Halving the cost of quantum addition

Craig Gidney

Google, Santa Barbara, CA 93117, USA

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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.

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► References

[1] 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.

[2] 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.

[3] 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.

[4] 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.

[5] S. B. Bravyi and A. Yu. Kitaev. Quantum codes on a lattice with boundary. arXiv:quant-ph/​9811052, 1998. URL https:/​/​arxiv.org/​abs/​quant-ph/​9811052.

[6] Richard P Brent and H-T_ Kung. A regular layout for parallel adders. IEEE transactions on Computers, (3): 260–264, 1982. 10.1109/​TC.1982.1675982.

[7] 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.

[8] 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.

[9] 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.

[10] Austin Fowler, Dmitri Maslov, Cody Jones, and Matt Amy. Private correspondence, Aug 2017.

[11] 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.

[12] 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.

[13] 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.

[14] 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.

[15] Cody Jones. Low-overhead constructions for the fault-tolerant toffoli gate. Physical Review A, 87 (2), feb 2013. 10.1103/​physreva.87.022328.

[16] Alexei Yu Kitaev, Alexander Shen, and Mikhail N Vyalyi. Classical and quantum computation. Number 47. American Mathematical Soc., 2002. 10.1090/​gsm/​047.

[17] 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.

[18] 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.

[19] Edgard Muñoz-Coreas and Himanshu Thapliyal. T-count optimized design of quantum integer multiplication, 2017. URL https:/​/​arxiv.org/​abs/​1706.05113.

[20] Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2009. 10.1017/​cbo9780511976667.

[21] 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.

[22] 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.

[23] 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.

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[1] 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).

[2] Andrew J. Daley, Immanuel Bloch, Christian Kokail, Stuart Flannigan, Natalie Pearson, Matthias Troyer, and Peter Zoller, "Practical quantum advantage in quantum simulation", Nature 607 7920, 667 (2022).

[3] 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).

[4] Craig Gidney and Martin Ekerå, "How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits", Quantum 5, 433 (2021).

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

[6] Joonho Lee, Dominic W. Berry, Craig Gidney, William J. Huggins, Jarrod R. McClean, Nathan Wiebe, and Ryan Babbush, "Even More Efficient Quantum Computations of Chemistry Through Tensor Hypercontraction", PRX Quantum 2 3, 030305 (2021).

[7] Craig Gidney and Martin Ekerå, "How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits", arXiv:1905.09749, (2019).

[8] 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).

[9] Alexander F. Shaw, Pavel Lougovski, Jesse R. Stryker, and Nathan Wiebe, "Quantum Algorithms for Simulating the Lattice Schwinger Model", Quantum 4, 306 (2020).

[10] 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).

[11] 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).

[12] 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).

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

[14] Yuan Su, Dominic W. Berry, Nathan Wiebe, Nicholas Rubin, and Ryan Babbush, "Fault-Tolerant Quantum Simulations of Chemistry in First Quantization", PRX Quantum 2 4, 040332 (2021).

[15] Sophia Simon, Raffaele Santagati, Matthias Degroote, Nikolaj Moll, Michael Streif, and Nathan Wiebe, "Improved Precision Scaling for Simulating Coupled Quantum-Classical Dynamics", PRX Quantum 5 1, 010343 (2024).

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

[17] 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).

[18] Annie Y. Wei, Preksha Naik, Aram W. Harrow, and Jesse Thaler, "Quantum algorithms for jet clustering", Physical Review D 101 9, 094015 (2020).

[19] Christoph Sünderhauf, Earl Campbell, and Joan Camps, "Block-encoding structured matrices for data input in quantum computing", Quantum 8, 1226 (2024).

[20] 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).

[21] Kianna Wan, Mario Berta, and Earl T. Campbell, "Randomized Quantum Algorithm for Statistical Phase Estimation", Physical Review Letters 129 3, 030503 (2022).

[22] Thomas E. O'Brien, Michael Streif, Nicholas C. Rubin, Raffaele Santagati, Yuan Su, William J. Huggins, Joshua J. Goings, Nikolaj Moll, Elica Kyoseva, Matthias Degroote, Christofer S. Tautermann, Joonho Lee, Dominic W. Berry, Nathan Wiebe, and Ryan Babbush, "Efficient quantum computation of molecular forces and other energy gradients", Physical Review Research 4 4, 043210 (2022).

[23] Dylan Herman, Ruslan Shaydulin, Yue Sun, Shouvanik Chakrabarti, Shaohan Hu, Pierre Minssen, Arthur Rattew, Romina Yalovetzky, and Marco Pistoia, "Constrained optimization via quantum Zeno dynamics", Communications Physics 6 1, 219 (2023).

[24] Alain Delgado, Pablo A. M. Casares, Roberto dos Reis, Modjtaba Shokrian Zini, Roberto Campos, Norge Cruz-Hernández, Arne-Christian Voigt, Angus Lowe, Soran Jahangiri, M. A. Martin-Delgado, Jonathan E. Mueller, and Juan Miguel Arrazola, "Simulating key properties of lithium-ion batteries with a fault-tolerant quantum computer", Physical Review A 106 3, 032428 (2022).

[25] 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).

[26] Cristian L. Cortes, Matthias Loipersberger, Robert M. Parrish, Sam Morley-Short, William Pol, Sukin Sim, Mark Steudtner, Christofer S. Tautermann, Matthias Degroote, Nikolaj Moll, Raffaele Santagati, and Michael Streif, "Fault-Tolerant Quantum Algorithm for Symmetry-Adapted Perturbation Theory", PRX Quantum 5 1, 010336 (2024).

[27] Zohreh Davoudi, Alexander F. Shaw, and Jesse R. Stryker, "General quantum algorithms for Hamiltonian simulation with applications to a non-Abelian lattice gauge theory", Quantum 7, 1213 (2023).

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

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

[30] Gregory D. Kahanamoku-Meyer and Norman Y. Yao, "Fast quantum integer multiplication with zero ancillas", arXiv:2403.18006, (2024).

[31] 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).

[32] Yunseong Nam, Yuan Su, and Dmitri Maslov, "Approximate quantum Fourier transform with O(n log(n)) T gates", npj Quantum Information 6, 26 (2020).

[33] S. Flannigan, N. Pearson, G. H. Low, A. Buyskikh, I. Bloch, P. Zoller, M. Troyer, and A. J. Daley, "Propagation of errors and quantitative quantum simulation with quantum advantage", Quantum Science and Technology 7 4, 045025 (2022).

[34] Hai-Sheng Li, Ping Fan, Haiying Xia, and Gui-Lu Long, "The circuit design and optimization of quantum multiplier and divider", Science China Physics, Mechanics, and Astronomy 65 6, 260311 (2022).

[35] F. Orts, E. Filatovas, G. Ortega, J. F. SanJuan-Estrada, and E. M. Garzón, "Improving the number of T gates and their spread in integer multipliers on quantum computing", Physical Review A 107 4, 042621 (2023).

[36] 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).

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

[38] Austin Gilliam, Stefan Woerner, and Constantin Gonciulea, "Grover Adaptive Search for Constrained Polynomial Binary Optimization", Quantum 5, 428 (2021).

[39] 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).

[40] Mark Steudtner, Sam Morley-Short, William Pol, Sukin Sim, Cristian L. Cortes, Matthias Loipersberger, Robert M. Parrish, Matthias Degroote, Nikolaj Moll, Raffaele Santagati, and Michael Streif, "Fault-tolerant quantum computation of molecular observables", Quantum 7, 1164 (2023).

[41] Michele Mosca and Priyanka Mukhopadhyay, "A polynomial time and space heuristic algorithm for T-count", Quantum Science and Technology 7 1, 015003 (2022).

[42] Koji Nagata and Tadao Nakamura, "Mathematical digital quantum computation by means of much more logical skills", Quantum Studies: Mathematics and Foundations (2024).

[43] Priyanka Mukhopadhyay, Torin F. Stetina, and Nathan Wiebe, "Quantum Simulation of the First-Quantized Pauli-Fierz Hamiltonian", PRX Quantum 5 1, 010345 (2024).

[44] Siyi Wang, Eugene Lim, and Anupam Chattopadhyay, "Boosting the Efficiency of Quantum Divider through Effective Design Space Exploration", arXiv:2403.01206, (2024).

[45] Priyanka Mukhopadhyay, Nathan Wiebe, and Hong Tao Zhang, "Synthesizing efficient circuits for Hamiltonian simulation", npj Quantum Information 9, 31 (2023).

[46] Qingfeng Wang, Ming Li, Christopher Monroe, and Yunseong Nam, "Resource-Optimized Fermionic Local-Hamiltonian Simulation on a Quantum Computer for Quantum Chemistry", Quantum 5, 509 (2021).

[47] Sean Greenaway, William Pol, and Sukin Sim, "A case study against QSVT: assessment of quantum phase estimation improved by signal processing techniques", arXiv:2404.01396, (2024).

[48] Johannes Bausch, "Fast Black-Box Quantum State Preparation", arXiv:2009.10709, (2020).

[49] Vadym Kliuchnikov, Kristin Lauter, Romy Minko, Adam Paetznick, and Christophe Petit, "Shorter quantum circuits via single-qubit gate approximation", Quantum 7, 1208 (2023).

[50] Mohammad Hassan Khatami, Udson C. Mendes, Nathan Wiebe, and Philip M. Kim, "Gate-based quantum computing for protein design", PLoS Computational Biology 19 4, e1011033 (2023).

[51] Aleks Kissinger, Neil J. Ross, and John van de Wetering, "Catalysing Completeness and Universality", arXiv:2404.09915, (2024).

[52] 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).

[53] Giulia Meuli, Mathias Soeken, and Giovanni De Micheli, "Xor-And-Inverter Graphs for Quantum Compilation", npj Quantum Information 8, 7 (2022).

[54] Yunseong Nam, Yuan Su, and Dmitri Maslov, "Approximate Quantum Fourier Transform with $O(n \log(n))$ T gates", arXiv:1803.04933, (2018).

[55] Earl T. Campbell, "Early fault-tolerant simulations of the Hubbard model", Quantum Science and Technology 7 1, 015007 (2022).

[56] Pablo A. M. Casares, Roberto Campos, and M. A. Martin-Delgado, "TFermion: A non-Clifford gate cost assessment library of quantum phase estimation algorithms for quantum chemistry", Quantum 6, 768 (2022).

[57] Johannes Bausch, "Fast Black-Box Quantum State Preparation", Quantum 6, 773 (2022).

[58] Shengbin Wang, Zhimin Wang, Guolong Cui, Shangshang Shi, Ruimin Shang, Lixin Fan, Wendong Li, Zhiqiang Wei, and Yongjian Gu, "Fast black-box quantum state preparation based on linear combination of unitaries", Quantum Information Processing 20 8, 270 (2021).

[59] Francisco Orts, Remigijus Paulavičius, and Ernestas Filatovas, "Improving the implementation of quantum blockchain based on hypergraphs", Quantum Information Processing 22 9, 330 (2023).

[60] Francisco Orts, Gloria Ortega, Elías F. Combarro, Ignacio F. Rúa, and Ester M. Garzón, "Optimized quantum leading zero detector circuits", Quantum Information Processing 22 1, 28 (2023).

[61] Angus Kan and Yunseong Nam, "Simulating lattice quantum electrodynamics on a quantum computer", Quantum Science and Technology 8 1, 015008 (2023).

[62] Shengbin Wang, Zhimin Wang, Runhong He, Shangshang Shi, Guolong Cui, Ruimin Shang, Jiayun Li, Yanan Li, Wendong Li, Zhiqiang Wei, and Yongjian Gu, "Inverse-coefficient black-box quantum state preparation", New Journal of Physics 24 10, 103004 (2022).

[63] Matthew Amy and Neil J. Ross, "Phase-state duality in reversible circuit design", Physical Review A 104 5, 052602 (2021).

[64] Gayathri S. S, R. Kumar, Majid Haghparast, and Samiappan Dhanalakshmi, "A Novel and Efficient square root Computation Quantum Circuit for Floating-point Standard", International Journal of Theoretical Physics 61 9, 234 (2022).

[65] S. S. Gayathri, R. Kumar, Samiappan Dhanalakshmi, and Brajesh Kumar Kaushik, "T-count optimized quantum circuit for floating point addition and multiplication", Quantum Information Processing 20 11, 378 (2021).

[66] G. A. L. White, C. D. Hill, and L. C. L. Hollenberg, "Truncated phase-based quantum arithmetic: Error propagation and resource reduction", Physical Review A 108 5, 052608 (2023).

[67] Michael Hanks and M. S. Kim, "Fault tolerance in qudit circuit design", Physical Review A 106 6, 062433 (2022).

[68] N. M. Guseynov and W. V. Pogosov, "Quantum simulation of fermionic systems using hybrid digital-analog quantum computing approach", Journal of Physics Condensed Matter 34 28, 285901 (2022).

[69] Shantao Zhao, Haisheng Li, Guiqiong Li, and Xiaohu Tang, "The implementation of the enhanced quantum floating-point adder", Modern Physics Letters A 37 26, 2250169-128 (2022).

[70] Gary J. Mooney, Charles D. Hill, and Lloyd C. L. Hollenberg, "Cost-optimal single-qubit gate synthesis in the Clifford hierarchy", Quantum 5, 396 (2021).

[71] Siyi Wang, Anubhab Baksi, and Anupam Chattopadhyay, "A Higher radix architecture for quantum carry-lookahead adder", Scientific Reports 13, 16338 (2023).

[72] 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).

[73] Sohrab Sajadimanesh and Ehsan Atoofian, "Implementation of a quantum division circuit on noisy intermediate-scale quantum devices using dynamic circuits and approximate computing", Physical Review A 109 5, 052601 (2024).

[74] Harashta Tatimma Larasati, Dedy Septono Catur Putranto, Rini Wisnu Wardhani, Jonguk Park, and Howon Kim, "Depth Optimization of FLT-Based Quantum Inversion Circuit", IEEE Access 11, 54910 (2023).

[75] Ping Fan and Hai-Sheng Li, "Designs of the divider and special multiplier optimizing T and CNOT gates", EPJ Quantum Technology 11 1, 13 (2024).

[76] Alexandru Paler, "Controlling distilleries in fault-tolerant quantum circuits: problem statement and analysis towards a solution", arXiv:1806.07266, (2018).

[77] Siyi Wang, Suman Deb, Ankit Mondal, and Anupam Chattopadhyay, "Optimal Toffoli-Depth Quantum Adder", arXiv:2405.02523, (2024).

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