Sparse quantum state preparation with improved Toffoli cost
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AbstractThe preparation of quantum states is one of the most fundamental tasks in quantum computing, and a key primitive in many quantum algorithms. Of particular interest to areas such as quantum simulation and linear-system solvers are sparse quantum states, which contain only a small number $s$ of non-zero computational basis states compared to a generic state. In this work, we present an approach that prepares $s$-sparse states on $n$ qubits, reducing the number of Toffoli gates required compared to prior art. We work in the established framework of first preparing a dense state on a $\lceil{\log(s)}\rceil$-qubit sub-register, and then mapping this state to the target state via an isometry, with the latter step dominating the cost of the full algorithm. The speed-up is achieved by designing an efficient algorithm for finding and implementing the isometry. The worst-case Toffoli cost of our isometry circuit, which may be viewed as a batched version of an approach by Malvetti et al., is essentially $2s$ for sufficiently large values of $n$, yielding roughly a $\log(s)/2$ improvement factor over the state-of-the-art. In numerical benchmarks on randomly chosen states, the cost is closer to $s$. With the improved isometry circuit, we examine the dense-state preparation step and present ways to optimize the joint cost of both steps, particularly in the case of target states with purely real coefficients, by outsourcing some sub-tasks from the dense-state preparation to the isometry.► BibTeX data@article{Rupprecht2026sparsequantumstate, doi = {10.22331/q-2026-09-10-2208}, url = {https://doi.org/10.22331/q-2026-09-10-2208}, title = {Sparse quantum state preparation with improved {T}offoli cost}, author = {Rupprecht, Felix and W{\"{o}}lk, Sabine}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2208}, month = sep, year = {2026} }► References [1] Morales, Mauro E. S. and Pira, Lirandë and Schleich, Philipp and Koor, Kelvin and Costa, Pedro C. S. and An, Dong and Aspuru-Guzik, Alán and Lin, Lin and Rebentrost, Patrick and Berry, Dominic W. ``Quantum linear system solvers: A survey of algorithms and applications''. Rev. Mod. Phys. 98, 025005 (2026). https://doi.org/10.1103/x6gh-d8gh [2] Maria Schuld, Ilya Sinayskiy, and Francesco Petruccione. ``An introduction to quantum machine learning''. Contemporary Physics 56, 172–185 (2015). https://doi.org/10.1080/00107514.2014.964942 [3] Dominic W. Berry, Yu Tong, Tanuj Khattar, Alec White, Tae In Kim, Guang Hao Low, Sergio Boixo, Zhiyan Ding, Lin Lin, Seunghoon Lee, Garnet Kin-Lic Chan, Ryan Babbush, and Nicholas C. Rubin. ``Rapid Initial-State Preparation for the Quantum Simulation of Strongly Correlated Molecules''. PRX Quantum 6, 020327 (2025). https://doi.org/10.1103/PRXQuantum.6.020327 [4] Stepan Fomichev, Kasra Hejazi, Modjtaba Shokrian Zini, Matthew Kiser, Joana Fraxanet, Pablo Antonio Moreno Casares, Alain Delgado, Joonsuk Huh, Arne-Christian Voigt, Jonathan E. Mueller, and Juan Miguel Arrazola. ``Initial State Preparation for Quantum Chemistry on Quantum Computers''. PRX Quantum 5, 040339 (2024). https://doi.org/10.1103/PRXQuantum.5.040339 [5] Guang Hao Low, Vadym Kliuchnikov, and Luke Schaeffer. ``Trading T gates for dirty qubits in state preparation and unitary synthesis''. Quantum 8, 1375 (2024). https://doi.org/10.22331/q-2024-06-17-1375 [6] Xiaoming Sun, Guojing Tian, Shuai Yang, Pei Yuan, and Shengyu Zhang. ``Asymptotically Optimal Circuit Depth for Quantum State Preparation and General Unitary Synthesis''. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 42, 3301–3314 (2023). https://doi.org/10.1109/TCAD.2023.3244885 [7] Xiao-Ming Zhang, Tongyang Li, and Xiao Yuan. ``Quantum State Preparation with Optimal Circuit Depth: Implementations and Applications''. Phys. Rev. Lett. 129, 230504 (2022). https://doi.org/10.1103/PhysRevLett.129.230504 [8] Kaiwen Gui, Alexander M. Dalzell, Alessandro Achille, Martin Suchara, and Frederic T. Chong. ``Spacetime-Efficient Low-Depth Quantum State Preparation with Applications''. Quantum 8, 1257 (2024). https://doi.org/10.22331/q-2024-02-15-1257 [9] C. Schön, E. Solano, F. Verstraete, J. I. Cirac, and M. M. Wolf. ``Sequential Generation of Entangled Multiqubit States''. Phys. Rev. Lett. 95, 110503 (2005). https://doi.org/10.1103/PhysRevLett.95.110503 [10] Daniel Malz, Georgios Styliaris, Zhi-Yuan Wei, and J. Ignacio Cirac. ``Preparation of Matrix Product States with Log-Depth Quantum Circuits''. Phys. Rev. Lett. 132, 040404 (2024). https://doi.org/10.1103/PhysRevLett.132.040404 [11] Kevin C. Smith, Abid Khan, Bryan K. Clark, S.M. Girvin, and Tzu-Chieh Wei. ``Constant-Depth Preparation of Matrix Product States with Adaptive Quantum Circuits''. PRX Quantum 5, 030344 (2024). https://doi.org/10.1103/PRXQuantum.5.030344 [12] Rui Mao, Guojing Tian, and Xiaoming Sun. ``Toward optimal circuit size for sparse quantum state preparation''. Phys. Rev. A 110, 032439 (2024). https://doi.org/10.1103/PhysRevA.110.032439 [13] Niels Gleinig and Torsten Hoefler. ``An Efficient Algorithm for Sparse Quantum State Preparation''. In 58th ACM/IEEE Design Automation Conference (DAC). Pages 433–438. (2021). https://doi.org/10.1109/DAC18074.2021.9586240 [14] Debora Ramacciotti, Andreea I. Lefterovici, and Antonio F. Rotundo. ``Simple quantum algorithm to efficiently prepare sparse states''. Phys. Rev. A 110, 032609 (2024). https://doi.org/10.1103/PhysRevA.110.032609 [15] Fereshte Mozafari, Giovanni De Micheli, and Yuxiang Yang. ``Efficient deterministic preparation of quantum states using decision diagrams''. Phys. Rev. A 106, 022617 (2022). https://doi.org/10.1103/PhysRevA.106.022617 [16] Lvzhou Li and Jingquan Luo. ``Nearly Optimal Circuit Size for Sparse Quantum State Preparation''. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Volume 334, pages 113:1–113:19. (2025). https://doi.org/10.4230/LIPIcs.ICALP.2025.113 [17] Daniel Litinski. ``Magic State Distillation: Not as Costly as You Think''. Quantum 3, 205 (2019). https://doi.org/10.22331/q-2019-12-02-205 [18] Craig Gidney, Noah Shutty, and Cody Jones. ``Magic state cultivation: growing T states as cheap as CNOT gates'' (2024). arXiv:2409.17595. arXiv:2409.17595 [19] Michael Beverland, Earl Campbell, Mark Howard, and Vadym Kliuchnikov. ``Lower bounds on the non-Clifford resources for quantum computations''. Quantum Science and Technology 5, 035009 (2020). https://doi.org/10.1088/2058-9565/ab8963 [20] Emanuel Malvetti, Raban Iten, and Roger Colbeck. ``Quantum Circuits for Sparse Isometries''. Quantum 5, 412 (2021). https://doi.org/10.22331/q-2021-03-15-412 [21] Norm M. Tubman, Carlos Mejuto-Zaera, Jeffrey M. Epstein, Diptarka Hait, Daniel S. Levine, William Huggins, Zhang Jiang, Jarrod R. McClean, Ryan Babbush, Martin Head-Gordon, and K. Birgitta Whaley. ``Postponing the orthogonality catastrophe: efficient state preparation for electronic structure simulations on quantum devices'' (2018). arXiv:1809.05523. arXiv:1809.05523 [22] Tiago M. L. de Veras, Leon D. da Silva, and Adenilton J. da Silva. ``Double sparse quantum state preparation''.
Quantum Information Processing 21, 204 (2022). https://doi.org/10.1007/s11128-022-03549-y [23] 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''. Phys. Rev. X 8, 041015 (2018). https://doi.org/10.1103/PhysRevX.8.041015 [24] Craig Gidney. ``Halving the cost of quantum addition''. Quantum 2, 74 (2018). https://doi.org/10.22331/q-2018-06-18-74 [25] Yuval R. Sanders, Dominic W. Berry, Pedro C.S. Costa, Louis W. Tessler, Nathan Wiebe, Craig Gidney, Hartmut Neven, and Ryan Babbush. ``Compilation of Fault-Tolerant Quantum Heuristics for Combinatorial Optimization''. PRX Quantum 1, 020312 (2020). https://doi.org/10.1103/PRXQuantum.1.020312 [26] Neil J. Ross and Peter Selinger. ``Optimal ancilla-free Clifford+T approximation of z-rotations''. Quantum Info. Comput. 16, 901–953 (2016). [27] Vadym Kliuchnikov, Kristin Lauter, Romy Minko, Adam Paetznick, and Christophe Petit. ``Shorter quantum circuits via single-qubit gate approximation''. Quantum 7, 1208 (2023). https://doi.org/10.22331/q-2023-12-18-1208 [28] Renaud Vilmart, Sunheang Ty, and Chetra Mang. ``Resource-Efficient Synthesis of Sparse Quantum States'' (2025). arXiv:2508.05386. arXiv:2508.05386 [29] Matthew P. Harrigan, Tanuj Khattar, Charles Yuan, Anurudh Peduri, Noureldin Yosri, Fionn D. Malone, Ryan Babbush, and Nicholas C. Rubin. ``Expressing and Analyzing Quantum Algorithms with Qualtran'' (2024). arXiv:2409.04643. arXiv:2409.04643 [30] Felix Rupprecht and Sabine Wölk. ``Code and Assets for: Sparse Quantum State Preparation with improved Toffoli cost''. Zenodo (2026). https://doi.org/10.5281/zenodo.18234600 [31] Daniel Litinski and Felix von Oppen. ``Lattice Surgery with a Twist: Simplifying Clifford Gates of Surface Codes''. Quantum 2, 62 (2018). https://doi.org/10.22331/q-2018-05-04-62 [32] Tanuj Khattar and Craig Gidney. ``Rise of conditionally clean ancillae for efficient quantum circuit constructions''. Quantum 9, 1752 (2025). https://doi.org/10.22331/q-2025-05-21-1752 [33] 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). https://doi.org/10.22331/q-2019-12-02-208 [34] Kaavya Sahay, Pei-Kai Tsai, Kathleen (Katie) Chang, Qile Su, Thomas B. Smith, Shraddha Singh, and Shruti Puri. ``Fold-transversal surface code cultivation''. PRX Quantum 7, 033006 (2026). https://doi.org/10.1103/gpvl-lg4c [35] Diego Ruiz, Jérémie Guillaud, Christophe Vuillot, and Mazyar Mirrahimi. ``Unfolded distillation: very low-cost magic state preparation for biased-noise qubits''. npj Quantum Information 12, 53 (2026). https://doi.org/10.1038/s41534-026-01197-z [36] William J. Huggins, Tanuj Khattar, and Nathan Wiebe. ``Productionizing Quantum Mass Production'' (2025). arXiv:2506.00132. arXiv:2506.00132 [37] Sam McArdle, Alexander M. Dalzell, Aleksander Kubica, and Fernando G. S. L. Brandão. ``The Fast for the Curious: How to accelerate fault-tolerant quantum applications'' (2025). arXiv:2510.26078. arXiv:2510.26078 [38] William J. Huggins, Tanuj Khattar, Amanda Xu, Matthew Harrigan, Christopher Kang, Guang Hao Low, Austin Fowler, Nicholas C. Rubin, and Ryan Babbush. ``The FLuid Allocation of Surface code Qubits (FLASQ) cost model for early fault-tolerant quantum algorithms'' (2025). arXiv:2511.08508. arXiv:2511.08508 [39] Tongyang Li, Fengning Ou, Xinzhao Wang, Penghui Yao, Pei Yuan, and Shengyu Zhang. ``Optimal T Counts under Sparsity: from QROM to State Preparation and Block Encoding'' (2026). arXiv:2607.28260. arXiv:2607.28260 [40] Jingquan Luo and Lvzhou Li. ``Sparse Quantum State Preparation with Sublinear T-Count'' (2026). arXiv:2608.00414. arXiv:2608.00414 [41] Benjamin Desef. ``Yquant: Typesetting quantum circuits in a human-readable language'' (2021). arXiv:2007.12931. arXiv:2007.12931Cited byCould not fetch Crossref cited-by data during last attempt 2026-09-10 09:57:32: Could not fetch cited-by data for 10.22331/q-2026-09-10-2208 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-09-10 09:57:33: Cannot retrieve data from ADS due to rate limitations.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. AbstractThe preparation of quantum states is one of the most fundamental tasks in quantum computing, and a key primitive in many quantum algorithms. Of particular interest to areas such as quantum simulation and linear-system solvers are sparse quantum states, which contain only a small number $s$ of non-zero computational basis states compared to a generic state. In this work, we present an approach that prepares $s$-sparse states on $n$ qubits, reducing the number of Toffoli gates required compared to prior art. We work in the established framework of first preparing a dense state on a $\lceil{\log(s)}\rceil$-qubit sub-register, and then mapping this state to the target state via an isometry, with the latter step dominating the cost of the full algorithm. The speed-up is achieved by designing an efficient algorithm for finding and implementing the isometry. The worst-case Toffoli cost of our isometry circuit, which may be viewed as a batched version of an approach by Malvetti et al., is essentially $2s$ for sufficiently large values of $n$, yielding roughly a $\log(s)/2$ improvement factor over the state-of-the-art. In numerical benchmarks on randomly chosen states, the cost is closer to $s$. With the improved isometry circuit, we examine the dense-state preparation step and present ways to optimize the joint cost of both steps, particularly in the case of target states with purely real coefficients, by outsourcing some sub-tasks from the dense-state preparation to the isometry.► BibTeX data@article{Rupprecht2026sparsequantumstate, doi = {10.22331/q-2026-09-10-2208}, url = {https://doi.org/10.22331/q-2026-09-10-2208}, title = {Sparse quantum state preparation with improved {T}offoli cost}, author = {Rupprecht, Felix and W{\"{o}}lk, Sabine}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2208}, month = sep, year = {2026} }► References [1] Morales, Mauro E. S. and Pira, Lirandë and Schleich, Philipp and Koor, Kelvin and Costa, Pedro C. S. and An, Dong and Aspuru-Guzik, Alán and Lin, Lin and Rebentrost, Patrick and Berry, Dominic W. ``Quantum linear system solvers: A survey of algorithms and applications''. Rev. Mod. Phys. 98, 025005 (2026). https://doi.org/10.1103/x6gh-d8gh [2] Maria Schuld, Ilya Sinayskiy, and Francesco Petruccione. ``An introduction to quantum machine learning''. Contemporary Physics 56, 172–185 (2015). https://doi.org/10.1080/00107514.2014.964942 [3] Dominic W. Berry, Yu Tong, Tanuj Khattar, Alec White, Tae In Kim, Guang Hao Low, Sergio Boixo, Zhiyan Ding, Lin Lin, Seunghoon Lee, Garnet Kin-Lic Chan, Ryan Babbush, and Nicholas C. Rubin. ``Rapid Initial-State Preparation for the Quantum Simulation of Strongly Correlated Molecules''. PRX Quantum 6, 020327 (2025). https://doi.org/10.1103/PRXQuantum.6.020327 [4] Stepan Fomichev, Kasra Hejazi, Modjtaba Shokrian Zini, Matthew Kiser, Joana Fraxanet, Pablo Antonio Moreno Casares, Alain Delgado, Joonsuk Huh, Arne-Christian Voigt, Jonathan E. Mueller, and Juan Miguel Arrazola. ``Initial State Preparation for Quantum Chemistry on Quantum Computers''. PRX Quantum 5, 040339 (2024). https://doi.org/10.1103/PRXQuantum.5.040339 [5] Guang Hao Low, Vadym Kliuchnikov, and Luke Schaeffer. ``Trading T gates for dirty qubits in state preparation and unitary synthesis''. Quantum 8, 1375 (2024). https://doi.org/10.22331/q-2024-06-17-1375 [6] Xiaoming Sun, Guojing Tian, Shuai Yang, Pei Yuan, and Shengyu Zhang. ``Asymptotically Optimal Circuit Depth for Quantum State Preparation and General Unitary Synthesis''. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 42, 3301–3314 (2023). https://doi.org/10.1109/TCAD.2023.3244885 [7] Xiao-Ming Zhang, Tongyang Li, and Xiao Yuan. ``Quantum State Preparation with Optimal Circuit Depth: Implementations and Applications''. Phys. Rev. Lett. 129, 230504 (2022). https://doi.org/10.1103/PhysRevLett.129.230504 [8] Kaiwen Gui, Alexander M. Dalzell, Alessandro Achille, Martin Suchara, and Frederic T. Chong. ``Spacetime-Efficient Low-Depth Quantum State Preparation with Applications''. Quantum 8, 1257 (2024). https://doi.org/10.22331/q-2024-02-15-1257 [9] C. Schön, E. Solano, F. Verstraete, J. I. Cirac, and M. M. Wolf. ``Sequential Generation of Entangled Multiqubit States''. Phys. Rev. Lett. 95, 110503 (2005). https://doi.org/10.1103/PhysRevLett.95.110503 [10] Daniel Malz, Georgios Styliaris, Zhi-Yuan Wei, and J. Ignacio Cirac. ``Preparation of Matrix Product States with Log-Depth Quantum Circuits''. Phys. Rev. Lett. 132, 040404 (2024). https://doi.org/10.1103/PhysRevLett.132.040404 [11] Kevin C. Smith, Abid Khan, Bryan K. Clark, S.M. Girvin, and Tzu-Chieh Wei. ``Constant-Depth Preparation of Matrix Product States with Adaptive Quantum Circuits''. PRX Quantum 5, 030344 (2024). https://doi.org/10.1103/PRXQuantum.5.030344 [12] Rui Mao, Guojing Tian, and Xiaoming Sun. ``Toward optimal circuit size for sparse quantum state preparation''. Phys. Rev. A 110, 032439 (2024). https://doi.org/10.1103/PhysRevA.110.032439 [13] Niels Gleinig and Torsten Hoefler. ``An Efficient Algorithm for Sparse Quantum State Preparation''. In 58th ACM/IEEE Design Automation Conference (DAC). Pages 433–438. (2021). https://doi.org/10.1109/DAC18074.2021.9586240 [14] Debora Ramacciotti, Andreea I. Lefterovici, and Antonio F. Rotundo. ``Simple quantum algorithm to efficiently prepare sparse states''. Phys. Rev. A 110, 032609 (2024). https://doi.org/10.1103/PhysRevA.110.032609 [15] Fereshte Mozafari, Giovanni De Micheli, and Yuxiang Yang. ``Efficient deterministic preparation of quantum states using decision diagrams''. Phys. Rev. A 106, 022617 (2022). https://doi.org/10.1103/PhysRevA.106.022617 [16] Lvzhou Li and Jingquan Luo. ``Nearly Optimal Circuit Size for Sparse Quantum State Preparation''. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Volume 334, pages 113:1–113:19. (2025). https://doi.org/10.4230/LIPIcs.ICALP.2025.113 [17] Daniel Litinski. ``Magic State Distillation: Not as Costly as You Think''. Quantum 3, 205 (2019). https://doi.org/10.22331/q-2019-12-02-205 [18] Craig Gidney, Noah Shutty, and Cody Jones. ``Magic state cultivation: growing T states as cheap as CNOT gates'' (2024). arXiv:2409.17595. arXiv:2409.17595 [19] Michael Beverland, Earl Campbell, Mark Howard, and Vadym Kliuchnikov. ``Lower bounds on the non-Clifford resources for quantum computations''. Quantum Science and Technology 5, 035009 (2020). https://doi.org/10.1088/2058-9565/ab8963 [20] Emanuel Malvetti, Raban Iten, and Roger Colbeck. ``Quantum Circuits for Sparse Isometries''. Quantum 5, 412 (2021). https://doi.org/10.22331/q-2021-03-15-412 [21] Norm M. Tubman, Carlos Mejuto-Zaera, Jeffrey M. Epstein, Diptarka Hait, Daniel S. Levine, William Huggins, Zhang Jiang, Jarrod R. McClean, Ryan Babbush, Martin Head-Gordon, and K. Birgitta Whaley. ``Postponing the orthogonality catastrophe: efficient state preparation for electronic structure simulations on quantum devices'' (2018). arXiv:1809.05523. arXiv:1809.05523 [22] Tiago M. L. de Veras, Leon D. da Silva, and Adenilton J. da Silva. ``Double sparse quantum state preparation''.
Quantum Information Processing 21, 204 (2022). https://doi.org/10.1007/s11128-022-03549-y [23] 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''. Phys. Rev. X 8, 041015 (2018). https://doi.org/10.1103/PhysRevX.8.041015 [24] Craig Gidney. ``Halving the cost of quantum addition''. Quantum 2, 74 (2018). https://doi.org/10.22331/q-2018-06-18-74 [25] Yuval R. Sanders, Dominic W. Berry, Pedro C.S. Costa, Louis W. Tessler, Nathan Wiebe, Craig Gidney, Hartmut Neven, and Ryan Babbush. ``Compilation of Fault-Tolerant Quantum Heuristics for Combinatorial Optimization''. PRX Quantum 1, 020312 (2020). https://doi.org/10.1103/PRXQuantum.1.020312 [26] Neil J. Ross and Peter Selinger. ``Optimal ancilla-free Clifford+T approximation of z-rotations''. Quantum Info. Comput. 16, 901–953 (2016). [27] Vadym Kliuchnikov, Kristin Lauter, Romy Minko, Adam Paetznick, and Christophe Petit. ``Shorter quantum circuits via single-qubit gate approximation''. Quantum 7, 1208 (2023). https://doi.org/10.22331/q-2023-12-18-1208 [28] Renaud Vilmart, Sunheang Ty, and Chetra Mang. ``Resource-Efficient Synthesis of Sparse Quantum States'' (2025). arXiv:2508.05386. arXiv:2508.05386 [29] Matthew P. Harrigan, Tanuj Khattar, Charles Yuan, Anurudh Peduri, Noureldin Yosri, Fionn D. Malone, Ryan Babbush, and Nicholas C. Rubin. ``Expressing and Analyzing Quantum Algorithms with Qualtran'' (2024). arXiv:2409.04643. arXiv:2409.04643 [30] Felix Rupprecht and Sabine Wölk. ``Code and Assets for: Sparse Quantum State Preparation with improved Toffoli cost''. Zenodo (2026). https://doi.org/10.5281/zenodo.18234600 [31] Daniel Litinski and Felix von Oppen. ``Lattice Surgery with a Twist: Simplifying Clifford Gates of Surface Codes''. Quantum 2, 62 (2018). https://doi.org/10.22331/q-2018-05-04-62 [32] Tanuj Khattar and Craig Gidney. ``Rise of conditionally clean ancillae for efficient quantum circuit constructions''. Quantum 9, 1752 (2025). https://doi.org/10.22331/q-2025-05-21-1752 [33] 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). https://doi.org/10.22331/q-2019-12-02-208 [34] Kaavya Sahay, Pei-Kai Tsai, Kathleen (Katie) Chang, Qile Su, Thomas B. Smith, Shraddha Singh, and Shruti Puri. ``Fold-transversal surface code cultivation''. PRX Quantum 7, 033006 (2026). https://doi.org/10.1103/gpvl-lg4c [35] Diego Ruiz, Jérémie Guillaud, Christophe Vuillot, and Mazyar Mirrahimi. ``Unfolded distillation: very low-cost magic state preparation for biased-noise qubits''. npj Quantum Information 12, 53 (2026). https://doi.org/10.1038/s41534-026-01197-z [36] William J. Huggins, Tanuj Khattar, and Nathan Wiebe. ``Productionizing Quantum Mass Production'' (2025). arXiv:2506.00132. arXiv:2506.00132 [37] Sam McArdle, Alexander M. Dalzell, Aleksander Kubica, and Fernando G. S. L. Brandão. ``The Fast for the Curious: How to accelerate fault-tolerant quantum applications'' (2025). arXiv:2510.26078. arXiv:2510.26078 [38] William J. Huggins, Tanuj Khattar, Amanda Xu, Matthew Harrigan, Christopher Kang, Guang Hao Low, Austin Fowler, Nicholas C. Rubin, and Ryan Babbush. ``The FLuid Allocation of Surface code Qubits (FLASQ) cost model for early fault-tolerant quantum algorithms'' (2025). arXiv:2511.08508. arXiv:2511.08508 [39] Tongyang Li, Fengning Ou, Xinzhao Wang, Penghui Yao, Pei Yuan, and Shengyu Zhang. ``Optimal T Counts under Sparsity: from QROM to State Preparation and Block Encoding'' (2026). arXiv:2607.28260. arXiv:2607.28260 [40] Jingquan Luo and Lvzhou Li. ``Sparse Quantum State Preparation with Sublinear T-Count'' (2026). arXiv:2608.00414. arXiv:2608.00414 [41] Benjamin Desef. ``Yquant: Typesetting quantum circuits in a human-readable language'' (2021). arXiv:2007.12931. arXiv:2007.12931Cited byCould not fetch Crossref cited-by data during last attempt 2026-09-10 09:57:32: Could not fetch cited-by data for 10.22331/q-2026-09-10-2208 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-09-10 09:57:33: Cannot retrieve data from ADS due to rate limitations.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.
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