The unitary group and examples from many-body physics

F.A. Matsen

Sten Rettrup

F. A. Matsen, M. Peredo

Yao Shen, Chi-Chun Zhou, Wu-Sheng Dai +1 more

The exact solution of the interacting many-body system is important and is difficult to solve. In this paper, we introduce a group method to solve the interacting many-body problem using the relation between the permutation group and the unitary group. We prove a group theorem first, then using the theorem, we represent the Hamiltonian of the interacting many-body system by the Casimir operators of unitary group. The eigenvalues of Casimir operators could give the exact values of energy and thus solve those problems exactly. This method maps the interacting many-body system onto an intermediate statistical representation. We give the relation between the conjugacy-class operator of permutation group and the Casimir operator of unitary group in the intermediate statistical representation, called the Gentile representation. Bose and Fermi cases are two limitations of the Gentile representation. We also discuss the representation space of symmetric and unitary group in the Gentile representation and give an example of the Heisenberg model to demonstrate this method. It is shown that this method is effective to solve interacting many-body problems.

Oxana Shaya, Zoë Holmes, Christoph Hirche +1 more

Abstract Understanding the complexity of quantum states and circuits is a central challenge in quantum information science, with broad implications in many-body physics, high-energy physics and quantum learning theory. A common way to model the behavior of typical states and circuits involves sampling unitary transformations from the Haar measure on the unitary group. In this work, we depart from this standard approach and instead study structured unitaries drawn from other compact connected groups, namely the symplectic and special orthogonal groups. By leveraging the concentration of measure phenomenon, we establish two main results. We show that random quantum states generated using symplectic or orthogonal unitaries typically exhibit an exponentially large strong state complexity, and are nearly orthogonal to one another. Similar behavior is observed for designs over these groups. Additionally, we demonstrate the average-case hardness of learning circuits composed of gates drawn from such classical groups of unitaries. Taken together, our results demonstrate that structured subgroups can exhibit a complexity comparable to that of the full unitary group.

F. A. Matsen