Open source Matrix Product States: Exact diagonalization and other entanglement-accurate methods revisited in quantum systems

TL;DR

This paper revisits entanglement-accurate methods for quantum many-body systems using open source Matrix Product States, emphasizing exact diagonalization techniques that preserve full entanglement for validation and exploration of complex quantum states.

Contribution

It introduces three algorithms for exact, non-truncated quantum evolution applicable to both closed and open systems, with implementation details and symmetry considerations.

Findings

Demonstrates convergence and efficiency of the algorithms

Shows application to the long-range quantum Ising model

Highlights the role of exact methods in studying highly entangled states

Abstract

Tensor network methods as presented in our open source Matrix Product States software have opened up the possibility to study many-body quantum physics in one and quasi-one-dimensional systems in an easily accessible package similar to density functional theory codes but for strongly correlated dynamics. Here, we address methods which allow one to capture the full entanglement without truncation of the Hilbert space. Such methods are suitable for validation of and comparisons to tensor network algorithms, but especially useful in the case of new kinds of quantum states with high entanglement violating the truncation in tensor networks. Quantum cellular automata are one example for such a system, characterized by tunable complexity, entanglement, and a large spread over the Hilbert space. Beyond the evolution of pure states as a closed system, we adapt the techniques for open quantum systems simulated via the Lindblad master equation. We present three algorithms for solving closed-system many-body time evolution without truncation of the Hilbert space. Exact diagonalization methods have the advantage that they not only keep the full entanglement but also require no approximations to the propagator. Seeking the limits of a maximal number of qubits on a single core, we use Trotter decompositions or Krylov approximation to the exponential of the Hamiltonian. All three methods are also implemented for open systems represented via the Lindblad master equation built from local channels. We show their convergence parameters and focus on efficient schemes for their implementations including Abelian symmetries, e.g., U(1) symmetry used for number conservation in the Bose-Hubbard model or discrete Z2 symmetries in the quantum Ising model. We present the thermalization timescale in the long-range quantum Ising model as a key example of how exact diagonalization contributes to novel physics.