Classical Representation and Manipulation of Quantum Many Body States and High Dimensional Data

Citation

Peng, Ruojing (2025) Classical Representation and Manipulation of Quantum Many Body States and High Dimensional Data. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/px31-5g53. https://resolver.caltech.edu/CaltechTHESIS:01282025-225116139

Abstract

This thesis contains several developments in extending the capability of classical simulations for representing and manipulating quantum many-body states and high dimensional data. In Chapter 1, we introduce the different types of problems considered in quantum chemistry (with ab initio molecular Hamiltonian) and condensed matter physics (with lattice model Hamiltonian) as well as a classical scenario of high-dimensional function integration. In each case, we briefly introduce a corresponding anstaz for representing either the quantum many-body wavefunction or the classical high-dimensional integrand, which provides context for more detailed discussion in subsequent chapters.

Chapter 2 describes a technical improvement on an existing formulation of the coupled cluster method, known as a popular wavefunction ansatz in quantum chemistry, for simulating finite-temperature non-equilibrium ab initio Hamiltonian dynamics. We adapt a technique from zero-temperature dynamics to the non-equilibrium finite-temperature coupled cluster method, thereby restoring conservation laws for 1-particle properties which were previously broken, and stabilizing the numerical behavior of the method for moderate time propagation. We demonstrated the capability of the method on both ab initio molecular systems such as field-driven H2 and electron transport in silicon cell, and model Hamiltonian such as the moderately interacting single impurity Anderson model (SIAM). We were able to perform stable dynamics simulation for sufficient amount of time to extract qualitatively correct physics, such as band population transport in silicon, and Kondo physics in the SIAM.

Chapter 3 and Chapter 4 introduce developments in tensor network state (TNS) methods for lattice model Hamiltonian. Typically, TNS are constructed to correctly represent the entanglement structure of target physical state, whose computation of e.g., amplitude and expectation value, can only be performed approximately. A representative example is the projected entangled pair state (PEPS) for representing ground states on 2-dimensional lattices. In Chapter 3, we describe several aspects in PEPS (and TNS in general) construction and computation, including approximate contraction and derivative computation, as well as encoding of Abelian symmetry and fermion statistics. We also introduce a change of perspective of TNS ansatz that restores its exact variationality which was hitherto considered only approximate due to the need of approximate contraction. With such new perspective on TNS ansatz, Chapter 4 then focuses on stochastic optimization of TNS using variational Monte Carlo (VMC). In particular, we investigate the convergence behavior of first- and second order update methods under stochastic noise, which was in turn affected by several factors such as sample size, system size, wavefunction quality and variational expressivity of the ansatz. We hope that the developments described in Chapter 3 and Chapter 4 can allow efficient large scale PEPS simulation of highly entangled states on 2-dimensional lattices, such as spin liquids, ground state of fermi-Hubbard model, and phases of uniform electron gas.

Chapter 5 introduces a constructive approach for representing high dimensional classical functions with tensor network, and perform integration with approximate contraction. Previous attempts of using tensor network for high dimensional integration typically fit a predetermined form of exactly contractable tensor network to the target function, where error is mainly due to the limited expressivity of the tensor network ansatz. On the other hand, our constructive approach is in principle free of representation error for any function that admits polynomial decomposition into small function blocks. The returned tensor network representation is of arbitrary geometry, where the error is mainly due to approximate contraction, which will benefit greatly from new developments in tenor network approximate contraction techniques.