Phase Transitions and Noise Robustness of Quantum Graph States
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AbstractGraph states are entangled states that are essential for quantum information processing. As experimental advances enable the realization of large-scale graph states, efficient fidelity estimation methods are crucial for assessing their robustness against noise. However, calculations of exact fidelity become intractable for large systems due to the exponential growth in the number of stabilizers. In this work, we show that the fidelity between any ideal graph state and its noisy counterpart under IID Pauli noise can be mapped to the partition function of a classical spin system, enabling efficient computation via statistical mechanical techniques. Using this approach, we analyze the fidelity for regular graph states under depolarizing noise and uncover the emergence of phase transitions in fidelity between the pure-state regime and the noise-dominated regime. Specifically, in 2D, phase transitions occur only when the degree satisfies $d\ge 6$, while in 3D they already appear at $d\ge 5$. However, for graph states with excessively high degree, such as fully connected graphs, the phase transition disappears. Robustness of graph states against noise is thus determined by their connectivity and spatial dimensionality. Graph states with lower degree and/or dimensionality, which exhibit a smooth crossover, demonstrate greater robustness, while highly connected or higher-dimensional graph states are more fragile. Extreme connectivity, as the fully connected graph state possesses, restores robustness. Furthermore, we show that the fidelity can be rewritten in the form of the partition function of a constraint-percolation problem. Within this picture, we discuss the qualitative difference between 2D regular graph states with $d=6$ and $d=5$ regarding the presence or absence of a phase transition, as well as the suppressed critical behavior of fully connected graph states.Featured image: Two-dimensional regular graph states with degree $d=4$ (square lattice) and $d=6$ (square lattice with diagonal bonds) and their specific heat $C/(n k_B)$ as a function of noise probability $p$ for system sizes $n_x=n_y=10,30,60$. For $d=4$, the peak remains smooth, indicating no phase transition. In contrast, for $d=6$, the peak sharpens and grows with system size around $p\!\approx\!0.5$, signaling a phase transition.Popular summaryGraph states are special quantum states that play a central role in quantum information processing. As experiments realize larger and more complex graph states, an important question arises: how stable are these states in the presence of noise? A standard way to address this question is to compute the fidelity, which measures how close a noisy quantum state remains to the ideal one. However, computing fidelity exactly becomes extremely challenging as the system size grows. In this work, we show that this problem can be translated into a corresponding classical spin system. This allows the efficient numerical calculation of fidelity using well-established methods in statistical physics. Using this approach, we find that as noise increases, the system can abruptly lose its quantum properties, similar to a phase transition such as water freezing or boiling.► BibTeX data@article{Numajiri2026phasetransitions, doi = {10.22331/q-2026-05-05-2094}, url = {https://doi.org/10.22331/q-2026-05-05-2094}, title = {Phase {T}ransitions and {N}oise {R}obustness of {Q}uantum {G}raph {S}tates}, author = {Numajiri, Tatsuya and Yamashika, Shion and Tanizawa, Tomonori and Yoshii, Ryosuke and Takeuchi, Yuki and Tsuchiya, Shunji}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2094}, month = may, year = {2026} }► References [1] N. Shettell and D. Markham, Graph States as a Resource for Quantum Metrology, Phys. Rev. Lett. 124, 110502 (2020). https://doi.org/10.1103/PhysRevLett.124.110502 [2] K. Azuma, K. Tamaki, and H.-K. Lo, All-photonic quantum repeaters, Nat. Commun. 6, 6787 (2015). https://doi.org/10.1038/ncomms7787 [3] D. Schlingemann and R.F. Werner, Quantum error-correcting codes associated with graphs, Phys. Rev. A 65, 012308 (2001). https://doi.org/10.1103/PhysRevA.65.012308 [4] R. Raussendorf and H. J. Briegel, A One-Way Quantum Computer, Phys. Rev. Lett. 86, 5188 (2001). https://doi.org/10.1103/PhysRevLett.86.5188 [5] R. Raussendorf, D. E. Browne, H. J. Briegel, Measurement-based quantum computation on cluster states, Phys. Rev. A 68, 022312 (2003). https://doi.org/10.1103/PhysRevA.68.022312 [6] Y. Zhou and A. Hamma, Entanglement of random hypergraph states, Phys. Rev. A 106, 012410 (2022). https://doi.org/10.1103/PhysRevA.106.012410 [7] J. Chen, Y. Yan, and Y. Zhou, Magic of quantum hypergraph states, Quantum 8, 1351 (2024). https://doi.org/10.22331/q-2024-05-21-1351 [8] S. Spilla, R. Migliore, M. Scala, and A. Napoli, GHZ state generation of three Josephson qubits in the presence of bosonic baths, J. Phys. B: At. Mol. Opt. Phys. 45, 065501 (2012). https://doi.org/10.1088/0953-4075/45/6/065501 [9] K. L. Brown, C. Horsman, V. Kendon, and W. J. Munro, Layer-by-layer generation of cluster states, Phys. Rev. A 85, 052305 (2012). https://doi.org/10.1103/PhysRevA.85.052305 [10] C. Monroe, R. Raussendorf, A. Ruthven, K. R. Brown, P. Maunz, L.-M. Duan, and J. Kim, Large-scale modular quantum-computer architecture with atomic memory and photonic interconnects, Phys. Rev. A 89, 022317 (2014). https://doi.org/10.1103/PhysRevA.89.022317 [11] K. Inaba, Y. Tokunaga, K. Tamaki, K. Igeta, and M. Yamashita, High-Fidelity Cluster State Generation for Ultracold Atoms in an Optical Lattice, Phys. Rev. Lett. 112, 110501 (2014). https://doi.org/10.1103/PhysRevLett.112.110501 [12] M. Gimeno-Segovia, P. Shadbolt, D. E. Browne, and T. Rudolph, From Three-Photon Greenberger-Horne-Zeilinger States to Ballistic Universal Quantum Computation, Phys. Rev. Lett. 115, 020502 (2015). https://doi.org/10.1103/PhysRevLett.115.020502 [13] C.-Y. Lu, X.-Q. Zhou, O. Gühne, W.-B. Gao, J. Zhang, Z.-S. Yuan, A. Goebel, T. Yang, and J.-W. Pan, Experimental entanglement of six photons in graph states, Nat. Phys. 3, 91 (2007). https://doi.org/10.1038/nphys507 [14] Y. Tokunaga, S. Kuwashiro, T. Yamamoto, M. Koashi, and N. Imoto, Generation of High-Fidelity Four-Photon Cluster State and Quantum-Domain Demonstration of One-Way Quantum Computing, Phys. Rev. Lett. 100, 210501 (2008). https://doi.org/10.1103/PhysRevLett.100.210501 [15] X.-C. Yao, T.-X. Wang, P. Xu, H. Lu, G.-S. Pan, X.-H. Bao, C.-Z. Peng, C.-Y. Lu, Y.-A. Chen, and J.-W. Pan, Observation of eight-photon entanglement, Nat. Photonics 6, 225 (2012). https://doi.org/10.1038/nphoton.2011.354 [16] X.-C. Yao, T.-X. Wang, H.-Z. Chen, W.-B. Gao, A. G. Fowler, R. Raussendorf, Z.-B. Chen, N.-L. Liu, C.-Y. Lu, Y.-J. Deng, Y.-A. Chen, and J.-W. Pan, Experimental demonstration of topological error correction, Nature (London) 482, 489 (2012). https://doi.org/10.1038/nature10770 [17] X.-L. Wang, L.-K. Chen, W. Li, H.-L. Huang, C. Liu, C. Chen, Y.-H. Luo, Z.-E. Su, D. Wu, Z.-D. Li, H. Lu, Y. Hu, X. Jiang, C.-Z. Peng, L. Li, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, and J.-W. Pan, Experimental Ten-Photon Entanglement, Phys. Rev. Lett. 117,210502 (2016). https://doi.org/10.1103/PhysRevLett.117.210502 [18] X.-L. Wang, Y.-H. Luo, H.-L. Huang, M.-C. Chen, Z.-E. Su, C. Liu, C. Chen, W. Li, Y.-Q. Fang, X. Jiang, J. Zhang, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, 18-Qubit Entanglement with Six Photons' Three Degrees of Freedom, Phys. Rev. Lett. 120, 260502 (2018). https://doi.org/10.1103/PhysRevLett.120.260502 [19] Y. Wang, Y. Li, Z.-q. Yin, and B. Zeng, 16-qubit IBM universal quantum computer can be fully entangled, npj Quantum Information 4, 46 (2018). https://doi.org/10.1038/s41534-018-0095-x [20] M. Gong, M.-C. Chen, Y. Zheng, S. Wang, C. Zha, H. Deng, Z. Yan, H. Rong, Y. Wu, S. Li, F. Chen, Y. Zhao, F. Liang, J. Lin, Y. Xu, C. Guo, L. Sun, A. D. Castellano, H. Wang, C. Peng, C.-Y. Lu, X. Zhu, and J.-W. Pan, Genuine 12-Qubit Entanglement on a Superconducting Quantum Processor, Phys. Rev. Lett. 122, 110501 (2019). https://doi.org/10.1103/PhysRevLett.122.110501 [21] G. J. Mooney, C. D. Hill, and L. C. L. Hollenberg, Entanglement in a 20-Qubit Superconducting Quantum Computer, Sci. Rep. 9, 13465 (2019). https://doi.org/10.1038/s41598-019-49805-7 [22] C. Roh, G. Gwak, Y.-D. Yoon, and Y.-S. Ra, Generation of three-dimensional cluster entangled state, Nat. Photon. 19, 526 (2025). https://doi.org/10.1038/s41566-025-01631-2 [23] M. Hayashi and T. Morimae, Verifiable Measurement-Only Blind Quantum Computing with Stabilizer Testing, Phys. Rev. Lett. 115, 220502 (2015). https://doi.org/10.1103/PhysRevLett.115.220502 [24] T. Morimae, D. Nagaj, and N. Schuch, Quantum proofs can be verified using only single-qubit measurements, Phys. Rev. A 93, 022326 (2016). https://doi.org/10.1103/PhysRevA.93.022326 [25] K. Fujii and M. Hayashi, Verifiable fault tolerance in measurement-based quantum computation, Phys. Rev. A 96, 030301(R) (2017). https://doi.org/10.1103/PhysRevA.96.030301 [26] S. Pallister, N. Linden, and A. Montanaro, Optimal Verification of Entangled States with Local Measurements, Phys. Rev. Lett. 120, 170502 (2018). https://doi.org/10.1103/PhysRevLett.120.170502 [27] M. Hayashi and M. Hajdušek, Self-guaranteed measurement-based quantum computation, Phys. Rev. A 97, 052308 (2018). https://doi.org/10.1103/PhysRevA.97.052308 [28] Y. Takeuchi and T. Morimae, Verification of Many-Qubit States, Phys. Rev. X 8, 021060 (2018). https://doi.org/10.1103/PhysRevX.8.021060 [29] Y. Takeuchi, A. Mantri, T. Morimae, A. Mizutani, and J. F. Fitzsimons, Resource-efficient verification of quantum computing using Serfling's bound, npj Quantum Information 5, 27 (2019). https://doi.org/10.1038/s41534-019-0142-2 [30] H. Zhu and M. Hayashi, Efficient Verification of Pure Quantum States in the Adversarial Scenario, Phys. Rev. Lett. 123, 260504 (2019). https://doi.org/10.1103/PhysRevLett.123.260504 [31] H. Zhu and M. Hayashi, General framework for verifying pure quantum states in the adversarial scenario, Phys. Rev. A 100, 062335 (2019). https://doi.org/10.1103/PhysRevA.100.062335 [32] D. Markham and A. Krause, A Simple Protocol for Certifying Graph States and Applications in Quantum Networks, Cryptography 4, 3 (2020). https://doi.org/10.3390/cryptography4010003 [33] N. Dangniam, Y.-G. Han, and H. Zhu, Optimal verification of stabilizer states, Phys. Rev. Research 2, 043323 (2020). https://doi.org/10.1103/PhysRevResearch.2.043323 [34] Z. Li, H. Zhu, and M. 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The list may be incomplete as not all publishers provide suitable and complete citation data.Could not fetch Crossref cited-by data during last attempt 2026-05-05 11:13:02: Could not fetch cited-by data for 10.22331/q-2026-05-05-2094 from Crossref. This is normal if the DOI was registered recently.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. AbstractGraph states are entangled states that are essential for quantum information processing. As experimental advances enable the realization of large-scale graph states, efficient fidelity estimation methods are crucial for assessing their robustness against noise. However, calculations of exact fidelity become intractable for large systems due to the exponential growth in the number of stabilizers. In this work, we show that the fidelity between any ideal graph state and its noisy counterpart under IID Pauli noise can be mapped to the partition function of a classical spin system, enabling efficient computation via statistical mechanical techniques. Using this approach, we analyze the fidelity for regular graph states under depolarizing noise and uncover the emergence of phase transitions in fidelity between the pure-state regime and the noise-dominated regime. Specifically, in 2D, phase transitions occur only when the degree satisfies $d\ge 6$, while in 3D they already appear at $d\ge 5$. However, for graph states with excessively high degree, such as fully connected graphs, the phase transition disappears. Robustness of graph states against noise is thus determined by their connectivity and spatial dimensionality. Graph states with lower degree and/or dimensionality, which exhibit a smooth crossover, demonstrate greater robustness, while highly connected or higher-dimensional graph states are more fragile. Extreme connectivity, as the fully connected graph state possesses, restores robustness. Furthermore, we show that the fidelity can be rewritten in the form of the partition function of a constraint-percolation problem. Within this picture, we discuss the qualitative difference between 2D regular graph states with $d=6$ and $d=5$ regarding the presence or absence of a phase transition, as well as the suppressed critical behavior of fully connected graph states.Featured image: Two-dimensional regular graph states with degree $d=4$ (square lattice) and $d=6$ (square lattice with diagonal bonds) and their specific heat $C/(n k_B)$ as a function of noise probability $p$ for system sizes $n_x=n_y=10,30,60$. For $d=4$, the peak remains smooth, indicating no phase transition. In contrast, for $d=6$, the peak sharpens and grows with system size around $p\!\approx\!0.5$, signaling a phase transition.Popular summaryGraph states are special quantum states that play a central role in quantum information processing. As experiments realize larger and more complex graph states, an important question arises: how stable are these states in the presence of noise? A standard way to address this question is to compute the fidelity, which measures how close a noisy quantum state remains to the ideal one. However, computing fidelity exactly becomes extremely challenging as the system size grows. In this work, we show that this problem can be translated into a corresponding classical spin system. This allows the efficient numerical calculation of fidelity using well-established methods in statistical physics. Using this approach, we find that as noise increases, the system can abruptly lose its quantum properties, similar to a phase transition such as water freezing or boiling.► BibTeX data@article{Numajiri2026phasetransitions, doi = {10.22331/q-2026-05-05-2094}, url = {https://doi.org/10.22331/q-2026-05-05-2094}, title = {Phase {T}ransitions and {N}oise {R}obustness of {Q}uantum {G}raph {S}tates}, author = {Numajiri, Tatsuya and Yamashika, Shion and Tanizawa, Tomonori and Yoshii, Ryosuke and Takeuchi, Yuki and Tsuchiya, Shunji}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2094}, month = may, year = {2026} }► References [1] N. Shettell and D. Markham, Graph States as a Resource for Quantum Metrology, Phys. Rev. Lett. 124, 110502 (2020). https://doi.org/10.1103/PhysRevLett.124.110502 [2] K. Azuma, K. Tamaki, and H.-K. Lo, All-photonic quantum repeaters, Nat. Commun. 6, 6787 (2015). https://doi.org/10.1038/ncomms7787 [3] D. Schlingemann and R.F. Werner, Quantum error-correcting codes associated with graphs, Phys. Rev. A 65, 012308 (2001). https://doi.org/10.1103/PhysRevA.65.012308 [4] R. Raussendorf and H. J. Briegel, A One-Way Quantum Computer, Phys. Rev. Lett. 86, 5188 (2001). https://doi.org/10.1103/PhysRevLett.86.5188 [5] R. Raussendorf, D. E. Browne, H. J. Briegel, Measurement-based quantum computation on cluster states, Phys. Rev. A 68, 022312 (2003). https://doi.org/10.1103/PhysRevA.68.022312 [6] Y. Zhou and A. Hamma, Entanglement of random hypergraph states, Phys. Rev. A 106, 012410 (2022). https://doi.org/10.1103/PhysRevA.106.012410 [7] J. Chen, Y. Yan, and Y. Zhou, Magic of quantum hypergraph states, Quantum 8, 1351 (2024). https://doi.org/10.22331/q-2024-05-21-1351 [8] S. Spilla, R. Migliore, M. Scala, and A. Napoli, GHZ state generation of three Josephson qubits in the presence of bosonic baths, J. Phys. B: At. Mol. Opt. Phys. 45, 065501 (2012). https://doi.org/10.1088/0953-4075/45/6/065501 [9] K. L. Brown, C. Horsman, V. Kendon, and W. J. Munro, Layer-by-layer generation of cluster states, Phys. Rev. A 85, 052305 (2012). https://doi.org/10.1103/PhysRevA.85.052305 [10] C. Monroe, R. Raussendorf, A. Ruthven, K. R. Brown, P. Maunz, L.-M. Duan, and J. Kim, Large-scale modular quantum-computer architecture with atomic memory and photonic interconnects, Phys. Rev. A 89, 022317 (2014). https://doi.org/10.1103/PhysRevA.89.022317 [11] K. Inaba, Y. Tokunaga, K. Tamaki, K. Igeta, and M. Yamashita, High-Fidelity Cluster State Generation for Ultracold Atoms in an Optical Lattice, Phys. Rev. Lett. 112, 110501 (2014). https://doi.org/10.1103/PhysRevLett.112.110501 [12] M. Gimeno-Segovia, P. Shadbolt, D. E. Browne, and T. Rudolph, From Three-Photon Greenberger-Horne-Zeilinger States to Ballistic Universal Quantum Computation, Phys. Rev. Lett. 115, 020502 (2015). https://doi.org/10.1103/PhysRevLett.115.020502 [13] C.-Y. Lu, X.-Q. Zhou, O. Gühne, W.-B. Gao, J. Zhang, Z.-S. Yuan, A. Goebel, T. Yang, and J.-W. Pan, Experimental entanglement of six photons in graph states, Nat. Phys. 3, 91 (2007). https://doi.org/10.1038/nphys507 [14] Y. Tokunaga, S. Kuwashiro, T. Yamamoto, M. Koashi, and N. Imoto, Generation of High-Fidelity Four-Photon Cluster State and Quantum-Domain Demonstration of One-Way Quantum Computing, Phys. Rev. Lett. 100, 210501 (2008). https://doi.org/10.1103/PhysRevLett.100.210501 [15] X.-C. Yao, T.-X. Wang, P. Xu, H. Lu, G.-S. Pan, X.-H. Bao, C.-Z. Peng, C.-Y. Lu, Y.-A. Chen, and J.-W. Pan, Observation of eight-photon entanglement, Nat. Photonics 6, 225 (2012). https://doi.org/10.1038/nphoton.2011.354 [16] X.-C. Yao, T.-X. Wang, H.-Z. Chen, W.-B. Gao, A. G. Fowler, R. Raussendorf, Z.-B. Chen, N.-L. Liu, C.-Y. Lu, Y.-J. Deng, Y.-A. Chen, and J.-W. Pan, Experimental demonstration of topological error correction, Nature (London) 482, 489 (2012). https://doi.org/10.1038/nature10770 [17] X.-L. Wang, L.-K. Chen, W. Li, H.-L. Huang, C. Liu, C. Chen, Y.-H. Luo, Z.-E. Su, D. Wu, Z.-D. Li, H. Lu, Y. Hu, X. Jiang, C.-Z. Peng, L. Li, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, and J.-W. Pan, Experimental Ten-Photon Entanglement, Phys. Rev. Lett. 117,210502 (2016). https://doi.org/10.1103/PhysRevLett.117.210502 [18] X.-L. Wang, Y.-H. Luo, H.-L. Huang, M.-C. Chen, Z.-E. Su, C. Liu, C. Chen, W. Li, Y.-Q. Fang, X. Jiang, J. Zhang, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, 18-Qubit Entanglement with Six Photons' Three Degrees of Freedom, Phys. Rev. Lett. 120, 260502 (2018). https://doi.org/10.1103/PhysRevLett.120.260502 [19] Y. Wang, Y. Li, Z.-q. Yin, and B. Zeng, 16-qubit IBM universal quantum computer can be fully entangled, npj Quantum Information 4, 46 (2018). https://doi.org/10.1038/s41534-018-0095-x [20] M. Gong, M.-C. Chen, Y. Zheng, S. Wang, C. Zha, H. Deng, Z. Yan, H. Rong, Y. Wu, S. Li, F. Chen, Y. Zhao, F. Liang, J. Lin, Y. Xu, C. Guo, L. Sun, A. D. Castellano, H. Wang, C. Peng, C.-Y. Lu, X. Zhu, and J.-W. Pan, Genuine 12-Qubit Entanglement on a Superconducting Quantum Processor, Phys. Rev. Lett. 122, 110501 (2019). https://doi.org/10.1103/PhysRevLett.122.110501 [21] G. J. Mooney, C. D. Hill, and L. C. L. Hollenberg, Entanglement in a 20-Qubit Superconducting Quantum Computer, Sci. Rep. 9, 13465 (2019). https://doi.org/10.1038/s41598-019-49805-7 [22] C. Roh, G. Gwak, Y.-D. Yoon, and Y.-S. Ra, Generation of three-dimensional cluster entangled state, Nat. Photon. 19, 526 (2025). https://doi.org/10.1038/s41566-025-01631-2 [23] M. Hayashi and T. Morimae, Verifiable Measurement-Only Blind Quantum Computing with Stabilizer Testing, Phys. Rev. Lett. 115, 220502 (2015). https://doi.org/10.1103/PhysRevLett.115.220502 [24] T. Morimae, D. Nagaj, and N. Schuch, Quantum proofs can be verified using only single-qubit measurements, Phys. Rev. A 93, 022326 (2016). https://doi.org/10.1103/PhysRevA.93.022326 [25] K. Fujii and M. Hayashi, Verifiable fault tolerance in measurement-based quantum computation, Phys. Rev. A 96, 030301(R) (2017). https://doi.org/10.1103/PhysRevA.96.030301 [26] S. Pallister, N. Linden, and A. Montanaro, Optimal Verification of Entangled States with Local Measurements, Phys. Rev. Lett. 120, 170502 (2018). https://doi.org/10.1103/PhysRevLett.120.170502 [27] M. Hayashi and M. Hajdušek, Self-guaranteed measurement-based quantum computation, Phys. Rev. A 97, 052308 (2018). https://doi.org/10.1103/PhysRevA.97.052308 [28] Y. Takeuchi and T. Morimae, Verification of Many-Qubit States, Phys. Rev. X 8, 021060 (2018). https://doi.org/10.1103/PhysRevX.8.021060 [29] Y. Takeuchi, A. Mantri, T. Morimae, A. Mizutani, and J. F. 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