This paper presents methods for replacing entangling operations with random local operations in quantum circuits, at the cost of increasing the number of required executions. The authors introduce the concept of "space-like cuts," where an entangling unitary is replaced with random local unitaries, and propose a new entanglement measure called the product extent, which bounds the cost of this replacement. They show that this procedure yields a quasiprobability decomposition with minimal 1-norm in certain cases, addressing an open question. As an application, they improve bounds on clustered Hamiltonian simulation, showing that interactions can be removed at a cost exponential in the sum of their strengths times the evolution time. They also improve the upper bound on the cost of replacing wires with measure-and-prepare channels using "time-like cuts." They prove a matching information-theoretic lower bound when estimating output probabilities. The paper discusses the theoretical implications of these results, including the relationship between classical and quantum resources required for different information processing tasks. The authors also introduce the concept of quasiprobability decompositions (QPDs) and show how they can be used to simulate quantum circuits with fewer qubits. They demonstrate that their methods can be applied to clustered Hamiltonian simulation, significantly improving the sample complexity compared to previous bounds. The paper also discusses the relationship between the product extent and the Choi-Jamiolkowski robustness, showing that the product extent provides a tighter bound in certain cases. The authors conclude that their methods offer a more efficient way to simulate quantum circuits, with potential applications in quantum computing and quantum simulation.This paper presents methods for replacing entangling operations with random local operations in quantum circuits, at the cost of increasing the number of required executions. The authors introduce the concept of "space-like cuts," where an entangling unitary is replaced with random local unitaries, and propose a new entanglement measure called the product extent, which bounds the cost of this replacement. They show that this procedure yields a quasiprobability decomposition with minimal 1-norm in certain cases, addressing an open question. As an application, they improve bounds on clustered Hamiltonian simulation, showing that interactions can be removed at a cost exponential in the sum of their strengths times the evolution time. They also improve the upper bound on the cost of replacing wires with measure-and-prepare channels using "time-like cuts." They prove a matching information-theoretic lower bound when estimating output probabilities. The paper discusses the theoretical implications of these results, including the relationship between classical and quantum resources required for different information processing tasks. The authors also introduce the concept of quasiprobability decompositions (QPDs) and show how they can be used to simulate quantum circuits with fewer qubits. They demonstrate that their methods can be applied to clustered Hamiltonian simulation, significantly improving the sample complexity compared to previous bounds. The paper also discusses the relationship between the product extent and the Choi-Jamiolkowski robustness, showing that the product extent provides a tighter bound in certain cases. The authors conclude that their methods offer a more efficient way to simulate quantum circuits, with potential applications in quantum computing and quantum simulation.