Low-frustration block encodings by patch routing

Abstract

Block-encoding has emerged as a core primitive in many quantum algorithms, including quantum simulation. In this work, we design efficient and extensible methods for generating low-frustration block encodings of frustrated lattice Hamiltonians built from repeated local interaction patches through a technique we call "patch routing". We construct block encodings for a full Hamiltonian by first block-encoding a small local patch and then using a SWAP-style routing network to coherently apply the local patch encoding across the lattice. The expensive, local block-encoding cost is paid only once per query and is independent of lattice size, whereas the comparatively inexpensive routing network grows linearly with the lattice size, dominating at large sizes. Our method allows for a hierarchical structure in which larger patches can be block-encoded with small normalization factors, albeit at a higher (local, non-extensive) block-encoding cost, thereby reducing overall query complexity in an algorithm. We exploit symmetries and algebraic structures in local interactions to construct efficient block encodings, and we apply these techniques to the Heisenberg, Fermi-Hubbard, and honeycomb spin-liquid models. For a $120\times 120$ square-lattice $J_1$-$J_2$ Heisenberg model near maximal frustration ($J_2/J_1=0.4-0.6$), our methods result in a $15-17\times$ reduction in the leading $T$ cost per qubitized-QPE run (ignoring initial state preparation and overlap) relative to prior state-of-the-art Pauli LCU constructions. Corresponding reductions are approximately $3\times$ for the $120\times120$ Fermi-Hubbard model at half-filling. For time evolution of the honeycomb spin-liquid model on $120\times120$ unit cells, the leading cost is reduced by $5\times$. Our framework provides a systematic way forward for larger and more accurate fault-tolerant simulations of real-world quantum systems.