Lattice Surgery with a Twist: Simplifying Clifford Gates of Surface Codes

Daniel Litinski and Felix von Oppen

Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany

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We present a planar surface-code-based scheme for fault-tolerant quantum computation which eliminates the time overhead of single-qubit Clifford gates, and implements long-range multi-target CNOT gates with a time overhead that scales only logarithmically with the control-target separation. This is done by replacing hardware operations for single-qubit Clifford gates with a classical tracking protocol. Inter-qubit communication is added via a modified lattice surgery protocol that employs twist defects of the surface code. The long-range multi-target CNOT gates facilitate magic state distillation, which renders our scheme fault-tolerant and universal.

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[1] J. Preskill, Reliable quantum computers, Proc. Roy. Soc. Lond. A 454, 385 (1998).

[2] A. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. 303, 2 (2003).

[3] B. M. Terhal, Quantum error correction for quantum memories, Rev. Mod. Phys. 87, 307 (2015).

[4] M. H. Devoret and R. J. Schoelkopf, Superconducting circuits for quantum information: An outlook, Science 339, 1169 (2013).

[5] D. Loss and D. P. DiVincenzo, Quantum computation with quantum dots, Phys. Rev. A 57, 120 (1998).

[6] R. M. Lutchyn, E. P. A. M. Bakkers, L. P. Kouwenhoven, P. Krogstrup, C. M. Marcus, and Y. Oreg, Realizing Majorana zero modes in superconductor-semiconductor heterostructures, arXiv:1707.04899 (2017).

[7] D. Gottesman, Stabilizer codes and quantum error correction, Ph.D. thesis, California Institute of Technology (1997).

[8] S. B. Bravyi and A. Y. Kitaev, Quantum codes on a lattice with boundary, arXiv:quant-ph/​9811052 (1998).

[9] E. T. Campbell, B. M. Terhal, and C. Vuillot, Roads towards fault-tolerant universal quantum computation, Nature 549, 172 (2017).

[10] D. S. Wang, A. G. Fowler, A. M. Stephens, and L. C. L. Hollenberg, Threshold error rates for the toric and planar codes, Quantum Info. Comput. 10, 456 (2010).

[11] R. S. Andrist, H. G. Katzgraber, H. Bombin, and M. A. Martin-Delgado, Error tolerance of topological codes with independent bit-flip and measurement errors, Phys. Rev. A 94, 012318 (2016).

[12] H. Bombin and M. A. Martin-Delgado, Topological quantum distillation, Phys. Rev. Lett. 97, 180501 (2006).

[13] A. J. Landahl, J. T. Anderson, and P. R. Rice, Fault-tolerant quantum computing with color codes, arXiv:1108.5738 (2011).

[14] A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Surface codes: Towards practical large-scale quantum computation, Phys. Rev. A 86, 032324 (2012).

[15] H. Bombin, Topological order with a twist: Ising anyons from an abelian model, Phys. Rev. Lett. 105, 030403 (2010).

[16] B. J. Brown, K. Laubscher, M. S. Kesselring, and J. R. Wootton, Poking holes and cutting corners to achieve Clifford gates with the surface code, Phys. Rev. X 7, 021029 (2017).

[17] M. B. Hastings and A. Geller, Reduced space-time and time costs using dislocation codes and arbitrary ancillas, Quantum Info. Comput. 15, 962 (2015).

[18] D. Gottesman, The Heisenberg representation of quantum computers, Proc. XXII Int. Coll. Group. Th. Meth. Phys. 1, 32 (1999).

[19] C. Horsman, A. G. Fowler, S. Devitt, and R. V. Meter, Surface code quantum computing by lattice surgery, New J. Phys. 14, 123011 (2012).

[20] S. Bravyi and A. Kitaev, Universal quantum computation with ideal Clifford gates and noisy ancillas, Phys. Rev. A 71, 022316 (2005).

[21] T. Karzig, C. Knapp, R. M. Lutchyn, P. Bonderson, M. B. Hastings, C. Nayak, J. Alicea, K. Flensberg, S. Plugge, Y. Oreg, C. M. Marcus, and M. H. Freedman, Scalable designs for quasiparticle-poisoning-protected topological quantum computation with Majorana zero modes, Phys. Rev. B 95, 235305 (2017).

[22] D. Litinski, M. S. Kesselring, J. Eisert, and F. von Oppen, Combining topological hardware and topological software: Color-code quantum computing with topological superconductor networks, Phys. Rev. X 7, 031048 (2017).

[23] D. Litinski and F. von Oppen, Braiding by Majorana tracking and long-range CNOT gates with color codes, Phys. Rev. B 96, 205413 (2017).

[24] G. Duclos-Cianci and D. Poulin, Fast decoders for topological quantum codes, Phys. Rev. Lett. 104, 050504 (2010).

[25] E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, Journal of Mathematical Physics 43, 4452 (2002).

[26] T. J. Yoder and I. H. Kim, The surface code with a twist, Quantum 1, 2 (2017).

[27] Y. Tomita and K. M. Svore, Low-distance surface codes under realistic quantum noise, Phys. Rev. A 90, 062320 (2014).

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[1] J. Pablo Bonilla Ataides, David K. Tuckett, Stephen D. Bartlett, Steven T. Flammia, and Benjamin J. Brown, "The XZZX surface code", Nature Communications 12 1, 2172 (2021).

[2] Noah Shutty and Christopher Chamberland, "Decoding Merged Color-Surface Codes and Finding Fault-Tolerant Clifford Circuits Using Solvers for Satisfiability Modulo Theories", Physical Review Applied 18 1, 014072 (2022).

[3] Markus S. Kesselring, Fernando Pastawski, Jens Eisert, and Benjamin J. Brown, "The boundaries and twist defects of the color code and their applications to topological quantum computation", Quantum 2, 101 (2018).

[4] Christophe Vuillot, Lingling Lao, Ben Criger, Carmen García Almudéver, Koen Bertels, and Barbara M Terhal, "Code deformation and lattice surgery are gauge fixing", New Journal of Physics 21 3, 033028 (2019).

[5] Ali Lavasani and Maissam Barkeshli, "Low overhead Clifford gates from joint measurements in surface, color, and hyperbolic codes", Physical Review A 98 5, 052319 (2018).

[6] B. David Clader, Alexander M. Dalzell, Nikitas Stamatopoulos, Grant Salton, Mario Berta, and William J. Zeng, "Quantum Resources Required to Block-Encode a Matrix of Classical Data", IEEE Transactions on Quantum Engineering 3, 1 (2022).

[7] Daniel Litinski, "A Game of Surface Codes: Large-Scale Quantum Computing with Lattice Surgery", Quantum 3, 128 (2019).

[8] Christopher Chamberland, Luis Goncalves, Prasahnt Sivarajah, Eric Peterson, and Sebastian Grimberg, "Techniques for combining fast local decoders with global decoders under circuit-level noise", Quantum Science and Technology 8 4, 045011 (2023).

[9] Modjtaba Shokrian Zini, Alain Delgado, Roberto dos Reis, Pablo Antonio Moreno Casares, Jonathan E. Mueller, Arne-Christian Voigt, and Juan Miguel Arrazola, "Quantum simulation of battery materials using ionic pseudopotentials", Quantum 7, 1049 (2023).

[10] Christopher Chamberland, Kyungjoo Noh, Patricio Arrangoiz-Arriola, Earl T. Campbell, Connor T. Hann, Joseph Iverson, Harald Putterman, Thomas C. Bohdanowicz, Steven T. Flammia, Andrew Keller, Gil Refael, John Preskill, Liang Jiang, Amir H. Safavi-Naeini, Oskar Painter, and Fernando G.S.L. Brandão, "Building a Fault-Tolerant Quantum Computer Using Concatenated Cat Codes", PRX Quantum 3 1, 010329 (2022).

[11] Daniel Herr, Alexandru Paler, Simon J Devitt, and Franco Nori, "Lattice surgery on the Raussendorf lattice", Quantum Science and Technology 3 3, 035011 (2018).

[12] Oscar Higgott, Thomas C. Bohdanowicz, Aleksander Kubica, Steven T. Flammia, and Earl T. Campbell, "Improved Decoding of Circuit Noise and Fragile Boundaries of Tailored Surface Codes", Physical Review X 13 3, 031007 (2023).

[13] Adam Holmes, Yongshan Ding, Ali Javadi-Abhari, Diana Franklin, Margaret Martonosi, and Frederic T. Chong, "Resource optimized quantum architectures for surface code implementations of magic-state distillation", Microprocessors and Microsystems 67, 56 (2019).

[14] Dong-Xiao Quan, Xiao-Jie Lü, and Wen-Fei Zhang, "Structure design and logical CNOT implementation of multi-logical-qubits surface code", Acta Physica Sinica 73 4, 040304 (2024).

[15] Yongshan Ding, Xin-Chuan Wu, Adam Holmes, Ash Wiseth, Diana Franklin, Margaret Martonosi, and Frederic T. Chong, 2020 ACM/IEEE 47th Annual International Symposium on Computer Architecture (ISCA) 570 (2020) ISBN:978-1-7281-4661-4.

[16] Michael Vasmer and Dan E. Browne, "Three-dimensional surface codes: Transversal gates and fault-tolerant architectures", Physical Review A 100 1, 012312 (2019).

[17] Dongxiao Quan, Chensong Liu, Xiaojie Lv, and Changxing Pei, "Implementation of Fault-Tolerant Encoding Circuit Based on Stabilizer Implementation and “Flag” Bits in Steane Code", Entropy 24 8, 1107 (2022).

[18] M. Gutiérrez, M. Müller, and A. Bermúdez, "Transversality and lattice surgery: Exploring realistic routes toward coupled logical qubits with trapped-ion quantum processors", Physical Review A 99 2, 022330 (2019).

[19] Hiroto Mukai, Keiichi Sakata, Simon J Devitt, Rui Wang, Yu Zhou, Yukito Nakajima, and Jaw-Shen Tsai, "Pseudo-2D superconducting quantum computing circuit for the surface code: proposal and preliminary tests", New Journal of Physics 22 4, 043013 (2020).

[20] Alexander M. Dalzell, B. David Clader, Grant Salton, Mario Berta, Cedric Yen-Yu Lin, David A. Bader, Nikitas Stamatopoulos, Martin J. A. Schuetz, Fernando G. S. L. Brandão, Helmut G. Katzgraber, and William J. Zeng, "End-To-End Resource Analysis for Quantum Interior-Point Methods and Portfolio Optimization", PRX Quantum 4 4, 040325 (2023).

[21] Arne L. Grimsmo and Shruti Puri, "Quantum Error Correction with the Gottesman-Kitaev-Preskill Code", PRX Quantum 2 2, 020101 (2021).

[22] Seok-Hyung Lee and Hyunseok Jeong, "Universal hardware-efficient topological measurement-based quantum computation via color-code-based cluster states", Physical Review Research 4 1, 013010 (2022).

[23] Mark Webber, Vincent Elfving, Sebastian Weidt, and Winfried K. Hensinger, "The impact of hardware specifications on reaching quantum advantage in the fault tolerant regime", AVS Quantum Science 4 1, 013801 (2022).

[24] Christopher Chamberland and Earl T. Campbell, "Circuit-level protocol and analysis for twist-based lattice surgery", Physical Review Research 4 2, 023090 (2022).

[25] X. Fu, L. Lao, K. Bertels, and C.G. Almudever, "A control microarchitecture for fault-tolerant quantum computing", Microprocessors and Microsystems 70, 21 (2019).

[26] Dongmoon Min, Junpyo Kim, Junhyuk Choi, Ilkwon Byun, Masamitsu Tanaka, Koji Inoue, and Jangwoo Kim, Proceedings of the 50th Annual International Symposium on Computer Architecture 1 (2023) ISBN:9798400700958.

[27] Soo-Cheol Oh and Gyu-Il Cha, "Logical qubit behavior model and fast simulation for surface code", Quantum Information Processing 22 7, 287 (2023).

[28] Filipa C. R. Peres, "Pauli-based model of quantum computation with higher-dimensional systems", Physical Review A 108 3, 032606 (2023).

[29] Principles of Superconducting Quantum Computers 327 (2022) ISBN:9781119750727.

[30] Yuval Oreg and Felix von Oppen, "Majorana Zero Modes in Networks of Cooper-Pair Boxes: Topologically Ordered States and Topological Quantum Computation", Annual Review of Condensed Matter Physics 11 1, 397 (2020).

[31] Daniel Litinski, "Magic State Distillation: Not as Costly as You Think", Quantum 3, 205 (2019).

[32] J. Eli Bourassa, Rafael N. Alexander, Michael Vasmer, Ashlesha Patil, Ilan Tzitrin, Takaya Matsuura, Daiqin Su, Ben Q. Baragiola, Saikat Guha, Guillaume Dauphinais, Krishna K. Sabapathy, Nicolas C. Menicucci, and Ish Dhand, "Blueprint for a Scalable Photonic Fault-Tolerant Quantum Computer", Quantum 5, 392 (2021).

[33] Thomas Häner and Mathias Soeken, "Lowering the T-depth of Quantum Circuits via Logic Network Optimization", ACM Transactions on Quantum Computing 3 2, 1 (2022).

[34] Tyler Leblond, Ryan S. Bennink, Justin G. Lietz, and Christopher M. Seck, Proceedings of the SC '23 Workshops of The International Conference on High Performance Computing, Network, Storage, and Analysis 1426 (2023) ISBN:9798400707858.

[35] Christopher Chamberland and Earl T. Campbell, "Universal Quantum Computing with Twist-Free and Temporally Encoded Lattice Surgery", PRX Quantum 3 1, 010331 (2022).

[36] Alexandre Blais, Steven M. Girvin, and William D. Oliver, "Quantum information processing and quantum optics with circuit quantum electrodynamics", Nature Physics 16 3, 247 (2020).

[37] Ryan Sweke, Markus S Kesselring, Evert P L van Nieuwenburg, and Jens Eisert, "Reinforcement learning decoders for fault-tolerant quantum computation", Machine Learning: Science and Technology 2 2, 025005 (2021).

[38] Élie Gouzien, Diego Ruiz, Francois-Marie Le Régent, Jérémie Guillaud, and Nicolas Sangouard, "Performance Analysis of a Repetition Cat Code Architecture: Computing 256-bit Elliptic Curve Logarithm in 9 Hours with 126 133 Cat Qubits", Physical Review Letters 131 4, 040602 (2023).

[39] Laura Ortiz Martín, Springer Theses 93 (2019) ISBN:978-3-030-23648-9.

[40] Austin G. Fowler and Craig Gidney, "Low overhead quantum computation using lattice surgery", arXiv:1808.06709, (2018).

[41] Daniel Litinski and Felix von Oppen, "Quantum computing with Majorana fermion codes", Physical Review B 97 20, 205404 (2018).

[42] Daniel Litinski and Felix von Oppen, "Braiding by Majorana tracking and long-range CNOT gates with color codes", Physical Review B 96 20, 205413 (2017).

[43] L. Ortiz, S. Varona, O. Viyuela, and M. A. Martin-Delgado, "Localization and oscillations of Majorana fermions in a two-dimensional electron gas coupled with d -wave superconductors", Physical Review B 97 6, 064501 (2018).

[44] Allan Tosta, Antônio C. Lourenço, Daniel Brod, Fernando Iemini, and Tiago Debarba, "Fermonic anyons: entanglement and quantum computation from a resource-theoretic perspective", arXiv:2306.00795, (2023).

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