Hyper-optimized Quantum Lego Contraction Schedules
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AbstractCalculating the quantum weight enumerator polynomial (WEP) is a valuable tool for characterizing quantum error-correcting (QEC) codes, but it is computationally hard for large or complex codes. The Quantum LEGO (QL) framework provides a tensor network approach for WEP calculation, in some cases offering superpolynomial speedups over brute-force methods, provided the code exhibits area law entanglement, that a good QL layout is used, and an efficient tensor network contraction schedule is found. We analyze the performance of a hyper-optimized contraction schedule framework across QL layouts for diverse stabilizer code families. We find that the intermediate tensors in the QL networks for stabilizer WEPs are often highly sparse, invalidating the dense-tensor assumption of standard cost functions. To address this, we introduce an exact, polynomial-time Sparse Stabilizer Tensor (SST) cost function based on the rank of the parity check matrices for intermediate tensors. The SST cost function correlates perfectly with the true contraction cost, providing a significant advantage over the default cost function, which exhibits large uncertainty. Optimizing contraction schedules using the SST cost function yields substantial performance gains, achieving up to orders of magnitude improvement in actual contraction cost compared to using the dense tensor cost function. Furthermore, the precise cost estimation from the SST function offers an efficient metric to decide whether the QL-based WEP calculation is computationally superior to brute force for a given QL layout. These results, enabled by PlanqTN, a new open-source QL implementation, validate hyper-optimized contraction as a crucial technique for leveraging the QL framework to explore the QEC code design space.Popular summaryQuantum error correction is one of the central tools for making large-scale quantum computers reliable. A major challenge, however, is that the space of possible error-correcting codes is enormous. To search this space effectively, researchers need tools that can build codes from smaller pieces, visualize their structure, and compute useful diagnostics without relying on brute-force methods that quickly become infeasible. In this work, we study how to accelerate calculations in the quantum LEGO framework, a tensor-network language for constructing and analyzing quantum error-correcting codes. In this picture, small stabilizer-code tensors are connected together like building blocks to form larger codes. One important diagnostic is the weight enumerator polynomial, which captures information about the distribution of stabilizers and can be used to infer code properties such as distance. While direct calculation scales exponentially in general, tensor-network contraction can be much faster when the code has a favorable structure and when a good contraction order is found. We combine quantum LEGO with Cotengra, a hyper-optimization package for finding efficient tensor-network contraction schedules. A key observation is that the tensors arising in stabilizer-code weight-enumerator calculations are often highly sparse. Standard tensor-network cost functions assume dense tensors, which can misestimate the true computational cost. We introduce a sparse stabilizer tensor cost function that uses stabilizer parity-check matrix ranks to exactly predict the number of polynomial multiplications needed during contraction. This gives more reliable contraction schedules and, in many examples, substantially reduces the cost of computing weight enumerators compared with dense-tensor optimization. The paper tests this approach across several code families, including concatenated repetition codes, holographic codes, rotated surface codes, Hamming codes, and bivariate bicycle codes. The calculations and figures are enabled by PlanqTN, an open-source Python library and interactive web studio for creating, manipulating, and analyzing tensor-network-based quantum error-correcting codes. PlanqTN implements the quantum LEGO framework, connects it with Cotengra-based contraction scheduling, and takes a unified approach to quantum LEGO, ZX calculus, and graph-state-style transformations. Through the web interface at https://planqtn.com, users can build tensor networks from smaller code tensors, transform layouts using LEGO and ZX-style moves, compute stabilizer parity-check matrices and weight enumerators, export constructions as Python code, and share or save tensor-network layouts. Overall, this work shows that combining stabilizer-specific structure, tensor-network optimization, and interactive software can make quantum LEGO more practical as a design and analysis tool for quantum error correction. It provides both an algorithmic improvement for weight-enumerator calculations and a software pathway for exploring new code constructions.► BibTeX data@article{Pato2026hyperoptimized, doi = {10.22331/q-2026-05-05-2092}, url = {https://doi.org/10.22331/q-2026-05-05-2092}, title = {Hyper-optimized {Q}uantum {L}ego {C}ontraction {S}chedules}, author = {Pato, Balint and Vanlerberghe, June and Brown, Kenneth R.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2092}, month = may, year = {2026} }► References [1] Daniel Gottesman. ``Stabilizer Codes and Quantum Error Correction'' (1997). arXiv:quant-ph/9705052. arXiv:quant-ph/9705052 [2] Andrew Cross and Drew Vandeth. ``Small Binary Stabilizer Subsystem Codes'' (2025). arXiv:2501.17447. arXiv:2501.17447 [3] H. Bombin and M. A. Martin-Delgado. ``Optimal Resources for Topological 2D Stabilizer Codes: Comparative Study''. Phys. Rev. A 76, 012305 (2007). arXiv:quant-ph/0703272. https://doi.org/10.1103/PhysRevA.76.012305 arXiv:quant-ph/0703272 [4] H. Bombin and M. A. Martin-Delgado. ``Topological Quantum Distillation''. Phys. Rev. Lett. 97, 180501 (2006). https://doi.org/10.1103/PhysRevLett.97.180501 [5] Jean-Pierre Tillich and Gilles Zemor. ``Quantum LDPC codes with positive rate and minimum distance proportional to n$\frac{1}{2}$''. In 2009 IEEE International Symposium on Information Theory. Pages 799–803. (2009). https://doi.org/10.1109/ISIT.2009.5205648 [6] Pavel Panteleev and Gleb Kalachev. ``Degenerate Quantum LDPC Codes With Good Finite Length Performance''. Quantum 5, 585 (2021). https://doi.org/10.22331/q-2021-11-22-585 [7] Ke Wang, Zhide Lu, Chuanyu Zhang, Gongyu Liu, Jiachen Chen, Yanzhe Wang, Yaozu Wu, Shibo Xu, Xuhao Zhu, Feitong Jin, Yu Gao, Ziqi Tan, Zhengyi Cui, Ning Wang, Yiren Zou, Aosai Zhang, Tingting Li, Fanhao Shen, Jiarun Zhong, Zehang Bao, Zitian Zhu, Yihang Han, Yiyang He, Jiayuan Shen, Han Wang, Jia-Nan Yang, Zixuan Song, Jinfeng Deng, Hang Dong, Zheng-Zhi Sun, Weikang Li, Qi Ye, Si Jiang, Yixuan Ma, Pei-Xin Shen, Pengfei Zhang, Hekang Li, Qiujiang Guo, Zhen Wang, Chao Song, H. Wang, and Dong-Ling Deng. ``Demonstration of low-overhead quantum error correction codes'' (2025) arXiv:2505.09684. https://doi.org/10.1038/s41567-025-03157-4 arXiv:2505.09684 [8] Sergey Bravyi, Andrew W. Cross, Jay M. Gambetta, Dmitri Maslov, Patrick Rall, and Theodore J. Yoder. ``High-threshold and low-overhead fault-tolerant quantum memory''. Nature 627, 778–782 (2024). https://doi.org/10.1038/s41586-024-07107-7 [9] Hayata Yamasaki and Masato Koashi. ``Time-Efficient Constant-Space-Overhead Fault-Tolerant Quantum Computation''. Nature Physics 20, 247–253 (2024). https://doi.org/10.1038/s41567-023-02325-8 [10] Fernando Pastawski, Beni Yoshida, Daniel Harlow, and John Preskill. ``Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence''. J. High Energ. Phys. 2015, 149 (2015). https://doi.org/10.1007/JHEP06(2015)149 [11] ChunJun Cao and Brad Lackey. ``Approximate Bacon-Shor code and holography''. J. High Energ. Phys. 2021, 127 (2021). https://doi.org/10.1007/JHEP05(2021)127 [12] Matthew Steinberg, Junyu Fan, Jens Eisert, Sebastian Feld, Alexander Jahn, and Chunjun Cao. ``Universal fault-tolerant logic with heterogeneous holographic codes'' (2025) arXiv:2504.10386. arXiv:2504.10386 [13] ChunJun Cao and Brad Lackey. ``Quantum Lego: Building Quantum Error Correction Codes from Tensor Networks''. PRX Quantum 3, 020332 (2022). https://doi.org/10.1103/PRXQuantum.3.020332 [14] Vincent Paul Su, ChunJun Cao, Hong-Ye Hu, Yariv Yanay, Charles Tahan, and Brian Swingle. ``Discovery of Optimal Quantum Error Correcting Codes via Reinforcement Learning'' (2023). arXiv:2305.06378. https://doi.org/10.1103/PhysRevApplied.23.034048 arXiv:2305.06378 [15] Peter Shor and Raymond Laflamme. ``Quantum Analog of the MacWilliams Identities for Classical Coding Theory''. Phys. Rev. Lett. 78, 1600–1602 (1997). https://doi.org/10.1103/PhysRevLett.78.1600 [16] ChunJun Cao and Brad Lackey. ``Quantum Weight Enumerators and Tensor Networks''. IEEE Trans. Inf. Theor. 70, 3512–3528 (2024). https://doi.org/10.1109/TIT.2023.3340503 [17] ChunJun Cao, Michael J. Gullans, Brad Lackey, and Zitao Wang. ``Quantum Lego Expansion Pack: Enumerators from Tensor Networks''. PRX Quantum 5, 030313 (2024). https://doi.org/10.1103/PRXQuantum.5.030313 [18] Carsten Damm, Markus Holzer, and Pierre McKenzie. ``The complexity of tensor calculus''. computational complexity 11, 54–89 (2002). https://doi.org/10.1007/s00037-000-0170-4 [19] L. G. Valiant. ``The complexity of computing the permanent''.
Theoretical Computer Science 8, 189–201 (1979). https://doi.org/10.1016/0304-3975(79)90044-6 [20] Johnnie Gray and Stefanos Kourtis. ``Hyper-optimized tensor network contraction''. Quantum 5, 410 (2021). https://doi.org/10.22331/q-2021-03-15-410 [21] Johnnie Gray. ``Cotengra: Hyper optimized contraction trees for large tensor networks and einsums''. GitHub (2025). url: https://github.com/jcmgray/cotengra. https://github.com/jcmgray/cotengra [22] Balint Pato, June Vanlerberghe, ChunJun Cao, Brad Lackey, and Kenneth Brown. ``PlanqTN, a Python library and interactive web app implementing the quantum LEGO framework''. Zenodo (2025). https://doi.org/10.5281/zenodo.16761072 [23] Leonid P. Pryadko, Vadim A. Shabashov, and Valerii K. Kozin. ``QDistRnd: A GAP package for computing the distance of quantum error-correcting codes''. J.
Open Source Softw. 7, 4120 (2022). https://doi.org/10.21105/joss.04120 [24] Y.-Y. Shi. ``Classical simulation of quantum many-body systems with a tree tensor network''. Phys. Rev. A 74 (2006). https://doi.org/10.1103/PhysRevA.74.022320 [25] Leslie G. Valiant. ``Holographic Algorithms''. SIAM Journal on Computing 37, 1565–1594 (2008). https://doi.org/10.1137/070682575 [26] Kazem Mahdavi and Deborah Koslover, editors. ``Advances in Quantum Computation''. Volume 482 of Contemporary Mathematics.
American Mathematical Society. Providence, Rhode Island (2009). https://doi.org/10.1090/conm/482 [27] S J Denny, J D Biamonte, D Jaksch, and S R Clark. ``Algebraically contractible topological tensor network states''. Journal of Physics A: Mathematical and Theoretical 45, 015309 (2011). https://doi.org/10.1088/1751-8113/45/1/015309 [28] Bryan O'Gorman. ``Parameterization of Tensor Network Contraction''. In Wim van Dam and Laura Mančinska, editors, 14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019). Volume 135 of Leibniz International Proceedings in Informatics (LIPIcs), pages 10:1–10:19. Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019). https://doi.org/10.4230/LIPIcs.TQC.2019.10 [29] Sebastian Schlag. ``High-quality hypergraph partitioning''. PhD thesis. Karlsruhe Institute of Technology, Germany. (2020). [30] Sebastian Schlag, Tobias Heuer, Lars Gottesbüren, Yaroslav Akhremtsev, Christian Schulz, and Peter Sanders. ``High-quality hypergraph partitioning''. ACM J. Exp. Algorithmics (2022). https://doi.org/10.1145/3529090 [31] P.W. Shor. ``Fault-tolerant quantum computation''. In Proceedings of 37th Conference on Foundations of Computer Science. Pages 56–65. (1996). https://doi.org/10.1109/SFCS.1996.548464 [32] Shaden Smith, Jongsoo Park, and George Karypis. ``Sparse Tensor Factorization on Many-Core Processors with High-Bandwidth Memory''. In 2017 IEEE International Parallel and Distributed Processing Symposium (IPDPS). Pages 1058–1067. (2017). https://doi.org/10.1109/IPDPS.2017.84 [33] Andrew Steane. ``Simple Quantum Error Correcting Codes''. Phys. Rev. A 54, 4741–4751 (1996). arXiv:quant-ph/9605021. https://doi.org/10.1103/PhysRevA.54.4741 arXiv:quant-ph/9605021 [34] Min Ye and Nicolas Delfosse. ``Quantum error correction for long chains of trapped ions'' (2025) arXiv:2503.22071. https://doi.org/10.22331/q-2025-11-27-1920 arXiv:2503.22071 [35] Balint Pato, June Vanlerberghe, and Kenneth Brown. ``Hyperoptimized quantum lego contraction schedules supplementary material''. Zenodo (2025). https://doi.org/10.5281/zenodo.17290495 [36] Wuxu Zhao, Menglong Fang, and Daiqin Su. ``A graph-based approach to entanglement entropy of quantum error correcting codes'' (2025). arXiv:2501.06407. arXiv:2501.06407 [37] Sergey Bravyi, Martin Suchara, and Alexander Vargo. ``Efficient algorithms for maximum likelihood decoding in the surface code''. Phys. Rev. A 90, 032326 (2014). https://doi.org/10.1103/PhysRevA.90.032326 [38] Terry Farrelly, Nicholas Milicevic, Robert J. Harris, Nathan A. McMahon, and Thomas M. Stace. ``Parallel decoding of multiple logical qubits in tensor-network codes''. Phys. Rev. A 105, 052446 (2022). https://doi.org/10.1103/PhysRevA.105.052446Cited byCould not fetch Crossref cited-by data during last attempt 2026-05-05 10:04:40: Could not fetch cited-by data for 10.22331/q-2026-05-05-2092 from Crossref. This is normal if the DOI was registered recently. On SAO/NASA ADS no data on citing works was found (last attempt 2026-05-05 10:04:41).This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractCalculating the quantum weight enumerator polynomial (WEP) is a valuable tool for characterizing quantum error-correcting (QEC) codes, but it is computationally hard for large or complex codes. The Quantum LEGO (QL) framework provides a tensor network approach for WEP calculation, in some cases offering superpolynomial speedups over brute-force methods, provided the code exhibits area law entanglement, that a good QL layout is used, and an efficient tensor network contraction schedule is found. We analyze the performance of a hyper-optimized contraction schedule framework across QL layouts for diverse stabilizer code families. We find that the intermediate tensors in the QL networks for stabilizer WEPs are often highly sparse, invalidating the dense-tensor assumption of standard cost functions. To address this, we introduce an exact, polynomial-time Sparse Stabilizer Tensor (SST) cost function based on the rank of the parity check matrices for intermediate tensors. The SST cost function correlates perfectly with the true contraction cost, providing a significant advantage over the default cost function, which exhibits large uncertainty. Optimizing contraction schedules using the SST cost function yields substantial performance gains, achieving up to orders of magnitude improvement in actual contraction cost compared to using the dense tensor cost function. Furthermore, the precise cost estimation from the SST function offers an efficient metric to decide whether the QL-based WEP calculation is computationally superior to brute force for a given QL layout. These results, enabled by PlanqTN, a new open-source QL implementation, validate hyper-optimized contraction as a crucial technique for leveraging the QL framework to explore the QEC code design space.Popular summaryQuantum error correction is one of the central tools for making large-scale quantum computers reliable. A major challenge, however, is that the space of possible error-correcting codes is enormous. To search this space effectively, researchers need tools that can build codes from smaller pieces, visualize their structure, and compute useful diagnostics without relying on brute-force methods that quickly become infeasible. In this work, we study how to accelerate calculations in the quantum LEGO framework, a tensor-network language for constructing and analyzing quantum error-correcting codes. In this picture, small stabilizer-code tensors are connected together like building blocks to form larger codes. One important diagnostic is the weight enumerator polynomial, which captures information about the distribution of stabilizers and can be used to infer code properties such as distance. While direct calculation scales exponentially in general, tensor-network contraction can be much faster when the code has a favorable structure and when a good contraction order is found. We combine quantum LEGO with Cotengra, a hyper-optimization package for finding efficient tensor-network contraction schedules. A key observation is that the tensors arising in stabilizer-code weight-enumerator calculations are often highly sparse. Standard tensor-network cost functions assume dense tensors, which can misestimate the true computational cost. We introduce a sparse stabilizer tensor cost function that uses stabilizer parity-check matrix ranks to exactly predict the number of polynomial multiplications needed during contraction. This gives more reliable contraction schedules and, in many examples, substantially reduces the cost of computing weight enumerators compared with dense-tensor optimization. The paper tests this approach across several code families, including concatenated repetition codes, holographic codes, rotated surface codes, Hamming codes, and bivariate bicycle codes. The calculations and figures are enabled by PlanqTN, an open-source Python library and interactive web studio for creating, manipulating, and analyzing tensor-network-based quantum error-correcting codes. PlanqTN implements the quantum LEGO framework, connects it with Cotengra-based contraction scheduling, and takes a unified approach to quantum LEGO, ZX calculus, and graph-state-style transformations. Through the web interface at https://planqtn.com, users can build tensor networks from smaller code tensors, transform layouts using LEGO and ZX-style moves, compute stabilizer parity-check matrices and weight enumerators, export constructions as Python code, and share or save tensor-network layouts. Overall, this work shows that combining stabilizer-specific structure, tensor-network optimization, and interactive software can make quantum LEGO more practical as a design and analysis tool for quantum error correction. It provides both an algorithmic improvement for weight-enumerator calculations and a software pathway for exploring new code constructions.► BibTeX data@article{Pato2026hyperoptimized, doi = {10.22331/q-2026-05-05-2092}, url = {https://doi.org/10.22331/q-2026-05-05-2092}, title = {Hyper-optimized {Q}uantum {L}ego {C}ontraction {S}chedules}, author = {Pato, Balint and Vanlerberghe, June and Brown, Kenneth R.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2092}, month = may, year = {2026} }► References [1] Daniel Gottesman. ``Stabilizer Codes and Quantum Error Correction'' (1997). arXiv:quant-ph/9705052. arXiv:quant-ph/9705052 [2] Andrew Cross and Drew Vandeth. ``Small Binary Stabilizer Subsystem Codes'' (2025). arXiv:2501.17447. arXiv:2501.17447 [3] H. Bombin and M. A. Martin-Delgado. ``Optimal Resources for Topological 2D Stabilizer Codes: Comparative Study''. Phys. Rev. A 76, 012305 (2007). arXiv:quant-ph/0703272. https://doi.org/10.1103/PhysRevA.76.012305 arXiv:quant-ph/0703272 [4] H. Bombin and M. A. Martin-Delgado. ``Topological Quantum Distillation''. Phys. Rev. Lett. 97, 180501 (2006). https://doi.org/10.1103/PhysRevLett.97.180501 [5] Jean-Pierre Tillich and Gilles Zemor. ``Quantum LDPC codes with positive rate and minimum distance proportional to n$\frac{1}{2}$''. In 2009 IEEE International Symposium on Information Theory. Pages 799–803. (2009). https://doi.org/10.1109/ISIT.2009.5205648 [6] Pavel Panteleev and Gleb Kalachev. ``Degenerate Quantum LDPC Codes With Good Finite Length Performance''. Quantum 5, 585 (2021). https://doi.org/10.22331/q-2021-11-22-585 [7] Ke Wang, Zhide Lu, Chuanyu Zhang, Gongyu Liu, Jiachen Chen, Yanzhe Wang, Yaozu Wu, Shibo Xu, Xuhao Zhu, Feitong Jin, Yu Gao, Ziqi Tan, Zhengyi Cui, Ning Wang, Yiren Zou, Aosai Zhang, Tingting Li, Fanhao Shen, Jiarun Zhong, Zehang Bao, Zitian Zhu, Yihang Han, Yiyang He, Jiayuan Shen, Han Wang, Jia-Nan Yang, Zixuan Song, Jinfeng Deng, Hang Dong, Zheng-Zhi Sun, Weikang Li, Qi Ye, Si Jiang, Yixuan Ma, Pei-Xin Shen, Pengfei Zhang, Hekang Li, Qiujiang Guo, Zhen Wang, Chao Song, H. Wang, and Dong-Ling Deng. ``Demonstration of low-overhead quantum error correction codes'' (2025) arXiv:2505.09684. https://doi.org/10.1038/s41567-025-03157-4 arXiv:2505.09684 [8] Sergey Bravyi, Andrew W. Cross, Jay M. Gambetta, Dmitri Maslov, Patrick Rall, and Theodore J. Yoder. ``High-threshold and low-overhead fault-tolerant quantum memory''. Nature 627, 778–782 (2024). https://doi.org/10.1038/s41586-024-07107-7 [9] Hayata Yamasaki and Masato Koashi. ``Time-Efficient Constant-Space-Overhead Fault-Tolerant Quantum Computation''. Nature Physics 20, 247–253 (2024). https://doi.org/10.1038/s41567-023-02325-8 [10] Fernando Pastawski, Beni Yoshida, Daniel Harlow, and John Preskill. ``Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence''. J. High Energ. Phys. 2015, 149 (2015). https://doi.org/10.1007/JHEP06(2015)149 [11] ChunJun Cao and Brad Lackey. ``Approximate Bacon-Shor code and holography''. J. High Energ. Phys. 2021, 127 (2021). https://doi.org/10.1007/JHEP05(2021)127 [12] Matthew Steinberg, Junyu Fan, Jens Eisert, Sebastian Feld, Alexander Jahn, and Chunjun Cao. ``Universal fault-tolerant logic with heterogeneous holographic codes'' (2025) arXiv:2504.10386. arXiv:2504.10386 [13] ChunJun Cao and Brad Lackey. ``Quantum Lego: Building Quantum Error Correction Codes from Tensor Networks''. PRX Quantum 3, 020332 (2022). https://doi.org/10.1103/PRXQuantum.3.020332 [14] Vincent Paul Su, ChunJun Cao, Hong-Ye Hu, Yariv Yanay, Charles Tahan, and Brian Swingle. ``Discovery of Optimal Quantum Error Correcting Codes via Reinforcement Learning'' (2023). arXiv:2305.06378. https://doi.org/10.1103/PhysRevApplied.23.034048 arXiv:2305.06378 [15] Peter Shor and Raymond Laflamme. ``Quantum Analog of the MacWilliams Identities for Classical Coding Theory''. Phys. Rev. Lett. 78, 1600–1602 (1997). https://doi.org/10.1103/PhysRevLett.78.1600 [16] ChunJun Cao and Brad Lackey. ``Quantum Weight Enumerators and Tensor Networks''. IEEE Trans. Inf. Theor. 70, 3512–3528 (2024). https://doi.org/10.1109/TIT.2023.3340503 [17] ChunJun Cao, Michael J. Gullans, Brad Lackey, and Zitao Wang. ``Quantum Lego Expansion Pack: Enumerators from Tensor Networks''. PRX Quantum 5, 030313 (2024). https://doi.org/10.1103/PRXQuantum.5.030313 [18] Carsten Damm, Markus Holzer, and Pierre McKenzie. ``The complexity of tensor calculus''. computational complexity 11, 54–89 (2002). https://doi.org/10.1007/s00037-000-0170-4 [19] L. G. Valiant. ``The complexity of computing the permanent''.
Theoretical Computer Science 8, 189–201 (1979). https://doi.org/10.1016/0304-3975(79)90044-6 [20] Johnnie Gray and Stefanos Kourtis. ``Hyper-optimized tensor network contraction''. Quantum 5, 410 (2021). https://doi.org/10.22331/q-2021-03-15-410 [21] Johnnie Gray. ``Cotengra: Hyper optimized contraction trees for large tensor networks and einsums''. GitHub (2025). url: https://github.com/jcmgray/cotengra. https://github.com/jcmgray/cotengra [22] Balint Pato, June Vanlerberghe, ChunJun Cao, Brad Lackey, and Kenneth Brown. ``PlanqTN, a Python library and interactive web app implementing the quantum LEGO framework''. Zenodo (2025). https://doi.org/10.5281/zenodo.16761072 [23] Leonid P. Pryadko, Vadim A. Shabashov, and Valerii K. Kozin. ``QDistRnd: A GAP package for computing the distance of quantum error-correcting codes''. J.
Open Source Softw. 7, 4120 (2022). https://doi.org/10.21105/joss.04120 [24] Y.-Y. Shi. ``Classical simulation of quantum many-body systems with a tree tensor network''. Phys. Rev. A 74 (2006). https://doi.org/10.1103/PhysRevA.74.022320 [25] Leslie G. Valiant. ``Holographic Algorithms''. SIAM Journal on Computing 37, 1565–1594 (2008). https://doi.org/10.1137/070682575 [26] Kazem Mahdavi and Deborah Koslover, editors. ``Advances in Quantum Computation''. Volume 482 of Contemporary Mathematics.
American Mathematical Society. Providence, Rhode Island (2009). https://doi.org/10.1090/conm/482 [27] S J Denny, J D Biamonte, D Jaksch, and S R Clark. ``Algebraically contractible topological tensor network states''. Journal of Physics A: Mathematical and Theoretical 45, 015309 (2011). https://doi.org/10.1088/1751-8113/45/1/015309 [28] Bryan O'Gorman. ``Parameterization of Tensor Network Contraction''. In Wim van Dam and Laura Mančinska, editors, 14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019). Volume 135 of Leibniz International Proceedings in Informatics (LIPIcs), pages 10:1–10:19. Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019). https://doi.org/10.4230/LIPIcs.TQC.2019.10 [29] Sebastian Schlag. ``High-quality hypergraph partitioning''. PhD thesis. Karlsruhe Institute of Technology, Germany. (2020). [30] Sebastian Schlag, Tobias Heuer, Lars Gottesbüren, Yaroslav Akhremtsev, Christian Schulz, and Peter Sanders. ``High-quality hypergraph partitioning''. ACM J. Exp. Algorithmics (2022). https://doi.org/10.1145/3529090 [31] P.W. Shor. ``Fault-tolerant quantum computation''. In Proceedings of 37th Conference on Foundations of Computer Science. Pages 56–65. (1996). https://doi.org/10.1109/SFCS.1996.548464 [32] Shaden Smith, Jongsoo Park, and George Karypis. ``Sparse Tensor Factorization on Many-Core Processors with High-Bandwidth Memory''. In 2017 IEEE International Parallel and Distributed Processing Symposium (IPDPS). Pages 1058–1067. (2017). https://doi.org/10.1109/IPDPS.2017.84 [33] Andrew Steane. ``Simple Quantum Error Correcting Codes''. Phys. Rev. A 54, 4741–4751 (1996). arXiv:quant-ph/9605021. https://doi.org/10.1103/PhysRevA.54.4741 arXiv:quant-ph/9605021 [34] Min Ye and Nicolas Delfosse. ``Quantum error correction for long chains of trapped ions'' (2025) arXiv:2503.22071. https://doi.org/10.22331/q-2025-11-27-1920 arXiv:2503.22071 [35] Balint Pato, June Vanlerberghe, and Kenneth Brown. ``Hyperoptimized quantum lego contraction schedules supplementary material''. Zenodo (2025). https://doi.org/10.5281/zenodo.17290495 [36] Wuxu Zhao, Menglong Fang, and Daiqin Su. ``A graph-based approach to entanglement entropy of quantum error correcting codes'' (2025). arXiv:2501.06407. arXiv:2501.06407 [37] Sergey Bravyi, Martin Suchara, and Alexander Vargo. ``Efficient algorithms for maximum likelihood decoding in the surface code''. Phys. Rev. A 90, 032326 (2014). https://doi.org/10.1103/PhysRevA.90.032326 [38] Terry Farrelly, Nicholas Milicevic, Robert J. Harris, Nathan A. McMahon, and Thomas M. Stace. ``Parallel decoding of multiple logical qubits in tensor-network codes''. Phys. Rev. A 105, 052446 (2022). https://doi.org/10.1103/PhysRevA.105.052446Cited byCould not fetch Crossref cited-by data during last attempt 2026-05-05 10:04:40: Could not fetch cited-by data for 10.22331/q-2026-05-05-2092 from Crossref. This is normal if the DOI was registered recently. On SAO/NASA ADS no data on citing works was found (last attempt 2026-05-05 10:04:41).This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.