Efficiently Learning, Testing, and Simulating Quantum Many-Body Systems

Abstract

This thesis focuses on quantum information and quantum computing, and their applications in studying quantum many-body systems. A remarkable interplay between computer science and quantum physics in the past few decades has revealed that a precise control and manipulation of interacting quantum systems enables us to process information and perform computations that go beyond the reach of conventional digital computers. This novel form of information processing has also resulted in a conceptually new toolkit for tackling fundamental questions about the physics of quantum many-body systems. This thesis studies new features of interacting quantum systems through the lens of computational complexity and information theory. We will see how using these new features in turn allows us to develop efficient classical and quantum algorithms for learning, testing, and simulating quantum many-body systems. Below are the main results of this thesis:

1. We develop an algorithm for reliably testing the amount of entanglement in a pure many-body quantum state. This algorithm tests whether a quantum state is a matrix product state of certain bond dimension in the property testing model. We provide both upper and lower bounds on the number of identical copies of the quantum state required by this algorithm.

2. We prove that a quantum information quantity, known as the entanglement spread, satisfies an area law in the ground state of any gapped local Hamiltonian with an arbitrary geometry. This new feature of ground-state entanglement is obtained using a connection to the seemingly different problem of finding the communication complexity of testing bipartite states.

3. We devise an algorithm for learning the local Hamiltonian that governs the interactions in a quantum many-body system. This algorithm uses the results of local measurements on the thermal state of the system, and provably only requires a number of samples that scales polynomially with the number of particles.

4. A quasi-polynomial time algorithm is developed that estimates the quantum partition function at temperatures above the phase transition point. We also study different characterizations of the thermal phase transition by connecting the exponential decay of correlations to the analyticity of the free energy in the high-temperature phase.

5. We rigorously bound the improvement that low-depth quantum circuits can provide over methods based on product states in estimating the ground-state energy of local Hamiltonians.