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 $D=\ensuremath{\lceil}{log}_{2}(\ensuremath{\chi})\ensuremath{\rceil}$ is the number of subsequently contracted qubits that scales logarithmically with the virtual bond dimension $\ensuremath{\chi}$. 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 $O(L{\ensuremath{\chi}}^{2})$. Moreover, block encodings generally require postselection. Our model exhibits a success probability decaying exponentially with $L$. Nevertheless, we present a way to mitigate this limitation for finite systems.