable solution for deriving the generator of the transformation, enabling accurate computation of effective Hamiltonians for arbitrary perturbative systems. The tool supports both time-independent and time-periodic Hamiltonians, extending beyond standard SWT to incorporate arbitrary coupling elimination, block-diagonalization and full-diagonalization routines, thus enabling precise handling of systems with intricate energy structures.
Combining non-Hermiticity and interactions yields unique effects in open quantum many-body systems. Here, we develop the generalized Schrieffer-Wolff transformation, discuss its symmetry properties, and derive the effective Hamiltonian suitable for various quasidegenerate non-Hermitian systems. We apply our results to an exemplary
PT-symmetric circuit QED composed of two non-Hermitian qubits embedded in a lossless resonator. We consider a resonant quantum circuit as $|{\ensuremath{\omega}}_{r}\ensuremath{-}\mathrm{\ensuremath{\Omega}}|\ensuremath{\ll}{\ensuremath{\omega}}_{r}$, where $\mathrm{\ensuremath{\Omega}}$ and ${\ensuremath{\omega}}_{r}$ are qubits and resonator frequencies, respectively, providing well-defined groups of quasidegenerate resonant states. For such a system, using direct numerical diagonalization we obtain the dependence of the low-lying eigenspectrum on the interaction strength between a single qubit and the resonator
g and the gain (loss) parameter $\ensuremath{\gamma}$, and compare that with the eigenvalues obtained analytically using the effective Hamiltonian of resonant states. We identify
PT-symmetry broken and unbroken phases, trace the formation of exceptional points of the second and the third order, and provide a complete phase diagram $g\ensuremath{-}\ensuremath{\gamma}$ of low-lying resonant states. We relate the formation of exceptional points to the additional
P-pseudo-Hermitian symmetry of the system and show that non-Hermiticity mixes the ``dark'' and the ``bright'' states, which has a direct experimental consequence.
Schrieffer-Wolff transformation (SWT) has been extensively used in quantum many-body physics to calculate the low energy effective Hamiltonian. It provides a perturbative method to comprehend the renormalization effects of strong correlations in the quantum many-body models. The generator for Schrieffer-Wolff transformation is calculated usually by heuristic methods. Recently, a systematic and elegant method for the calculation of this extremely significant transformation has been reported [1]. Given the huge significance of SWT for many areas including quantum condensed matter physics, quantum optics and quantum cavity electrodynamics, it is imperative to develop quantum algorithm for carrying out SWT on quantum computer. In this paper, we put forward this quantum algorithm and demonstrate it for single impurity Anderson model (SIAM), thereby arriving at Kondo model as effective Hamiltonian. We implement our quantum algorithm in QisKit and carry out SWT for SIAM on IBM Quantum computers. To the best of our knowledge, this work is the first of its kind to obtain Kondo model from Anderson impurity model using a quantum algorithm.