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arXiv:2202.12082

Algebraic-Dynamical Theory for Quantum Many-Body Hamiltonians: A Formalized Approach to Strongly Interacting Systems

Wenxin Ding

Published Jan 1, 2024• 2 citations
arXivHTMLPDF
DOI
10.2139/ssrn.4700439

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Recently, an {\it algebraic-dynamical theory} (ADT) for strongly interacting many-body quantum Hamiltonians in W. Ding, arXiv: 2202.12082 (2022). By introducing the complete operator basis set, ADT proposes a generic framework for systematically constructing dynamical theories for interacting quantum Hamiltonians, using quantum entanglement as the organizing principle. In this work, we study exact ADT solutions of interacting two-spin problems which can be used as "free theories" for perturbation study on relevant solvable limits. Then we perform ADT perturbation calculations and obtain the correct ground state under the perturbation, which shows that the ADT framework is capable of constructing correct dynamical perturbation theories for strongly interacting quantum spin models. We also discuss the implication for relevant lattice models.
Dynamical decoupling and Hamiltonian engineering are well-established techniques that have been used to control qubit systems. However, designing the corresponding methods for qudit systems has been challenging due to the lack of a Bloch sphere representation, more complex interactions, and additional control constraints. By identifying several general structures associated with such problems, we develop a formalism for the robust dynamical decoupling and Hamiltonian engineering of strongly interacting qudit systems. Our formalism significantly simplifies qudit pulse-sequence design while naturally incorporating robustness conditions necessary for experimental practicality. We experimentally demonstrate these techniques in a strongly interacting, disordered ensemble of spin-1 nitrogen-vacancy centers, achieving more than an order-of-magnitude improvement in coherence time over existing pulse sequences. We further describe how our techniques enable the engineering of exotic many-body phenomena such as quantum many-body scars, and open up new opportunities for quantum metrology with enhanced sensitivities. These results enable wide-reaching new applications for dynamical decoupling and Hamiltonian engineering in many-body physics and quantum metrology. Published by the American Physical Society 2024
In this paper, we discuss the extension of the recently introduced subsystem embedding subalgebra coupled cluster (SES-CC) formalism to unitary CC formalisms. In analogy to the standard single-reference SES-CC formalism, its unitary CC extension allows one to include the dynamical (outside the active space) correlation effects in an SES induced complete active space (CAS) effective Hamiltonian. In contrast to the standard single-reference SES-CC theory, the unitary CC approach results in a Hermitian form of the effective Hamiltonian. Additionally, for the double unitary CC (DUCC) formalism, the corresponding CAS eigenvalue problem provides a rigorous separation of external cluster amplitudes that describe dynamical correlation effects—used to define the effective Hamiltonian—from those corresponding to the internal (inside the active space) excitations that define the components of eigenvectors associated with the energy of the entire system. The proposed formalism can be viewed as an efficient way of downfolding many-electron Hamiltonian to the low-energy model represented by a particular choice of CAS. In principle, this technique can be extended to any type of CAS representing an arbitrary energy window of a quantum system. The Hermitian character of low-dimensional effective Hamiltonians makes them an ideal target for several types of full configuration interaction type eigensolvers. As an example, we also discuss the algebraic form of the perturbative expansions of the effective DUCC Hamiltonians corresponding to composite unitary CC theories and discuss possible algorithms for hybrid classical and quantum computing. Given growing interest in quantum computing, we provide energies for H2 and Be systems obtained with the quantum phase estimator algorithm available in the Quantum Development Kit for the approximate DUCC Hamiltonians.