Dynamical cluster-based strategy for improving tensor network algorithms in quantum circuit simulations

We optimize matrix-product state-based algorithms for simulating quantum circuits with finite fidelity, specifically the time-evolving block decimation (TEBD) and the density-matrix renormalization group (DMRG) algorithms, by exploiting the irregular arrangement of entangling operations in circuits. We introduce a variation of the standard TEBD algorithm, we termed "cluster-TEBD", which dynamically arranges qubits into entanglement clusters, enabling the exact contraction of multiple circuit layers in a single time step. Moreover, we enhance the DMRG algorithm by introducing an adaptive protocol, which analyzes the entanglement distribution within each circuit section to be contracted, dynamically adjusting the qubit grouping at each iteration. We analyze the performances of these enhanced algorithms in simulating both stabilizer and nonstabilizer random-structured quantum circuits, with up to 1000 qubits and 100 layers of Clifford and non-Clifford gates, and in simulating Shor's quantum algorithm with up to hundreds of thousands of layers. Our findings show that, even with reasonable computational resources per task, cluster-based approaches can significantly speed up simulations of large-sized quantum circuits and improve the fidelity of the final states.