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arXiv:2404.05936

Learning Symmetric Hamiltonian

Jing Zhou, D. L. Zhou

Published Apr 9, 2024• 0 citations
arXivHTMLPDF

Abstract

Hamiltonian Learning is a process of recovering system Hamiltonian from measurements, which is a fundamental problem in quantum information processing. In this study, we investigate the problem of learning the symmetric Hamiltonian from its eigenstate. Inspired by the application of group theory in block diagonal secular determination, we have derived a method to determine the number of linearly independent equations about the Hamiltonian unknowns obtained from an eigenstate. This number corresponds to the degeneracy of the associated irreducible representation of the Hamiltonian symmetry group. To illustrate our approach, we examine the XXX Hamiltonian and the XXZ Hamiltonian. We first determine the Hamiltonian symmetry group, then work out the decomposition of irreducible representation, which serves as foundation for analyzing the uniqueness of recovered Hamiltonian. Our numerical findings consistently align with our theoretical analysis.

DOI
10.48550/arxiv.2404.05936

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Hamiltonian Learning is a process of recovering system Hamiltonian from measurements, which is a fundamental problem in quantum information processing. In this study, we investigate the problem of learning the symmetric Hamiltonian from its eigenstate. Inspired by the application of group theory in block diagonal secular determination, we have derived a method to determine the number of linearly independent equations about the Hamiltonian unknowns obtained from an eigenstate. This number corresponds to the degeneracy of the associated irreducible representation of the Hamiltonian symmetry group. To illustrate our approach, we examine the XXX Hamiltonian and the XXZ Hamiltonian. We first determine the Hamiltonian symmetry group, then work out the decomposition of irreducible representation, which serves as foundation for analyzing the uniqueness of recovered Hamiltonian. Our numerical findings consistently align with our theoretical analysis.
Recovering an unknown Hamiltonian from measurements is an increasingly important task for certification of noisy quantum devices and simulators. Recent works have succeeded in recovering the Hamiltonian of an isolated quantum system with local interactions from long-ranged correlators of a single eigenstate. Here, we show that such Hamiltonians can be recovered from local observables alone, using computational and measurement resources scaling linearly with the system size. In fact, to recover the Hamiltonian acting on each finite spatial domain, only observables within that domain are required. The observables can be measured in a Gibbs state as well as a single eigenstate; furthermore, they can be measured in a state evolved by the Hamiltonian for a long time, allowing us to recover a large family of time-dependent Hamiltonians. We derive an estimate for the statistical recovery error due to approximation of expectation values using a finite number of samples, which agrees well with numerical simulations.
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A quantum state for being an eigenstate of some local Hamiltonian should be constrained by zero energy variance and consequently, the constraint is rather strong that a single eigenstate may uniquely determine the Hamiltonian. For non-Hermitian systems, it is natural to expect that determining the Hamiltonian requires a pair of both left and right eigenstates. Here, we observe that it can be sufficient to determine a non-Hermitian Hamiltonian from a single right or left eigenstate. Our approach is based on the quantum covariance matrix, where the solution of Hamiltonian corresponds to the complex null vector. Our scheme favors non-Hermitian Hamiltonian learning on experimental quantum systems, as only the right eigenstates there can be accessed. Furthermore, we use numerical simulations to examine the effects of measurement errors and show the stability of our scheme.
The exploration of neural network quantum states has become widespread in studies of complicated quantum many-body systems. However, achieving high precision remains challenging due to the exponential growth of Hilbert space size and the intricate sign structures. Utilizing symmetries of the physical system, we propose a method to evaluate and sample the variational Ansatz within a symmetric subspace. This approach isolates different symmetry sectors, reducing the relevant Hilbert space size by a factor approximately proportional to the size of the symmetry group. It is inspired by exact diagonalization techniques and the work of Choo et al., [Phys. Rev. Lett. 121, 167204 (2018)]. We validate our method using the frustrated spin-$\frac{1}{2}\phantom{\rule{4pt}{0ex}}{J}_{1}\text{\ensuremath{-}}{J}_{2}$ antiferromagnetic Heisenberg chain, and we compare its performance to the case without symmetrization. The results indicate that our symmetric subspace approach achieves a substantial improvement over the full Hilbert space on optimizing the Ansatz, reducing the energy error by orders of magnitude. We also compare the results on degenerate eigenstates with different quantum numbers, highlighting the advantage of operating within a smaller Hilbert subspace.
We present two methods to study model Hamiltonians defined on clusters. One is the symmetrization of the Hilbert space basis according to irreducible representations of the symmetry group of the Hamiltonian. The other one adapts the Lanczos diagonalization method directly to the second quantized form of the Hamiltonian studied. By the symmetrization of the basis, one takes advantage of the symmetries of the Hamiltonian, and uses the group theory to find the invariant subspaces of the Hamiltonian in the complete Hilbert space. Then one needs only to work within the invariant subspaces. This method cuts down substantially the size of the matrix representation of the Hamiltonian to be diagonalized. Therefore, one could study relatively larger systems. As a result, all the eigenvectors and eigenvalues are obtained. Consequently, one could calculate thermodynamical quantities and various correlation functions, such as, the spin correlation, the charge correlation, the superconductivity pairing correlation, and the photoemission spectra. One the other hand, the Lanczos method, mentioned here in particular, is a specially developed computer program for the study of the model Hamiltonians. This program uses the key feature of the Lanczos method in diagonalization, allows one to use the second quantized form of the Hamiltonian directly without pregeneration or storing the matrix elements of the Hamiltonian. Two methods could be combined to take the full advantage of the symmetries of the Hamiltonian by input of the symmetrized wavefunction as the starting trial wavefunction in the Lanczos method. Here we illustrate the above two methods by applying to the Hubbard, and extended Hubbard models on a simple cubic cluster, and the lattice Anderson model on the two dimensional square lattice. Other interesting results for different systems are also briefly discussed here.