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The Journal of Supercomputing

A community detection-based parallel algorithm for quantum circuit simulation using tensor networks

Alfred M. Pastor, José M. Badía, Maribel Castillo

Published Jan 31, 2025• 4 citations
Publisher

Abstract

Abstract Quantum computing holds significant promise for solving complex problems, but simulating quantum circuits on classical computers remains essential due to the current limitations of quantum hardware. Efficient simulation is crucial for the development and validation of quantum algorithms and quantum computers. This paper explores and compares various strategies to leverage different levels of parallelism to accelerate the contraction of tensor networks representing large quantum circuits. We propose a new parallel multistage algorithm based on communities. The original tensor network is partitioned into several communities, which are then contracted in parallel. The pairs of tensors of the resulting network can be contracted in parallel using a GPU. We use the Girvan–Newman algorithm to obtain the communities and the contraction plans. We compare the new algorithm with two other parallelisation strategies: one based on contracting all the pairs of tensors in the GPU and another one that uses slicing to cut some indexes of the tensor network and then MPI processes to contract the resulting slices in parallel. The new parallel algorithm gets the best results with different well-known quantum circuits with a high degree of entanglement, including random quantum circuits. In conclusion, the results show that the main factor that limits the simulation is the space cost. However, the parallel multistage algorithm manages to reduce the cost of sequential simulation for circuits with a high number of qubits and allows simulating larger circuits.

DOI
10.1007/s11227-025-06918-3

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Abstract Quantum computing holds significant promise for solving complex problems, but simulating quantum circuits on classical computers remains essential due to the current limitations of quantum hardware. Efficient simulation is crucial for the development and validation of quantum algorithms and quantum computers. This paper explores and compares various strategies to leverage different levels of parallelism to accelerate the contraction of tensor networks representing large quantum circuits. We propose a new parallel multistage algorithm based on communities. The original tensor network is partitioned into several communities, which are then contracted in parallel. The pairs of tensors of the resulting network can be contracted in parallel using a GPU. We use the Girvan–Newman algorithm to obtain the communities and the contraction plans. We compare the new algorithm with two other parallelisation strategies: one based on contracting all the pairs of tensors in the GPU and another one that uses slicing to cut some indexes of the tensor network and then MPI processes to contract the resulting slices in parallel. The new parallel algorithm gets the best results with different well-known quantum circuits with a high degree of entanglement, including random quantum circuits. In conclusion, the results show that the main factor that limits the simulation is the space cost. However, the parallel multistage algorithm manages to reduce the cost of sequential simulation for circuits with a high number of qubits and allows simulating larger circuits.
Efficient simulation of quantum circuits has become indispensable with the rapid development of quantum hardware. The primary simulation methods are based on state vectors and tensor networks. As the number of qubits and quantum gates grows larger in current quantum devices, traditional state-vector-based quantum circuit simulation methods prove inadequate due to the overwhelming size of the Hilbert space and extensive entanglement. Consequently, brutal force tensor network simulation algorithms become the only viable solution in such scenarios. The two main challenges faced in tensor network simulation algorithms are optimal contraction path finding and efficient execution on modern computing devices, with the latter determining the actual efficiency. In this study, we investigate the optimization of such tensor network simulations on modern GPUs and propose general optimization strategies from two aspects: computational efficiency and accuracy. First, we propose to transform critical Einstein summation operations into GEMM operations, leveraging the specific features of tensor network simulations to amplify the efficiency of GPUs. Second, by analyzing the data characteristics of quantum circuits, we employ extended precision to ensure the accuracy of simulation results and mixed precision to fully exploit the potential of GPUs, resulting in faster and more precise simulations. Our numerical experiments demonstrate that our approach can achieve a 3.96× reduction in verification time for random quantum circuit samples in the 18-cycle case of Sycamore, with sustained performance exceeding 21 TFLOPS on one A100. This method can be easily extended to the 20-cycle case, maintaining the same performance, accelerating by 12.5× compared to the state-of-the-art CPU-based results and 4.48–6.78× compared to the state-of-the-art GPU-based results reported in the literature. Furthermore, the strategies presented in this article hold general applicability for accelerating problems that can be tackled by contracting tensor networks such as graphical models and combinatorial optimizations.
Tensors are a natural generalization of matrices, and tensor networks are a natural generalization of matrix products. Despite the simple definition of tensor networks, they are versatile enough to represent many different kinds of "products" that arise in various theoretical and practical problems. In particular, the powerful computational model of quantum computation can be defined almost entirely in terms of matrix products and tensor products, both of which are special cases of tensor networks. As such, (classical) algorithms for evaluating tensor networks have profound importance in the study of quantum computation. In this thesis, we design and implement a parallel algorithm for tensor network contraction. In addition to finding efficient contraction orders for a tensor network, we also dynamically slice it into multiple sub-tasks with lower space and time costs, in order to evaluate the tensor network in parallel. We refer to such an evaluation strategy as a contraction scheme for the tensor network. In addition, we introduce a local optimization procedure that improves the efficiency of the contraction schemes we find. We also investigate the applications of our parallel tensor network contraction algorithm in quantum computation. The most ready application is the simulation of random quantum supremacy circuits, where we benchmark our algorithm to demonstrate its advantage over other similar tensor network based simulators. Other applications we found include evaluating the energy function of a Quantum Approximate Optimization Algorithm (QAOA), and simulating surface codes under a realistic error model with crosstalk.
Abstract Quantum computing holds the potential to solve complex problems currently beyond the capabilities of classical computers by exploiting the principles of quantum mechanics. However, due to the immaturity of such computers and the inability to know the state of the circuit during execution, efficient quantum simulators on classical computers are needed. This paper compares different methods for the contraction of quantum circuits represented as tensor networks. Tensor Decision Diagrams (TDD) are used to implement two contraction ordering methods that exploit the structure of the circuits to reduce temporal and spatial costs. Experimental results show that these methods improve other well established approaches and that they are faster that other well-known simulation tools based on other circuit representations. In addition, an analysis of different implementations of the same quantum algorithm highlights the impact of gate sets on contraction efficiency.
Abstract We develop an algorithmic framework for contracting tensor networks and demonstrate its power by classically simulating quantum computation of sizes previously deemed out of reach. Our main contribution, index slicing, is a method that efficiently parallelizes the contraction by breaking it down into much smaller and identically structured subtasks, which can then be executed in parallel without dependencies. We benchmark our algorithm on a class of random quantum circuits, achieving greater than 105 times acceleration over the original estimate of the simulation cost. We then demonstrate applications of the simulation framework for aiding the development of quantum algorithms and quantum error correction. As tensor networks are widely used in computational science, our simulation framework may find further applications.
Parallel tensor network contraction algorithms have emerged as the pivotal benchmarks for assessing the classical limits of computation, exemplified by Google's demonstration of quantum supremacy through random circuit sampling. However, the massive parallelization of the algorithm makes it vulnerable to computer node failures. In this work, we apply coded computing to a practical parallel tensor network contraction algorithm. To the best of our knowledge, this is the first attempt to code tensor network contractions. Inspired by matrix multiplication codes, we provide two coding schemes: 2-node code for practicality in quantum simulation and hyperedge code for generality. Our 2-node code successfully achieves significant gain for f-resilient number compared to naive replication, proportional to both the number of node failures and the dimension product of sliced indices. Our hyperedge code can cover tensor networks out of the scope of quantum, with degraded gain in the exchange of its generality.
Tensor networks are the main building blocks in various computational sciences, ranging from many-body theory and quantum computing to probability and machine learning. Here we propose a parallel algorithm for the contraction of tensor networks using probabilistic graphical models. Given a tensor network, this algorithm splits the contraction task almost optimally into subtasks, which can be contracted independently in parallel. Our approach is based on the heuristic solution of the $\ensuremath{\mu}$-treewidth deletion problem in graph theory. We apply the resulting algorithm to the simulation of random quantum circuits and discuss the extensions for general tensor network contractions. According to our estimates, the lower bound on the simulation time of 1 million amplitudes of a 49-qubit circuit of depth 40 on the Summit supercomputer is >5>5>5 seconds.