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arXiv:2512.13908 · Science

Magic state cultivation on a superconducting quantum processor

Emma Rosenfeld, Craig Gidney, Gabrielle Roberts, Alexis Morvan, Nathan Lacroix, Dvir Kafri, Jeffrey Marshall, Ming Li, Volodymyr Sivak, Dmitry Abanin, Amira Abbas, Rajeev Acharya, Laleh Aghababaie Beni, Georg Aigeldinger, Ross Alcaraz, Sayra Alcaraz, Trond I. Andersen, Markus Ansmann, Frank Arute, Kunal Arya, Walt Askew, Nikita Astrakhantsev, Juan Atalaya, Ryan Babbush, Brian Ballard, Joseph C. Bardin, Hector Bates, Andreas Bengtsson, Majid Bigdeli Karimi, Alexander Bilmes, Simon Bilodeau, Felix Borjans, Jenna Bovaird, Dylan Bowers, Leon Brill, Peter Brooks, Michael Broughton, David A. Browne, Brett Buchea, Bob B. Buckley, Tim Burger, Brian Burkett, Nicholas Bushnell, Jamal Busnaina, Anthony Cabrera, Juan Campero, Hung-Shen Chang, Silas Chen, Zijun Chen, Ben Chiaro, Liang-Ying Chih, Agnetta Y. Cleland, Bryan Cochrane, Matt Cockrell, Josh Cogan, Paul Conner, Harold Cook, Rodrigo G. Cortiñas, William Courtney, Alexander L. Crook, Ben Curtin, Martin Damyanov, Sayan Das, Dripto M. Debroy, Sean Demura, Paul Donohoe, Ilya Drozdov, Andrew Dunsworth, Valerie Ehimhen, Alec Eickbusch, Aviv Moshe Elbag, Lior Ella, Mahmoud Elzouka, David Enriquez, Catherine Erickson, Lara Faoro, Vinicius S. Ferreira, Marcos Flores, Leslie Flores Burgos, Sam Fontes, Ebrahim Forati, Jeremiah Ford, Brooks Foxen, Masaya Fukami, Alan Wing Lun Fung, Lenny Fuste, Suhas Ganjam, Gonzalo Garcia, Christopher Garrick, Robert Gasca, Helge Gehring, Robert Geiger, Élie Genois, William Giang, Dar Gilboa, James E. Goeders, Edward C. Gonzales, Raja Gosula, Stijn J. de Graaf, Alejandro Grajales Dau, Dietrich Graumann, Joel Grebel, Alex Greene, Jonathan A. Gross, Jose Guerrero, Loïck Le Guevel, Tan Ha, Steve Habegger, Tanner Hadick, Ali Hadjikhani, Michael C. Hamilton, Monica Hansen, Matthew P. Harrigan, Sean D. Harrington, Jeanne Hartshorn, Stephen Heslin, Paula Heu, Oscar Higgott, Reno Hiltermann, Jeremy Hilton, Hsin-Yuan Huang, Mike Hucka, Christopher Hudspeth, Ashley Huff, William J. Huggins, Lev B. Ioffe, Evan Jeffrey, Shaun Jevons, Zhang Jiang, Xiaoxuan Jin, Chaitali Joshi, Pavol Juhas, Andreas Kabel, Hui Kang, Kiseo Kang, Amir H. Karamlou, Ryan Kaufman, Kostyantyn Kechedzhi, Tanuj Khattar, Mostafa Khezri, Seon Kim, Paul V. Klimov, Can M. Knaut, Bryce Kobrin, Alexander N. Korotkov, Fedor Kostritsa, John Mark Kreikebaum, Ryuho Kudo, Ben Kueffler, Arun Kumar, Vladislav D. Kurilovich, Vitali Kutsko, Tiano Lange-Dei, Brandon W. Langley, Pavel Laptev, Kim-Ming Lau, Emma Leavell, Justin Ledford, Joy Lee, Kenny Lee, Brian J. Lester, Wendy Leung, Lily Li, Wing Yan Li, Alexander T. Lill, William P. Livingston, Matthew T. Lloyd, Aditya Locharla, Laura De Lorenzo, Erik Lucero, Daniel Lundahl, Aaron Lunt, Sid Madhuk, Aniket Maiti, Ashley Maloney, Salvatore Mandrà, Leigh S. Martin, Orion Martin, Eric Mascot, Paul Masih Das, Dmitri Maslov, Melvin Mathews, Cameron Maxfield, Jarrod R. McClean, Matt McEwen, Seneca Meeks, Anthony Megrant, Kevin C. Miao, Zlatko K. Minev, Reza Molavi, Sebastian Molina, Shirin Montazeri, Charles Neill, Michael Newman, Anthony Nguyen, Murray Nguyen, Chia-Hung Ni, Murphy Yuezhen Niu, Nicholas Noll, Logan Oas, William D. Oliver, Raymond Orosco, Kristoffer Ottosson, Alice Pagano, Agustin Di Paolo, Sherman Peek, David Peterson, Alex Pizzuto, Elias Portoles, Rebecca Potter, Orion Pritchard, Michael Qian, Chris Quintana, Ganesh Ramachandran, Arpit Ranadive, Matthew J. Reagor, Rachel Resnick, David M. Rhodes, Daniel Riley, Roberto Rodriguez, Emma Ropes, Lucia B. De Rose, Eliott Rosenberg, Dario Rosenstock, Elizabeth Rossi, Pedram Roushan, David A. Rower, Robert Salazar, Kannan Sankaragomathi, Murat Can Sarihan, Max Schaefer, Sebastian Schroeder, Henry F. Schurkus, Aria Shahingohar, Michael J. Shearn, Aaron Shorter, Noah Shutty, Vladimir Shvarts, Spencer Small, W. Clarke Smith, David A. Sobel, Barrett Spells, Sofia Springer, George Sterling, Jordan Suchard, Aaron Szasz, Alexander Sztein, Madeline Taylor, Jothi Priyanka Thiruraman, Douglas Thor, Dogan Timucin, Eifu Tomita, Alfredo Torres, M. Mert Torunbalci, Hao Tran, Abeer Vaishnav, Justin Vargas, Sergey Vdovichev, Guifre Vidal, Benjamin Villalonga, Catherine Vollgraff Heidweiller, Meghan Voorhees, Steven Waltman, Jonathan Waltz, Shannon X. Wang, Danni Wang, Brayden Ware, James D. Watson, Yonghua Wei, Travis Weidel, Theodore White, Kristi Wong, Bryan W. K. Woo, Christopher J. Wood, Maddy Woodson, Cheng Xing, Z. Jamie Yao, Ping Yeh, Bicheng Ying, Juhwan Yoo, Noureldin Yosri, Elliot Young, Grayson Young, Adam Zalcman, Ran Zhang, Yaxing Zhang, Ningfeng Zhu, Nicholas Zobrist, Zhenjie Zou, Hartmut Neven, Sergio Boixo, Cody Jones, Julian Kelly, Alexandre Bourassa, Kevin J. Satzinger

quant-phSubmitted Dec 15, 2025• Announced Dec 17, 2025• 0 citations
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Abstract

Fault-tolerant quantum computing requires a universal gate set, but the necessary non-Clifford gates represent a significant resource cost for most quantum error correction architectures. Magic state cultivation offers an efficient alternative to resource-intensive distillation protocols; however, testing the proposal's assumptions represents a challenging departure from quantum memory experiments. We present an experimental study of magic state cultivation on a superconducting quantum processor. We implement cultivation, including code-switching into a surface code, and develop a fault-tolerant measurement protocol to bound the magic state fidelity. Cultivation reduces the error by a factor of 40, with a state fidelity of 0.9999(1) (retaining 8% of attempts). Our results experimentally establish magic state cultivation as a viable solution to one of quantum computing's most significant challenges.

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Magic-state preparation is a central component of fault-tolerant quantum computing. Recent theoretical and experimental successes in code-switch-based magic-state preparation have underscored the promise of these methods for quantum error correction. Similarly, magic-state cultivation has likewise been demonstrated in both numerical and experimental settings. However, a thorough comparison between magic-state cultivation and code-switch-based magic-state factories is still missing. In this work, we carry out end-to-end simulations of magic-state preparation using code switching and compare its resource requirements and performance against magic-state cultivation. As part of this analysis, we develop a lattice-surgery protocol for transfer between the doubled color code and the rotated surface code. We extend the complete code-switching protocol to the $d=5$ doubled color code and perform the corresponding end-to-end simulations. Finally, we propose two fault-tolerant magic-state preparation protocols that combine phase-kickback checks with a transversal non-Clifford gate.

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Fault-tolerant quantum computing requires a universal gate set, but the necessary non-Clifford gates represent a significant resource cost for most quantum error correction architectures. Magic state cultivation offers an efficient alternative to resource-intensive distillation protocols; however, testing the proposal's assumptions represents a challenging departure from quantum memory experiments. We present an experimental study of magic state cultivation on a superconducting quantum processor. We implement cultivation, including code-switching into a surface code, and develop a fault-tolerant measurement protocol to bound the magic state fidelity. Cultivation reduces the error by a factor of 40, with a state fidelity of 0.9999(1) (retaining 8% of attempts). Our results experimentally establish magic state cultivation as a viable solution to one of quantum computing's most significant challenges.
Magic-state preparation is a central component of fault-tolerant quantum computing. Recent theoretical and experimental successes in code-switch-based magic-state preparation have underscored the promise of these methods for quantum error correction. Similarly, magic-state cultivation has likewise been demonstrated in both numerical and experimental settings. However, a thorough comparison between magic-state cultivation and code-switch-based magic-state factories is still missing. In this work, we carry out end-to-end simulations of magic-state preparation using code switching and compare its resource requirements and performance against magic-state cultivation. As part of this analysis, we develop a lattice-surgery protocol for transfer between the doubled color code and the rotated surface code. We extend the complete code-switching protocol to the d=5d=5d=5 doubled color code and perform the corresponding end-to-end simulations. Finally, we propose two fault-tolerant magic-state preparation protocols that combine phase-kickback checks with a transversal non-Clifford gate.
Fault-tolerant quantum computing based on surface code has emerged as an attractive candidate for practical large-scale quantum computers to achieve robust noise resistance. To achieve universality, magic states preparation is a commonly approach for introducing non-Clifford gates. Here, we present a hardware-efficient and scalable protocol for arbitrary logical state preparation for the rotated surface code, and further experimentally implement it on the Zuchongzhi 2.1 superconducting quantum processor. An average of 0.8983±0.0002 logical fidelity at different logical states with distance three is achieved, taking into account both state preparation and measurement errors. In particular, the logical magic states |A^{π/4}⟩_{L}, |H⟩_{L}, and |T⟩_{L} are prepared nondestructively with logical fidelities of 0.8771±0.0009, 0.9090±0.0009, and 0.8890±0.0010, respectively, which are higher than the state distillation protocol threshold, 0.859 (for H-type magic state) and 0.827 (for T-type magic state). Our work provides a viable and efficient avenue for generating high-fidelity raw logical magic states, which is essential for realizing non-Clifford logical gates in the surface code.
Fault-tolerant implementation of non-Clifford gates is a major challenge for achieving universal fault-tolerant quantum computing with quantum error-correcting codes. Magic state distillation is the most well-studied method for this but requires significant resources. Hence, it is crucial to tailor and optimize magic state distillation for specific codes from both logical- and physical-level perspectives. In this work, we perform such optimization for two-dimensional color codes, which are promising due to their higher encoding rates compared to surface codes, transversal implementation of Clifford gates, and efficient lattice surgery. We propose two carefully designed distillation schemes based on the 15-to-1 distillation circuit and lattice surgery, differing in their methods for handling faulty rotations. Our first scheme employs faulty T measurement, achieving infidelities of O ( p 3 ) for physical noise strength p . To achieve lower infidelities, our second scheme integrates distillation with “cultivation” (a distillation-free approach to fault tolerantly prepare magic states through transversal Clifford measurements). Our second scheme achieves significantly lower infidelities (e.g., approximately 2 × 10 − 16 at p = 10 − 3 ), surpassing the capabilities of both cultivation and single-level distillation. Notably, to reach a given target infidelity, our schemes require approximately 2 orders of magnitude fewer resources than the previous best magic-state-distillation schemes for color codes.
Efficient Magic State Cultivation for T\sqrt{T}T​ Gates
Recently, experimental and theoretical quantum error correction methodology has seen remarkable breakthroughs. In particular, magic state cultivation has been shown to simplify magic-state preparation and make it feasible for near-term devices. However, recent research on magic state cultivation has focused primarily on the cultivation of T∣+>LT\left| + \right>_LT∣+⟩L​. Only a few other magic state cultivation methods beyond T∣+>LT\left| + \right>_LT∣+⟩L​ have been investigated. Here, we generalize phase kickback checks for magic states at arbitrary Clifford hierarchy levels in specific codes. We provide an example of cultivation of T∣+>L\sqrt{T}\left| + \right>_LT​∣+⟩L​ in the doubled color code and the corresponding escape strategy using lattice surgery from the color code to large rotated surface codes. Using state vector simulation for un-grown cultivation, we observe a strong consistence between S∣+>LS\left| + \right>_LS∣+⟩L​ and T∣+>L\sqrt{T}\left| + \right>_LT​∣+⟩L​ cultivation's performance on the doubled color code. Finally, we discuss the application of the corresponding T∣+>L\sqrt{T}\left| + \right>_LT​∣+⟩L​ cultivation, incorporating the STAR architecture and TTT gates, for early fault-tolerant quantum computing and its potential to shorten gate synthesis in the fully fault-tolerant quantum computing era.
Magic state cultivation is a promising method for preparing high-fidelity non-Clifford resource states - a central component of fault-tolerant quantum computing - with low spacetime overhead. In benchmarking cultivation, existing resource estimates assume access to noiseless transversal measurement of the output magic state to allow for postselection during the final escape stage of the cultivation process. However, such measurements would destroy the magic state and are therefore unavailable during computation. We develop an open-boundary postselection method, Caliper, that uses only nondestructive mid-circuit syndrome information and substantially outperforms existing methods operating under these constraints. In some regimes, our postselection method uniquely recovers the resource efficiency of prior estimates. However, in other cases, we observe a tradeoff. At spacetime costs reported in previous works, the achievable logical error rates can be orders of magnitude higher, and recovering the reported logical error rate requires substantially larger codes and longer escape stages. Our results highlight how practical considerations involving mid-circuit postselection can impact the design and performance of magic state cultivation protocols.
Magic state cultivation is a leading approach for generating the resource states required for fault-tolerant quantum computation. Here we present a cultivation protocol that increases the success probability of magic state generation in platforms with flexible and nonlocal connectivity. Our method implements cultivation directly on the surface code, avoiding the detour through alternative, less efficient error-correcting codes used in prior approaches. This both improves the acceptance rate and preserves compatibility with the geometry of the code and the hardware. Numerical simulations show that our protocols improve success probabilities and reduce output error rates compared with protocols tailored to locally connected platforms. Under realistic noise models for cold-atom and trapped-ion systems, we improve the rate of magic state generation by more than a factor of 20. Finally, we study qubit loss and erasure and show that very low error rates can be achieved with minimal overhead, reaching below 10 − 6 infidelity using only nine physical erasure qubits.