The paper gives algorithms to decompose structured sparse matrices into polylogarithmically many sigma-basis operators and builds quantum circuits for both variational and fault-tolerant use, cutting term counts exponentially versus Pauli decompositions.
In this case, k = 1 as the second term in the tensor product is I
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Efficient Quantum Access Model for Sparse Structured Matrices using Linear Combination of Things
The paper gives algorithms to decompose structured sparse matrices into polylogarithmically many sigma-basis operators and builds quantum circuits for both variational and fault-tolerant use, cutting term counts exponentially versus Pauli decompositions.