Optimal schedule of multi-channel quantum Zeno dragging with application to solving the k-SAT problem

Yipei Zhang; Alain Sarlette; Philippe Lewalle; Tathagata Karmakar; Thilo Scharnhorst; K. Birgitta Whaley

doi:10.22331/q-2026-07-20-2167

Optimal schedule of multi-channel quantum Zeno dragging with application to solving the k-SAT problem

Yipei Zhang^1,2, Alain Sarlette³, Philippe Lewalle^1,2, Tathagata Karmakar^1,2, Thilo Scharnhorst^1,2,4, and K. Birgitta Whaley^1,2

¹Berkeley Center for Quantum Information and Computation, Berkeley, California 94720, USA
²Department of Chemistry, University of California, Berkeley, California 94720, USA
³Laboratoire de Physique de l’École Normale Supérieure Inria, ENS, Mines ParisTech, Université PSL, Sorbonne Université Paris France
⁴Department of Physics, University of California, Berkeley, California 94720, USA

Published:	2026-07-20, volume 10, page 2167
Editor:	Milad Marvian
Eprint:	arXiv:2507.16128v2
Doi:	https://doi.org/10.22331/q-2026-07-20-2167
Citation:	Quantum 10, 2167 (2026).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

Quantum Zeno dragging enables the preparation of common eigenstates of a set of observables by frequent measurement and adiabatic-like modulation of the measurement basis. In this work, we present a deeper analysis of multi-channel Zeno dragging using generalized measurements, i.e. simultaneously measuring a set of non-commuting observables that vary slowly in time, to drag the state towards a target subspace. For concreteness, we will focus on a measurement-driven approach to solving k-SAT problems as examples. We first compute some analytical upper bounds on the convergence time, including the effect of finite measurement time resolution. We then apply optimal control theory to obtain the optimal dragging schedule that lower bounds the convergence time, for low-dimensional settings. This study provides a theoretical foundation for multi-channel Zeno dragging and its optimization, and also serves as a guide for designing optimal dragging schedules for quantum information tasks including measurement-driven quantum algorithms.

► BibTeX data

@article{Zhang2026optimalscheduleof,
  doi = {10.22331/q-2026-07-20-2167},
  url = {https://doi.org/10.22331/q-2026-07-20-2167},
  title = {Optimal schedule of multi-channel quantum {Z}eno dragging with application to solving the k-{SAT} problem},
  author = {Zhang, Yipei and Sarlette, Alain and Lewalle, Philippe and Karmakar, Tathagata and Scharnhorst, Thilo and Whaley, K. Birgitta},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {10},
  pages = {2167},
  month = jul,
  year = {2026}
}

► References

[1] Christiane P. Koch, Ugo Boscain, Tommaso Calarco, Gunther Dirr, Stefan Filipp, Steffen J. Glaser, Ronnie Kosloff, Simone Montangero, Thomas Schulte-Herbrüggen, Dominique Sugny, and Frank K. Wilhelm. ``Quantum optimal control in quantum technologies. Strategic report on current status, visions and goals for research in Europe''. EPJ Quantum Technol. 9, 19 (2022).
https://doi.org/10.1140/epjqt/s40507-022-00138-x

[2] D. D'Alessandro. ``Introduction to quantum control and dynamics''. Chapman and Hall/CRC. (2021).
https://doi.org/10.1201/9781003051268

[3] U. Boscain, M. Sigalotti, and D. Sugny. ``Introduction to the Pontryagin maximum principle for quantum optimal control''. PRX Quantum 2, 030203 (2021).
https://doi.org/10.1103/PRXQuantum.2.030203

[4] Sebastian Deffner and Steve Campbell. ``Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control''. Journal of Physics A: Mathematical and Theoretical 50, 453001 (2017).
https://doi.org/10.1088/1751-8121/aa86c6

[5] Christiane P Koch. ``Controlling open quantum systems: tools, achievements, and limitations''. Journal of Physics: Condensed Matter 28, 213001 (2016).
https://doi.org/10.1088/0953-8984/28/21/213001

[6] Patrick M Harrington, Erich J Mueller, and Kater W Murch. ``Engineered dissipation for quantum information science''. Nature Reviews Physics 4, 660–671 (2022).
https://doi.org/10.1038/s42254-022-00494-8

[7] B. Misra and E. C. G. Sudarshan. ``The Zeno’s paradox in quantum theory''. Journal of Mathematical Physics 18, 756–763 (1977).
https://doi.org/10.1063/1.523304

[8] Jay Gambetta, Alexandre Blais, M. Boissonneault, A. A. Houck, D. I. Schuster, and S. M. Girvin. ``Quantum trajectory approach to circuit QED: Quantum jumps and the Zeno effect''. Phys. Rev. A 77, 012112 (2008).
https://doi.org/10.1103/PhysRevA.77.012112

[9] Rusko Ruskov, Alexander N. Korotkov, and Ari Mizel. ``Quantum zeno stabilization in weak continuous measurement of two qubits''. Phys. Rev. B 73, 085317 (2006).
https://doi.org/10.1103/PhysRevB.73.085317

[10] Jean-Michel Raimond, Clément Sayrin, S Gleyzes, Igor Dotsenko, Michel Brune, Serge Haroche, Paolo Facchi, and Saverio Pascazio. ``Phase space tweezers for tailoring cavity fields by quantum zeno dynamics''. Physical review letters 105, 213601 (2010).
https://doi.org/10.1103/PhysRevLett.105.213601

[11] Jean-Michel Raimond, Paolo Facchi, Bruno Peaudecerf, Saverio Pascazio, Clément Sayrin, Igor Dotsenko, Sébastien Gleyzes, Michel Brune, and Serge Haroche. ``Quantum zeno dynamics of a field in a cavity''. Physical Review A—Atomic, Molecular, and Optical Physics 86, 032120 (2012).
https://doi.org/10.1103/PhysRevA.86.032120

[12] Adrien Signoles, Adrien Facon, Dorian Grosso, Igor Dotsenko, Serge Haroche, Jean-Michel Raimond, Michel Brune, and Sébastien Gleyzes. ``Confined quantum zeno dynamics of a watched atomic arrow''. Nature Physics 10, 715–719 (2014).
https://doi.org/10.1038/nphys3076

[13] S. Hacohen-Gourgy, L. P. García-Pintos, L. S. Martin, J. Dressel, and I. Siddiqi. ``Incoherent qubit control using the quantum zeno effect''. Phys. Rev. Lett. 120, 020505 (2018).
https://doi.org/10.1103/PhysRevLett.120.020505

[14] Frank Verstraete, Michael M Wolf, and J Ignacio Cirac. ``Quantum computation and quantum-state engineering driven by dissipation''. Nature physics 5, 633–636 (2009).
https://doi.org/10.1038/nphys1342

[15] Y. Liu, S. Shankar, N. Ofek, M. Hatridge, A. Narla, K. M. Sliwa, L. Frunzio, R. J. Schoelkopf, and M. H. Devoret. ``Comparing and combining measurement-based and driven-dissipative entanglement stabilization''. Phys. Rev. X 6, 011022 (2016).
https://doi.org/10.1103/PhysRevX.6.011022

[16] Florentin Reiter, David Reeb, and Anders S. Sørensen. ``Scalable dissipative preparation of many-body entanglement''. Phys. Rev. Lett. 117, 040501 (2016).
https://doi.org/10.1103/PhysRevLett.117.040501

[17] Florentin Reiter, Anders Søndberg Sørensen, Peter Zoller, and CA Muschik. ``Dissipative quantum error correction and application to quantum sensing with trapped ions''. Nature communications 8, 1822 (2017).
https://doi.org/10.1038/s41467-017-01895-5

[18] Leo Zhou, Soonwon Choi, and Mikhail D. Lukin. ``Symmetry-protected dissipative preparation of matrix product states''. Phys. Rev. A 104, 032418 (2021).
https://doi.org/10.1103/PhysRevA.104.032418

[19] E. Doucet, F. Reiter, L. Ranzani, and A. Kamal. ``High fidelity dissipation engineering using parametric interactions''. Phys. Rev. Res. 2, 023370 (2020).
https://doi.org/10.1103/PhysRevResearch.2.023370

[20] T. Brown, E. Doucet, D. Ristè, G. Ribeill, K. Cicak, J. Aumentado, R. Simmonds, L. Govia, A. Kamal, and L. Ranzani. ``Trade off-free entanglement stabilization in a superconducting qutrit-qubit system''. Nat. Commun. 13, 3994 (2022).
https://doi.org/10.1038/s41467-022-31638-0

[21] E. Doucet, L. C. G. Govia, and A. Kamal. ``Scalable entanglement stabilization with modular reservoir engineering'' (2023). arXiv:2301.05725.
arXiv:2301.05725

[22] Eliya Blumenthal, Chen Mor, Asaf A Diringer, Leigh S Martin, Philippe Lewalle, Daniel Burgarth, K Birgitta Whaley, and Shay Hacohen-Gourgy. ``Demonstration of universal control between non-interacting qubits using the quantum zeno effect''. npj Quantum Information 8, 88 (2022).
https://doi.org/10.1038/s41534-022-00594-4

[23] Philippe Lewalle, Leigh S. Martin, Emmanuel Flurin, Song Zhang, Eliya Blumenthal, Shay Hacohen-Gourgy, Daniel Burgarth, and K. Birgitta Whaley. ``A Multi-Qubit Quantum Gate Using the Zeno Effect''. Quantum 7, 1100 (2023).
https://doi.org/10.22331/q-2023-09-07-1100

[24] Philippe Lewalle, Yipei Zhang, and K. Birgitta Whaley. ``Optimal zeno dragging for quantum control: A shortcut to zeno with action-based scheduling optimization''. PRX Quantum 5, 020366 (2024).
https://doi.org/10.1103/PRXQuantum.5.020366

[25] Mazyar Mirrahimi, Zaki Leghtas, Victor V Albert, Steven Touzard, Robert J Schoelkopf, Liang Jiang, and Michel H Devoret. ``Dynamically protected cat-qubits: a new paradigm for universal quantum computation''. New Journal of Physics 16, 045014 (2014).
https://doi.org/10.1088/1367-2630/16/4/045014

[26] Steven Touzard, Alexander Grimm, Zaki Leghtas, Shantanu O Mundhada, Philip Reinhold, Christopher Axline, Matt Reagor, Kevin Chou, Jacob Blumoff, Katrina M Sliwa, et al. ``Coherent oscillations inside a quantum manifold stabilized by dissipation''. Physical Review X 8, 021005 (2018).
https://doi.org/10.1103/PhysRevX.8.021005

[27] Jérémie Guillaud and Mazyar Mirrahimi. ``Repetition cat qubits for fault-tolerant quantum computation''. Phys. Rev. X 9, 041053 (2019).
https://doi.org/10.1103/PhysRevX.9.041053

[28] Ronan Gautier, Mazyar Mirrahimi, and Alain Sarlette. ``Designing high-fidelity zeno gates for dissipative cat qubits''. PRX Quantum 4, 040316 (2023).
https://doi.org/10.1103/PRXQuantum.4.040316

[29] Simon C Benjamin, Liming Zhao, and Joseph F Fitzsimons. ``Measurement-driven quantum computing: Performance of a 3-sat solver'' (2017). arXiv:1711.02687.
arXiv:1711.02687

[30] Liming Zhao, Carlos A. Pérez-Delgado, Simon C. Benjamin, and Joseph F. Fitzsimons. ``Measurement-driven analog of adiabatic quantum computation for frustration-free hamiltonians''. Phys. Rev. A 100, 032331 (2019).
https://doi.org/10.1103/PhysRevA.100.032331

[31] András Pál Gilyén and Or Sattath. ``On preparing ground states of gapped hamiltonians: An efficient quantum lovász local lemma''. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS). Pages 439–450. (2017).
https://doi.org/10.1109/FOCS.2017.47

[32] Toby S Cubitt. ``Dissipative ground state preparation and the dissipative quantum eigensolver'' (2023) arXiv:2303.11962.
arXiv:2303.11962

[33] Yipei Zhang, Philippe Lewalle, and K Birgitta Whaley. ``Solving k–sat problems with generalized quantum measurement''. npj Quantum Information 11, 170 (2025).
https://doi.org/10.1038/s41534-025-01069-y

[34] Jesse Berwald, Nicholas Chancellor, and Raouf Dridi. ``Grover Speedup from Many Forms of the Zeno Effect''. Quantum 8, 1532 (2024).
https://doi.org/10.22331/q-2024-11-20-1532

[35] Jesse Berwald, Nicholas Chancellor, and Raouf Dridi. ``Zeno-effect computation: Opportunities and challenges''. Phys. Rev. A 111, 042623 (2025).
https://doi.org/10.1103/PhysRevA.111.042623

[36] Andrew M. Childs, Enrico Deotto, Edward Farhi, Jeffrey Goldstone, Sam Gutmann, and Andrew J. Landahl. ``Quantum search by measurement''. Phys. Rev. A 66, 032314 (2002).
https://doi.org/10.1103/PhysRevA.66.032314

[37] Y. Aharonov and M. Vardi. ``Meaning of an individual "feynman path"''. Phys. Rev. D 21, 2235–2240 (1980).
https://doi.org/10.1103/PhysRevD.21.2235

[38] Dominic Shea and Alessandro Romito. ``Action formalism for geometric phases from self-closing quantum trajectories''. Journal of Physics A: Mathematical and Theoretical 57, 315303 (2024).
https://doi.org/10.1088/1751-8121/ad5e4b

[39] Tameem Albash and Daniel A. Lidar. ``Adiabatic quantum computation''. Rev. Mod. Phys. 90, 015002 (2018).
https://doi.org/10.1103/RevModPhys.90.015002

[40] D. Guéry-Odelin, A. Ruschhaupt, A. Kiely, E. Torrontegui, S. Martínez-Garaot, and J. G. Muga. ``Shortcuts to adiabaticity: Concepts, methods, and applications''. Rev. Mod. Phys. 91, 045001 (2019).
https://doi.org/10.1103/RevModPhys.91.045001

[41] Satoshi Morita and Hidetoshi Nishimori. ``Mathematical foundation of quantum annealing''. Journal of Mathematical Physics 49, 125210 (2008).
https://doi.org/10.1063/1.2995837

[42] Arnab Das and Bikas K. Chakrabarti. ``Colloquium: Quantum annealing and analog quantum computation''. Rev. Mod. Phys. 80, 1061–1081 (2008).
https://doi.org/10.1103/RevModPhys.80.1061

[43] M. S. Sarandy and D. A. Lidar. ``Adiabatic approximation in open quantum systems''. Phys. Rev. A 71, 012331 (2005).
https://doi.org/10.1103/PhysRevA.71.012331

[44] G. Vacanti, R. Fazio, S. Montangero, G. M. Palma, M. Paternostro, and V. Vedral. ``Transitionless quantum driving in open quantum systems''. New Journal of Physics 16, 053017 (2014).
https://doi.org/10.1088/1367-2630/16/5/053017

[45] Lorenzo Campos Venuti, Tameem Albash, Daniel A. Lidar, and Paolo Zanardi. ``Adiabaticity in open quantum systems''. Phys. Rev. A 93, 032118 (2016).
https://doi.org/10.1103/PhysRevA.93.032118

[46] Jérémie Roland and Nicolas J. Cerf. ``Quantum search by local adiabatic evolution''. Phys. Rev. A 65, 042308 (2002).
https://doi.org/10.1103/PhysRevA.65.042308

[47] Constantin Brif, Matthew D Grace, Mohan Sarovar, and Kevin C Young. ``Exploring adiabatic quantum trajectories via optimal control''. New Journal of Physics 16, 065013 (2014).
https://doi.org/10.1088/1367-2630/16/6/065013

[48] Pratibha Raghupati Hegde, Gianluca Passarelli, Annarita Scocco, and Procolo Lucignano. ``Genetic optimization of quantum annealing''. Phys. Rev. A 105, 012612 (2022).
https://doi.org/10.1103/PhysRevA.105.012612

[49] Mogens Dalgaard, Felix Motzoi, Jens Jakob Sørensen, and Jacob Sherson. ``Global optimization of quantum dynamics with alphazero deep exploration''. NPJ quantum information 6, 6 (2020).
https://doi.org/10.1038/s41534-019-0241-0

[50] Xiaodong Yang, Ran Liu, Jun Li, and Xinhua Peng. ``Optimizing adiabatic quantum pathways via a learning algorithm''. Phys. Rev. A 102, 012614 (2020).
https://doi.org/10.1103/PhysRevA.102.012614

[51] Yu-Qin Chen, Yu Chen, Chee-Kong Lee, Shengyu Zhang, and Chang-Yu Hsieh. ``Optimizing quantum annealing schedules with monte carlo tree search enhanced with neural networks''. Nature Machine Intelligence 4, 269–278 (2022).
https://doi.org/10.1038/s42256-022-00446-y

[52] A. Chantasri, J. Dressel, and A. N. Jordan. ``Action principle for continuous quantum measurement''. Phys. Rev. A 88, 042110 (2013).
https://doi.org/10.1103/PhysRevA.88.042110

[53] Areeya Chantasri and Andrew N. Jordan. ``Stochastic path-integral formalism for continuous quantum measurement''. Phys. Rev. A 92, 032125 (2015).
https://doi.org/10.1103/PhysRevA.92.032125

[54] Wirawat Kokaew, Thiparat Chotibut, and Areeya Chantasri. ``Quantum state-preparation control in noisy environment via most-likely paths'' (2024). arXiv:2209.13164.
https://doi.org/10.1007/s11128-025-05034-8
arXiv:2209.13164

[55] Tathagata Karmakar and Andrew N Jordan. ``Cdj-pontryagin optimal control for general continuously monitored quantum systems'' (2025) arXiv:2504.08173.
https://doi.org/10.22331/q-2026-03-24-2043
arXiv:2504.08173

[56] Stephen A. Cook. ``The complexity of theorem-proving procedures''. In Proceedings of the Third Annual ACM Symposium on Theory of Computing. Page 151–158. STOC '71New York, NY, USA (1971). Association for Computing Machinery.
https://doi.org/10.1145/800157.805047

[57] Richard M. Karp. ``Reducibility among combinatorial problems''. Pages 85–103. Springer US. Boston, MA (1972).
https://doi.org/10.1007/978-1-4684-2001-2_9

[58] L. A. Levin. ``Universal sequential search problems''. Problems Inform. Transmission 9, 265 (1973). url: http://mathscinet.ams.org/mathscinet-getitem?mr=340042.
http://mathscinet.ams.org/mathscinet-getitem?mr=340042

[59] Chris Calabro, Russell Impagliazzo, Valentine Kabanets, and Ramamohan Paturi. ``The complexity of unique k-sat: An isolation lemma for k-cnfs''. Journal of Computer and System Sciences 74, 386–393 (2008).
https://doi.org/10.1016/j.jcss.2007.06.015

[60] L G Valiant and V V Vazirani. ``Np is as easy as detecting unique solutions''. In Proceedings of the Seventeenth Annual ACM Symposium on Theory of Computing. Page 458–463. STOC '85New York, NY, USA (1985). Association for Computing Machinery.
https://doi.org/10.1145/22145.22196

[61] Alexander N. Korotkov. ``Selective quantum evolution of a qubit state due to continuous measurement''. Phys. Rev. B 63, 115403 (2001).
https://doi.org/10.1103/PhysRevB.63.115403

[62] Alexander N. Korotkov. ``Continuous quantum measurement of a double dot''. Phys. Rev. B 60, 5737–5742 (1999).
https://doi.org/10.1103/PhysRevB.60.5737

[63] Alexander N. Korotkov. ``Quantum bayesian approach to circuit qed measurement with moderate bandwidth''. Phys. Rev. A 94, 042326 (2016).
https://doi.org/10.1103/PhysRevA.94.042326

[64] John Steinmetz, Debmalya Das, Irfan Siddiqi, and Andrew N. Jordan. ``Continuous measurement of a qudit using dispersively coupled radiation''. Phys. Rev. A 105, 052229 (2022).
https://doi.org/10.1103/PhysRevA.105.052229

[65] Markus Aspelmeyer, Tobias J. Kippenberg, and Florian Marquardt. ``Cavity optomechanics''. Rev. Mod. Phys. 86, 1391–1452 (2014).
https://doi.org/10.1103/RevModPhys.86.1391

[66] H. M. Wiseman and G. J. Milburn. ``Quantum theory of field-quadrature measurements''. Phys. Rev. A 47, 642–662 (1993).
https://doi.org/10.1103/PhysRevA.47.642

[67] G. J. Milburn, K. Jacobs, and D. F. Walls. ``Quantum-limited measurements with the atomic force microscope''. Phys. Rev. A 50, 5256–5263 (1994).
https://doi.org/10.1103/PhysRevA.50.5256

[68] Andrew N. Jordan and Irfan A. Siddiqi. ``Quantum measurement: Theory and practice''. Cambridge University Press. (2024).
https://doi.org/10.1017/9781009103909

[69] R.L. Hudson and K.R. Parthasarathy. ``Quantum Ito's formula and stochastic evolutions''. Commun. Math. Phys. 93, 301–323 (1984).
https://doi.org/10.1007/BF01258530

[70] V.P. Belavkin. ``A new wave equation for a continuous nondemolition measurement''. Physics letters A 140, 355–358 (1989).
https://doi.org/10.1016/0375-9601(89)90066-2

[71] Howard M. Wiseman. ``Quantum trajectories and quantum measurement theory''. Quantum Semiclass. Opt. 8, 205 (1996).
https://doi.org/10.1088/1355-5111/8/1/015

[72] Todd A. Brun. ``A simple model of quantum trajectories''. American Journal of Physics 70, 719 (2002).
https://doi.org/10.1119/1.1475328

[73] Luc Bouten, Madalin Guta, and Hans Maassen. ``Stochastic schrödinger equations''. Journal of Physics A: Mathematical and General 37, 3189 (2004).
https://doi.org/10.1088/0305-4470/37/9/010

[74] C. Gardiner and P. Zoller. ``Quantum noise: A handbook of markovian and non-markovian quantum stochastic methods with applications to quantum optics''. Springer Series in Synergetics. Springer. (2004). url: https://books.google.fr/books?id=a_xsT8oGhdgC.
https://books.google.fr/books?id=a_xsT8oGhdgC

[75] Kurt Jacobs and Daniel A. Steck. ``A straightforward introduction to continuous quantum measurement''. Contemporary Physics 47, 279–303 (2006).
https://doi.org/10.1080/00107510601101934

[76] Alberto Barchielli and Matteo Gregoratti. ``Quantum trajectories and measurements in continuous time: the diffusive case''. Volume 782. Springer Science & Business Media. (2009).
https://doi.org/10.1007/978-3-642-01298-3

[77] Howard M Wiseman and Gerard J Milburn. ``Quantum measurement and control''. Cambridge university press. (2009).
https://doi.org/10.1017/CBO9780511813948

[78] Kurt Jacobs. ``Quantum measurement theory and its applications''. Cambridge University Press. (2014).
https://doi.org/10.1017/CBO9781139179027

[79] Andrew N. Jordan and Markus Büttiker. ``Continuous quantum measurement with independent detector cross correlations''. Phys. Rev. Lett. 95, 220401 (2005).
https://doi.org/10.1103/PhysRevLett.95.220401

[80] Rusko Ruskov, Alexander N. Korotkov, and Klaus Mølmer. ``Qubit state monitoring by measurement of three complementary observables''. Phys. Rev. Lett. 105, 100506 (2010).
https://doi.org/10.1103/PhysRevLett.105.100506

[81] Rusko Ruskov, Joshua Combes, Klaus Mølmer, and Howard M. Wiseman. ``Qubit purification speed-up for three complementary continuous measurements''. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 370, 5291–5307 (2012).
https://doi.org/10.1098/rsta.2011.0516

[82] Pierre Rouchon and Jason F. Ralph. ``Efficient quantum filtering for quantum feedback control''. Phys. Rev. A 91, 012118 (2015).
https://doi.org/10.1103/PhysRevA.91.012118

[83] Shay Hacohen-Gourgy, Leigh S Martin, Emmanuel Flurin, Vinay V Ramasesh, K Birgitta Whaley, and Irfan Siddiqi. ``Quantum dynamics of simultaneously measured non-commuting observables''. Nature 538, 491–494 (2016).
https://doi.org/10.1038/nature19762

[84] Q. Ficheux, S. Jezouin, Z. Leghtas, and B. Huard. ``Dynamics of a qubit while simultaneously monitoring its relaxation and dephasing''. Nat. Comm. 9, 1926 (2018).
https://doi.org/10.1038/s41467-018-04372-9

[85] Philippe Lewalle, Sreenath K. Manikandan, Cyril Elouard, and Andrew N. Jordan and. ``Measuring fluorescence to track a quantum emitter's state: a theory review''. Contemporary Physics 61, 26–50 (2020).
https://doi.org/10.1080/00107514.2020.1747201

[86] Tathagata Karmakar, Philippe Lewalle, and Andrew N. Jordan. ``Stochastic path-integral analysis of the continuously monitored quantum harmonic oscillator''. PRX Quantum 3, 010327 (2022).
https://doi.org/10.1103/PRXQuantum.3.010327

[87] Christopher S. Jackson and Carlton M. Caves. ``Simultaneous measurements of noncommuting observables: Positive transformations and instrumental lie groups''. Entropy 25 (2023).
https://doi.org/10.3390/e25091254

[88] Nattaphong Wonglakhon, Howard M. Wiseman, and Areeya Chantasri. ``Completely positive trace-preserving maps for higher-order unraveling of lindblad master equations''. Phys. Rev. A 110, 062207 (2024).
https://doi.org/10.1103/PhysRevA.110.062207

[89] Pierre Guilmin, Pierre Rouchon, and Antoine Tilloy. ``Time-averaged continuous quantum measurement'' (2025). arXiv:2505.20382.
arXiv:2505.20382

[90] E. Arthurs and J. L. Kelly. ``B.s.t.j. briefs: On the simultaneous measurement of a pair of conjugate observables''. The Bell System Technical Journal 44, 725–729 (1965).
https://doi.org/10.1002/j.1538-7305.1965.tb01684.x

[91] J. Shapiro and S. Wagner. ``Phase and amplitude uncertainties in heterodyne detection''. IEEE Journal of Quantum Electronics 20, 803–813 (1984).
https://doi.org/10.1109/JQE.1984.1072470

[92] J.H. Shapiro. ``The Quantum Theory of Optical Communications''. IEEE Journal of Selected Topics in Quantum Electronics 15, 1547–1569 (2009).
https://doi.org/10.1109/JSTQE.2009.2024959

[93] Carlton M. Caves, Joshua Combes, Zhang Jiang, and Shashank Pandey. ``Quantum limits on phase-preserving linear amplifiers''. Phys. Rev. A 86, 063802 (2012).
https://doi.org/10.1103/PhysRevA.86.063802

[94] Maicol A Ochoa, Wolfgang Belzig, and Abraham Nitzan. ``Simultaneous weak measurement of non-commuting observables: a generalized Arthurs-Kelly protocol''. Scientific reports 8, 15781 (2018).
https://doi.org/10.1038/s41598-018-33562-0

[95] Goran Lindblad. ``On the generators of quantum dynamical semigroups''. Communications in mathematical physics 48, 119–130 (1976).
https://doi.org/10.1007/BF01608499

[96] Crispin Gardiner. ``Stochastic methods: a handbook for the natural and social sciences''. Springer-Verlag. Berlin Heidelberg (2009).

[97] N. G. van Kampen. ``Stochastic processes in physics and chemistry''. Elsevier Sci. and Tech. (Amsterdam). (2007).
https://doi.org/10.1016/B978-0-444-52965-7.X5000-4

[98] Peter E. Kloeden and Eckhard Platen. ``Numerical solution of stochastic differential equations''. Springer-Verlag. Berlin, Heidelberg (1992).
https://doi.org/10.1007/978-3-662-12616-5

[99] Philippe Lewalle. ``Quantum trajectories and their extremal–probability paths: New phenomena and applications''. PhD Dissertation, University of Rochester (2021).

[100] Francesco Ciccarello, Salvatore Lorenzo, Vittorio Giovannetti, and G. Massimo Palma. ``Quantum collision models: Open system dynamics from repeated interactions''. Physics Reports 954, 1–70 (2022).
https://doi.org/10.1016/j.physrep.2022.01.001

[101] Stuart S. Szigeti, Andre R. R. Carvalho, James G. Morley, and Michael R. Hush. ``Ignorance is bliss: General and robust cancellation of decoherence via no-knowledge quantum feedback''. Phys. Rev. Lett. 113, 020407 (2014).
https://doi.org/10.1103/PhysRevLett.113.020407

[102] Jirawat Saiphet, Sujin Suwanna, André R. R. Carvalho, and Areeya Chantasri. ``Time-delayed quantum feedback and incomplete decoherence suppression with a no-knowledge measurement''. Phys. Rev. A 103, 022208 (2021).
https://doi.org/10.1103/PhysRevA.103.022208

[103] Kater W Murch, SJ Weber, Christopher Macklin, and Irfan Siddiqi. ``Observing single quantum trajectories of a superconducting quantum bit''. Nature 502, 211–214 (2013).
https://doi.org/10.1038/nature12539

[104] Philippe Campagne-Ibarcq, Pierre Six, Landry Bretheau, Alain Sarlette, Mazyar Mirrahimi, Pierre Rouchon, and Benjamin Huard. ``Observing quantum state diffusion by heterodyne detection of fluorescence''. Physical Review X 6, 011002 (2016).
https://doi.org/10.1103/PhysRevX.6.011002

[105] Philippe Campagne-Ibarcq, Sébastien Jezouin, Nathanaël Cottet, Pierre Six, Landry Bretheau, François Mallet, Alain Sarlette, Pierre Rouchon, and Benjamin Huard. ``Using spontaneous emission of a qubit as a resource for feedback control''. Physical Review Letters 117, 060502 (2016). arXiv:1602.05479.
https://doi.org/10.1103/PhysRevLett.117.060502
arXiv:1602.05479

[106] N. Roch, M. E. Schwartz, F. Motzoi, C. Macklin, R. Vijay, A. W. Eddins, A. N. Korotkov, K. B. Whaley, M. Sarovar, and I. Siddiqi. ``Observation of measurement-induced entanglement and quantum trajectories of remote superconducting qubits''. Phys. Rev. Lett. 112, 170501 (2014).
https://doi.org/10.1103/PhysRevLett.112.170501

[107] E. Flurin, V. V. Ramasesh, S. Hacohen-Gourgy, L. S. Martin, N. Y. Yao, and I. Siddiqi. ``Observing topological invariants using quantum walks in superconducting circuits''. Phys. Rev. X 7, 031023 (2017).
https://doi.org/10.1103/PhysRevX.7.031023

[108] R. D. Somma and S. Boixo. ``Spectral gap amplification''. SIAM Journal on Computing 42, 593–610 (2013). arXiv:https://doi.org/10.1137/120871997.
https://doi.org/10.1137/120871997
arXiv:https://doi.org/10.1137/120871997

[109] Donald E Kirk. ``Optimal control theory: an introduction''. Courier Corporation. (2004).
https://doi.org/10.1002/aic.690170452

[110] Domenico d’Alessandro. ``Introduction to quantum control and dynamics''. Chapman and hall/CRC. (2021).
https://doi.org/10.1201/9781003051268

[111] Lev Semenovich Pontryagin. ``Mathematical theory of optimal processes''. Routledge. (2018).
https://doi.org/10.1201/9780203749319

[112] Heinz Schättler and Urszula Ledzewicz. ``Optimal control for mathematical models of cancer therapies: An application of geometric methods''. Volume 42 of Interdisciplinary Applied Mathematics, pages 1–496. Springer New York. New York, NY (2015).
https://doi.org/10.1007/978-1-4614-3834-2

[113] V. I. Arnold. ``Mathematical Methods of Classical Mechanics''. Springer. New York (1989).
https://doi.org/10.1007/978-1-4757-1693-1

[114] Lorenzo Campos Venuti, Domenico D'Alessandro, and Daniel A. Lidar. ``Optimal control for quantum optimization of closed and open systems''. Phys. Rev. Appl. 16, 054023 (2021).
https://doi.org/10.1103/PhysRevApplied.16.054023

[115] Ronan Gautier, Élie Genois, and Alexandre Blais. ``Optimal control in large open quantum systems: The case of transmon readout and reset''. Phys. Rev. Lett. 134, 070802 (2025).
https://doi.org/10.1103/PhysRevLett.134.070802

[116] Navin Khaneja, Timo Reiss, Cindie Kehlet, Thomas Schulte-Herbrüggen, and Steffen J. Glaser. ``Optimal control of coupled spin dynamics: design of nmr pulse sequences by gradient ascent algorithms''. Journal of Magnetic Resonance 172, 296–305 (2005).
https://doi.org/10.1016/j.jmr.2004.11.004

[117] Y. Nesterov. ``A method for solving the convex programming problem with convergence rate o(1/k2)''. Dokl Akad Nauk SSSR 269, 543 (1983). url: https://cir.nii.ac.jp/crid/1370862715914709505.
https://cir.nii.ac.jp/crid/1370862715914709505

[118] Yurii Nesterov. ``Introductory lectures on convex optimization: A basic course''. Volume 87. Springer Science & Business Media. (2013).
https://doi.org/10.1007/978-1-4419-8853-9

[119] Alberto Carlini, Akio Hosoya, Tatsuhiko Koike, and Yosuke Okudaira. ``Time-optimal quantum evolution''. Phys. Rev. Lett. 96, 060503 (2006).
https://doi.org/10.1103/PhysRevLett.96.060503

[120] Troels F. Rønnow, Zhihui Wang, Joshua Job, Sergio Boixo, Sergei V. Isakov, David Wecker, John M. Martinis, Daniel A. Lidar, and Matthias Troyer. ``Defining and detecting quantum speedup''. Science 345, 420–424 (2014).
https://doi.org/10.1126/science.1252319

[121] Tameem Albash and Daniel A. Lidar. ``Demonstration of a scaling advantage for a quantum annealer over simulated annealing''. Phys. Rev. X 8, 031016 (2018).
https://doi.org/10.1103/PhysRevX.8.031016

[122] Arianna C. Cylke, Philippe Lewalle, Tanawut Noungneaw, Howard M. Wiseman, Andrew N. Jordan, and Areeya Chantasri. ``Stochastic action functionals for diffusive quantum trajectories (in preparation)'' (2025).

[123] Juan Atalaya, Alexander N. Korotkov, and K. Birgitta Whaley. ``Error-correcting bacon-shor code with continuous measurement of noncommuting operators''. Phys. Rev. A 102, 022415 (2020).
https://doi.org/10.1103/PhysRevA.102.022415

[124] Juan Atalaya, Mohammad Bahrami, Leonid P. Pryadko, and Alexander N. Korotkov. ``Bacon-shor code with continuous measurement of noncommuting operators''. Phys. Rev. A 95, 032317 (2017).
https://doi.org/10.1103/PhysRevA.95.032317

[125] Dimitris Achlioptas, Assaf Naor, and Yuval Peres. ``Rigorous location of phase transitions in hard optimization problems''. Nature 435, 759–764 (2005).
https://doi.org/10.1038/nature03602

[126] Jing Zhang, Yu xi Liu, Re-Bing Wu, Kurt Jacobs, and Franco Nori. ``Quantum feedback: Theory, experiments, and applications''. Physics Reports 679, 1–60 (2017).
https://doi.org/10.1016/j.physrep.2017.02.003

[127] Charlene Ahn, Andrew C. Doherty, and Andrew J. Landahl. ``Continuous quantum error correction via quantum feedback control''. Phys. Rev. A 65, 042301 (2002).
https://doi.org/10.1103/PhysRevA.65.042301

[128] Charlene Ahn, H. M. Wiseman, and G. J. Milburn. ``Quantum error correction for continuously detected errors''. Phys. Rev. A 67, 052310 (2003).
https://doi.org/10.1103/PhysRevA.67.052310

[129] Charlene Ahn, Howard Wiseman, and Kurt Jacobs. ``Quantum error correction for continuously detected errors with any number of error channels per qubit''. Phys. Rev. A 70, 024302 (2004).
https://doi.org/10.1103/PhysRevA.70.024302

[130] Mohan Sarovar, Charlene Ahn, Kurt Jacobs, and Gerard J. Milburn. ``Practical scheme for error control using feedback''. Phys. Rev. A 69, 052324 (2004).
https://doi.org/10.1103/PhysRevA.69.052324

[131] Ramon van Handel and Hideo Mabuchi. ``Optimal error tracking via quantum coding and continuous syndrome measurement'' (2005). arXiv:quant-ph/0511221.
arXiv:quant-ph/0511221

[132] Ognyan Oreshkov and Todd A. Brun. ``Continuous quantum error correction for non-markovian decoherence''. Phys. Rev. A 76, 022318 (2007).
https://doi.org/10.1103/PhysRevA.76.022318

[133] Eduardo Mascarenhas, Breno Marques, Marcelo Terra Cunha, and Marcelo França Santos. ``Continuous quantum error correction through local operations''. Phys. Rev. A 82, 032327 (2010).
https://doi.org/10.1103/PhysRevA.82.032327

[134] Razieh Mohseninia, Jing Yang, Irfan Siddiqi, Andrew N. Jordan, and Justin Dressel. ``Always-On Quantum Error Tracking with Continuous Parity Measurements''. Quantum 4, 358 (2020).
https://doi.org/10.22331/q-2020-11-04-358

[135] J. Atalaya, S. Zhang, M. Y. Niu, A. Babakhani, H. C. H. Chan, J. M. Epstein, and K. B. Whaley. ``Continuous quantum error correction for evolution under time-dependent hamiltonians''. Phys. Rev. A 103, 042406 (2021).
https://doi.org/10.1103/PhysRevA.103.042406

[136] Ian Convy and K. Birgitta Whaley. ``A Logarithmic Bayesian Approach to Quantum Error Detection''. Quantum 6, 680 (2022).
https://doi.org/10.22331/q-2022-04-04-680

[137] William P. Livingston, Machiel S. Blok, Emmanuel Flurin, Justin Dressel, Andrew N. Jordan, and Irfan Siddiqi. ``Experimental demonstration of continuous quantum error correction''. Nat. Commun. 13, 2307 (2022).
https://doi.org/10.1038/s41467-022-29906-0

[138] Ian Convy, Haoran Liao, Song Zhang, Sahil Patel, William P Livingston, Ho Nam Nguyen, Irfan Siddiqi, and K Birgitta Whaley. ``Machine learning for continuous quantum error correction on superconducting qubits''. New Journal of Physics 24, 063019 (2022).
https://doi.org/10.1088/1367-2630/ac66f9

[139] Alicia B. Magann, Kenneth M. Rudinger, Matthew D. Grace, and Mohan Sarovar. ``Feedback-based quantum optimization''. Phys. Rev. Lett. 129, 250502 (2022).
https://doi.org/10.1103/PhysRevLett.129.250502

[140] Joshua Combes and Kurt Jacobs. ``Rapid state reduction of quantum systems using feedback control''. Phys. Rev. Lett. 96, 010504 (2006).
https://doi.org/10.1103/PhysRevLett.96.010504

[141] Tathagata Karmakar, Philippe Lewalle, Yipei Zhang, and K. Birgitta Whaley. ``Noise-canceling quantum feedback: non-hermitian dynamics with applications to state preparation and magic state distillation'' (2025). arXiv:2507.05611.
https://doi.org/10.1103/cnxj-m1wn
arXiv:2507.05611

[142] T. Schöning. ``A probabilistic algorithm for k-sat and constraint satisfaction problems''. In 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039). Pages 410–414. (1999).
https://doi.org/10.1109/SFFCS.1999.814612

[143] Edward Farhi, Shelby Kimmel, and Kristan Temme. ``A quantum version of Schöning's algorithm applied to quantum 2-SAT'' (2016). arXiv:1603.06985.
arXiv:1603.06985

[144] A. T. Rezakhani, W.-J. Kuo, A. Hamma, D. A. Lidar, and P. Zanardi. ``Quantum adiabatic brachistochrone''. Phys. Rev. Lett. 103, 080502 (2009).
https://doi.org/10.1103/PhysRevLett.103.080502

[145] Joachim Cohen and Mazyar Mirrahimi. ``Dissipation-induced continuous quantum error correction for superconducting circuits''. Phys. Rev. A 90, 062344 (2014).
https://doi.org/10.1103/PhysRevA.90.062344

[146] Jae-Mo Lihm, Kyungjoo Noh, and Uwe R. Fischer. ``Implementation-independent sufficient condition of the knill-laflamme type for the autonomous protection of logical qudits by strong engineered dissipation''. Phys. Rev. A 98, 012317 (2018).
https://doi.org/10.1103/PhysRevA.98.012317

[147] José Lebreuilly, Kyungjoo Noh, Chiao-Hsuan Wang, Steven M. Girvin, and Liang Jiang. ``Autonomous quantum error correction and quantum computation'' (2021). arXiv:2103.05007.
arXiv:2103.05007

[148] Jeffrey M. Gertler, Brian Baker, Juliang Li, Shruti Shirol, Jens Koch, and Chen Wang. ``Protecting a bosonic qubit with autonomous quantum error correction''. Nature 590, 243–248 (2021).
https://doi.org/10.1038/s41586-021-03257-0

[149] Qian Xu, Guo Zheng, Yu-Xin Wang, Peter Zoller, Aashish A. Clerk, and Liang Jiang. ``Autonomous quantum error correction and fault-tolerant quantum computation with squeezed cat qubits''. npj Quantum Information 9, 1–11 (2023).
https://doi.org/10.1038/s41534-023-00746-0

[150] Zhenyu Cai, Ryan Babbush, Simon C. Benjamin, Suguru Endo, William J. Huggins, Ying Li, Jarrod R. McClean, and Thomas E. O'Brien. ``Quantum error mitigation''. Rev. Mod. Phys. 95, 045005 (2023).
https://doi.org/10.1103/RevModPhys.95.045005

[151] Hang Ren, Yipei Zhang, Wendy M Billings, Rebecca Tomann, Nikolay V Tkachenko, Martin Head-Gordon, and K Birgitta Whaley. ``An error mitigated non-orthogonal quantum eigensolver via shadow tomography'' (2025). arXiv:2504.16008.
https://doi.org/10.1103/7c5b-3v56
arXiv:2504.16008

[152] Hang Ren, Yipei Zhang, Ze Zheng, Cuifeng Ying, Lei Xu, Mohsen Rahmani, and K. Birgitta Whaley. ``Error mitigated metasurface-based randomized measurement schemes''. Phys. Rev. Res. 6, 033310 (2024).
https://doi.org/10.1103/PhysRevResearch.6.033310

[153] Chris N Self, Marcello Benedetti, and David Amaro. ``Protecting expressive circuits with a quantum error detection code''. Nature Physics 20, 219–224 (2024).
https://doi.org/10.1038/s41567-023-02282-2

[154] Hong-Ye Hu, Andi Gu, Swarnadeep Majumder, Hang Ren, Yipei Zhang, Derek S Wang, Yi-Zhuang You, Zlatko Minev, Susanne F Yelin, and Alireza Seif. ``Demonstration of robust and efficient quantum property learning with shallow shadows''. Nature Communications 16, 2943 (2025).
https://doi.org/10.1038/s41467-025-57349-w

Cited by

[1] Franz J. Schreiber, Maximilian J. Kramer, Alexander Nietner, and Jens Eisert, "A measurement-driven quantum algorithm for SAT: Performance guarantees via spectral gaps and measurement parallelization", arXiv:2511.09647, (2025).

[2] Tim Pokart, Frank Pollmann, and Jan Carl Budich, "Entanglement Barriers from Computational Complexity: Matrix-Product-State Approach to Satisfiability", arXiv:2602.20299, (2026).

The above citations are from SAO/NASA ADS (last updated successfully 2026-07-21 00:25:12). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2026-07-21 00:25:09).

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.