Preparing Hamiltonians for Quantum Simulation: A Computational Framework for Cartan Decomposition via Lax Dynamics

The juxtaposition of Feynman's conjecture that, “if we could build a quantum simulator at our disposal, composed of spin-½ particles that we could manipulate at will, then we would be able to engineer the interaction between those particles according to the one we want to simulate, and thus predict the value of physical quantities by simply performing the appropriate measurements on the quantum simulator,” and Lloyd's article in Science affirming that, “quantum computers can be programmed to simulate any local quantum system,” evinces the profound gravity of the Hamiltonian simulation problem and its applications. To prepare for such a simulation, it is essential to convert the unitary operators described mathematically to the unitary operators recognizable as quantum circuits, yet most current techniques are prone to approximation errors which affect the simulation authenticity. This project tackles the difficulties from an innovative avenue of the Cartan decomposition via the Lax dynamics. Not only is this process numerically feasible, but also produces a genuine unitary synthesis that is optimal in both the precision and the usage of minimally required synthesis components. This project aims to establish theoretic and algorithmic foundations and develop numerical methods. Upon completion, the theory and the experiments are expected to find applicability extending from quantum simulation to other areas such as gauging the quality of other approaches or evaluating the robustness of a given system. The gains from this work will solidify the many studies under one standard framework. Preliminary results show promising potential of this project. Training of at least one graduate student on the topics of the project is expected.To simulate the time evolution of a quantum system on a classical computer is hard - The computational power required to even describe a quantum system scales exponentially with the number of its constituents, let alone integrate its equations of motion. Hamiltonian simulation on a quantum machine is a possible solution to this challenge - Assuming that a quantum system composing of spin-½ particles can be manipulated at will, then it is tenable to engineer the interaction between those particles according to the one that is to be simulated, and thus predict the value of physical quantities by simply performing the appropriate measurements on the system. Establishing a linkage between the unitary operators described mathematically as a logic solution and the unitary operators recognizable as quantum circuits for execution is therefore essential for algorithm design and circuit implementation. Most current techniques are prone to approximation errors. This project is to tackle the Cartan decomposition via the notion of Lax dynamics, which not only is numerically feasible, but also produces a genuine unitary synthesis that is optimal in both the precision with controllable integration errors and the usage of only minimally required synthesis components. This project aims at establishing theoretic and algorithmic foundations of the goals: 1) exploit the geometric properties of Hamiltonian subalgebras; 2) describe a common mechanism for deriving the Lax dynamics; 3) develop a decomposition-based quantum algorithm; and 4) experiment the algorithm on the IBM Quantum Hub systems. Six specific tasks will be undertaken to derive the theory and numerical methods to reach these goals. Upon completion, the theory and the experiments are expected to find applicability extending from quantum simulation to many-body problem, and to various research endeavors in quantum information science.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.