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

TL;DR

The paper tackles the exponential difficulty of classically simulating large quantum circuits under finite fidelity by introducing entanglement clustering: cluster-TEBD contracts multiple gate layers exactly by forming entanglement clusters, and a dynamical adaptive grouping routine for DMRG adapts qubit grouping to the circuit's entanglement pattern. These methods leverage bond-dimension concepts $\chi_b$, entanglement entropy $S_b$, and fidelity measures $f_i$ to manage memory and accuracy, enabling larger-scale simulations than standard TEBD/DMRG. Benchmarking on random-structured Clifford and non-Clifford circuits up to $N$ qubits and $L$ layers, as well as Shor's algorithm with hundreds of thousands of layers, shows substantial runtime speedups (often by factors of 2–20) and improved final state fidelities when using the cluster-based strategies. The framework is generalizable to other tensor-network forms and 2D geometries, with prospects for parallelization and applicability to quantum-device emulation and digital twins.

Abstract

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.

Figures (12)

• Figure 1: Quantum circuit with eight qubits and seven layers modeled as a tensor network. The initial state is an outer product of rank-1 tensors (vectors), as in this case it is a fully separable state. The gates are modeled as rank-2 tensors (matrices) for single-qubit operations, and rank-4 tensors for two-qubit operations, and they are labeled according to the qubits they act on and the layer they belong to.
• Figure 2: Entangled $N$-qubit quantum state $\ket{\psi}$, initially represented as a rank-$N$ tensor with total size $2^N$, decomposed into an $N$-site matrix-product state with successive singular value decompositions.
• Figure 3: Truncating singular values in an SVD step. The matrix of singular values $S \in \mathbb{R}^{\chi_i \times \chi_i}$ is reduced to a matrix $\tilde{S} \in \mathbb{R}^{\tilde{\chi}_i \times \tilde{\chi}_i}$ with $\tilde{\chi} \leq \chi$.
• Figure 4: Single iteration of TEBD on a two-qubit gate. The red node symbolizes the orthogonality center of the MPS.
• Figure 5: Contraction step of the DMRG algorithm for quantum circuits, setting $L_{\mathrm{max}} = 4$.