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Cambridge University Press eBooks

Approximate tensor network contraction

Alexander M. Dalzell, Sam McArdle, Mario Berta, Przemyslaw Bienias, Chi-Fang Chen, András Gilyén, Connor T. Hann, Michael J. Kastoryano, Emil T. Khabiboulline, Aleksander Kubica, Grant Salton, Samson Wang, Fernando G. S. L. Brandão

Published Apr 24, 2025• 0 citations
Publisher

Abstract

This chapter covers the quantum algorithmic primitive of approximate tensor network contraction. Tensor networks are a powerful classical method for representing complex classical data as a network of individual tensor objects. To evaluate the tensor network, it must be contracted, which can be computationally challenging. A quantum algorithm for approximate tensor network contraction can provide a quantum speedup for contracting tensor networks that satisfy certain conditions.

DOI
10.1017/9781009639651.027

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This chapter covers the quantum algorithmic primitive of approximate tensor network contraction. Tensor networks are a powerful classical method for representing complex classical data as a network of individual tensor objects. To evaluate the tensor network, it must be contracted, which can be computationally challenging. A quantum algorithm for approximate tensor network contraction can provide a quantum speedup for contracting tensor networks that satisfy certain conditions.
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.
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.
Tensor network algorithms seek to minimize correlations to compress the classical data representing quantum states. Tensor network algorithms and similar tools—called tensor network methods—form the backbone of modern numerical methods used to simulate many-body physics and have a further range of applications in machine learning. Finding and contracting tensor network states is a computational task, which may be accelerated by quantum computing. We present a quantum algorithm that returns a classical description of a rank-r tensor network state satisfying an area law and approximating an eigenvector given black-box access to a unitary matrix. Our work creates a bridge between several contemporary approaches, including tensor networks, the variational quantum eigensolver (VQE), quantum approximate optimization algorithm (QAOA), and quantum computation.
Tensor network contractions are widely used in statistical physics, quantum computing, and computer science. We introduce a method to efficiently approximate tensor network contractions using low-rank approximations, where each intermediate tensor generated during the contractions is approximated as a low-rank binary tree tensor network. The proposed algorithm has the flexibility to incorporate a large portion of the environment when performing low-rank approximations, which can lead to high accuracy for a given rank. Here, the environment refers to the remaining set of tensors in the network, and low-rank approximations with larger environments can generally provide higher accuracy. For contracting tensor networks defined on lattices, the proposed algorithm can be viewed as a generalization of the standard boundary-based algorithms. In addition, the algorithm includes a cost-efficient density matrix algorithm for approximating a tensor network with a general graph structure into a tree structure, whose computational cost is asymptotically upper-bounded by that of the standard algorithm that uses canonicalization. Experimental results indicate that the proposed technique outperforms previously proposed approximate tensor network contraction algorithms for multiple problems in terms of both accuracy and efficiency.
Tensor network contraction is a fundamental operation for dealing with high-dimensional data, which is now widely used in classical simulations of quantum circuits. To improve the performance of simulations, existing methods are implemented by optimizing contraction methods. However, with the increasing size of the tensor network, the contraction efficiency of these methods is difficult to satisfy the performance requirements and also suffers from the challenge of contracting tensor tasks that exceed GPU memory. In this paper, we propose a high-performance chunking method for tensor contraction. By chunking the given tensor, we obtain small tensors or vectors assign the corresponding threads, and contract them in parallel through the GPU. Solving high-order tensor contractions using distributed tensor contraction methods. Compared to the Einsum, our method achieves an average speedup of 22.28×. We also have experiments for large-scale tensor network contractions and our method achieves an average speedup of 13.56× compared to Tensordot.