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 \texttt{PREP} and local \texttt{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.

arXiv:2606.19083 [quant-ph]

(or arXiv:2606.19083v2 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2606.19083 Focus to learn more arXiv-issued DOI via DataCite

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