Efficient Block Encoding of Structured Hamiltonians by Separating Where and What

Abstract

Fault-tolerant block encodings of structured Hamiltonians can reduce their non-Clifford cost in SELECT by selecting where an operator acts separately from what is applied. We construct permute-act-unpermute circuits whose CSWAP networks exploit the support geometry: a selected support is brought to a fixed target register, a shared local circuit acts there, and the permutation is undone. For supports of fixed size, the $T$ count of SELECT then grows with the system size rather than with the number of terms, without requiring translational symmetry or factorised coefficients. For two-site supports, we establish and attain the minimum CSWAP count within address-controlled networks of site transpositions. Compiled for Pauli terms, these circuits use $8SN+O(\log N)$ $T$ gates with $O(\log N)$ permutation work qubits, where $N$ is the number of system qubits and $S$ is determined by the support geometry, with $S=1$ for nearest-neighbour interactions and $S=3/2$ for all-to-all pairs. We also introduce a bridged CSWAP that retains and repairs a temporary AND across the target action. When the repair is Clifford, it halves the $T$ count of a matched forward-inverse CSWAP pair at the cost of one retained work qubit. For a Heisenberg ring, the complete block-encoding query reduces the $T$ count by a factor approaching 3 as the system size grows. For a five-orbital Anderson impurity model, which combines nearest-neighbour and all-to-all support geometries, the reduction is about 1.7 at large bath sizes. Both comparisons use the lowest-cost compiled baselines considered, at unchanged block-encoding normalisation and comparable peak ancilla counts.