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arXiv:2201.13304 · Physical Review Research

Quantum algorithms for Schrieffer-Wolff transformation

Zongkang Zhang, Yongdan Yang, Xiaosi Xu, Ying Li

Published Oct 12, 2022• 6 citations
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Abstract

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.

DOI
10.1103/physrevresearch.4.043023

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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.
We derive recursive relations for the Schrieffer-Wolff (SW) transformation applied to the half-filled Hubbard dimer. While the standard SW transformation is set to block-diagonalize the transformed Hamiltonian solely at the first order of perturbation, we infer from recursive relations two types of modifications, variational or iterative, that approximate, or even enforce for the homogeneous case, the desired block-diagonalization at infinite order of perturbation. The modified SW unitary transformations are then used to design an test quantum algorithms adapted to the noisy and fault-tolerant era. This work paves the way toward the design of alternative quantum algorithms for the general Hubbard Hamiltonian.
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.
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.
Schrieffer-Wolff transformation is a very important transformation in Quantum Many Body physics. Yet, there isn't an explicit method in the literature to calculate the generator of this unitary transformation directly from the hamiltonian. In this paper we present an explicit method to compute the generator of the Schrieffer-Wolff transformation in general and then calculate the same for two very important hamiltonians in condensed matter physics, the Single Impurity Anderson model (SIAM) and its lattice generalization , the Periodic Anderson Model (PAM). Our method can be used to carry out the transformation directly from the hamiltonian and the generator of the transformation can be obtained without any recourse to perturbative expansion or intuitive guesswork. In fact here, the full generator is calculated rather than its expansion to first two orders which is usually the case.
An open question in designing superconducting quantum circuits is how best to reduce the full circuit Hamiltonian which describes their dynamics to an effective two-level qubit Hamiltonian which is appropriate for manipulation of quantum information. Despite advances in numerical methods to simulate the spectral properties of multi-element superconducting circuits (Yurke B and Denker J S 1984 Phys. Rev. A 29 1419, Reiter F and Sørensen A S 2012 Phys. Rev. A 85 032111 and Amin M H et al 2012 Phys. Rev. A 86 052314), the literature lacks a consistent and effective method of determining the effective qubit Hamiltonian. Here we address this problem by introducing a novel local basis reduction method. This method does not require any ad hoc assumption on the structure of the Hamiltonian such as its linear response to applied fields. We numerically benchmark the local basis reduction method against other Hamiltonian reduction methods in the literature and report specific examples of superconducting qubits, including the capacitively-shunted flux qubit, where the standard reduction approaches fail. By combining the local basis reduction method with the Schrieffer–Wolff transformation we further extend its applicability to systems of interacting qubits and use it to extract both non-stoquastic two-qubit Hamiltonians and three-local interaction terms in three-qubit Hamiltonians.
Abstract An open question in designing superconducting quantum circuits is how best to reduce the full circuit Hamiltonian which describes their dynamics to an effective two-level qubit Hamiltonian which is appropriate for manipulation of quantum information. Despite advances in numerical methods to simulate the spectral properties of multi-element superconducting circuits (Yurke B and Denker J S 1984 Phys. Rev. A 29 1419, Reiter F and Sørensen A S 2012 Phys. Rev. A 85 032111 and Amin M H et al 2012 Phys. Rev. A 86 052314), the literature lacks a consistent and effective method of determining the effective qubit Hamiltonian. Here we address this problem by introducing a novel local basis reduction method . This method does not require any ad hoc assumption on the structure of the Hamiltonian such as its linear response to applied fields. We numerically benchmark the local basis reduction method against other Hamiltonian reduction methods in the literature and report specific examples of superconducting qubits, including the capacitively-shunted flux qubit, where the standard reduction approaches fail. By combining the local basis reduction method with the Schrieffer–Wolff transformation we further extend its applicability to systems of interacting qubits and use it to extract both non-stoquastic two-qubit Hamiltonians and three-local interaction terms in three-qubit Hamiltonians.