Programmatic Representation of Quantum Many Body Systems

Abstract

The problem of simulating quantum many-body systems is fundamental in condensed matter physics, quantum computing, and quantum chemistry. The exact simulation of quantum many-body systems is generally intractable on classical computers, and developing efficient simulation methods is crucial for understanding and utilizing quantum systems. Meanwhile, from the computer science community, the development of formal languages has dramatically improved programming and software efficiency. Thus, it is natural to ask whether we can develop and utilize such representations to simulate quantum many-body systems. We propose so-called \textit{programmatic representations} for simulating quantum many-body systems on computational devices.

We begin with introducing the programmatic representations for quantum circuits, quantum operators, quantum states, pulse sequences, and more general quantum programs with control flows are discussed. We further introduce the transformation of and between these representations, which leads to the development of several software frameworks, including \texttt{Yao} and \texttt{Bloqade}, which achieved state-of-the-art performance in simulating quantum circuits and Rydberg atom array dynamics. We introduce the transformation for automatic differentiation and show that by utilizing the reversibility of the quantum circuits, only constant memory overhead is needed for the automatic differentiation of quantum circuits in simulators. As a result, we report the differentiation of 10,000-layer quantum circuits that no previous software can achieve.

On top of these technical developments in exact simulation, hardware modeling, and automatic differentiation, we generalize the numerical renormalization group formulations from Wilson and White, namely Wilson's NRG and White's DMRG, which we call the \textit{operator learning renormalization group (OLRG)}. OLRG allows solving general quantum many-body problems with arbitrary operator maps in lieu of a state ansatz. We introduce a theory framework guiding the design of OLRG loss functions, providing a rigorous error bound for real-time evolution. We further show OLRG can solve the quantum many-body problems with arbitrary operator maps such as neural networks using the Operator Matrix Map (OMM), and can be used to generate control parameters for a quantum device using the Hamiltonian Expression Map (HEM). We explore different hyperparameters for both OMM and HEM for a 1D transverse field Ising model and show that our theoretical loss function correctly guides both the OMM and HEM to ground truth using differentiable programming.

We conclude by discussing the future directions of applying programmatic representations to quantum many-body systems and the future directions of quantum many-body system simulation.