Tensor Networks as an Explicit Interface for Quantum Block-Encodings

Abstract:Tensor networks (TNs) give explicit classical descriptions of structured finite linear maps, while block-encodings (BEs) are the standard quantum access model for such maps. We establish TNs as a universal data interface for quantum algorithms: any explicitly specified TN for an arbitrary finite linear map compiles directly to an explicit qubit BE, with no penalty on the selected-block round trip under faithful compression. Along a chosen sweep, the compiler handles local non-unitarity by exact local dilations and aggregates the resulting post-selection conditions online using logarithmically many additional flag qubits. The chosen sweep incurs three exact, sweep-dependent costs (the accumulated scale, the frontier memory, and the number of genuinely dilated local steps), which the main theorems are stated in terms of. In the bounded-local explicit arithmetic model, this yields linear-time compilation to linear-size circuits with constant-size local gadgets, and hence a constant-factor size correspondence between bounded-local TNs and bounded-local BEs. The same construction gives a selected-block round trip BE->TN->BE: a BE canonically yields a TN for its selected block rather than for an arbitrary unitary extension, which can then be compressed or approximated classically before recompilation. Faithful Schmidt-rank compression transfers monotonically (the scale cost and frontier memory cannot increase, with operator error bounded by the discarded weight), while arbitrary restructuring admits no general scale guarantee, a limitation we show is unavoidable. As consequences and limits of this scale accounting, we characterize exact scale optimality, show that bridge-hourglass forests admit scale-optimal sweeps after exact recursive local bond compression, and prove that certifying unrestricted exact scale optimality is already hard for diagonal MPOs on a path unless P=NP.

arXiv:2510.00935 [quant-ph]

(or arXiv:2510.00935v3 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2510.00935 Focus to learn more arXiv-issued DOI via DataCite

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.