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A 3D lattice defect and efficient computations in topological MBQC

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A 3D lattice defect and efficient computations in topological MBQC

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AbstractWe describe an efficient, fully fault-tolerant implementation of Measurement-Based Quantum Computation (MBQC) in the 3D cluster state. The two key novelties are (i) the introduction of a lattice defect in the underlying cluster state and (ii) the use of the Rudolph-Grover rebit encoding. Concretely, (i) allows for a topological implementation of the Hadamard gate, while (ii) does the same for the phase gate. Furthermore, we develop general ideas towards circuit compaction and algorithmic circuit verification, which we implement for the Reed-Muller code used for magic state distillation. Our performance analysis highlights the overall improvements provided by the new methods. Popular summaryMeasurement-Based Quantum Computation (MBQC) is a protocol in which quantum computation is carried out through local measurements on entangled resource states. In this work, we focus on MBQC using the two-dimensional Toric code with holes, which provides a fault-tolerant way to store logical qubits. Computation is achieved by locally entangling multiple Toric code layers and performing measurements that teleport and transform quantum information from an input code to an output code. Specifically, the pre-measurement state is a three-dimensional cluster state (known in this context as the RHG state), and the extra dimension can be interpreted as simulated time evolution. We formalize this framework and rigorously show how measurements of individual physical qubit implement logical computations. This approach admits a graphical representation in which measurement patterns define hole lines and correlation surfaces carry stabilizers that commute with the measurements. We prove a theorem linking these correlation surfaces to the resulting logical transformation between input and output Toric codes. Since the accessible set of operations is limited, we introduce a structural defect in the otherwise regular 3D lattice supporting the RHG state that enables additional, fault-tolerant logical gates. These include the Hadamard and the so-called S gate. We analyze the resource overhead required to implement these operations and show a gain of one full order of magnitude. We also propose graphical simplifications to further reduce their cost. Finally, we present an algorithmic verifier that determines whether a given measurement pattern realizes a desired logical circuit.► BibTeX data@article{Tournaire2026dlatticedefect, doi = {10.22331/q-2026-02-06-1997}, url = {https://doi.org/10.22331/q-2026-02-06-1997}, title = {A 3{D} lattice defect and efficient computations in topological {MBQC}}, author = {Tournaire, Gabrielle and Schwiering, Marvin and Raussendorf, Robert and Bachmann, Sven}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {1997}, month = feb, year = {2026} }► References [1] J. M. Raimond and S. Haroche. ``Quantum Computing: Dream Or Nightmare''. In 31st Rencontres de Moriond: Dark Matter and Cosmology, Quantum Measurements and Experimental Gravitation. Pages 341–346. Ed. Frontieres (1996). url: https://doi.org/10.1063/1.881512. https://doi.org/10.1063/1.881512 [2] A. R. Calderbank and Peter W. Shor. ``Good quantum error-correcting codes exist''. Physical Review A 54, 1098–1105 (1996). https://doi.org/10.1103/PhysRevA.54.1098 [3] A. M. Steane. ``Simple quantum error-correcting codes''. Physical Review A 54, 4741–4751 (1996). https://doi.org/10.1103/PhysRevA.54.4741 [4] D. Gottesman. ``Stabilizer codes and quantum error correction''. PhD thesis. California Institute of Technology. (1997). url: https://doi.org/10.48550/arXiv.quant-ph/9705052. https://doi.org/10.48550/arXiv.quant-ph/9705052 arXiv:quant-ph/9705052 [5] D. Aharonov and M. Ben-Or. ``Fault-tolerant quantum computation with constant error''. In Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing. Page 176–188. STOC '97. Association for Computing Machinery (1997). https://doi.org/10.1145/258533.258579 [6] P. Aliferis, D. Gottesman, and J. Preskill. ``Quantum accuracy threshold for concatenated distance-3 codes''. Quantum Information and Computation 6, 97–165 (2005). https://doi.org/10.26421/QIC6.2-1 [7] D. Aharonov, A. Kitaev, and J. Preskill. ``Fault-tolerant quantum computation with long-range correlated noise''.

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Hostetter. ``Galois: A performant NumPy extension for Galois fields''. https://github.com/mhostetter/galois (2020). https://github.com/mhostetter/galoisCited byCould not fetch Crossref cited-by data during last attempt 2026-02-06 13:17:22: Could not fetch cited-by data for 10.22331/q-2026-02-06-1997 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-02-06 13:17:28: No response from ADS or unable to decode the received json data when getting the list of citing works.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. AbstractWe describe an efficient, fully fault-tolerant implementation of Measurement-Based Quantum Computation (MBQC) in the 3D cluster state. The two key novelties are (i) the introduction of a lattice defect in the underlying cluster state and (ii) the use of the Rudolph-Grover rebit encoding. Concretely, (i) allows for a topological implementation of the Hadamard gate, while (ii) does the same for the phase gate. Furthermore, we develop general ideas towards circuit compaction and algorithmic circuit verification, which we implement for the Reed-Muller code used for magic state distillation. Our performance analysis highlights the overall improvements provided by the new methods. Popular summaryMeasurement-Based Quantum Computation (MBQC) is a protocol in which quantum computation is carried out through local measurements on entangled resource states. In this work, we focus on MBQC using the two-dimensional Toric code with holes, which provides a fault-tolerant way to store logical qubits. Computation is achieved by locally entangling multiple Toric code layers and performing measurements that teleport and transform quantum information from an input code to an output code. Specifically, the pre-measurement state is a three-dimensional cluster state (known in this context as the RHG state), and the extra dimension can be interpreted as simulated time evolution. We formalize this framework and rigorously show how measurements of individual physical qubit implement logical computations. This approach admits a graphical representation in which measurement patterns define hole lines and correlation surfaces carry stabilizers that commute with the measurements. We prove a theorem linking these correlation surfaces to the resulting logical transformation between input and output Toric codes. Since the accessible set of operations is limited, we introduce a structural defect in the otherwise regular 3D lattice supporting the RHG state that enables additional, fault-tolerant logical gates. These include the Hadamard and the so-called S gate. We analyze the resource overhead required to implement these operations and show a gain of one full order of magnitude. We also propose graphical simplifications to further reduce their cost. Finally, we present an algorithmic verifier that determines whether a given measurement pattern realizes a desired logical circuit.► BibTeX data@article{Tournaire2026dlatticedefect, doi = {10.22331/q-2026-02-06-1997}, url = {https://doi.org/10.22331/q-2026-02-06-1997}, title = {A 3{D} lattice defect and efficient computations in topological {MBQC}}, author = {Tournaire, Gabrielle and Schwiering, Marvin and Raussendorf, Robert and Bachmann, Sven}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {1997}, month = feb, year = {2026} }► References [1] J. M. Raimond and S. Haroche. ``Quantum Computing: Dream Or Nightmare''. In 31st Rencontres de Moriond: Dark Matter and Cosmology, Quantum Measurements and Experimental Gravitation. Pages 341–346. Ed. Frontieres (1996). url: https://doi.org/10.1063/1.881512. https://doi.org/10.1063/1.881512 [2] A. R. Calderbank and Peter W. Shor. ``Good quantum error-correcting codes exist''. Physical Review A 54, 1098–1105 (1996). https://doi.org/10.1103/PhysRevA.54.1098 [3] A. M. Steane. ``Simple quantum error-correcting codes''. Physical Review A 54, 4741–4751 (1996). https://doi.org/10.1103/PhysRevA.54.4741 [4] D. Gottesman. ``Stabilizer codes and quantum error correction''. PhD thesis. California Institute of Technology. (1997). url: https://doi.org/10.48550/arXiv.quant-ph/9705052. https://doi.org/10.48550/arXiv.quant-ph/9705052 arXiv:quant-ph/9705052 [5] D. Aharonov and M. Ben-Or. ``Fault-tolerant quantum computation with constant error''. In Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing. Page 176–188. STOC '97. Association for Computing Machinery (1997). https://doi.org/10.1145/258533.258579 [6] P. Aliferis, D. Gottesman, and J. Preskill. ``Quantum accuracy threshold for concatenated distance-3 codes''. Quantum Information and Computation 6, 97–165 (2005). https://doi.org/10.26421/QIC6.2-1 [7] D. Aharonov, A. Kitaev, and J. Preskill. ``Fault-tolerant quantum computation with long-range correlated noise''.

In Luca Aceto, Ivan Damgård, Leslie Ann Goldberg, Magnús M. Halldórsson, Anna Ingólfsdóttir, and Igor Walukiewicz, editors, Automata, Languages and Programming. Pages 298–310.

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A 3D lattice defect and efficient computations in topological MBQC

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