Recent work has shown that it can be advantageous to implement a composite
channel that partitions the Hamiltonian H for a given simulation problem into
subsets A and B such that H=A+B, where the terms in A are simulated
with a Trotter-Suzuki channel and the B terms are randomly sampled via the
QDrift 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→1 norm on both imaginary-time
QDrift and Composite channels. Another recent result demonstrated that
simulations of Hamiltonians containing geometrically-local interactions for
systems defined on finite lattices can be improved by decomposing H into
subsets that contain only terms supported on that subset of the lattice using a
Lieb-Robinson argument. Here, we provide a quantum algorithm by unifying this
result with the composite approach into ``local composite channels" and we
upper bound the diamond distance. We provide exact numerical simulations of
algorithmic cost by counting the number of gates of the form e−iHj​t and
e−Hj​β to meet a certain error tolerance ϵ. We show constant
factor advantages for a variety of interesting Hamiltonians, the maximum of
which is a ≈20 fold speedup that occurs for a simulation of Jellium.Comment: 49 pages, 13 figure