  We study methods to replace entangling operations with random local
operations in a quantum computation, at the cost of increasing the number of
required executions. First, we consider "space-like cuts" where an entangling
unitary is replaced with random local unitaries. We propose an entanglement
measure for quantum dynamics, the product extent, which bounds the cost in a
procedure for this replacement based on two copies of the Hadamard test. In the
terminology of prior work, this procedure yields a quasiprobability
decomposition with minimal 1-norm in a number of cases, which addresses an open
question of Piveteau and Sutter. As an application, we give dramatically
improved bounds on clustered Hamiltonian simulation. Specifically we show that
interactions can be removed at a cost exponential in the sum of their strengths
times the evolution time.
  We also give an improved upper bound on the cost of replacing wires with
measure-and-prepare channels using "time-like cuts". We prove a matching
information-theoretic lower bound when estimating output probabilities.
