We improve the number of T gates needed to perform an n-bit adder from $8n + O(1)$ to $4n + O(1)$. We do so via a "temporary logical-AND" construction which uses four T gates to store the logical-AND of two qubits into an ancilla and zero T gates to later erase the ancilla. This construction is equivalent to one by Jones, except that our framing makes it clear that the technique is far more widely applicable than previously realized. Temporary logical-ANDs can be applied to integer arithmetic, modular arithmetic, rotation synthesis, the quantum Fourier transform, Shor's algorithm, Grover oracles, and many other circuits. Because T gates dominate the cost of quantum computation based on the surface code, and temporary logical-ANDs are widely applicable, this represents a significant reduction in projected costs of quantum computation. In addition to our n-bit adder, we present an n-bit controlled adder circuit with T-count of $8n + O(1)$, a temporary adder that can be computed for the same cost as the normal adder but whose result can be kept until it is later uncomputed without using T gates, and discuss some other constructions whose T-count is improved by the temporary logical-AND.
 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 (6): 818–830, jun 2013. 10.1109/tcad.2013.2244643.
 Ryan Babbush, Craig Gidney, Dominic W Berry, Nathan Wiebe, Jarrod McClean, Alexandru Paler, Austin Fowler, and Hartmut Neven. Encoding electronic spectra in quantum circuits with linear T complexity. arXiv preprint arXiv:1805.03662, 2018. URL https://arxiv.org/abs/1805.03662.
 Adriano Barenco, Charles H. Bennett, Richard Cleve, David P. DiVincenzo, Norman Margolus, Peter Shor, Tycho Sleator, John A. Smolin, and Harald Weinfurter. Elementary gates for quantum computation. Physical Review A, 52 (5): 3457–3467, nov 1995. 10.1103/physreva.52.3457.
 R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O'Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and John M. Martinis. Superconducting quantum circuits at the surface code threshold for fault tolerance. Nature, 508: 500–503, 2014. 10.1038/nature13171. arXiv:1402.4848.
 Steven A. Cuccaro, Thomas G. Draper, Samuel A. Kutin, and David Petrie Moulton. A new quantum ripple-carry addition circuit, 2004. URL https://arxiv.org/abs/quant-ph/0410184.
 E. Dennis, A. Kitaev, A. Landahl, and J. Preskill. Topological quantum memory. J. Math. Phys., 43: 4452–4505, 2002. 10.1063/1.1499754. arXiv:quant-ph/0110143.
 Thomas G. Draper, Samuel A. Kutin, Eric M. Rains, and Krysta M. Svore. A logarithmic-depth quantum carry-lookahead adder. 2004. URL https://arxiv.org/abs/quant-ph/0406142.
 Austin Fowler, Dmitri Maslov, Cody Jones, and Matt Amy. Private correspondence, Aug 2017.
 Austin G. Fowler, Matteo Mariantoni, John M. Martinis, and Andrew N. Cleland. Surface codes: Towards practical large-scale quantum computation. Physical Review A, 86 (3), sep 2012. 10.1103/physreva.86.032324.
 J. M. Gambetta, J. M. Chow, and M. Steffen. Building logical qubits in a superconducting quantum computing system. npj Quantum Information, 3 (2), 2017. 10.1038/s41534-016-0004-0. arXiv:1510.04375.
 Clare Horsman, Austin G Fowler, Simon Devitt, and Rodney Van Meter. Surface code quantum computing by lattice surgery. New Journal of Physics, 14 (12): 123011, 2012. 10.1088/1367-2630/14/12/123011.
 Mark Howard and Earl Campbell. Application of a resource theory for magic states to fault-tolerant quantum computing. Physical review letters, 118 (9): 090501, 2017. 10.1103/PhysRevLett.118.090501.
 V. Lahtinen and J. K. Pachos. A short introduction to topological quantum computation. 10.21468/SciPostPhys.3.3.021. URL https://arxiv.org/abs/1705.04103.
 B. Lekitsch, S. Weidt, A. G. Fowler, K. Mølmer, S. J. Devitt, C. Wunderlich, and W. K. Hensinger. Blueprint for a microwave trapped-ion quantum computer. Science Advances, 3 (2): e1601540, 2017. 10.1126/sciadv.1601540. arXiv:1508.00420.
 R. Raussendorf and J. Harrington. Fault-tolerant quantum computation with high threshold in two dimensions. Phys. Rev. Lett., 98: 190504, 2007. 10.1103/PhysRevLett.98.190504. arXiv:quant-ph/0610082.
 Robert Raussendorf, Jim Harrington, and Kovid Goyal. Topological fault-tolerance in cluster state quantum computation. New Journal of Physics, 9 (6): 199, 2007. 10.1088/1367-2630/9/6/199.
 Malte Schlosser, Sascha Tichelmann, Jens Kruse, and Gerhard Birkl. Scalable architecture for quantum information processing with atoms in optical micro-structures. Quantum Information Processing, 10 (6): 907, 2011. 10.1007/s11128-011-0297-z. 1108.5136.
 Alexander F. Shaw, Pavel Lougovski, Jesse R. Stryker, and Nathan Wiebe, "Quantum Algorithms for Simulating the Lattice Schwinger Model", Quantum 4, 306 (2020).
 Alexandru Paler, Oumarou Oumarou, and Robert Basmadjian, "Parallelizing the queries in a bucket-brigade quantum random access memory", Physical Review A 102 3, 032608 (2020).
 Jesse R. Stryker, "Oracles for Gauss's law on digital quantum computers", Physical Review A 99 4, 042301 (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).
 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).
 Shengbin Wang, Zhimin Wang, Wendong Li, Lixin Fan, Guolong Cui, Zhiqiang Wei, and Yongjian Gu, "Quantum circuits design for evaluating transcendental functions based on a function-value binary expansion method", Quantum Information Processing 19 10, 347 (2020).
 Miguel-Angel Sicilia, Salvador Sánchez-Alonso, Marçal Mora-Cantallops, and Elena García-Barriocanal, Communications in Computer and Information Science 1266, 292 (2020) ISBN:978-3-030-58792-5.
 Ryan Babbush, Dominic W. Berry, and Hartmut Neven, "Quantum simulation of the Sachdev-Ye-Kitaev model by asymmetric qubitization", Physical Review A 99 4, 040301 (2019).
 Hyeongrak Choi, Mihir Pant, Saikat Guha, and Dirk Englund, "Percolation-based architecture for cluster state creation using photon-mediated entanglement between atomic memories", npj Quantum Information 5 1, 104 (2019).
 Thomas Häner, Samuel Jaques, Michael Naehrig, Martin Roetteler, and Mathias Soeken, Lecture Notes in Computer Science 12100, 425 (2020) ISBN:978-3-030-44222-4.
 Earl Campbell, "Random Compiler for Fast Hamiltonian Simulation", Physical Review Letters 123 7, 070503 (2019).
 Yuval R. Sanders, Guang Hao Low, Artur Scherer, and Dominic W. Berry, "Black-Box Quantum State Preparation without Arithmetic", Physical Review Letters 122 2, 020502 (2019).
 Edgard Munoz-Coreas and Himanshu Thapliyal, 2019 IEEE Computer Society Annual Symposium on VLSI (ISVLSI) 360 (2019) ISBN:978-1-7281-3391-1.
 Giulia Meuli, Mathias Soeken, Martin Roetteler, and Giovanni De Micheli, 2020 IEEE International Symposium on Circuits and Systems (ISCAS) 1 (2020) ISBN:978-1-7281-3320-1.
 Michael Beverland, Earl Campbell, Mark Howard, and Vadym Kliuchnikov, "Lower bounds on the non-Clifford resources for quantum computations", Quantum Science and Technology 5 3, 035009 (2020).
 Annie Y. Wei, Preksha Naik, Aram W. Harrow, and Jesse Thaler, "Quantum algorithms for jet clustering", Physical Review D 101 9, 094015 (2020).
 Daniel Litinski, "Magic State Distillation: Not as Costly as You Think", Quantum 3, 205 (2019).
 Yunseong Nam and Dmitri Maslov, "Low-cost quantum circuits for classically intractable instances of the Hamiltonian dynamics simulation problem", npj Quantum Information 5 1, 44 (2019).
 Craig Gidney and Austin G. Fowler, "Efficient magic state factories with a catalyzed |CCZ⟩ to 2|T⟩ transformation", Quantum 3, 135 (2019).
 Yudong Cao, Jonathan Romero, Jonathan P. Olson, Matthias Degroote, Peter D. Johnson, Mária Kieferová, Ian D. Kivlichan, Tim Menke, Borja Peropadre, Nicolas P. D. Sawaya, Sukin Sim, Libor Veis, and Alán Aspuru-Guzik, "Quantum Chemistry in the Age of Quantum Computing", Chemical Reviews 119 19, 10856 (2019).
 Daniel Litinski, "A Game of Surface Codes: Large-Scale Quantum Computing with Lattice Surgery", Quantum 3, 128 (2019).
 Andreas Fischer and Alexandra Paler, Proceedings of the 34th ACM/SIGAPP Symposium on Applied Computing 1378 (2019) ISBN:9781450359337.
 Ryan Babbush, Craig Gidney, Dominic W. Berry, Nathan Wiebe, Jarrod McClean, Alexandru Paler, Austin Fowler, and Hartmut Neven, "Encoding Electronic Spectra in Quantum Circuits with Linear T Complexity", Physical Review X 8 4, 041015 (2018).
 Niel de Beaudrap, Xiaoning Bian, and Quanlong Wang, "Techniques to Reduce $\pi/4$-Parity-Phase Circuits, Motivated by the ZX Calculus", arXiv:1911.09039.
 Niel de Beaudrap, Xiaoning Bian, and Quanlong Wang, "Fast and effective techniques for T-count reduction via spider nest identities", arXiv:2004.05164.
 Hai-Sheng Li, Ping Fan, Haiying Xia, Huiling Peng, and Gui-Lu Long, "Efficient quantum arithmetic operation circuits for quantum image processing", Science China Physics, Mechanics, and Astronomy 63 8, 280311 (2020).
 Alexandru Paler, "Controlling distilleries in fault-tolerant quantum circuits: problem statement and analysis towards a solution", arXiv:1806.07266.
The above citations are from Crossref's cited-by service (last updated successfully 2020-10-21 23:30:19) and SAO/NASA ADS (last updated successfully 2020-10-21 23:30:25). 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.