# What do coupling coefficients mean in terms of Pauli gates, and why are they time dependent?

I am reading this error mitigation paper by the IBM team and I am slightly confused about the meaning of "coupling coefficients" when describing multi-qubit Hamiltonian. I have only seen coupling coefficients in terms describing interaction between atom and external electric field. What do these coefficients mean in terms of pauli gates and why are they time dependent?

• Pauli matrices can be used to decompose arbitrary Hamiltonians. If the Hamiltonian is time-dependent, the coefficients must be time-dependent. Does that answer the question?
– glS
Jun 6, 2020 at 13:31

The set $$G_N$$ of all $$N$$-fold tensor products of the identity and Pauli gates $$X$$, $$Y$$ and $$Z$$ is a basis of the real vector space $$L_H$$ of Hermitian operators on $$N$$ qubits. Therefore, $$K(t)\in L_H$$ can be written as a linear combination of elements in $$G_N$$
$$K(t) = \sum_\alpha J_\alpha(t) P_\alpha\tag1$$
where $$P_\alpha\in G_N$$ and $$J_\alpha(t)\in\mathbb{R}$$. The coefficients $$J_\alpha$$ are time-dependent, because $$P_\alpha$$ are not and $$K$$ is. In other words, if coefficients $$J_\alpha$$ were not time-dependent then the right-hand side of $$(1)$$ would be constant, unlike the left-hand side.