Approximate Contraction of Arbitrary Tensor Networks with a Flexible and Efficient Density Matrix Algorithm
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...
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| Format: | Article |
| Language: | English |
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Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften
2024-12-01
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| Series: | Quantum |
| Online Access: | https://quantum-journal.org/papers/q-2024-12-27-1580/pdf/ |
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| _version_ | 1850058449206902784 |
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| author | Linjian Ma Matthew Fishman Edwin Miles Stoudenmire Edgar Solomonik |
| author_facet | Linjian Ma Matthew Fishman Edwin Miles Stoudenmire Edgar Solomonik |
| author_sort | Linjian Ma |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-ce07d3401ce34bdc82033dcea600e008 |
| institution | DOAJ |
| issn | 2521-327X |
| language | English |
| publishDate | 2024-12-01 |
| publisher | Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften |
| record_format | Article |
| series | Quantum |
| spelling | doaj-art-ce07d3401ce34bdc82033dcea600e0082025-08-20T02:51:10ZengVerein zur Förderung des Open Access Publizierens in den QuantenwissenschaftenQuantum2521-327X2024-12-018158010.22331/q-2024-12-27-158010.22331/q-2024-12-27-1580Approximate Contraction of Arbitrary Tensor Networks with a Flexible and Efficient Density Matrix AlgorithmLinjian MaMatthew FishmanEdwin Miles StoudenmireEdgar SolomonikTensor 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.https://quantum-journal.org/papers/q-2024-12-27-1580/pdf/ |
| spellingShingle | Linjian Ma Matthew Fishman Edwin Miles Stoudenmire Edgar Solomonik Approximate Contraction of Arbitrary Tensor Networks with a Flexible and Efficient Density Matrix Algorithm Quantum |
| title | Approximate Contraction of Arbitrary Tensor Networks with a Flexible and Efficient Density Matrix Algorithm |
| title_full | Approximate Contraction of Arbitrary Tensor Networks with a Flexible and Efficient Density Matrix Algorithm |
| title_fullStr | Approximate Contraction of Arbitrary Tensor Networks with a Flexible and Efficient Density Matrix Algorithm |
| title_full_unstemmed | Approximate Contraction of Arbitrary Tensor Networks with a Flexible and Efficient Density Matrix Algorithm |
| title_short | Approximate Contraction of Arbitrary Tensor Networks with a Flexible and Efficient Density Matrix Algorithm |
| title_sort | approximate contraction of arbitrary tensor networks with a flexible and efficient density matrix algorithm |
| url | https://quantum-journal.org/papers/q-2024-12-27-1580/pdf/ |
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