A Game of Surface Codes: Large-Scale Quantum Computing with Lattice Surgery

Daniel Litinski

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

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Given a quantum gate circuit, how does one execute it in a fault-tolerant architecture with as little overhead as possible? In this paper, we discuss strategies for surface-code quantum computing on small, intermediate and large scales. They are strategies for space-time trade-offs, going from slow computations using few qubits to fast computations using many qubits. Our schemes are based on surface-code patches, which not only feature a low space cost compared to other surface-code schemes, but are also conceptually simple~--~simple enough that they can be described as a tile-based game with a small set of rules. Therefore, no knowledge of quantum error correction is necessary to understand the schemes in this paper, but only the concepts of qubits and measurements.

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Useful, classically intractable quantum computations can be very long, potentially consisting of billions of quantum gates. Some of these computations use as few as 100 qubits, but these need to be logical error-corrected qubits, rather than physical qubits, as the coherence times of currently available physical qubits are many orders of magnitude shorter than the execution times of these computations. With logical qubits, the set of available operation is determined by the error-correcting code, independently of the underlying physical hardware. For full-scale quantum computations on hundreds of logical qubits (i.e., potentially hundreds of thousands of physical qubits), it is useful to have a framework that describes logical qubits and operations without keeping track of the details of the physical hardware and error-correction protocols. This paper introduces such a framework with a focus on surface codes, but can also be applied to other topological codes. It uses lattice surgery to describe all logical operations via the easy-to-understand concepts of qubits and measurements, avoiding anyons and topological braiding diagrams. Using this framework, a complete full-scale quantum computer is constructed, consisting of qubit blocks that perform magic state distillation and blocks that consume magic states to advance the computation. Finally, space-time trade-offs are discussed, i.e., how to use more qubits to compute faster.

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[1] M. Reiher, N. Wiebe, K. M. Svore, D. Wecker, and M. Troyer, Elucidating reaction mechanisms on quantum computers, PNAS 114, 7555 (2017).

[2] R. Babbush, C. Gidney, D. W. Berry, N. Wiebe, J. McClean, A. Paler, A. Fowler, and H. Neven, Encoding electronic spectra in quantum circuits with linear T complexity, Phys. Rev. X 8, 041015 (2018a).

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

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

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

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

[7] 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).

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

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

[10] 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).

[11] D. Litinski and F. v. Oppen, Lattice Surgery with a Twist: Simplifying Clifford Gates of Surface Codes, Quantum 2, 62 (2018).

[12] A. G. Fowler and C. Gidney, Low overhead quantum computation using lattice surgery, arXiv:1808.06709 (2018).

[13] A. J. Landahl and C. Ryan-Anderson, Quantum computing by color-code lattice surgery, arXiv:1407.5103 (2014).

[14] Y. Li, A magic state’s fidelity can be superior to the operations that created it, New J. Phys. 17, 023037 (2015).

[15] D. Herr, F. Nori, and S. J. Devitt, Optimization of lattice surgery is NP-hard, npj Quant. Inf. 3, 35 (2017a).

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

[17] J. Haah and M. B. Hastings, Codes and Protocols for Distilling $T$, controlled-$S$, and Toffoli Gates, Quantum 2, 71 (2018).

[18] S. Bravyi and J. Haah, Magic-state distillation with low overhead, Phys. Rev. A 86, 052329 (2012).

[19] C. Jones, Multilevel distillation of magic states for quantum computing, Phys. Rev. A 87, 042305 (2013a).

[20] A. G. Fowler, S. J. Devitt, and C. Jones, Surface code implementation of block code state distillation, Scientific Rep. 3, 1939 (2013).

[21] A. G. Fowler, Time-optimal quantum computation, arXiv:1210.4626 (2012).

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

[23] V. Kliuchnikov, D. Maslov, and M. Mosca, Fast and efficient exact synthesis of single-qubit unitaries generated by Clifford and $T$ gates, Quantum Info. Comput. 13, 607 (2013a).

[24] V. Kliuchnikov, D. Maslov, and M. Mosca, Asymptotically optimal approximation of single qubit unitaries by Clifford and $T$ circuits using a constant number of ancillary qubits, Phys. Rev. Lett. 110, 190502 (2013b).

[25] D. Gosset, V. Kliuchnikov, M. Mosca, and V. Russo, An algorithm for the $T$-count, arXiv:1308.4134 (2013).

[26] L. E. Heyfron and E. T. Campbell, An efficient quantum compiler that reduces $T$ count, Quantum Sci. Technol. 4, 015004 (2018).

[27] M. Amy, D. Maslov, M. Mosca, and M. Roetteler, A meet-in-the-middle algorithm for fast synthesis of depth-optimal quantum circuits, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 32, 818 (2013).

[28] P. Selinger, Quantum circuits of $T$-depth one, Phys. Rev. A 87, 042302 (2013).

[29] M. Amy, D. Maslov, and M. Mosca, Polynomial-time $T$-depth optimization of Clifford+$T$ circuits via matroid partitioning, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 33, 1476 (2014).

[30] D. Litinski and F. von Oppen, Quantum computing with Majorana fermion codes, Phys. Rev. B 97, 205404 (2018).

[31] A. Lavasani and M. Barkeshli, Low overhead Clifford gates from joint measurements in surface, color, and hyperbolic codes, Phys. Rev. A 98, 052319 (2018).

[32] J. I. Hall, Notes on Coding Theory Chapter 6: Modifying Codes, https:/​/​users.math.msu.edu/​users/​jhall/​classes/​ codenotes/​Mod.pdf, accessed: 2019-01-30.

[33] E. T. Campbell and M. Howard, Magic state parity-checker with pre-distilled components, Quantum 2, 56 (2018).

[34] A. M. Meier, B. Eastin, and E. Knill, Magic-state distillation with the four-qubit code, Quant. Inf. Comp. 13, 195 (2013).

[35] E. T. Campbell and J. O'Gorman, An efficient magic state approach to small angle rotations, Quantum Sci. Technol. 1, 015007 (2016).

[36] D. Herr, F. Nori, and S. J. Devitt, Lattice surgery translation for quantum computation, New J. Phys. 19, 013034 (2017b).

[37] A. G. Fowler and S. J. Devitt, A bridge to lower overhead quantum computation, arXiv:1209.0510 (2012).

[38] C. Gidney and A. G. Fowler, Efficient magic state factories with a catalyzed $|CCZ\rangle$ to $2|T\rangle$ transformation, arXiv:1812.01238 (2018).

[39] C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, Purification of noisy entanglement and faithful teleportation via noisy channels, Phys. Rev. Lett. 76, 722 (1996a).

[40] C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, Concentrating partial entanglement by local operations, Phys. Rev. A 53, 2046 (1996b).

[41] C. Dickel, J. J. Wesdorp, N. K. Langford, S. Peiter, R. Sagastizabal, A. Bruno, B. Criger, F. Motzoi, and L. DiCarlo, Chip-to-chip entanglement of transmon qubits using engineered measurement fields, Phys. Rev. B 97, 064508 (2018).

[42] P. Campagne-Ibarcq, E. Zalys-Geller, A. Narla, S. Shankar, P. Reinhold, L. Burkhart, C. Axline, W. Pfaff, L. Frunzio, R. J. Schoelkopf, and M. H. Devoret, Deterministic remote entanglement of superconducting circuits through microwave two-photon transitions, Phys. Rev. Lett. 120, 200501 (2018).

[43] C. J. Axline, L. D. Burkhart, W. Pfaff, M. Zhang, K. Chou, P. Campagne-Ibarcq, P. Reinhold, L. Frunzio, S. Girvin, L. Jiang, et al., On-demand quantum state transfer and entanglement between remote microwave cavity memories, Nat. Phys. 14, 705 (2018).

[44] N. J. Ross and P. Selinger, Optimal ancilla-free Clifford+T approximation of z-rotations, arXiv:1403.2975 (2014).

[45] G. Duclos-Cianci and D. Poulin, Reducing the quantum-computing overhead with complex gate distillation, Phys. Rev. A 91, 042315 (2015).

[46] A. W. Harrow, B. Recht, and I. L. Chuang, Efficient discrete approximations of quantum gates, Journal of Mathematical Physics 43, 4445 (2002).

[47] G. Duclos-Cianci and K. M. Svore, Distillation of nonstabilizer states for universal quantum computation, Phys. Rev. A 88, 042325 (2013).

[48] A. Bocharov, Y. Gurevich, and K. M. Svore, Efficient decomposition of single-qubit gates into $v$ basis circuits, Phys. Rev. A 88, 012313 (2013).

[49] N. C. Jones, J. D. Whitfield, P. L. McMahon, M.-H. Yung, R. V. Meter, A. Aspuru-Guzik, and Y. Yamamoto, Faster quantum chemistry simulation on fault-tolerant quantum computers, New J. Phys. 14, 115023 (2012).

[50] G. H. Low and I. L. Chuang, Hamiltonian simulation by qubitization, arXiv:1610.06546 (2016).

[51] G. H. Low and I. L. Chuang, Optimal Hamiltonian simulation by quantum signal processing, Phys. Rev. Lett. 118, 010501 (2017).

[52] R. Babbush, D. W. Berry, J. R. McClean, and H. Neven, Quantum simulation of chemistry with sublinear scaling to the continuum, arXiv:1807.09802 (2018b).

[53] C. Jones, Low-overhead constructions for the fault-tolerant Toffoli gate, Phys. Rev. A 87, 022328 (2013b).

[54] C. Gidney, Halving the cost of quantum addition, Quantum 2, 74 (2018).

[55] E. T. Campbell and M. Howard, Unified framework for magic state distillation and multiqubit gate synthesis with reduced resource cost, Phys. Rev. A 95, 022316 (2017).

[56] J. O'Gorman and E. T. Campbell, Quantum computation with realistic magic-state factories, Phys. Rev. A 95, 032338 (2017).

[57] K. K. Likharev and V. K. Semenov, RSFQ logic/​memory family: A new Josephson-junction technology for sub-terahertz-clock-frequency digital systems, IEEE Transactions on Applied Superconductivity 1, 3 (1991).

[58] A. G. Fowler, S. J. Devitt, and C. Jones, Synthesis of arbitrary quantum circuits to topological assembly: Systematic, online and compact, Scientific Rep. 7, 10414 (2017).

[59] A. Paler, I. Polian, K. Nemoto, and S. J. Devitt, Fault-tolerant, high-level quantum circuits: form, compilation and description, Quantum Sci. Technol. 2, 025003 (2017).

[60] L. Lao, B. van Wee, I. Ashraf, J. van Someren, N. Khammassi, K. Bertels, and C. G. Almudever, Mapping of lattice surgery-based quantum circuits on surface code architectures, Quantum Sci. Technol. 4, 015005 (2018).

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

[62] M. S. Kesselring, F. Pastawski, J. Eisert, and B. J. Brown, The boundaries and twist defects of the color code and their applications to topological quantum computation, Quantum 2, 101 (2018).

[63] H. P. Nautrup, N. Friis, and H. J. Briegel, Fault-tolerant interface between quantum memories and quantum processors, Nat. Commun. 8, 1321 (2017).

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

[65] IBM doubling qubits every 8 months, https:/​/​www.nextbigfuture.com/​2018/​02/​ibm-doubling-qubits-every-8-months-and-ecommerce-cryptography-at-risk-in-7-15-years.html, accessed: 2018-08-01.

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[72] Craig Gidney and Austin G. Fowler, "Efficient magic state factories with a catalyzed|CCZ⟩to2|T⟩transformation", Quantum 3, 135 (2019).

[73] J. Haferkamp, D. Hangleiter, A. Bouland, B. Fefferman, J. Eisert, and J. Bermejo-Vega, "Closing Gaps of a Quantum Advantage with Short-Time Hamiltonian Dynamics", Physical Review Letters 125 25, 250501 (2020).

[74] Rajeev Acharya, Igor Aleiner, Richard Allen, Trond I. Andersen, Markus Ansmann, Frank Arute, Kunal Arya, Abraham Asfaw, Juan Atalaya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, Joao Basso, Andreas Bengtsson, Sergio Boixo, Gina Bortoli, Alexandre Bourassa, Jenna Bovaird, Leon Brill, Michael Broughton, Bob B. Buckley, David A. Buell, Tim Burger, Brian Burkett, Nicholas Bushnell, Yu Chen, Zijun Chen, Ben Chiaro, Josh Cogan, Roberto Collins, Paul Conner, William Courtney, Alexander L. Crook, Ben Curtin, Dripto M. Debroy, Alexander Del Toro Barba, Sean Demura, Andrew Dunsworth, Daniel Eppens, Catherine Erickson, Lara Faoro, Edward Farhi, Reza Fatemi, Leslie Flores Burgos, Ebrahim Forati, Austin G. Fowler, Brooks Foxen, William Giang, Craig Gidney, Dar Gilboa, Marissa Giustina, Alejandro Grajales Dau, Jonathan A. Gross, Steve Habegger, Michael C. Hamilton, Matthew P. Harrigan, Sean D. Harrington, Oscar Higgott, Jeremy Hilton, Markus Hoffmann, Sabrina Hong, Trent Huang, Ashley Huff, William J. Huggins, Lev B. Ioffe, Sergei V. Isakov, Justin Iveland, Evan Jeffrey, Zhang Jiang, Cody Jones, Pavol Juhas, Dvir Kafri, Kostyantyn Kechedzhi, Julian Kelly, Tanuj Khattar, Mostafa Khezri, Mária Kieferová, Seon Kim, Alexei Kitaev, Paul V. Klimov, Andrey R. Klots, Alexander N. Korotkov, Fedor Kostritsa, John Mark Kreikebaum, David Landhuis, Pavel Laptev, Kim-Ming Lau, Lily Laws, Joonho Lee, Kenny Lee, Brian J. Lester, Alexander Lill, Wayne Liu, Aditya Locharla, Erik Lucero, Fionn D. Malone, Jeffrey Marshall, Orion Martin, Jarrod R. McClean, Trevor McCourt, Matt McEwen, Anthony Megrant, Bernardo Meurer Costa, Xiao Mi, Kevin C. Miao, Masoud Mohseni, Shirin Montazeri, Alexis Morvan, Emily Mount, Wojciech Mruczkiewicz, Ofer Naaman, Matthew Neeley, Charles Neill, Ani Nersisyan, Hartmut Neven, Michael Newman, Jiun How Ng, Anthony Nguyen, Murray Nguyen, Murphy Yuezhen Niu, Thomas E. O’Brien, Alex Opremcak, John Platt, Andre Petukhov, Rebecca Potter, Leonid P. Pryadko, Chris Quintana, Pedram Roushan, Nicholas C. Rubin, Negar Saei, Daniel Sank, Kannan Sankaragomathi, Kevin J. Satzinger, Henry F. Schurkus, Christopher Schuster, Michael J. Shearn, Aaron Shorter, Vladimir Shvarts, Jindra Skruzny, Vadim Smelyanskiy, W. Clarke Smith, George Sterling, Doug Strain, Marco Szalay, Alfredo Torres, Guifre Vidal, Benjamin Villalonga, Catherine Vollgraff Heidweiller, Theodore White, Cheng Xing, Z. Jamie Yao, Ping Yeh, Juhwan Yoo, Grayson Young, Adam Zalcman, Yaxing Zhang, and Ningfeng Zhu, "Suppressing quantum errors by scaling a surface code logical qubit", Nature 614 7949, 676 (2023).

[75] Sergey Bravyi, David Gosset, Robert König, and Marco Tomamichel, "Quantum advantage with noisy shallow circuits", Nature Physics 16 10, 1040 (2020).

[76] Ivan B. Djordjevic, 2020 22nd International Conference on Transparent Optical Networks (ICTON) 1 (2020) ISBN:978-1-7281-8423-4.

[77] Kaavya Sahay, Junlan Jin, Jahan Claes, Jeff D. Thompson, and Shruti Puri, "High-Threshold Codes for Neutral-Atom Qubits with Biased Erasure Errors", Physical Review X 13 4, 041013 (2023).

[78] Qian Xu, J. Pablo Bonilla Ataides, Christopher A. Pattison, Nithin Raveendran, Dolev Bluvstein, Jonathan Wurtz, Bane Vasić, Mikhail D. Lukin, Liang Jiang, and Hengyun Zhou, "Constant-overhead fault-tolerant quantum computation with reconfigurable atom arrays", Nature Physics (2024).

[79] Louis Paletta, Anthony Leverrier, Alain Sarlette, Mazyar Mirrahimi, and Christophe Vuillot, "Robust sparse IQP sampling in constant depth", Quantum 8, 1337 (2024).

[80] Ivan B. Djordjevic, Quantum Information Processing, Quantum Computing, and Quantum Error Correction 469 (2021) ISBN:9780128219829.

[81] Jing Hao Chai and Hui Khoon Ng, "On the Fault‐Tolerance Threshold for Surface Codes with General Noise", Advanced Quantum Technologies 5 10, 2200008 (2022).

[82] Junyu Liu, Connor T. Hann, and Liang Jiang, "Data centers with quantum random access memory and quantum networks", Physical Review A 108 3, 032610 (2023).

[83] Xiao-Ming Zhang, Tongyang Li, and Xiao Yuan, "Quantum State Preparation with Optimal Circuit Depth: Implementations and Applications", Physical Review Letters 129 23, 230504 (2022).

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

[85] Aleks Kissinger and John van de Wetering, "Simulating quantum circuits with ZX-calculus reduced stabiliser decompositions", Quantum Science and Technology 7 4, 044001 (2022).

[86] Narayanan Rengaswamy, Robert Calderbank, Swanand Kadhe, and Henry D. Pfister, "Logical Clifford Synthesis for Stabilizer Codes", IEEE Transactions on Quantum Engineering 1, 1 (2020).

[87] Ivan B. Djordjevic, Quantum Communication, Quantum Networks, and Quantum Sensing 407 (2023) ISBN:9780128229422.

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

[89] Jinyoung Ha, Jonghyun Lee, and Jun Heo, "Resource analysis and modifications of quantum computing with noisy qubits for elliptic curve discrete logarithms", Scientific Reports 14 1, 3927 (2024).

[90] Hammam Qassim, Joel J. Wallman, and Joseph Emerson, "Clifford recompilation for faster classical simulation of quantum circuits", Quantum 3, 170 (2019).

[91] 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).

[92] Ivan B. Djordjevic, Quantum Communication, Quantum Networks, and Quantum Sensing 563 (2022) ISBN:9780128229422.

[93] Alexandru Paler, Oumarou Oumarou, and Robert Basmadjian, "On the Realistic Worst-Case Analysis of Quantum Arithmetic Circuits", IEEE Transactions on Quantum Engineering 3, 1 (2022).

[94] Sebastian Leontica and David Amaro, "Exploring the neighborhood of 1-layer QAOA with instantaneous quantum polynomial circuits", Physical Review Research 6 1, 013071 (2024).

[95] Sam McArdle, "Learning from Physics Experiments with Quantum Computers: Applications in Muon Spectroscopy", PRX Quantum 2 2, 020349 (2021).

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

[97] Zhenyu Cai, Adam Siegel, and Simon Benjamin, "Looped Pipelines Enabling Effective 3D Qubit Lattices in a Strictly 2D Device", PRX Quantum 4 2, 020345 (2023).

[98] Giovanni De Micheli, Jie-Hong R. Jiang, Robert Rand, Kaitlin Smith, and Mathias Soeken, "Advances in Quantum Computation and Quantum Technologies: A Design Automation Perspective", IEEE Journal on Emerging and Selected Topics in Circuits and Systems 12 3, 584 (2022).

[99] Jason Cong, 2023 60th ACM/IEEE Design Automation Conference (DAC) 1 (2023) ISBN:979-8-3503-2348-1.

[100] Héctor Bombín, Mihir Pant, Sam Roberts, and Karthik I. Seetharam, "Fault-Tolerant Postselection for Low-Overhead Magic State Preparation", PRX Quantum 5 1, 010302 (2024).

[101] Jonghyun Lee, Yujin Kang, Jinyoung Ha, and Jun Heo, "Lattice surgery-based Surface Code architecture using remote logical CNOT operation", Quantum Information Processing 21 6, 217 (2022).

[102] Shilin Huang, Michael Newman, and Kenneth R. Brown, "Fault-tolerant weighted union-find decoding on the toric code", Physical Review A 102 1, 012419 (2020).

[103] Hasan Sayginel, Francois Jamet, Abhishek Agarwal, Dan E Browne, and Ivan Rungger, "A fault-tolerant variational quantum algorithm with limited T-depth", Quantum Science and Technology 9 1, 015015 (2024).

[104] Héctor Bombín, Chris Dawson, Ryan V. Mishmash, Naomi Nickerson, Fernando Pastawski, and Sam Roberts, "Logical Blocks for Fault-Tolerant Topological Quantum Computation", PRX Quantum 4 2, 020303 (2023).

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

[106] 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).

[107] Benjamin J. Brown and Sam Roberts, "Universal fault-tolerant measurement-based quantum computation", Physical Review Research 2 3, 033305 (2020).

[108] Ivan Djordjevic, "Surface-Codes-Based Quantum Communication Networks", Entropy 22 9, 1059 (2020).

[109] William J. Huggins and Jarrod R. McClean, "Accelerating Quantum Algorithms with Precomputation", Quantum 8, 1264 (2024).

[110] Simon Burton and Dan Browne, "Limitations on Transversal Gates for Hypergraph Product Codes", IEEE Transactions on Information Theory 68 3, 1772 (2022).

[111] 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).

[112] Soronzonbold Otgonbaatar and Dieter Kranzlmüller, "Exploiting the Quantum Advantage for Satellite Image Processing: Review and Assessment", IEEE Transactions on Quantum Engineering 5, 1 (2024).

[113] Antonio D. Corcoles, Abhinav Kandala, Ali Javadi-Abhari, Douglas T. McClure, Andrew W. Cross, Kristan Temme, Paul D. Nation, Matthias Steffen, and Jay M. Gambetta, "Challenges and Opportunities of Near-Term Quantum Computing Systems", Proceedings of the IEEE 108 8, 1338 (2020).

[114] Tomas Jochym-O'Connor and Theodore J. Yoder, "Four-dimensional toric code with non-Clifford transversal gates", Physical Review Research 3 1, 013118 (2021).

[115] Utkarsh Azad, Aleksandra Lipińska, Shilpa Mahato, Rijul Sachdeva, Debasmita Bhoumik, and Ritajit Majumdar, "Surface code design for asymmetric error channels", IET Quantum Communication 3 3, 174 (2022).

[116] Sengthai Heng, Sovanmonynuth Heng, Dongmin Kim, and Youngsun Han, "Robust hybrid qubit plane architecture for enhancing lattice-surgery-based surface codes", Physical Review A 109 2, 022440 (2024).

[117] 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).

[118] György P. Gehér, Ophelia Crawford, and Earl T. Campbell, "Tangling Schedules Eases Hardware Connectivity Requirements for Quantum Error Correction", PRX Quantum 5 1, 010348 (2024).

[119] Yujin Kang, Jonghyun Lee, Jinyoung Ha, and Jun Heo, "Fault-tolerant quantum computation using low-cost joint measurements", Quantum Information Processing 23 5, 190 (2024).

[120] Alexander Erhard, Hendrik Poulsen Nautrup, Michael Meth, Lukas Postler, Roman Stricker, Martin Stadler, Vlad Negnevitsky, Martin Ringbauer, Philipp Schindler, Hans J. Briegel, Rainer Blatt, Nicolai Friis, and Thomas Monz, "Entangling logical qubits with lattice surgery", Nature 589 7841, 220 (2021).

[121] 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.

[122] Yutaro Akahoshi, Kazunori Maruyama, Hirotaka Oshima, Shintaro Sato, and Keisuke Fujii, "Partially Fault-Tolerant Quantum Computing Architecture with Error-Corrected Clifford Gates and Space-Time Efficient Analog Rotations", PRX Quantum 5 1, 010337 (2024).

[123] 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).

[124] Longcheng Li, Cheng Guo, Qian Li, and Xiaoming Sun, "Fast exact synthesis of two-qubit unitaries using a near-minimum number of T gates", Physical Review A 107 4, 042424 (2023).

[125] Qian Xu, Nam Mannucci, Alireza Seif, Aleksander Kubica, Steven T. Flammia, and Liang Jiang, "Tailored XZZX codes for biased noise", Physical Review Research 5 1, 013035 (2023).

[126] Isaac H. Kim, Ye-Hua Liu, Sam Pallister, William Pol, Sam Roberts, and Eunseok Lee, "Fault-tolerant resource estimate for quantum chemical simulations: Case study on Li-ion battery electrolyte molecules", Physical Review Research 4 2, 023019 (2022).

[127] Yiting Liu, Zhi Ma, Lan Luo, Chao Du, Yangyang Fei, Hong Wang, Qianheng Duan, and Jing Yang, "Magic state distillation and cost analysis in fault-tolerant universal quantum computation", Quantum Science and Technology 8 4, 043001 (2023).

[128] Simone Bordoni and Stefano Giagu, "Convolutional neural network based decoders for surface codes", Quantum Information Processing 22 3, 151 (2023).

[129] Madhav Krishnan Vijayan, Alexandru Paler, Jason Gavriel, Casey R. Myers, Peter Rohde, and Simon J Devitt, "Compilation of algorithm-specific graph states for quantum circuits", Quantum Science and Technology 9 2, 025005 (2024).

[130] Alexander Cowtan, Silas Dilkes, Ross Duncan, Will Simmons, and Seyon Sivarajah, "Phase Gadget Synthesis for Shallow Circuits", Electronic Proceedings in Theoretical Computer Science 318, 213 (2020).

[131] 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).

[132] 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.

[133] Daniele Cuomo, Marcello Caleffi, Kevin Krsulich, Filippo Tramonto, Gabriele Agliardi, Enrico Prati, and Angela Sara Cacciapuoti, "Optimized Compiler for Distributed Quantum Computing", ACM Transactions on Quantum Computing 4 2, 1 (2023).

[134] Nick S. Blunt, Joan Camps, Ophelia Crawford, Róbert Izsák, Sebastian Leontica, Arjun Mirani, Alexandra E. Moylett, Sam A. Scivier, Christoph Sünderhauf, Patrick Schopf, Jacob M. Taylor, and Nicole Holzmann, "Perspective on the Current State-of-the-Art of Quantum Computing for Drug Discovery Applications", Journal of Chemical Theory and Computation 18 12, 7001 (2022).

[135] Arpit Dua, Tomas Jochym-O'Connor, and Guanyu Zhu, "Quantum error correction with fractal topological codes", Quantum 7, 1122 (2023).

[136] Daan Camps, Ermal Rrapaj, Katherine Klymko, Brian Austin, and Nicholas J. Wright, ISC High Performance 2024 Research Paper Proceedings (39th International Conference) 1 (2024) ISBN:978-3-9826336-0-2.

[137] 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).

[138] William J. Huggins, Sam McArdle, Thomas E. O’Brien, Joonho Lee, Nicholas C. Rubin, Sergio Boixo, K. Birgitta Whaley, Ryan Babbush, and Jarrod R. McClean, "Virtual Distillation for Quantum Error Mitigation", Physical Review X 11 4, 041036 (2021).

[139] Mark A Webster, Armanda O Quintavalle, and Stephen D Bartlett, "Transversal diagonal logical operators for stabiliser codes", New Journal of Physics 25 10, 103018 (2023).

[140] Jin‐Ho On, Chei‐Yol Kim, Soo‐Cheol Oh, Sang‐Min Lee, and Gyu‐Il Cha, "A multilayered Pauli tracking architecture for lattice surgery‐based logical qubits", ETRI Journal 45 3, 462 (2023).

[141] Alexander Cowtan and Shahn Majid, "Algebraic aspects of boundaries in the Kitaev quantum double model", Journal of Mathematical Physics 64 10, 102203 (2023).

[142] Markus S. Kesselring, Julio C. Magdalena de la Fuente, Felix Thomsen, Jens Eisert, Stephen D. Bartlett, and Benjamin J. Brown, "Anyon Condensation and the Color Code", PRX Quantum 5 1, 010342 (2024).

[143] Matthew Otten, Byeol Kang, Dmitry Fedorov, Joo-Hyoung Lee, Anouar Benali, Salman Habib, Stephen K. Gray, and Yuri Alexeev, "QREChem: quantum resource estimation software for chemistry applications", Frontiers in Quantum Science and Technology 2, 1232624 (2023).

[144] Aleksander Kubica and Michael Vasmer, "Single-shot quantum error correction with the three-dimensional subsystem toric code", Nature Communications 13 1, 6272 (2022).

[145] 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).

[146] Simon Cichy, Paul K. Faehrmann, Sumeet Khatri, and Jens Eisert, "Perturbative gadgets for gate-based quantum computing: Nonrecursive constructions without subspace restrictions", Physical Review A 109 5, 052624 (2024).

[147] Ethan Hansen, Sanskriti Joshi, and Hannah Rarick, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 199 (2023) ISBN:979-8-3503-4323-6.

[148] 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).

[149] Mingzheng Zhu, Hao Fu, Jun Wu, Chi Zhang, Wei Xie, and Xiang-Yang Li, 2024 IEEE/ACM International Symposium on Code Generation and Optimization (CGO) 158 (2024) ISBN:979-8-3503-9509-9.

[150] Aleksei V. Ivanov, Christoph Sünderhauf, Nicole Holzmann, Tom Ellaby, Rachel N. Kerber, Glenn Jones, and Joan Camps, "Quantum computation for periodic solids in second quantization", Physical Review Research 5 1, 013200 (2023).

[151] Earl Campbell, Ankur Khurana, and Ashley Montanaro, "Applying quantum algorithms to constraint satisfaction problems", Quantum 3, 167 (2019).

[152] Nick S. Blunt, György P. Gehér, and Alexandra E. Moylett, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 290 (2023) ISBN:979-8-3503-4323-6.

[153] Alexander M. Dalzell, Sam McArdle, Mario Berta, Przemyslaw Bienias, Chi-Fang Chen, András Gilyén, Connor T. Hann, Michael J. Kastoryano, Emil T. Khabiboulline, Aleksander Kubica, Grant Salton, Samson Wang, and Fernando G. S. L. Brandão, "Quantum algorithms: A survey of applications and end-to-end complexities", arXiv:2310.03011, (2023).

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

[155] Daniel Bochen Tan, Murphy Yuezhen Niu, and Craig Gidney, "A SAT Scalpel for Lattice Surgery: Representation and Synthesis of Subroutines for Surface-Code Fault-Tolerant Quantum Computing", arXiv:2404.18369, (2024).

[156] Guang Hao Low, Vadym Kliuchnikov, and Luke Schaeffer, "Trading T-gates for dirty qubits in state preparation and unitary synthesis", arXiv:1812.00954, (2018).

[157] Ryan Sweke, Markus S. Kesselring, Evert P. L. van Nieuwenburg, and Jens Eisert, "Reinforcement Learning Decoders for Fault-Tolerant Quantum Computation", arXiv:1810.07207, (2018).

[158] Yang Wang, Selwyn Simsek, Thomas M. Gatterman, Justin A. Gerber, Kevin Gilmore, Dan Gresh, Nathan Hewitt, Chandler V. Horst, Mitchell Matheny, Tanner Mengle, Brian Neyenhuis, and Ben Criger, "Fault-Tolerant One-Bit Addition with the Smallest Interesting Colour Code", arXiv:2309.09893, (2023).

[159] Craig Gidney and Austin G. Fowler, "Flexible layout of surface code computations using AutoCCZ states", arXiv:1905.08916, (2019).

[160] Vlad Gheorghiu and Michele Mosca, "Benchmarking the quantum cryptanalysis of symmetric, public-key and hash-based cryptographic schemes", arXiv:1902.02332, (2019).

[161] Travis L. Scholten, Carl J. Williams, Dustin Moody, Michele Mosca, William Hurley, William J. Zeng, Matthias Troyer, and Jay M. Gambetta, "Assessing the Benefits and Risks of Quantum Computers", arXiv:2401.16317, (2024).

[162] Niel de Beaudrap, Xiaoning Bian, and Quanlong Wang, "Fast and effective techniques for T-count reduction via spider nest identities", arXiv:2004.05164, (2020).

[163] Alex Townsend-Teague, Julio Magdalena de la Fuente, and Markus Kesselring, "Floquetifying the Colour Code", arXiv:2307.11136, (2023).

[164] Mason Rhodes, Michael Kreshchuk, and Shivesh Pathak, "Exponential improvements in the simulation of lattice gauge theories using near-optimal techniques", arXiv:2405.10416, (2024).

[165] Craig Gidney, Michael Newman, Peter Brooks, and Cody Jones, "Yoked surface codes", arXiv:2312.04522, (2023).

[166] Andrew J. Landahl and Benjamin C. A. Morrison, "Logical fermions for fault-tolerant quantum simulation", arXiv:2110.10280, (2021).

[167] Samuel J. Elman, Jason Gavriel, and Ryan L. Mann, "Optimal Scheduling of Graph States via Path Decompositions", arXiv:2403.04126, (2024).

[168] Niel de Beaudrap, Xiaoning Bian, and Quanlong Wang, "Techniques to Reduce $\pi/4$-Parity-Phase Circuits, Motivated by the ZX Calculus", arXiv:1911.09039, (2019).

[169] Jason D. Chadwick, Christopher Kang, Joshua Viszlai, Sophia Fuhui Lin, and Frederic T. Chong, "Averting multi-qubit burst errors in surface code magic state factories", arXiv:2405.00146, (2024).

[170] Hiroki Hamaguchi, Kou Hamada, and Nobuyuki Yoshioka, "Handbook for Efficiently Quantifying Robustness of Magic", arXiv:2311.01362, (2023).

[171] Narayanan Rengaswamy, Robert Calderbank, Swanand Kadhe, and Henry D. Pfister, "Logical Clifford Synthesis for Stabilizer Codes", arXiv:1907.00310, (2019).

[172] Olivia Di Matteo, Vlad Gheorghiu, and Michele Mosca, "Fault tolerant resource estimation of quantum random-access memories", arXiv:1902.01329, (2019).

[173] Alexandru Paler, "SurfBraid: A concept tool for preparing and resource estimating quantum circuits protected by the surface code", arXiv:1902.02417, (2019).

[174] Prithviraj Prabhu and Christopher Chamberland, "New magic state distillation factories optimized by temporally encoded lattice surgery", arXiv:2210.15814, (2022).

[175] Alexandru Paler and Austin G. Fowler, "OpenSurgery for Topological Assemblies", arXiv:1906.07994, (2019).

[176] Sitong Liu, Naphan Benchasattabuse, Darcy QC Morgan, Michal Hajdušek, Simon J. Devitt, and Rodney Van Meter, "A Substrate Scheduler for Compiling Arbitrary Fault-tolerant Graph States", arXiv:2306.03758, (2023).

[177] Daniel Herr, Alexandru Paler, Simon J. Devitt, and Franco Nori, "Time versus Hardware: Reducing Qubit Counts with a (Surface Code) Data Bus", arXiv:1902.08117, (2019).

[178] Brendan Reid, "A simple method for compiling quantum stabilizer circuits", arXiv:2404.19408, (2024).

[179] Aleks Kissinger and John van de Wetering, "Scalable spider nests (...or how to graphically grok transversal non-Clifford gates)", arXiv:2404.07828, (2024).

[180] Yutaka Hirano, Tomohiro Itogawa, and Keisuke Fujii, "Leveraging Zero-Level Distillation to Generate High-Fidelity Magic States", arXiv:2404.09740, (2024).

The above citations are from Crossref's cited-by service (last updated successfully 2024-05-26 08:18:11) and SAO/NASA ADS (last updated successfully 2024-05-26 08:18:12). The list may be incomplete as not all publishers provide suitable and complete citation data.