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Recent work has shown that it can be advantageous to implement a compositechannel that partitions the Hamiltonian $H$ for a given simulation problem intosubsets $A$ and $B$ such that $H=A+B$, where the terms in $A$ are simulatedwith a Trotter-Suzuki channel and the $B$ terms are randomly sampled via theQDrift algorithm. Here we show that this approach holds in imaginary time,making it a candidate classical algorithm for quantum Monte-Carlo calculations.We upper-bound the induced Schatten-$1 \to 1$ norm on both imaginary-timeQDrift and Composite channels. Another recent result demonstrated thatsimulations of Hamiltonians containing geometrically-local interactions forsystems defined on finite lattices can be improved by decomposing $H$ intosubsets that contain only terms supported on that subset of the lattice using aLieb-Robinson argument. Here, we provide a quantum algorithm by unifying thisresult with the composite approach into ``local composite channels" and weupper bound the diamond distance. We provide exact numerical simulations ofalgorithmic cost by counting the number of gates of the form $e^{-iH_j t}$ and$e^{-H_j \beta}$ to meet a certain error tolerance $\epsilon$. We show constantfactor advantages for a variety of interesting Hamiltonians, the maximum ofwhich is a $\approx 20$ fold speedup that occurs for a simulation of Jellium.

Composite QDrift-Product Formulas for Quantum and Classical Simulations in Real and Imaginary Time