Commuting Pauli Hamiltonians as Maps between Free Modules

Commuting Pauli Hamiltonians as Maps between Free Modules

We study unfrustrated spin Hamiltonians that consist of commuting tensor products of Pauli matrices. Assuming translation-invariance, we observe that the Hamiltonian is described by a map between modules over the translation group algebra, so homological methods are applicable. We show universal properties of topologically ordered phases in low spatial dimensions. Particularly, we prove that in...

Convexity and Commuting Hamiltonians

The converse was proved by A. Horn [5], so that all points in this convex hull occur as diagonals of some matrix A with the given eigenvalues. Kostant [7] generalized these results to any compact Lie group G in the following manner. We consider the adjoint action of G on its Lie algebra L(G). If T is a maximal torus of G and W its Weyl group, then it is well known that W-orbits in L(T) correspo...

Thermalization Time Bounds for Pauli Stabilizer Hamiltonians

We prove a general lower bound to the spectral gap of the Davies generator for Hamiltonians that can be written as the sum of commuting Pauli operators. These Hamiltonians, defined on the Hilbert space of N -qubits, serve as one of the most frequently considered candidates for a self-correcting quantum memory. A spectral gap bound on the Davies generator establishes an upper limit on the life t...

Commuting Idempotents, Square-free Modules, and the Exchange Property

We give a criterion for when idempotents of a ring R which commute modulo the Jacobson radical J(R) can be lifted to commuting idempotents of R. If such lifting is possible, we give extra information about the lifts. A “half-commuting” analogue is also proven, and this is used to give sufficient conditions for a ring to have the internal exchange property. In particular, we show that if R/J(R) ...

Bohr-Sommerfeld conditions for several commuting Hamiltonians