quantum-computing

Impact of Clifford operations on non-stabilizing power and quantum chaos

Quantum Journal

44 min read

0 likes

⚡ Quantum Brief

Researchers have established a direct mathematical relationship between the total non-stabilizing power in quantum circuits and the individual contributions of non-Clifford gates when interspersed with random Clifford operations. The study reveals that non-stabilizerness thermalizes to a universal Haar-averaged value in generic circuits, approaching equilibrium exponentially with circuit depth—mirroring how physical systems reach thermal equilibrium. Two-qubit gate analysis demonstrates this thermalization in tractable systems, while operator-space non-stabilizerness behavior is explored in physical models, providing concrete examples of the theoretical framework. The work links non-stabilizing power to quantum chaos emergence in brick-wall circuits, showing how magic and entanglement jointly govern chaotic dynamics, offering potential control mechanisms for quantum algorithms. Practical applications include quantum device benchmarking, error-correction optimization, and algorithm design, with insights into fine-tuning chaotic evolution for enhanced randomness generation.

Impact of Clifford operations on non-stabilizing power and quantum chaos

Summarize this article with:

AbstractNon-stabilizerness, alongside entanglement, is a crucial ingredient for fault-tolerant quantum computation and achieving a genuine quantum advantage. Despite recent progress, a complete understanding of the generation and thermalization of non-stabilizerness in circuits that mix Clifford and non-Clifford operations remains elusive. While Clifford operations do not generate non-stabilizerness, their interplay with non-Clifford gates can strongly impact the overall non-stabilizing dynamics of generic quantum circuits. In this work, we establish a direct relationship between the final non-stabilizing power and the individual powers of the non-Clifford gates, in circuits where these gates are interspersed with random Clifford operations. By leveraging this result, we unveil the thermalization of non-stabilizing power to its Haar-averaged value in generic circuits. As a precursor, we analyze two-qubit gates and illustrate this thermalization in analytically tractable systems. Extending this, we explore the operator-space non-stabilizing power and demonstrate its behavior in physical models. Finally, we examine the role of non-stabilizing power in the emergence of quantum chaos in brick-wall quantum circuits. Our work elucidates how non-stabilizing dynamics evolve and thermalize in quantum circuits and thus contributes to a better understanding of quantum computational resources and of their role in quantum chaos.Popular summaryQuantum technologies promise capacities widely beyond the grasp of classical hardware. To achieve this, specifically quantum mechanical “resources” are required, a prime example being entanglement. Another equally important resource is non-stabilizerness, often called “magic”, i.e., the capacity to produce non-Clifford dynamics in quantum circuits. Its presence makes quantum circuits exceedingly difficult to simulate on classical computers and is therefore essential for achieving genuine quantum advantage. However, how non-stabilizerness builds up and evolves in realistic quantum circuits remains an outstanding question. Even though Clifford operations cannot create non-stabilizerness, they play a crucial role in shaping how non-stabilizerness develops when combined with non-Clifford gates. Surprisingly, random Clifford operations can enhance, redistribute, or even suppress the build-up of magic, depending on the situation. This work identifies a simple and elegant rule that determines the total amount of non-stabilizerness directly from the individual contributions of the non-Clifford gates, in circuits where these gates appear alongside random Clifford operations. As quantum circuits grow deeper, non-stabilizerness naturally approaches a universal equilibrium value, similar to how physical systems relax toward thermal equilibrium. This relaxation occurs exponentially with circuit depth. This work further provides evidence that the emergence of quantum chaos, a hallmark of quantum complexity, is governed by the mutual influence of non-stabilizerness and entanglement. This behavior provides means to fine tune quantum circuit dynamics, for example, to control chaotic evolution and optimize randomness generation. The insights presented here have practical relevance for benchmarking quantum devices, quantum error-correction strategies, and guiding the design of quantum algorithms.► BibTeX data@article{Varikuti2026impactofclifford, doi = {10.22331/q-2026-03-10-2017}, url = {https://doi.org/10.22331/q-2026-03-10-2017}, title = {Impact of {C}lifford operations on non-stabilizing power and quantum chaos}, author = {Varikuti, Naga Dileep and Bandyopadhyay, Soumik and Hauke, Philipp}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2017}, month = mar, year = {2026} }► References [1] Alán Aspuru-Guzik, Anthony D Dutoi, Peter J Love, and Martin Head-Gordon. Simulated quantum computation of molecular energies. Science, 309 (5741): 1704–1707, 2005. 10.1126/science.1113479. https://doi.org/10.1126/science.1113479 [2] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. Quantum entanglement. Rev. Mod. Phys., 81: 865–942, 2009. 10.1103/RevModPhys.81.865. https://doi.org/10.1103/RevModPhys.81.865 [3] Animesh Datta, Anil Shaji, and Carlton M Caves. Quantum discord and the power of one qubit. Physical review letters, 100 (5): 050502, 2008. 10.1103/PhysRevLett.100.050502. https://doi.org/10.1103/PhysRevLett.100.050502 [4] Earl T Campbell, Barbara M Terhal, and Christophe Vuillot. Roads towards fault-tolerant universal quantum computation. Nature, 549 (7671): 172–179, 2017. 10.1038/nature23460. https://doi.org/10.1038/nature23460 [5] John Preskill. Quantum computing in the nisq era and beyond. Quantum, 2: 79, 2018. 10.22331/q-2018-08-06-79. https://doi.org/10.22331/q-2018-08-06-79 [6] Eric Chitambar and Gilad Gour. Quantum resource theories. Rev. Mod. Phys., 91: 025001, 2019. 10.1103/RevModPhys.91.025001. https://doi.org/10.1103/RevModPhys.91.025001 [7] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. Quantum-enhanced measurements: beating the standard quantum limit. Science, 306 (5700): 1330–1336, 2004. 10.1126/science.1104149. https://doi.org/10.1126/science.1104149 [8] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. Quantum metrology. Physical review letters, 96 (1): 010401, 2006. 10.1103/PhysRevLett.96.010401. https://doi.org/10.1103/PhysRevLett.96.010401 [9] C. L. Degen, F. Reinhard, and P. Cappellaro. Quantum sensing. Rev. Mod. Phys., 89: 035002, 2017. 10.1103/RevModPhys.89.035002. https://doi.org/10.1103/RevModPhys.89.035002 [10] Philipp Hauke, Markus Heyl, Luca Tagliacozzo, and Peter Zoller. Measuring multipartite entanglement through dynamic susceptibilities. Nature Physics, 12 (8): 778–782, 2016. 10.1038/nphys3700. https://doi.org/10.1038/nphys3700 [11] Luca Pezzè, Augusto Smerzi, Markus K. Oberthaler, Roman Schmied, and Philipp Treutlein. Quantum metrology with nonclassical states of atomic ensembles. Rev. Mod. Phys., 90: 035005, 2018. 10.1103/RevModPhys.90.035005. https://doi.org/10.1103/RevModPhys.90.035005 [12] Charles H Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K Wootters. Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels. Physical review letters, 70 (13): 1895, 1993. 10.1103/PhysRevLett.70.1895. https://doi.org/10.1103/PhysRevLett.70.1895 [13] Peter W Shor. Scheme for reducing decoherence in quantum computer memory. Physical review A, 52 (4): R2493, 1995. 10.1103/PhysRevA.52.R2493. https://doi.org/10.1103/PhysRevA.52.R2493 [14] Barbara M. Terhal. Quantum error correction for quantum memories. Rev. Mod. Phys., 87: 307–346, 2015. 10.1103/RevModPhys.87.307. https://doi.org/10.1103/RevModPhys.87.307 [15] Joschka Roffe. Quantum error correction: an introductory guide. Contemporary Physics, 60 (3): 226–245, 2019. 10.1080/00107514.2019.1667078. https://doi.org/10.1080/00107514.2019.1667078 [16] Philipp Hauke, Lars Bonnes, Markus Heyl, and Wolfgang Lechner. Probing entanglement in adiabatic quantum optimization with trapped ions. Frontiers in Physics, 3: 21, 2015. 10.3389/fphy.2015.00021. https://doi.org/10.3389/fphy.2015.00021 [17] Gopal Chandra Santra, Sudipto Singha Roy, Daniel J. Egger, and Philipp Hauke. Genuine multipartite entanglement in quantum optimization. Phys. Rev. A, 111: 022434, 2025a. 10.1103/PhysRevA.111.022434. https://doi.org/10.1103/PhysRevA.111.022434 [18] Gopal Chandra Santra, Fred Jendrzejewski, Philipp Hauke, and Daniel J. Egger. Squeezing and quantum approximate optimization. Phys. Rev. A, 109: 012413, 2024. 10.1103/PhysRevA.109.012413. https://doi.org/10.1103/PhysRevA.109.012413 [19] Yanzhu Chen, Linghua Zhu, Nicholas J. Mayhall, Edwin Barnes, and Sophia E. Economou. How much entanglement do quantum optimization algorithms require? In Quantum 2.0 Conference and Exhibition, page QM4A.2.

Optica Publishing Group, 2022. 10.1364/QUANTUM.2022.QM4A.2. https://doi.org/10.1364/QUANTUM.2022.QM4A.2 [20] Daniel Gottesman. The heisenberg representation of quantum computers. arXiv preprint quant-ph/9807006, 1998. 10.48550/arXiv.quant-ph/9807006. https://doi.org/10.48550/arXiv.quant-ph/9807006 arXiv:quant-ph/9807006 [21] Scott Aaronson and Daniel Gottesman. Improved simulation of stabilizer circuits. Physical Review Aâ€”Atomic, Molecular, and Optical Physics, 70 (5): 052328, 2004. 10.1103/PhysRevA.70.052328. https://doi.org/10.1103/PhysRevA.70.052328 [22] Sergey Bravyi and Alexei Kitaev. Universal quantum computation with ideal clifford gates and noisy ancillas. Physical Review Aâ€”Atomic, Molecular, and Optical Physics, 71 (2): 022316, 2005. 10.1103/PhysRevA.71.022316. https://doi.org/10.1103/PhysRevA.71.022316 [23] Sergey Bravyi and Jeongwan Haah. Magic-state distillation with low overhead. Physical Review Aâ€”Atomic, Molecular, and Optical Physics, 86 (5): 052329, 2012. 10.1103/PhysRevA.86.052329. https://doi.org/10.1103/PhysRevA.86.052329 [24] Shiyu Zhou, Zhicheng Yang, Alioscia Hamma, and Claudio Chamon. Single t gate in a clifford circuit drives transition to universal entanglement spectrum statistics. SciPost Physics, 9 (6): 087, 2020. 10.21468/SciPostPhys.9.6.087. https://doi.org/10.21468/SciPostPhys.9.6.087 [25] Zi-Wen Liu and Andreas Winter. Many-body quantum magic. PRX Quantum, 3 (2): 020333, 2022. 10.1103/PRXQuantum.3.020333. https://doi.org/10.1103/PRXQuantum.3.020333 [26] Pradeep Niroula, Christopher David White, Qingfeng Wang, Sonika Johri, Daiwei Zhu, Christopher Monroe, Crystal Noel, and Michael J Gullans. Phase transition in magic with random quantum circuits. Nature physics, 20 (11): 1786–1792, 2024. 10.1038/s41567-024-02637-3. https://doi.org/10.1038/s41567-024-02637-3 [27] Alberto Giuseppe Catalano, Jovan Odavić, Gianpaolo Torre, Alioscia Hamma, Fabio Franchini, and Salvatore Marco Giampaolo. Magic phase transition and non-local complexity in generalized w state. SciPost Physics Core, 8 (4): 078, 2025. 10.21468/SciPostPhysCore.8.4.078. https://doi.org/10.21468/SciPostPhysCore.8.4.078 [28] Kaifeng Bu, Roy J Garcia, Arthur Jaffe, Dax Enshan Koh, and Lu Li. Complexity of quantum circuits via sensitivity, magic, and coherence. Communications in Mathematical Physics, 405 (7): 161, 2024. 10.1007/s00220-024-05030-6. https://doi.org/10.1007/s00220-024-05030-6 [29] Earl T Campbell. Catalysis and activation of magic states in fault-tolerant architectures. Physical Review Aâ€”Atomic, Molecular, and Optical Physics, 83 (3): 032317, 2011. 10.1103/PhysRevA.83.032317. https://doi.org/10.1103/PhysRevA.83.032317 [30] Xhek Turkeshi, Marco Schirò, and Piotr Sierant. Measuring nonstabilizerness via multifractal flatness. Physical Review A, 108 (4): 042408, 2023. 10.1103/PhysRevA.108.042408. https://doi.org/10.1103/PhysRevA.108.042408 [31] Tobias Haug and Lorenzo Piroli. Quantifying nonstabilizerness of matrix product states. Physical Review B, 107 (3): 035148, 2023. 10.1103/PhysRevB.107.035148. https://doi.org/10.1103/PhysRevB.107.035148 [32] Guglielmo Lami and Mario Collura. Nonstabilizerness via perfect pauli sampling of matrix product states. Phys. Rev. Lett., 131: 180401, 2023. 10.1103/PhysRevLett.131.180401. https://doi.org/10.1103/PhysRevLett.131.180401 [33] Poetri Sonya Tarabunga, Emanuele Tirrito, Mari Carmen Bañuls, and Marcello Dalmonte. Nonstabilizerness via matrix product states in the pauli basis.

Physical Review Letters, 133 (1): 010601, 2024. 10.1103/PhysRevLett.133.010601. https://doi.org/10.1103/PhysRevLett.133.010601 [34] Poetri Sonya Tarabunga and Claudio Castelnovo. Magic in generalized rokhsar-kivelson wavefunctions. Quantum, 8: 1347, 2024. 10.22331/q-2024-05-14-1347. https://doi.org/10.22331/q-2024-05-14-1347 [35] Xhek Turkeshi, Emanuele Tirrito, and Piotr Sierant. Magic spreading in random quantum circuits. Nature Communications, 16 (1): 2575, 2025. 10.1038/s41467-025-57704-x. https://doi.org/10.1038/s41467-025-57704-x [36] Jordi Arnau Montana López and Pavel Kos. Exact solution of long-range stabilizer rényi entropy in the dual-unitary xxz model. Journal of Physics A: Mathematical and Theoretical, 57 (47): 475301, 2024. 10.1088/1751-8121/ad85b0. https://doi.org/10.1088/1751-8121/ad85b0 [37] J. Odavić, M. Viscardi, and A. Hamma. Stabilizer entropy in nonintegrable quantum evolutions. Phys. Rev. B, 112: 104301, 2025. 10.1103/y9r6-dx7p. https://doi.org/10.1103/y9r6-dx7p [38] Zong-Yue Hou, ChunJun Cao, and Zhi-Cheng Yang. Stabilizer entanglement enhances magic injection. arXiv preprint arXiv:2503.20873, 2025. arXiv:2503.20873 [39] Faidon Andreadakis and Paolo Zanardi. Exact link between nonlocal nonstabilizerness and operator entanglement. Phys. Rev. A, 113: L010404, 2026. 10.1103/9x56-2b45. https://doi.org/10.1103/9x56-2b45 [40] Neil Dowling, Kavan Modi, and Gregory A. L. White. Bridging entanglement and magic resources within operator space. Phys. Rev. Lett., 135: 160201, 2025a. 10.1103/c7k1-xcwy. https://doi.org/10.1103/c7k1-xcwy [41] Caroline E. P. Robin. Stabilizer-accelerated quantum many-body ground-state estimation. Phys. Rev. A, 112: 052408, 2025. 10.1103/5qr5-7jkz. https://doi.org/10.1103/5qr5-7jkz [42] Caroline E. P. Robin and Martin J. Savage. Quantum complexity fluctuations from nuclear and hypernuclear forces. Phys. Rev. C, 112: 044004, 2025. 10.1103/r8rq-y9tb. https://doi.org/10.1103/r8rq-y9tb [43] Emanuele Tirrito, Xhek Turkeshi, and Piotr Sierant. Anticoncentration and nonstabilizerness spreading under ergodic quantum dynamics. Phys. Rev. Lett., 135: 220401, 2025. 10.1103/1jzy-sk9r. https://doi.org/10.1103/1jzy-sk9r [44] Mircea Bejan, Campbell McLauchlan, and Benjamin Béri. Dynamical magic transitions in monitored clifford+$t$ circuits. PRX Quantum, 5: 030332, 2024. 10.1103/PRXQuantum.5.030332. https://doi.org/10.1103/PRXQuantum.5.030332 [45] Bhargavi Jonnadula, Prabha Mandayam, Karol Życzkowski, and Arul Lakshminarayan. Entanglement measures of bipartite quantum gates and their thermalization under arbitrary interaction strength.

Physical Review Research, 2 (4): 043126, 2020. 10.1103/PhysRevResearch.2.043126. https://doi.org/10.1103/PhysRevResearch.2.043126 [46] Luca D'Alessio and Marcos Rigol. Long-time behavior of isolated periodically driven interacting lattice systems. Phys. Rev. X, 4: 041048, 2014. 10.1103/PhysRevX.4.041048. https://doi.org/10.1103/PhysRevX.4.041048 [47] Paolo Zanardi, Christof Zalka, and Lara Faoro. Entangling power of quantum evolutions. Physical Review A, 62 (3): 030301, 2000. 10.1103/PhysRevA.62.030301. https://doi.org/10.1103/PhysRevA.62.030301 [48] Michael A Nielsen, Christopher M Dawson, Jennifer L Dodd, Alexei Gilchrist, Duncan Mortimer, Tobias J Osborne, Michael J Bremner, Aram W Harrow, and Andrew Hines. Quantum dynamics as a physical resource. Physical Review A, 67 (5): 052301, 2003. 10.1103/PhysRevA.67.052301. https://doi.org/10.1103/PhysRevA.67.052301 [49] Bhargavi Jonnadula, Prabha Mandayam, Karol Życzkowski, and Arul Lakshminarayan. Impact of local dynamics on entangling power. Physical Review A, 95 (4): 040302, 2017. 10.1103/PhysRevA.95.040302. https://doi.org/10.1103/PhysRevA.95.040302 [50] Arul Lakshminarayan. Entangling power of quantized chaotic systems. Physical Review E, 64 (3): 036207, 2001. 10.1103/PhysRevE.64.036207. https://doi.org/10.1103/PhysRevE.64.036207 [51] Rajarshi Pal and Arul Lakshminarayan. Entangling power of time-evolution operators in integrable and nonintegrable many-body systems. Physical Review B, 98 (17): 174304, 2018. 10.1103/PhysRevB.98.174304. https://doi.org/10.1103/PhysRevB.98.174304 [52] Georgios Styliaris, Namit Anand, and Paolo Zanardi. Information scrambling over bipartitions: Equilibration, entropy production, and typicality. Phys. Rev. Lett., 126: 030601, 2021. 10.1103/PhysRevLett.126.030601. https://doi.org/10.1103/PhysRevLett.126.030601 [53] Naga Dileep Varikuti and Vaibhav Madhok. Out-of-time ordered correlators in kicked coupled tops: Information scrambling in mixed phase space and the role of conserved quantities. Chaos, 34: 063124, 2024. 10.1063/5.0191140. https://doi.org/10.1063/5.0191140 [54] Lorenzo Leone, Salvatore FE Oliviero, and Alioscia Hamma. Stabilizer rényi entropy.

Physical Review Letters, 128 (5): 050402, 2022. 10.1103/PhysRevLett.128.050402. https://doi.org/10.1103/PhysRevLett.128.050402 [55] Frank Verstraete, Valentin Murg, and J Ignacio Cirac. Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems. Advances in physics, 57 (2): 143–224, 2008. 10.1080/14789940801912366. https://doi.org/10.1080/14789940801912366 [56] Ulrich Schollwöck. The density-matrix renormalization group in the age of matrix product states. Annals of physics, 326 (1): 96–192, 2011. 10.1016/j.aop.2010.09.012. https://doi.org/10.1016/j.aop.2010.09.012 [57] D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac. Matrix product state representations. Quantum Info. Comput., 7 (5): 401â€“430, 2007. 10.26421/QIC7.5-6-1. https://doi.org/10.26421/QIC7.5-6-1 [58] Sergey Bravyi and David Gosset. Improved classical simulation of quantum circuits dominated by clifford gates. Phys. Rev. Lett., 116: 250501, 2016. 10.1103/PhysRevLett.116.250501. https://doi.org/10.1103/PhysRevLett.116.250501 [59] Yifan Zhang and Yuxuan Zhang. Classical simulability of quantum circuits with shallow magic depth. PRX Quantum, 6: 010337, 2025. 10.1103/PRXQuantum.6.010337. https://doi.org/10.1103/PRXQuantum.6.010337 [60] Zejun Liu and Bryan K Clark. Classical simulability of clifford+ t circuits with clifford-augmented matrix product states. arXiv preprint arXiv:2412.17209, 2024. arXiv:2412.17209 [61] Andi Gu, Salvatore F.E. Oliviero, and Lorenzo Leone. Magic-induced computational separation in entanglement theory. PRX Quantum, 6: 020324, 2025. 10.1103/PRXQuantum.6.020324. https://doi.org/10.1103/PRXQuantum.6.020324 [62] Michele Viscardi, Marcello Dalmonte, Alioscia Hamma, and Emanuele Tirrito. Interplay of entanglement structures and stabilizer entropy in spin models. arXiv preprint arXiv:2503.08620, 2025. arXiv:2503.08620 [63] Gerald E Fux, Emanuele Tirrito, Marcello Dalmonte, and Rosario Fazio. Entanglement–nonstabilizerness separation in hybrid quantum circuits.

Physical Review Research, 6 (4): L042030, 2024. 10.1103/PhysRevResearch.6.L042030. https://doi.org/10.1103/PhysRevResearch.6.L042030 [64] M Frau, PS Tarabunga, M Collura, M Dalmonte, and E Tirrito. Nonstabilizerness versus entanglement in matrix product states. Physical Review B, 110 (4): 045101, 2024. 10.1103/PhysRevB.110.045101. https://doi.org/10.1103/PhysRevB.110.045101 [65] Rohit Kumar Shukla, Arul Lakshminarayan, and Sunil Kumar Mishra. Out-of-time-order correlators of nonlocal block-spin and random observables in integrable and nonintegrable spin chains. Physical Review B, 105 (22): 224307, 2022. 10.1103/PhysRevB.105.224307. https://doi.org/10.1103/PhysRevB.105.224307 [66] Gopal Chandra Santra, Alex Windey, Soumik Bandyopadhyay, Andrea Legramandi, and Philipp Hauke. Complexity transitions in chaotic quantum systems: Nonstabilizerness, entanglement, and fractal dimension in syk and random matrix models. arXiv preprint arXiv:2505.09707, 2025b. arXiv:2505.09707 [67] Ivan Chernyshev, Caroline E. P. Robin, and Martin J. Savage. Quantum magic and computational complexity in the neutrino sector. Phys. Rev. Res., 7: 023228, 2025. 10.1103/PhysRevResearch.7.023228. https://doi.org/10.1103/PhysRevResearch.7.023228 [68] Florian Brökemeier, S Momme Hengstenberg, James WT Keeble, Caroline EP Robin, Federico Rocco, and Martin J Savage. Quantum magic and multipartite entanglement in the structure of nuclei. Physical Review C, 111 (3): 034317, 2025. 10.1103/PhysRevC.111.034317. https://doi.org/10.1103/PhysRevC.111.034317 [69] Easwar Magesan, Jay M. Gambetta, B. R. Johnson, Colm A. Ryan, Jerry M. Chow, Seth T. Merkel, Marcus P. da Silva, George A. Keefe, Mary B. Rothwell, Thomas A. Ohki, Mark B. Ketchen, and M. Steffen. Efficient measurement of quantum gate error by interleaved randomized benchmarking. Phys. Rev. Lett., 109: 080505, 2012. 10.1103/PhysRevLett.109.080505. https://doi.org/10.1103/PhysRevLett.109.080505 [70] Joseph M. Renes, Robin Blume-Kohout, A. J. Scott, and Carlton M. Caves. Symmetric informationally complete quantum measurements. J. Math. Phys., 45 (6): 2171–2180, 2004. 10.1063/1.1737053. https://doi.org/10.1063/1.1737053 [71] Pavan Hosur, Xiao-Liang Qi, Daniel A. Roberts, and Beni Yoshida. Chaos in quantum channels. J. High Energ. Phys., 2016 (2): 4, 2016. 10.1007/JHEP02(2016)004. https://doi.org/10.1007/JHEP02(2016)004 [72] Sergey Bravyi, Dan Browne, Padraic Calpin, Earl Campbell, David Gosset, and Mark Howard. Simulation of quantum circuits by low-rank stabilizer decompositions. Quantum, 3: 181, 2019. 10.22331/q-2019-09-02-181. https://doi.org/10.22331/q-2019-09-02-181 [73] Sergey Bravyi, Graeme Smith, and John A Smolin. Trading classical and quantum computational resources. Physical Review X, 6 (2): 021043, 2016. 10.1103/PhysRevX.6.021043. https://doi.org/10.1103/PhysRevX.6.021043 [74] Victor Veitch, SA Hamed Mousavian, Daniel Gottesman, and Joseph Emerson. The resource theory of stabilizer quantum computation. New Journal of Physics, 16 (1): 013009, 2014. 10.1088/1367-2630/16/1/013009. https://doi.org/10.1088/1367-2630/16/1/013009 [75] Hakop Pashayan, Joel J. Wallman, and Stephen D. Bartlett. Estimating outcome probabilities of quantum circuits using quasiprobabilities. Phys. Rev. Lett., 115: 070501, 2015. 10.1103/PhysRevLett.115.070501. https://doi.org/10.1103/PhysRevLett.115.070501 [76] Neil Dowling, Pavel Kos, and Xhek Turkeshi. Magic resources of the heisenberg picture. Phys. Rev. Lett., 135: 050401, 2025b. 10.1103/p7xt-s9nz. https://doi.org/10.1103/p7xt-s9nz [77] Roy J Garcia, Kaifeng Bu, and Arthur Jaffe. Resource theory of quantum scrambling. Proceedings of the National Academy of Sciences, 120 (17): e2217031120, 2023. 10.1073/pnas.2217031120. https://doi.org/10.1073/pnas.2217031120 [78] Emanuele Tirrito, Poetri Sonya Tarabunga, Gugliemo Lami, Titas Chanda, Lorenzo Leone, Salvatore FE Oliviero, Marcello Dalmonte, Mario Collura, and Alioscia Hamma. Quantifying nonstabilizerness through entanglement spectrum flatness. Physical Review A, 109 (4): L040401, 2024. 10.1103/PhysRevA.109.L040401. https://doi.org/10.1103/PhysRevA.109.L040401 [79] Lorenzo Leone and Lennart Bittel. Stabilizer entropies are monotones for magic-state resource theory. Physical Review A, 110 (4): L040403, 2024. 10.1103/PhysRevA.110.L040403. https://doi.org/10.1103/PhysRevA.110.L040403 [80] Salvatore FE Oliviero, Lorenzo Leone, Alioscia Hamma, and Seth Lloyd. Measuring magic on a quantum processor. npj Quantum Information, 8 (1): 148, 2022. 10.1038/s41534-022-00666-5. https://doi.org/10.1038/s41534-022-00666-5 [81] Poetri Sonya Tarabunga, Emanuele Tirrito, Titas Chanda, and Marcello Dalmonte. Many-body magic via pauli-markov chainsâ€”from criticality to gauge theories. PRX Quantum, 4 (4): 040317, 2023. 10.1103/PRXQuantum.4.040317. https://doi.org/10.1103/PRXQuantum.4.040317 [82] Lorenzo Leone, Salvatore FE Oliviero, You Zhou, and Alioscia Hamma. Quantum chaos is quantum. Quantum, 5: 453, 2021. 10.22331/q-2021-05-04-453. https://doi.org/10.22331/q-2021-05-04-453 [83] Hyungwon Kim, Tatsuhiko N. Ikeda, and David A. Huse. Testing whether all eigenstates obey the eigenstate thermalization hypothesis. Phys. Rev. E, 90: 052105, 2014. 10.1103/PhysRevE.90.052105. https://doi.org/10.1103/PhysRevE.90.052105 [84] Naga Dileep Varikuti and Soumik Bandyopadhyay. Unraveling the emergence of quantum state designs in systems with symmetry. Quantum, 8: 1456, 2024. 10.22331/q-2024-08-29-1456. https://doi.org/10.22331/q-2024-08-29-1456 [85] Bruno Bertini, Pavel Kos, and Tomaž Prosen. Exact spectral form factor in a minimal model of many-body quantum chaos. Physical review letters, 121 (26): 264101, 2018. 10.1103/PhysRevLett.121.264101. https://doi.org/10.1103/PhysRevLett.121.264101 [86] Pavel Kos, Marko Ljubotina, and Tomaž Prosen. Many-body quantum chaos: Analytic connection to random matrix theory. Physical Review X, 8 (2): 021062, 2018. 10.1103/PhysRevX.8.021062. https://doi.org/10.1103/PhysRevX.8.021062 [87] Bruno Bertini, Pavel Kos, and Tomaž Prosen. Exact correlation functions for dual-unitary lattice models in 1+ 1 dimensions. Physical review letters, 123 (21): 210601, 2019. 10.1103/PhysRevLett.123.210601. https://doi.org/10.1103/PhysRevLett.123.210601 [88] Amos Chan, Andrea De Luca, and JT Chalker. Spectral statistics in spatially extended chaotic quantum many-body systems. Physical review letters, 121 (6): 060601, 2018. 10.1103/PhysRevLett.121.060601. https://doi.org/10.1103/PhysRevLett.121.060601 [89] Alessio Lerose, Michael Sonner, and Dmitry A Abanin. Influence matrix approach to many-body floquet dynamics. Physical Review X, 11 (2): 021040, 2021. 10.1103/PhysRevX.11.021040. https://doi.org/10.1103/PhysRevX.11.021040 [90] Naga Dileep Varikuti, Abinash Sahu, Arul Lakshminarayan, and Vaibhav Madhok. Probing dynamical sensitivity of a non-kolmogorov-arnold-moser system through out-of-time-order correlators. Phys. Rev. E, 109: 014209, 2024. 10.1103/PhysRevE.109.014209. https://doi.org/10.1103/PhysRevE.109.014209 [91] Shashi C. L. Srivastava, Steven Tomsovic, Arul Lakshminarayan, Roland Ketzmerick, and Arnd Bäcker. Universal scaling of spectral fluctuation transitions for interacting chaotic systems. Phys. Rev. Lett., 116: 054101, 2016. 10.1103/PhysRevLett.116.054101. https://doi.org/10.1103/PhysRevLett.116.054101 [92] Tanay Nag, Sthitadhi Roy, Amit Dutta, and Diptiman Sen. Dynamical localization in a chain of hard core bosons under periodic driving. Physical Review B, 89 (16): 165425, 2014. 10.1103/PhysRevB.89.165425. https://doi.org/10.1103/PhysRevB.89.165425 [93] Dominik Hahn and Luis Colmenarez. Absence of localization in weakly interacting floquet circuits. Phys. Rev. B, 109: 094207, 2024. 10.1103/PhysRevB.109.094207. https://doi.org/10.1103/PhysRevB.109.094207 [94] Philipp Hauke, Fernando M Cucchietti, Luca Tagliacozzo, Ivan Deutsch, and Maciej Lewenstein. Can one trust quantum simulators? Reports on Progress in Physics, 75 (8): 082401, 2012. 10.1088/0034-4885/75/8/082401. https://doi.org/10.1088/0034-4885/75/8/082401 [95] Pablo M. Poggi, Nathan K. Lysne, Kevin W. Kuper, Ivan H. Deutsch, and Poul S. Jessen. Quantifying the sensitivity to errors in analog quantum simulation. PRX Quantum, 1: 020308, 2020. 10.1103/PRXQuantum.1.020308. https://doi.org/10.1103/PRXQuantum.1.020308 [96] Karthik Chinni, Pablo M. Poggi, and Ivan H. Deutsch. Effect of chaos on the simulation of quantum critical phenomena in analog quantum simulators. Phys. Rev. Res., 3: 033145, 2021. 10.1103/PhysRevResearch.3.033145. https://doi.org/10.1103/PhysRevResearch.3.033145 [97] Abinash Sahu, Naga Dileep Varikuti, and Vaibhav Madhok. Quantum signatures of chaos in noisy tomography. arXiv preprint arXiv:2211.11221, 2022. arXiv:2211.11221 [98] Markus Heyl, Philipp Hauke, and Peter Zoller. Quantum localization bounds trotter errors in digital quantum simulation. Science advances, 5 (4): eaau8342, 2019. 10.1126/sciadv.aau8342. https://doi.org/10.1126/sciadv.aau8342 [99] Lukas M Sieberer, Tobias Olsacher, Andreas Elben, Markus Heyl, Philipp Hauke, Fritz Haake, and Peter Zoller. Digital quantum simulation, trotter errors, and quantum chaos of the kicked top. npj Quantum Information, 5 (1): 78, 2019. 10.1038/s41534-019-0192-5. https://doi.org/10.1038/s41534-019-0192-5 [100] Karthik Chinni, Manuel H Muñoz-Arias, Ivan H Deutsch, and Pablo M Poggi. Trotter errors from dynamical structural instabilities of floquet maps in quantum simulation. PRX Quantum, 3 (1): 010351, 2022. 10.1103/PRXQuantum.3.010351. https://doi.org/10.1103/PRXQuantum.3.010351 [101] Nick Sauerwein, Francesca Orsi, Philipp Uhrich, Soumik Bandyopadhyay, Francesco Mattiotti, Tigrane Cantat-Moltrecht, Guido Pupillo, Philipp Hauke, and Jean-Philippe Brantut. Engineering random spin models with atoms in a high-finesse cavity. Nature Physics, 19 (8): 1128–1134, 2023. 10.1038/s41567-023-02033-3. https://doi.org/10.1038/s41567-023-02033-3 [102] Philipp Uhrich, Soumik Bandyopadhyay, Nick Sauerwein, Julian Sonner, Jean-Philippe Brantut, and Philipp Hauke. A cavity quantum electrodynamics implementation of the sachdev–ye–kitaev model. arXiv preprint arXiv:2303.11343, 2023. arXiv:2303.11343 [103] Rahel Baumgartner, Pietro Pelliconi, Soumik Bandyopadhyay, Francesca Orsi, Nick Sauerwein, Philipp Hauke, Jean-Philippe Brantut, and Julian Sonner. Quantum simulation of the sachdev-ye-kitaev model using time-dependent disorder in optical cavities. arXiv preprint arXiv:2411.17802, 2024. arXiv:2411.17802 [104] Cahit Kargi, Juan Pablo Dehollain, Fabio Henriques, Lukas M. Sieberer, Tobias Olsacher, Philipp Hauke, Markus Heyl, Peter Zoller, and Nathan K. Langford. Quantum chaos and universal trotterisation performance behaviours in digital quantum simulation.

In Quantum Information and Measurement VI 2021, page W3A.1.

Optica Publishing Group, 2021. 10.1364/QIM.2021.W3A.1. https://doi.org/10.1364/QIM.2021.W3A.1 [105] A. Smith, C. A. Riofrío, B. E. Anderson, H. Sosa-Martinez, I. H. Deutsch, and P. S. Jessen. Quantum state tomography by continuous measurement and compressed sensing. Phys. Rev. A, 87: 030102, 2013. 10.1103/PhysRevA.87.030102. https://doi.org/10.1103/PhysRevA.87.030102 [106] Seth T. Merkel, Carlos A. Riofrío, Steven T. Flammia, and Ivan H. Deutsch. Random unitary maps for quantum state reconstruction. Phys. Rev. A, 81: 032126, 2010. 10.1103/PhysRevA.81.032126. https://doi.org/10.1103/PhysRevA.81.032126 [107] Patrick Hayden and John Preskill. Black holes as mirrors: quantum information in random subsystems. J. High Energ. Phys., 2007 (09): 120, 2007. 10.1088/1126-6708/2007/09/120. https://doi.org/10.1088/1126-6708/2007/09/120 [108] Beni Yoshida and Alexei Kitaev. Efficient decoding for the Hayden-Preskill protocol. arXiv:1710.03363 [hep-th], 2017. arXiv:1710.03363 [109] Aram W. Harrow and Richard A. Low. Random quantum circuits are approximate 2-designs. Commun. Math. Phys., 291 (1): 257–302, 2009. 10.1007/s00220-009-0873-6. https://doi.org/10.1007/s00220-009-0873-6 [110] Winton G. Brown and Lorenza Viola. Convergence rates for arbitrary statistical moments of random quantum circuits. Phys. Rev. Lett., 104: 250501, 2010. 10.1103/PhysRevLett.104.250501. https://doi.org/10.1103/PhysRevLett.104.250501 [111] Jordan S. Cotler, Daniel K. Mark, Hsin-Yuan Huang, Felipe Hernández, Joonhee Choi, Adam L. Shaw, Manuel Endres, and Soonwon Choi. Emergent quantum state designs from individual many-body wave functions. PRX Quantum, 4: 010311, 2023. 10.1103/PRXQuantum.4.010311. https://doi.org/10.1103/PRXQuantum.4.010311 [112] Joonhee Choi, Adam L. Shaw, Ivaylo S. Madjarov, Xin Xie, Ran Finkelstein, Jacob P. Covey, Jordan S. Cotler, Daniel K. Mark, Hsin-Yuan Huang, Anant Kale, Hannes Pichler, Fernando G. S. L. Brandão, Soonwon Choi, and Manuel Endres. Preparing random states and benchmarking with many-body quantum chaos. Nature, 613: 468–473, 2023. 10.1038/s41586-022-05442-1. https://doi.org/10.1038/s41586-022-05442-1 [113] Joseph Emerson, Robert Alicki, and Karol Å»yczkowski. Scalable noise estimation with random unitary operators. J. Opt. B: Quantum and Semiclass. Opt., 7: S347, 2005. 10.1088/1464-4266/7/10/021. https://doi.org/10.1088/1464-4266/7/10/021 [114] E. Knill, D. Leibfried, R. Reichle, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland. Randomized benchmarking of quantum gates. Phys. Rev. A, 77: 012307, 2008. 10.1103/PhysRevA.77.012307. https://doi.org/10.1103/PhysRevA.77.012307 [115] Christoph Dankert, Richard Cleve, Joseph Emerson, and Etera Livine. Exact and approximate unitary 2-designs and their application to fidelity estimation. Phys. Rev. A, 80: 012304, 2009. 10.1103/PhysRevA.80.012304. https://doi.org/10.1103/PhysRevA.80.012304 [116] B. Vermersch, A. Elben, L. M. Sieberer, N. Y. Yao, and P. Zoller. Probing scrambling using statistical correlations between randomized measurements. Phys. Rev. X, 9: 021061, 2019. 10.1103/PhysRevX.9.021061. https://doi.org/10.1103/PhysRevX.9.021061 [117] Andreas Elben, Steven T. Flammia, Hsin-Yuan Huang, Richard Kueng, John Preskill, Benoı̂t Vermersch, and Peter Zoller. The randomized measurement toolbox. Nat. Rev. Phys., 5 (1): 9–24, 2023. 10.1038/s42254-022-00535-2. https://doi.org/10.1038/s42254-022-00535-2 [118] Navin Khaneja, Roger Brockett, and Steffen J Glaser. Time optimal control in spin systems. Physical Review A, 63 (3): 032308, 2001. 10.1103/PhysRevA.63.032308. https://doi.org/10.1103/PhysRevA.63.032308 [119] Barbara Kraus and J Ignacio Cirac. Optimal creation of entanglement using a two-qubit gate. Physical Review A, 63 (6): 062309, 2001. 10.1103/PhysRevA.63.062309. https://doi.org/10.1103/PhysRevA.63.062309 [120] Jun Zhang, Jiri Vala, Shankar Sastry, and K Birgitta Whaley. Geometric theory of nonlocal two-qubit operations. Physical Review A, 67 (4): 042313, 2003. 10.1103/PhysRevA.67.042313. https://doi.org/10.1103/PhysRevA.67.042313 [121] Vadim Oganesyan and David A Huse. Localization of interacting fermions at high temperature. Physical Review Bâ€”Condensed Matter and Materials Physics, 75 (15): 155111, 2007. 10.1103/PhysRevB.75.155111. https://doi.org/10.1103/PhysRevB.75.155111 [122] YY Atas, Eugene Bogomolny, O Giraud, and G Roux. Distribution of the ratio of consecutive level spacings in random matrix ensembles. Physical review letters, 110 (8): 084101, 2013. 10.1103/PhysRevLett.110.084101. https://doi.org/10.1103/PhysRevLett.110.084101 [123] Freeman J Dyson. The threefold way. algebraic structure of symmetry groups and ensembles in quantum mechanics. Journal of Mathematical Physics, 3 (6): 1199–1215, 1962. 10.1063/1.1703863. https://doi.org/10.1063/1.1703863 [124] Tomasz Tkocz, Marek Smaczyński, Marek Kuś, Ofer Zeitouni, and Karol Życzkowski. Tensor products of random unitary matrices. Random Matrices: Theory and Applications, 1 (04): 1250009, 2012. 10.1142/S2010326312500098. https://doi.org/10.1142/S2010326312500098 [125] N. D. Varikuti, S. Bandyopadhyay, and P. Hauke. Dataset for "impact of clifford operations on non-stabilizing power and quantum chaos". 2026. 10.5281/zenodo.18745429. https://doi.org/10.5281/zenodo.18745429 [126] Andrew J Scott. Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions. Physical Review Aâ€”Atomic, Molecular, and Optical Physics, 69 (5): 052330, 2004. 10.1103/PhysRevA.69.052330. https://doi.org/10.1103/PhysRevA.69.052330 [127] Suhail Ahmad Rather, Adam Burchardt, Wojciech Bruzda, Grzegorz Rajchel-Mieldzioć, Arul Lakshminarayan, and Karol Życzkowski. Thirty-six entangled officers of euler: Quantum solution to a classically impossible problem.

Physical Review Letters, 128 (8): 080507, 2022. 10.1103/PhysRevLett.128.080507. https://doi.org/10.1103/PhysRevLett.128.080507 [128] 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. https://doi.org/10.1103/PhysRevLett.118.090501 [129] David Gross. Hudsonâ€™s theorem for finite-dimensional quantum systems. Journal of mathematical physics, 47 (12), 2006. 10.1063/1.2393152. https://doi.org/10.1063/1.2393152 [130] Gadi Aleksandrowicz, Thomas Alexander, Panagiotis Barkoutsos, Luciano Bello, Yael Ben-Haim, David Bucher, Francisco Jose Cabrera-HernÃ¡ndez, Jorge Carballo-Franquis, Adrian Chen, Chun-Fu Chen, Jerry M. Chow, Antonio D. CÃ³rcoles-Gonzales, Abigail J. Cross, Andrew Cross, Juan Cruz-Benito, Chris Culver, Salvador De La Puente GonzÃ¡lez, Enrique De La Torre, Delton Ding, Eugene Dumitrescu, Ivan Duran, Pieter Eendebak, Mark Everitt, Ismael Faro Sertage, Albert Frisch, Andreas Fuhrer, Jay Gambetta, Borja Godoy Gago, Juan Gomez-Mosquera, Donny Greenberg, Ikko Hamamura, Vojtech Havlicek, Joe Hellmers, Åukasz Herok, Hiroshi Horii, Shaohan Hu, Takashi Imamichi, Toshinari Itoko, Ali Javadi-Abhari, Naoki Kanazawa, Anton Karazeev, Kevin Krsulich, Peng Liu, Yang Luh, Yunho Maeng, Manoel Marques, Francisco Jose MartÃn-FernÃ¡ndez, Douglas T. McClure, David McKay, Srujan Meesala, Antonio Mezzacapo, Nikolaj Moll, Diego Moreda RodrÃguez, Giacomo Nannicini, Paul Nation, Pauline Ollitrault, Lee James O'Riordan, Hanhee Paik, JesÃºs PÃ©rez, Anna Phan, Marco Pistoia, Viktor Prutyanov, Max Reuter, Julia Rice, AbdÃ³n RodrÃguez Davila, Raymond Harry Putra Rudy, Mingi Ryu, Ninad Sathaye, Chris Schnabel, Eddie Schoute, Kanav Setia, Yunong Shi, Adenilton Silva, Yukio Siraichi, Seyon Sivarajah, John A. Smolin, Mathias Soeken, Hitomi Takahashi, Ivano Tavernelli, Charles Taylor, Pete Taylour, Kenso Trabing, Matthew Treinish, Wes Turner, Desiree Vogt-Lee, Christophe Vuillot, Jonathan A. Wildstrom, Jessica Wilson, Erick Winston, Christopher Wood, Stephen Wood, Stefan WÃ¶rner, Ismail Yunus Akhalwaya, and Christa Zoufal. Qiskit: An open-source framework for quantum computing. 2019. 10.5281/zenodo.2562111. https://doi.org/10.5281/zenodo.2562111Cited byCould not fetch Crossref cited-by data during last attempt 2026-03-10 13:53:04: Could not fetch cited-by data for 10.22331/q-2026-03-10-2017 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-03-10 13:53:04: No response from ADS or unable to decode the received json data when getting the list of citing works.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. AbstractNon-stabilizerness, alongside entanglement, is a crucial ingredient for fault-tolerant quantum computation and achieving a genuine quantum advantage. Despite recent progress, a complete understanding of the generation and thermalization of non-stabilizerness in circuits that mix Clifford and non-Clifford operations remains elusive. While Clifford operations do not generate non-stabilizerness, their interplay with non-Clifford gates can strongly impact the overall non-stabilizing dynamics of generic quantum circuits. In this work, we establish a direct relationship between the final non-stabilizing power and the individual powers of the non-Clifford gates, in circuits where these gates are interspersed with random Clifford operations. By leveraging this result, we unveil the thermalization of non-stabilizing power to its Haar-averaged value in generic circuits. As a precursor, we analyze two-qubit gates and illustrate this thermalization in analytically tractable systems. Extending this, we explore the operator-space non-stabilizing power and demonstrate its behavior in physical models. Finally, we examine the role of non-stabilizing power in the emergence of quantum chaos in brick-wall quantum circuits. Our work elucidates how non-stabilizing dynamics evolve and thermalize in quantum circuits and thus contributes to a better understanding of quantum computational resources and of their role in quantum chaos.Popular summaryQuantum technologies promise capacities widely beyond the grasp of classical hardware. To achieve this, specifically quantum mechanical “resources” are required, a prime example being entanglement. Another equally important resource is non-stabilizerness, often called “magic”, i.e., the capacity to produce non-Clifford dynamics in quantum circuits. Its presence makes quantum circuits exceedingly difficult to simulate on classical computers and is therefore essential for achieving genuine quantum advantage. However, how non-stabilizerness builds up and evolves in realistic quantum circuits remains an outstanding question. Even though Clifford operations cannot create non-stabilizerness, they play a crucial role in shaping how non-stabilizerness develops when combined with non-Clifford gates. Surprisingly, random Clifford operations can enhance, redistribute, or even suppress the build-up of magic, depending on the situation. This work identifies a simple and elegant rule that determines the total amount of non-stabilizerness directly from the individual contributions of the non-Clifford gates, in circuits where these gates appear alongside random Clifford operations. As quantum circuits grow deeper, non-stabilizerness naturally approaches a universal equilibrium value, similar to how physical systems relax toward thermal equilibrium. This relaxation occurs exponentially with circuit depth. This work further provides evidence that the emergence of quantum chaos, a hallmark of quantum complexity, is governed by the mutual influence of non-stabilizerness and entanglement. This behavior provides means to fine tune quantum circuit dynamics, for example, to control chaotic evolution and optimize randomness generation. The insights presented here have practical relevance for benchmarking quantum devices, quantum error-correction strategies, and guiding the design of quantum algorithms.► BibTeX data@article{Varikuti2026impactofclifford, doi = {10.22331/q-2026-03-10-2017}, url = {https://doi.org/10.22331/q-2026-03-10-2017}, title = {Impact of {C}lifford operations on non-stabilizing power and quantum chaos}, author = {Varikuti, Naga Dileep and Bandyopadhyay, Soumik and Hauke, Philipp}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2017}, month = mar, year = {2026} }► References [1] Alán Aspuru-Guzik, Anthony D Dutoi, Peter J Love, and Martin Head-Gordon. Simulated quantum computation of molecular energies. Science, 309 (5741): 1704–1707, 2005. 10.1126/science.1113479. https://doi.org/10.1126/science.1113479 [2] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. Quantum entanglement. Rev. Mod. Phys., 81: 865–942, 2009. 10.1103/RevModPhys.81.865. https://doi.org/10.1103/RevModPhys.81.865 [3] Animesh Datta, Anil Shaji, and Carlton M Caves. Quantum discord and the power of one qubit. Physical review letters, 100 (5): 050502, 2008. 10.1103/PhysRevLett.100.050502. https://doi.org/10.1103/PhysRevLett.100.050502 [4] Earl T Campbell, Barbara M Terhal, and Christophe Vuillot. Roads towards fault-tolerant universal quantum computation. Nature, 549 (7671): 172–179, 2017. 10.1038/nature23460. https://doi.org/10.1038/nature23460 [5] John Preskill. Quantum computing in the nisq era and beyond. Quantum, 2: 79, 2018. 10.22331/q-2018-08-06-79. https://doi.org/10.22331/q-2018-08-06-79 [6] Eric Chitambar and Gilad Gour. Quantum resource theories. Rev. Mod. Phys., 91: 025001, 2019. 10.1103/RevModPhys.91.025001. https://doi.org/10.1103/RevModPhys.91.025001 [7] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. Quantum-enhanced measurements: beating the standard quantum limit. Science, 306 (5700): 1330–1336, 2004. 10.1126/science.1104149. https://doi.org/10.1126/science.1104149 [8] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. Quantum metrology. Physical review letters, 96 (1): 010401, 2006. 10.1103/PhysRevLett.96.010401. https://doi.org/10.1103/PhysRevLett.96.010401 [9] C. L. Degen, F. Reinhard, and P. Cappellaro. Quantum sensing. Rev. Mod. Phys., 89: 035002, 2017. 10.1103/RevModPhys.89.035002. https://doi.org/10.1103/RevModPhys.89.035002 [10] Philipp Hauke, Markus Heyl, Luca Tagliacozzo, and Peter Zoller. Measuring multipartite entanglement through dynamic susceptibilities. Nature Physics, 12 (8): 778–782, 2016. 10.1038/nphys3700. https://doi.org/10.1038/nphys3700 [11] Luca Pezzè, Augusto Smerzi, Markus K. Oberthaler, Roman Schmied, and Philipp Treutlein. Quantum metrology with nonclassical states of atomic ensembles. Rev. Mod. Phys., 90: 035005, 2018. 10.1103/RevModPhys.90.035005. https://doi.org/10.1103/RevModPhys.90.035005 [12] Charles H Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K Wootters. Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels. Physical review letters, 70 (13): 1895, 1993. 10.1103/PhysRevLett.70.1895. https://doi.org/10.1103/PhysRevLett.70.1895 [13] Peter W Shor. Scheme for reducing decoherence in quantum computer memory. Physical review A, 52 (4): R2493, 1995. 10.1103/PhysRevA.52.R2493. https://doi.org/10.1103/PhysRevA.52.R2493 [14] Barbara M. Terhal. Quantum error correction for quantum memories. Rev. Mod. Phys., 87: 307–346, 2015. 10.1103/RevModPhys.87.307. https://doi.org/10.1103/RevModPhys.87.307 [15] Joschka Roffe. Quantum error correction: an introductory guide. Contemporary Physics, 60 (3): 226–245, 2019. 10.1080/00107514.2019.1667078. https://doi.org/10.1080/00107514.2019.1667078 [16] Philipp Hauke, Lars Bonnes, Markus Heyl, and Wolfgang Lechner. Probing entanglement in adiabatic quantum optimization with trapped ions. Frontiers in Physics, 3: 21, 2015. 10.3389/fphy.2015.00021. https://doi.org/10.3389/fphy.2015.00021 [17] Gopal Chandra Santra, Sudipto Singha Roy, Daniel J. Egger, and Philipp Hauke. Genuine multipartite entanglement in quantum optimization. Phys. Rev. A, 111: 022434, 2025a. 10.1103/PhysRevA.111.022434. https://doi.org/10.1103/PhysRevA.111.022434 [18] Gopal Chandra Santra, Fred Jendrzejewski, Philipp Hauke, and Daniel J. Egger. Squeezing and quantum approximate optimization. Phys. Rev. A, 109: 012413, 2024. 10.1103/PhysRevA.109.012413. https://doi.org/10.1103/PhysRevA.109.012413 [19] Yanzhu Chen, Linghua Zhu, Nicholas J. Mayhall, Edwin Barnes, and Sophia E. Economou. How much entanglement do quantum optimization algorithms require? In Quantum 2.0 Conference and Exhibition, page QM4A.2.

In Quantum Information and Measurement VI 2021, page W3A.1.

Read Original

Source Information

Source: Quantum Journal

Website: https://quantum-journal.org/feed/

Impact of Clifford operations on non-stabilizing power and quantum chaos

Tags

Source Information