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arXiv:2010.14962 · Physical Review A

Simulation of quantum computing on classical supercomputers with tensor-network edge cutting

Ya-Qian Zhao, Ren-Gang Li, Jin-Zhe Jiang, Chen Li, Hong-Zhen Li, En-Dong Wang, Wei-Feng Gong, Xin Zhang, Zhi-Qiang Wei

Published Sep 2, 2021• 14 citations
arXivHTMLPDF

Abstract

Simulation of quantum computing on supercomputers is a significant research topic, which plays a vital role in quantum algorithm verification, error-tolerant verification, and other applications. Tensor-network contraction based on density matrix is an important single-amplitude simulation strategy, but it is hard to execute on the distributed computing systems. In this paper, we studied the problem in detail, and propose a scheme based on cutting edges of undirected graphs. This scheme cuts edges of undirected graphs with large tree width to obtain many undirected subgraphs with small tree width, and these subgraphs contracted on different computing cores. The contraction results of slave cores are summarized in the master node, which is consistent with the original tensor-network contraction. Thus, we can simulate the larger scale quantum circuit than single core. Moreover, it is an NP-hard problem to find the global optimum cutting edges, and we propose a search strategy based on a heuristic algorithm to approach it. In order to verify the effectiveness of our scheme, we conduct tests based on the quantum approximation optimization algorithm (QAOA) and Shor's algorithm, and it can simulate a 120-qubit three-regular QAOA algorithm on a 4096-core supercomputer, which greatly exceeds the simulation scale on a single core of 100 qubits.

DOI
10.1103/physreva.104.032603

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Simulation of quantum computing on supercomputers is a significant research topic, which plays a vital role in quantum algorithm verification, error-tolerant verification, and other applications. Tensor-network contraction based on density matrix is an important single-amplitude simulation strategy, but it is hard to execute on the distributed computing systems. In this paper, we studied the problem in detail, and propose a scheme based on cutting edges of undirected graphs. This scheme cuts edges of undirected graphs with large tree width to obtain many undirected subgraphs with small tree width, and these subgraphs contracted on different computing cores. The contraction results of slave cores are summarized in the master node, which is consistent with the original tensor-network contraction. Thus, we can simulate the larger scale quantum circuit than single core. Moreover, it is an NP-hard problem to find the global optimum cutting edges, and we propose a search strategy based on a heuristic algorithm to approach it. In order to verify the effectiveness of our scheme, we conduct tests based on the quantum approximation optimization algorithm (QAOA) and Shor's algorithm, and it can simulate a 120-qubit three-regular QAOA algorithm on a 4096-core supercomputer, which greatly exceeds the simulation scale on a single core of 100 qubits.
Classical simulation of quantum computation is necessary for studying the numerical behavior of quantum algorithms, as there does not yet exist a large viable quantum computer on which to perform numerical tests. Tensor network (TN) contraction is an algorithmic method that can efficiently simulate some quantum circuits, often greatly reducing the computational cost over methods that simulate the full Hilbert space. In this study we implement a tensor network contraction program for simulating quantum circuits using multi-core compute nodes. We show simulation results for the Max-Cut problem on 3- through 7-regular graphs using the quantum approximate optimization algorithm (QAOA), successfully simulating up to 100 qubits. We test two different methods for generating the ordering of tensor index contractions: one is based on the tree decomposition of the line graph, while the other generates ordering using a straight-forward stochastic scheme. Through studying instances of QAOA circuits, we show the expected result that as the treewidth of the quantum circuit's line graph decreases, TN contraction becomes significantly more efficient than simulating the whole Hilbert space. The results in this work suggest that tensor contraction methods are superior only when simulating Max-Cut/QAOA with graphs of regularities approximately five and below. Insight into this point of equal computational cost helps one determine which simulation method will be more efficient for a given quantum circuit. The stochastic contraction method outperforms the line graph based method only when the time to calculate a reasonable tree decomposition is prohibitively expensive. Finally, we release our software package, qTorch (Quantum TensOR Contraction Handler), intended for general quantum circuit simulation. For a nontrivial subset of these quantum circuits, 50 to 100 qubits can easily be simulated on a single compute node.
One of the key problems in tensor network based quantum circuit simulation is the construction of a contraction tree which minimizes the cost of the simulation, where the cost can be expressed in the number of operations as a proxy for the simulation running time. This same problem arises in a variety of application areas, such as combinatorial scientific computing, marginalization in probabilistic graphical models, and solving constraint satisfaction problems. In this paper, we reduce the computationally hard portion of this problem to one of graph linear ordering, and demonstrate how existing approaches in this area can be utilized to achieve results up to several orders of magnitude better than existing state of the art methods for the same running time. To do so, we introduce a novel polynomial time algorithm for constructing an optimal contraction tree from a given order. Furthermore, we introduce a fast and high quality linear ordering solver, and demonstrate its applicability as a heuristic for providing orderings for contraction trees. Finally, we compare our solver with competing methods for constructing contraction trees in quantum circuit simulation on a collection of randomly generated Quantum Approximate Optimization Algorithm Max Cut circuits and show that our method achieves superior results on a majority of tested quantum circuits. Reproducibility: Our source code and data are available at https://github.com/cameton/HPEC2022_ContractionTrees.
In this work, we present a new large-scale quantum circuit simulator. It is based on the tensor network contraction technique to represent quantum circuits. We propose a novel parallelization algorithm based on step-dependent slicing . In this paper, we push the requirement on the size of a quantum computer that will be needed to demonstrate the advantage of quantum computation with Quantum Approximate Optimization Algorithm (QAOA). We computed a single amplitude of QAOA ansatz state on 210 qubits. The simulation involved 1,785 gates on 1,024 nodes of the Cray XC 40 supercomputer Theta. To the best of our knowledge, this constitutes the largest simulation of QAOA ansatz simulations reported to this date.
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.
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.
The study of quantum circuit simulation using classical computers is a key research topic that helps define the boundary of verifiable quantum advantage, solve quantum many-body problems, and inform development of quantum hardware and software. Tensor networks have become forefront mathematical tools for these tasks. Here we introduce a method to approximately simulate quantum circuits using sparsely-populated tensors. We describe a sparse tensor data structure that can represent quantum states with no underlying symmetry, and outline algorithms to efficiently contract and truncate these tensors. We show that the data structure and contraction algorithm are efficient, leading to expected runtime scalings versus qubit number and circuit depth. Our results motivate future research in optimization of sparse tensor networks for quantum simulation.