Matrix Product Operators In The Age of Block Encoding

Abstract

We develop a block-encoding compiler that treats matrix product operators as compressed, virtual-path linear combination of unitaries programs. The compiler constructs conditional PREP and local SELECT stages directly from a parent matrix product operator, establishing tensor networks as structured quantum intermediate representations that can be efficiently compiled to block-encoded circuits. We apply the construction to real-time evolution in the Heisenberg chain and two perturbed Heisenberg-family models. Across the regimes studied, the compressed, approximately unitary propagator MPOs retain mild bond dimension and LCU normalization. Relative to an LCU that explicitly lists the $\mathcal{O}(N^K)$ Pauli-product branches of an order-K truncated Taylor polynomial, our virtual-transition implementation replaces combinatorial branch enumeration by a circuit complexity scaling as $\mathcal{O}(\alpha_{\rm MPO}N\chi^2)$, approaching $\mathcal{O}(N\chi^2)$ when $\alpha_{\rm MPO}$ remains mild. We numerically characterize how truncation order, bond-dimension budget, and system size affect approximation error, normalization, and compiler cost. These results demonstrate how classically compressed tensor-network representations can serve as quantum compiler intermediate representations for block encoding and opens new avenues to accelerate quantum algorithms.