Block encoding of matrix product operators

Quantum signal processing combined with quantum eigenvalue transformation has recently emerged as a unifying framework for several quantum algorithms. In its standard form, it consists of two separate routines: block encoding, which encodes a Hamiltonian in a larger unitary, and signal processing, which achieves an almost arbitrary polynomial transformation of such a Hamiltonian using rotation gates. The bottleneck of the entire operation is typically constituted by block encoding and, in recent years, several problem-specific techniques have been introduced to overcome this problem. Within this framework, we present a procedure to block-encode a Hamiltonian based on its matrix product operator (MPO) representation. More specifically, we encode every MPO tensor in a larger unitary of dimension $D+2$, where $D = \lceil\log(\chi)\rceil$ is the number of subsequently contracted qubits that scales logarithmically with the virtual bond dimension $\chi$. Given any system of size $L$, our method requires $L+D$ ancillary qubits in total, while the number of one- and two-qubit gates decomposing the block encoding circuit scales as $\mathcal{O}(L\cdot\chi^2)$.