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.
Video of the QIP talk about this paper:
 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).
 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).
 H. Bombin, Topological order with a twist: Ising anyons from an abelian model, Phys. Rev. Lett. 105, 030403 (2010).
 C. Horsman, A. G. Fowler, S. Devitt, and R. V. Meter, Surface code quantum computing by lattice surgery, New J. Phys. 14, 123011 (2012).
 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).
 Y. Li, A magic state’s fidelity can be superior to the operations that created it, New J. Phys. 17, 023037 (2015).
 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).
 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).
 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).
 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).
 A. Lavasani and M. Barkeshli, Low overhead Clifford gates from joint measurements in surface, color, and hyperbolic codes, Phys. Rev. A 98, 052319 (2018).
 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.
 E. T. Campbell and J. O'Gorman, An efficient magic state approach to small angle rotations, Quantum Sci. Technol. 1, 015007 (2016).
 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).
 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).
 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).
 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).
 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).
 G. H. Low and I. L. Chuang, Optimal Hamiltonian simulation by quantum signal processing, Phys. Rev. Lett. 118, 010501 (2017).
 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).
 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).
 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).
 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).
 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).
 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).
 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.
 Sam McArdle, Suguru Endo, Alán Aspuru-Guzik, Simon C. Benjamin, and Xiao Yuan, "Quantum computational chemistry", Reviews of Modern Physics 92 1, 015003 (2020).
 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).
 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).
 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).
 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).
 Daniel Litinski, "Magic State Distillation: Not as Costly as You Think", Quantum 3, 205 (2019).
 Sorin Hoară, Roxana-Mariana Beiu, and Valeriu Beiu, Advances in Intelligent Systems and Computing 1243, 57 (2021) ISBN:978-3-030-53650-3.
 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).
 Michael Vasmer and Dan E. Browne, "Three-dimensional surface codes: Transversal gates and fault-tolerant architectures", Physical Review A 100 1, 012312 (2019).
 Michael Hanks, Marta P. Estarellas, William J. Munro, and Kae Nemoto, "Effective Compression of Quantum Braided Circuits Aided by ZX-Calculus", Physical Review X 10 4, 041030 (2020).
 Iskren Vankov, Daniel Mills, Petros Wallden, and Elham Kashefi, "Methods for classically simulating noisy networked quantum architectures", Quantum Science and Technology 5 1, 014001 (2019).
 Casey Duckering, Jonathan M. Baker, David I. Schuster, and Frederic T. Chong, 2020 53rd Annual IEEE/ACM International Symposium on Microarchitecture (MICRO) 173 (2020) ISBN:978-1-7281-7383-2.
 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).
 Dominic W. Berry, Craig Gidney, Mario Motta, Jarrod R. McClean, and Ryan Babbush, "Qubitization of Arbitrary Basis Quantum Chemistry Leveraging Sparsity and Low Rank Factorization", Quantum 3, 208 (2019).
 Ian D. Kivlichan, Craig Gidney, Dominic W. Berry, Nathan Wiebe, Jarrod McClean, Wei Sun, Zhang Jiang, Nicholas Rubin, Austin Fowler, Alán Aspuru-Guzik, Hartmut Neven, and Ryan Babbush, "Improved Fault-Tolerant Quantum Simulation of Condensed-Phase Correlated Electrons via Trotterization", Quantum 4, 296 (2020).
 Niel de Beaudrap and Dominic Horsman, "The ZX calculus is a language for surface code lattice surgery", Quantum 4, 218 (2020).
 Craig Gidney and Austin G. Fowler, "Efficient magic state factories with a catalyzed|CCZ⟩to2|T⟩transformation", Quantum 3, 135 (2019).
 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).
 Sergey Bravyi, David Gosset, Robert König, and Marco Tomamichel, "Quantum advantage with noisy shallow circuits", Nature Physics 16 10, 1040 (2020).
 Christopher Chamberland and Andrew W. Cross, "Fault-tolerant magic state preparation with flag qubits", Quantum 3, 143 (2019).
 Benjamin J. Brown and Sam Roberts, "Universal fault-tolerant measurement-based quantum computation", Physical Review Research 2 3, 033305 (2020).
 Ivan B. Djordjevic, "Surface-Codes-Based Quantum Communication Networks", Entropy 22 9, 1059 (2020).
 Ryan Sweke, Markus S. Kesselring, Evert P. L. van Nieuwenburg, and Jens Eisert, "Reinforcement Learning Decoders for Fault-Tolerant Quantum Computation", arXiv:1810.07207.
 Austin G. Fowler and Craig Gidney, "Low overhead quantum computation using lattice surgery", arXiv:1808.06709.
 Guang Hao Low, Vadym Kliuchnikov, and Luke Schaeffer, "Trading T-gates for dirty qubits in state preparation and unitary synthesis", arXiv:1812.00954.
 Vlad Gheorghiu and Michele Mosca, "Benchmarking the quantum cryptanalysis of symmetric, public-key and hash-based cryptographic schemes", arXiv:1902.02332.
 Craig Gidney and Austin G. Fowler, "Flexible layout of surface code computations using AutoCCZ states", arXiv:1905.08916.
 Olivia Di Matteo, Vlad Gheorghiu, and Michele Mosca, "Fault tolerant resource estimation of quantum random-access memories", arXiv:1902.01329.
 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.
 Alexandru Paler, "SurfBraid: A concept tool for preparing and resource estimating quantum circuits protected by the surface code", arXiv:1902.02417.
 J. Pablo Bonilla Ataides, David K. Tuckett, Stephen D. Bartlett, Steven T. Flammia, and Benjamin J. Brown, "The XZZX Surface Code", arXiv:2009.07851.
 Niel de Beaudrap, Xiaoning Bian, and Quanlong Wang, "Techniques to Reduce $\pi/4$-Parity-Phase Circuits, Motivated by the ZX Calculus", arXiv:1911.09039.
 Alexandru Paler and Austin G. Fowler, "OpenSurgery for Topological Assemblies", arXiv:1906.07994.
 Niel de Beaudrap, Xiaoning Bian, and Quanlong Wang, "Fast and effective techniques for T-count reduction via spider nest identities", arXiv:2004.05164.
The above citations are from Crossref's cited-by service (last updated successfully 2021-01-26 05:19:38) and SAO/NASA ADS (last updated successfully 2021-01-26 05:19:39). The list may be incomplete as not all publishers provide suitable and complete citation data.
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.