Abstract A non-Abelian group of 16 symmetry operations on (generally) non-Hermitian discrete Hamiltonians represented by matrices is studied. The symmetry operations are described by unitary/antiunitary superoperators that arise when combining three basic generating operations with simple ‘geometric’ interpretations. The corresponding Hamiltonian symmetries occur when the Hamiltonian remains invariant under the superoperator action. These symmetries include PT-symmetry and Hermiticity as particular cases. The interplay between the group of symmetry operations and Hamiltonian symmetries is analyzed systematically by introducing the concepts of equivalent operations and associated symmetries. Spectral properties implied by some of the symmetries are described.
We discuss Hamiltonian symmetries and invariants for quantum systems driven by non-Hermitian Hamiltonians. For time-independent Hermitian Hamiltonians, a unitary or antiunitary transformation
AHA† that leaves the Hamiltonian
H unchanged represents a symmetry of the Hamiltonian, which implies the commutativity
[H,A]=0, and a conservation law, namely the invariance of expectation values of
A. For non-Hermitian Hamiltonians,
H† comes into play as a distinct operator that complements
H in generalized unitarity relations. The above description of symmetries has to be extended to include also
A-pseudohermiticity relations of the form
AH=H†A. A superoperator formulation of Hamiltonian symmetries is provided and exemplified for Hamiltonians of a particle moving in one-dimension considering the set of
A operators forming Klein's 4-group: parity, time-reversal, parity\&time-reversal, and unity. The link between symmetry and conservation laws is discussed and shown to be more subtle for non-Hermitian than for Hermitian Hamiltonians.
It is shown that if a Hamiltonian H is Hermitian, then there always exists an operator having the following properties: (i) is linear and Hermitian; (ii) commutes with H ; (iii) 2 = 1; (iv) the n th eigenstate of H is also an eigenstate of with eigenvalue (−1) n . Given these properties, it is appropriate to refer to as the parity operator and to say that H has parity symmetry, even though may not refer to spatial reflection. Thus, if the Hamiltonian has the form H = p 2 + V ( x ), where V ( x ) is real (so that H possesses time-reversal symmetry), then it immediately follows that H has symmetry. This shows that symmetry is a generalization of Hermiticity: all Hermitian Hamiltonians of the form H = p 2 + V ( x ) have symmetry, but not all -symmetric Hamiltonians of this form are Hermitian.
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.