Symmetries and Symmetry Groups in Quantum Physics

Florian A. Scheck

Asım Orhan Barut

Saumya Shah, Patrick Rebentrost

The computational cost of simulating quantum many-body systems can often be reduced by taking advantage of physical symmetries. While methods exist for specific symmetry classes, a general algorithm to find the full permutation symmetry group of an arbitrary Pauli Hamiltonian is notably lacking. This paper introduces a new method that identifies this symmetry group by establishing an isomorphism between the Hamiltonian's permutation symmetry group and the automorphism group of a coloured bipartite graph constructed from the Hamiltonian. We formally prove this isomorphism and show that for physical Hamiltonians with bounded locality and interaction degree, the resulting graph has a bounded degree, reducing the computational problem of finding the automorphism group to polynomial time. The algorithm's validity is empirically confirmed on various physical models with known symmetries. We further show that the problem of deciding whether two Hamiltonians are permutation-equivalent is polynomial-time reducible to the graph isomorphism problem using our graph representation. This work provides a general, structurally exact tool for algorithmic symmetry finding, enabling the scalable application of these symmetries to Hamiltonian simulation problems.

Fl. Stancu

Abstract Group theory is a branch of mathematics appropriate for studying symmetries of physical systems. A symmetry is a natural attribute of the physical world and therefore is the starting point of any physical law. The foundation of dynamics is in Lagrangian theory or equivalently in Hamiltonian theory. A symmetry of a physical system is expressed by the invariance of its Lagrangian or Hamiltonian, or ultimately of its equations of motion, under certain transformations. When these transformations form groups it is convenient and useful to study symmetries through group theory. Group theory can explain observed regularities and helps to simplify and unify the physics of apparently different systems. It is a valuable tool that helps in predicting the behaviour of systems for which a symmetry has been found.

Bipin R. Desai

I. Schmelzer

A non-uniqueness result for the canonical structure in quantum theory shows that the classical part of the Copenhagen interpretation contains physically important information not contained in its quantum part. As a consequence, we cannot compute the symmetry group of a quantum theory considering only the quantum part. The unavoidable vagueness of the classical part therefore leads to a similar vagueness in the definition of the symmetry group. This makes it at least problematic, if not impossible, to establish the true symmetry group of a quantum theory in the Copenhagen interpretation. Different from the old measurement problem, the symmetry group is to important physically to be rejected as a metaphysical pseudoproblem.