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Write the local Hamiltonian as $H={\sum }_{j=1}^m{h}_j$. Since its terms commute, their matrix exponentials factor exactly:Each factor acts on at most two qubits and can be compiled over a fixed universal quantum gate set to operator norm error at most $\epsilon /m$. The telescoping bound for products of operators then bounds the total error by the sum of the factor errors, at most $\epsilon$. Because $m$ is polynomial in $n$ and the Solovay--Kitaev theorem gives gate count polynomial in $\log (m/\epsilon )$ for each fixed-dimensional factor, this is an efficient commuting local Hamiltonian simulation. Finally, the eigenvalue equation $H|\Psi ⟩=\lambda |\Psi ⟩$ impliesso $|\Psi ⟩$ remains an eigenstate and its eigenvalue is $e^{-i\lambda t}$.

For a suitable finite inverse-closed universal quantum gate set, the Solovay--Kitaev theorem approximates a fixed-dimensional unitary to operator norm error $\epsilon$ with a gate sequence of length polynomial in $\log (1/\epsilon )$.

A two-qubit gate is a unitary operator on two qubit lines, extended by the identity on the others. It may be entangling, as a CNOT gate is, or a product of local operations. In a Hamiltonian simulation model allowing arbitrary two-qubit gates, the exponential of a Hermitian term supported on two lines counts as one such gate. Compiling it into a fixed discrete universal quantum gate set introduces a separate approximation cost.