Implementing any Linear Combination of Unitaries on Intermediate-term Quantum Computers

Shantanav Chakraborty

doi:10.22331/q-2024-10-10-1496

Implementing any Linear Combination of Unitaries on Intermediate-term Quantum Computers

Shantanav Chakraborty

CQST and CSTAR, International Institute of Information Technology Hyderabad, Telangana, India

Published:	2024-10-10, volume 8, page 1496
Eprint:	arXiv:2302.13555v4
Doi:	https://doi.org/10.22331/q-2024-10-10-1496
Citation:	Quantum 8, 1496 (2024).

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

Abstract

We develop three new methods to implement any Linear Combination of Unitaries (LCU), a powerful quantum algorithmic tool with diverse applications. While the standard LCU procedure requires several ancilla qubits and sophisticated multi-qubit controlled operations, our methods consume significantly fewer quantum resources. The first method ($\textit{Single-Ancilla LCU}$) estimates expectation values of observables with respect to any quantum state prepared by an LCU procedure while requiring only a single ancilla qubit, and no multi-qubit controlled operations. The second approach ($\textit{Analog LCU}$) is a simple, physically motivated, continuous-time analogue of LCU, tailored to hybrid qubit-qumode systems. The third method ($\textit{Ancilla-free LCU}$) requires no ancilla qubit at all and is useful when we are interested in the projection of a quantum state (prepared by the LCU procedure) in some subspace of interest. We apply the first two techniques to develop new quantum algorithms for a wide range of practical problems, ranging from Hamiltonian simulation, ground state preparation and property estimation, and quantum linear systems. Remarkably, despite consuming fewer quantum resources they retain a provable quantum advantage. The third technique allows us to connect discrete and continuous-time quantum walks with their classical counterparts. It also unifies the recently developed optimal quantum spatial search algorithms in both these frameworks, and leads to the development of new ones that require fewer ancilla qubits. Overall, our results are quite generic and can be readily applied to other problems, even beyond those considered here.

Featured image: The three different techniques for implementing any LCU on intermediate-term quantum computers, and their applications.

Popular summary

There are only a few quantum algorithmic primitives that have found broad applicability in the development of novel quantum algorithms. Linear Combination of Unitaries (LCU), a generic procedure by which any operator (not necessarily unitary) that can be expressed as a weighted sum of unitaries can be implemented on a quantum computer, is a rare exception in this regard. Indeed, LCU has been applied to a diverse range of problems of practical interest: it has been central to the development of optimal quantum algorithms for Hamiltonian simulation, quantum linear algebra and machine learning, ground state preparation and quantum optimization, quantum walks, and several others. However, the standard method for implementing any LCU is extremely resource-heavy, requiring several ancilla qubits and complicated multiqubit controlled operations. This makes it suitable for only fully programmable, fault-tolerant quantum computers that are decades away. However, quantum machines that will become available in the next few years won’t not have such capabilities. These devices have a limited number of logical qubits (severely restricting the ancilla space available) and can only run short-depth circuits with no multi-qubit controlled gates. In this work, we develop three novel techniques to implement any LCU on such intermediate-term quantum computers.

The first method, known as Single-ancilla LCU, implements any LCU using only a single ancilla qubit, no multi-qubit controlled logic, and quantum circuits of shorter depth than the standard method. The key idea is to substitute complicated controlled operations (in the standard LCU approach) with importance sampling and make several independent runs of short-depth quantum circuits to estimate the expectation value of observables vis-à-vis any state prepared by LCU. We apply this to develop new algorithms for ground state property estimation, Hamiltonian simulation, and quantum linear systems that are suitable for early fault-tolerant quantum computers. We characterize the end-to-end complexities of our algorithms, which, remarkably in some regimes, have shorter gate depths as compared to even state-of-the-art methods, despite requiring significantly fewer resources.

Our second method is a physically motivated approach for running LCU on hybrid qubit-qumode systems, a promising candidate for intermediate-term quantum computation. This involves coupling a discrete system with a continuous variable one (such as a one-dimensional quantum harmonic oscillator), and simply evolving under the resulting interaction Hamiltonian. This simple technique, Analog LCU, leads to a new quantum linear systems algorithm by engineering only Gaussian states, which are considerably simpler to implement in such devices.

We also prove that for certain problems it is possible to implement LCU without using any ancilla qubits at all. This is particularly relevant for quantum walks on a graph where we are often interested in the average number of steps after which the overlap of the walker with a marked subset of nodes is high. Our third technique, Ancilla-free LCU, shows that in such settings, quantum walk algorithms (which rely on standard LCU) can be modified to not require any ancilla qubits. This leads to new quantum walk hitting time (spatial search) algorithms with significantly fewer ancilla qubits and helps relate different frameworks of classical and quantum walks.

Overall, our methods are quite general and can be readily applied to any problem where LCU is employed, even beyond those considered here. Our work opens the possibility of developing alternative variants of widely applicable state-of-the-art quantum algorithmic primitives tailored to quantum computers that are currently being developed.

► BibTeX data

@article{Chakraborty2024implementingany,
  doi = {10.22331/q-2024-10-10-1496},
  url = {https://doi.org/10.22331/q-2024-10-10-1496},
  title = {Implementing any {L}inear {C}ombination of {U}nitaries on {I}ntermediate-term {Q}uantum {C}omputers},
  author = {Chakraborty, Shantanav},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1496},
  month = oct,
  year = {2024}
}

► References

[1] Google Quantum AI. Quantum supremacy using a programmable superconducting processor. Nature, 574(7779):505–510, 2019. doi:10.1038/s41586-019-1666-5.
https://doi.org/10.1038/s41586-019-1666-5

[2] Han-Sen Zhong, Hui Wang, Yu-Hao Deng, Ming-Cheng Chen, Li-Chao Peng, Yi-Han Luo, Jian Qin, Dian Wu, Xing Ding, Yi Hu, et al. Quantum computational advantage using photons. Science, 370(6523):1460–1463, 2020. doi:10.1126/science.abe8770.
https://doi.org/10.1126/science.abe8770

[3] Philippe Campagne-Ibarcq, Alec Eickbusch, Steven Touzard, Evan Zalys-Geller, Nicholas E Frattini, Volodymyr V Sivak, Philip Reinhold, Shruti Puri, Shyam Shankar, Robert J Schoelkopf, et al. Quantum error correction of a qubit encoded in grid states of an oscillator. Nature, 584(7821):368–372, 2020. doi:10.1038/s41586-020-2603-3.
https://doi.org/10.1038/s41586-020-2603-3

[4] Lars S Madsen, Fabian Laudenbach, Mohsen Falamarzi Askarani, Fabien Rortais, Trevor Vincent, Jacob FF Bulmer, Filippo M Miatto, Leonhard Neuhaus, Lukas G Helt, Matthew J Collins, et al. Quantum computational advantage with a programmable photonic processor. Nature, 606(7912):75–81, 2022. doi:10.1038/s41586-022-04725-x.
https://doi.org/10.1038/s41586-022-04725-x

[5] Google Quantum AI. Suppressing quantum errors by scaling a surface code logical qubit. Nature, 614:676–681, 2023. doi:10.1038/s41586-022-05434-1.
https://doi.org/10.1038/s41586-022-05434-1

[6] John Preskill. Quantum computing in the NISQ era and beyond. Quantum, 2:79, 2018. doi:10.22331/q-2018-08-06-79.
https://doi.org/10.22331/q-2018-08-06-79

[7] Kishor Bharti, Alba Cervera-Lierta, Thi Ha Kyaw, Tobias Haug, Sumner Alperin-Lea, Abhinav Anand, Matthias Degroote, Hermanni Heimonen, Jakob S Kottmann, Tim Menke, et al. Noisy intermediate-scale quantum algorithms. Reviews of Modern Physics, 94(1):015004, 2022. doi:10.1103/RevModPhys.94.015004.
https://doi.org/10.1103/RevModPhys.94.015004

[8] Earl T Campbell. Early fault-tolerant simulations of the hubbard model. Quantum Science and Technology, 7(1):015007, 2021. doi:10.1088/2058-9565/ac3110.
https://doi.org/10.1088/2058-9565/ac3110

[9] Paul K. Faehrmann, Mark Steudtner, Richard Kueng, Maria Kieferova, and Jens Eisert. Randomizing multi-product formulas for Hamiltonian simulation. Quantum, 6:806, 2022. doi:10.22331/q-2022-09-19-806.
https://doi.org/10.22331/q-2022-09-19-806

[10] Yulong Dong, Lin Lin, and Yu Tong. Ground-state preparation and energy estimation on early fault-tolerant quantum computers via quantum eigenvalue transformation of unitary matrices. PRX Quantum, 3(4):040305, 2022. doi:10.1103/PRXQuantum.3.040305.
https://doi.org/10.1103/PRXQuantum.3.040305

[11] Ruizhe Zhang, Guoming Wang, and Peter Johnson. Computing ground state properties with early fault-tolerant quantum computers. Quantum, 6:761, 2022. doi:10.22331/q-2022-07-11-761.
https://doi.org/10.22331/q-2022-07-11-761

[12] Lin Lin and Yu Tong. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum, 3(1):010318, 2022. doi:10.1103/PRXQuantum.3.010318.
https://doi.org/10.1103/PRXQuantum.3.010318

[13] Guoming Wang, Daniel Stilck-França, Ruizhe Zhang, Shuchen Zhu, and Peter D Johnson. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum, 7:1167, 2023. doi:10.22331/q-2023-11-06-1167.
https://doi.org/10.22331/q-2023-11-06-1167

[14] Andreas Wallraff, David I Schuster, Alexandre Blais, Luigi Frunzio, R-S Huang, Johannes Majer, Sameer Kumar, Steven M Girvin, and Robert J Schoelkopf. Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics. Nature, 431(7005):162–167, 2004. doi:10.1038/nature02851.
https://doi.org/10.1038/nature02851

[15] J-M Pirkkalainen, SU Cho, Jian Li, GS Paraoanu, PJ Hakonen, and MA Sillanpää. Hybrid circuit cavity quantum electrodynamics with a micromechanical resonator. Nature, 494(7436):211–215, 2013. doi:10.1038/nature11821.
https://doi.org/10.1038/nature11821

[16] Gershon Kurizki, Patrice Bertet, Yuimaru Kubo, Klaus Mølmer, David Petrosyan, Peter Rabl, and Jörg Schmiedmayer. Quantum technologies with hybrid systems. Proceedings of the National Academy of Sciences, 112(13):3866–3873, 2015. doi:10.1073/pnas.1419326112.
https://doi.org/10.1073/pnas.1419326112

[17] Ulrik L Andersen, Jonas S Neergaard-Nielsen, Peter Van Loock, and Akira Furusawa. Hybrid discrete-and continuous-variable quantum information. Nature Physics, 11(9):713–719, 2015. doi:10.1038/nphys3410.
https://doi.org/10.1038/nphys3410

[18] HCJ Gan, Gleb Maslennikov, Ko-Wei Tseng, Chihuan Nguyen, and Dzmitry Matsukevich. Hybrid quantum computing with conditional beam splitter gate in trapped ion system. Physical review letters, 124(17):170502, 2020. doi:10.1103/PhysRevLett.124.170502.
https://doi.org/10.1103/PhysRevLett.124.170502

[19] Nicolas PD Sawaya, Tim Menke, Thi Ha Kyaw, Sonika Johri, Alán Aspuru-Guzik, and Gian Giacomo Guerreschi. Resource-efficient digital quantum simulation of d-level systems for photonic, vibrational, and spin-s hamiltonians. npj Quantum Information, 6(1):49, 2020. doi:10.1038/s41534-020-0278-0.
https://doi.org/10.1038/s41534-020-0278-0

[20] Dietrich Leibfried, Rainer Blatt, Christopher Monroe, and David Wineland. Quantum dynamics of single trapped ions. Reviews of Modern Physics, 75(1):281, 2003. doi:10.1103/RevModPhys.75.281.
https://doi.org/10.1103/RevModPhys.75.281

[21] Daniel Gottesman, Alexei Kitaev, and John Preskill. Encoding a qubit in an oscillator. Physical Review A, 64(1):012310, 2001. doi:10.1103/PhysRevA.64.012310.
https://doi.org/10.1103/PhysRevA.64.012310

[22] Andrew M Childs and Nathan Wiebe. Hamiltonian simulation using linear combinations of unitary operations. Quantum Information & Computation, 12(11-12):901–924, 2012. URL: https://dl.acm.org/doi/10.5555/2481569.2481570.
https://dl.acm.org/doi/10.5555/2481569.2481570

[23] Dominic W. Berry, Andrew M. Childs, Richard Cleve, Robin Kothari, and Rolando D. Somma. Exponential improvement in precision for simulating sparse hamiltonians. In Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing, STOC '14, page 283–292, New York, NY, USA, 2014. Association for Computing Machinery. doi:10.1145/2591796.2591854.
https://doi.org/10.1145/2591796.2591854

[24] Dominic W. Berry, Andrew M. Childs, and Robin Kothari. Hamiltonian simulation with nearly optimal dependence on all parameters. In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, pages 792–809, 2015. doi:10.1109/FOCS.2015.54.
https://doi.org/10.1109/FOCS.2015.54

[25] Dominic W Berry, Andrew M Childs, Richard Cleve, Robin Kothari, and Rolando D Somma. Simulating Hamiltonian dynamics with a truncated Taylor series. Physical review letters, 114(9):090502, 2015. doi:10.1103/PhysRevLett.114.090502.
https://doi.org/10.1103/PhysRevLett.114.090502

[26] Andrew M. Childs, Robin Kothari, and Rolando D. Somma. Quantum algorithm for systems of linear equations with exponentially improved dependence on precision. SIAM Journal on Computing, 46(6):1920–1950, 2017. doi:10.1137/16M1087072.
https://doi.org/10.1137/16M1087072

[27] Shantanav Chakraborty, András Gilyén, and Stacey Jeffery. The Power of Block-Encoded Matrix Powers: Improved Regression Techniques via Faster Hamiltonian Simulation. In Christel Baier, Ioannis Chatzigiannakis, Paola Flocchini, and Stefano Leonardi, editors, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019), volume 132 of Leibniz International Proceedings in Informatics (LIPIcs), pages 33:1–33:14, Dagstuhl, Germany, 2019. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik. doi:10.4230/LIPIcs.ICALP.2019.33.
https://doi.org/10.4230/LIPIcs.ICALP.2019.33

[28] Dominic W Berry, Andrew M Childs, Aaron Ostrander, and Guoming Wang. Quantum algorithm for linear differential equations with exponentially improved dependence on precision. Communications in Mathematical Physics, 356:1057–1081, 2017. doi:10.1007/s00220-017-3002-y.
https://doi.org/10.1007/s00220-017-3002-y

[29] Jin-Peng Liu, Herman Øie Kolden, Hari K Krovi, Nuno F Loureiro, Konstantina Trivisa, and Andrew M Childs. Efficient quantum algorithm for dissipative nonlinear differential equations. Proceedings of the National Academy of Sciences, 118(35):e2026805118, 2021. doi:10.1073/pnas.2026805118.
https://doi.org/10.1073/pnas.2026805118

[30] Andrew M Childs, Jin-Peng Liu, and Aaron Ostrander. High-precision quantum algorithms for partial differential equations. Quantum, 5:574, 2021. doi:10.22331/q-2021-11-10-574.
https://doi.org/10.22331/q-2021-11-10-574

[31] Simon Apers and Alain Sarlette. Quantum fast-forwarding: Markov chains and graph property testing. Quantum Information & Computation, 19(3-4):181–213, 2019. URL: https://dl.acm.org/doi/10.5555/3370245.3370246.
https://dl.acm.org/doi/10.5555/3370245.3370246

[32] Andris Ambainis, András Gilyén, Stacey Jeffery, and Martins Kokainis. Quadratic speedup for finding marked vertices by quantum walks. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, page 412–424, New York, NY, USA, 2020. Association for Computing Machinery. doi:10.1145/3357713.3384252.
https://doi.org/10.1145/3357713.3384252

[33] Simon Apers, Shantanav Chakraborty, Leonardo Novo, and Jérémie Roland. Quadratic speedup for spatial search by continuous-time quantum walk. Physical Review Letters, 129(16):160502, 2022. doi:10.1103/PhysRevLett.129.160502.
https://doi.org/10.1103/PhysRevLett.129.160502

[34] Yimin Ge, Jordi Tura, and J Ignacio Cirac. Faster ground state preparation and high-precision ground energy estimation with fewer qubits. Journal of Mathematical Physics, 60(2):022202, 2019. doi:10.1063/1.5027484.
https://doi.org/10.1063/1.5027484

[35] Trevor Keen, Eugene Dumitrescu, and Yan Wang. Quantum algorithms for ground-state preparation and Green's function calculation. arXiv preprint arXiv:2112.05731, 2021. doi:10.48550/arXiv.2112.05731.
https://doi.org/10.48550/arXiv.2112.05731
arXiv:2112.05731

[36] Min-Quan He, Dan-Bo Zhang, and Z. D. Wang. Quantum Gaussian filter for exploring ground-state properties. Phys. Rev. A, 106:032420, 2022. doi:10.1103/PhysRevA.106.032420.
https://doi.org/10.1103/PhysRevA.106.032420

[37] Anirban Narayan Chowdhury and Rolando D Somma. Quantum algorithms for gibbs sampling and hitting-time estimation. Quantum Information & Computation, 17(1-2):41–64, 2017. URL: https://dl.acm.org/doi/10.5555/3179483.3179486.
https://dl.acm.org/doi/10.5555/3179483.3179486

[38] Joran Van Apeldoorn, András Gilyén, Sander Gribling, and Ronald de Wolf. Quantum SDP-solvers: Better upper and lower bounds. Quantum, 4:230, 2020. doi:doi.org/10.22331/q-2020-02-14-230.
https://doi.org/10.22331/q-2020-02-14-230

[39] M. Szegedy. Quantum speed-up of markov chain based algorithms. In 45th Annual IEEE Symposium on Foundations of Computer Science, pages 32–41, 2004. doi:10.1109/FOCS.2004.53.
https://doi.org/10.1109/FOCS.2004.53

[40] Frédéric Magniez, Ashwin Nayak, Jérémie Roland, and Miklos Santha. Search via quantum walk. In Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, pages 575–584, 2007. doi:10.1137/090745854.
https://doi.org/10.1137/090745854

[41] Hari Krovi, Frédéric Magniez, Maris Ozols, and Jérémie Roland. Quantum walks can find a marked element on any graph. Algorithmica, 74(2):851–907, 2016. doi:10.1007/s00453-015-9979-8.
https://doi.org/10.1007/s00453-015-9979-8

[42] Simon Apers, András Gilyén, and Stacey Jeffery. A Unified Framework of Quantum Walk Search. In Markus Bläser and Benjamin Monmege, editors, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021), volume 187 of Leibniz International Proceedings in Informatics (LIPIcs), pages 6:1–6:13, Dagstuhl, Germany, 2021. Schloss Dagstuhl – Leibniz-Zentrum für Informatik. doi:10.4230/LIPIcs.STACS.2021.6.
https://doi.org/10.4230/LIPIcs.STACS.2021.6

[43] Guang Hao Low and Isaac L Chuang. Hamiltonian simulation by qubitization. Quantum, 3:163, 2019. doi:10.22331/q-2019-07-12-163.
https://doi.org/10.22331/q-2019-07-12-163

[44] András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe. Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics. arXiv preprint arXiv:1806.01838, 2018. doi:10.48550/arXiv.1806.01838.
https://doi.org/10.48550/arXiv.1806.01838
arXiv:1806.01838

[45] András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe. Quantum singular value transformation and beyond: Exponential improvements for quantum matrix arithmetics. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, page 193–204, New York, NY, USA, 2019. Association for Computing Machinery. doi:10.1145/3313276.3316366.
https://doi.org/10.1145/3313276.3316366

[46] John M Martyn, Zane M Rossi, Andrew K Tan, and Isaac L Chuang. Grand unification of quantum algorithms. PRX Quantum, 2(4):040203, 2021. doi:10.1103/PRXQuantum.2.040203.
https://doi.org/10.1103/PRXQuantum.2.040203

[47] Gilles Brassard, Peter Hoyer, Michele Mosca, and Alain Tapp. Quantum amplitude amplification and estimation. Contemporary Mathematics, 305:53–74, 2002. doi:10.1090/conm/305.
https://doi.org/10.1090/conm/305

[48] Andrew M Childs, Yuan Su, Minh C Tran, Nathan Wiebe, and Shuchen Zhu. Theory of Trotter error with commutator scaling. Physical Review X, 11(1):011020, 2021. doi:10.1103/PhysRevX.11.011020.
https://doi.org/10.1103/PhysRevX.11.011020

[49] Andrew M Childs, Dmitri Maslov, Yunseong Nam, Neil J Ross, and Yuan Su. Toward the first quantum simulation with quantum speedup. Proceedings of the National Academy of Sciences, 115(38):9456–9461, 2018. doi:10.1073/pnas.1801723115.
https://doi.org/10.1073/pnas.1801723115

[50] Qi Zhao, You Zhou, Alexander F Shaw, Tongyang Li, and Andrew M Childs. Hamiltonian Simulation with random inputs. Physical Review Letters, 129(27):270502, 2022. doi:10.1103/PhysRevLett.129.270502.
https://doi.org/10.1103/PhysRevLett.129.270502

[51] Pedro CS Costa, Dong An, Yuval R Sanders, Yuan Su, Ryan Babbush, and Dominic W Berry. Optimal scaling quantum linear-systems solver via discrete adiabatic theorem. PRX Quantum, 3(4):040303, 2022. doi:10.1103/PRXQuantum.3.040303.
https://doi.org/10.1103/PRXQuantum.3.040303

[52] Guang Hao Low and Isaac L Chuang. Hamiltonian simulation by uniform spectral amplification. arXiv preprint arXiv:1707.05391, 2017. doi:10.48550/arXiv.1707.05391.
https://doi.org/10.48550/arXiv.1707.05391
arXiv:1707.05391

[53] Subir Sachdev. Quantum Phase Transitions. Cambridge University Press, 2nd edition, 2011. doi:10.1017/CBO9780511973765.
https://doi.org/10.1017/CBO9780511973765

[54] Sam McArdle, Suguru Endo, Alán Aspuru-Guzik, Simon C Benjamin, and Xiao Yuan. Quantum computational chemistry. Reviews of Modern Physics, 92(1):015003, 2020. doi:10.1103/RevModPhys.92.015003.
https://doi.org/10.1103/RevModPhys.92.015003

[55] David J Griffiths and Darrell F Schroeter. Introduction to quantum mechanics. Cambridge University Press, 2018. doi:doi.org/10.1017/9781316995433.
https://doi.org/10.1017/9781316995433

[56] Christian Weedbrook, Stefano Pirandola, Raúl García-Patrón, Nicolas J Cerf, Timothy C Ralph, Jeffrey H Shapiro, and Seth Lloyd. Gaussian quantum information. Reviews of Modern Physics, 84(2):621, 2012. doi:10.1103/RevModPhys.84.621.
https://doi.org/10.1103/RevModPhys.84.621

[57] Andrew M. Childs and Jeffrey Goldstone. Spatial search by quantum walk. Phys. Rev. A, 70:022314, 2004. doi:10.1103/PhysRevA.70.022314.
https://doi.org/10.1103/PhysRevA.70.022314

[58] Andrew M Childs. On the relationship between continuous-and discrete-time quantum walk. Communications in Mathematical Physics, 294(2):581–603, 2010. doi:10.1007/s00220-009-0930-1.
https://doi.org/10.1007/s00220-009-0930-1

[59] Oleksandr Kyriienko. Quantum inverse iteration algorithm for programmable quantum simulators. npj Quantum Information, 6(1):7, 2020. doi:10.1038/s41534-019-0239-7.
https://doi.org/10.1038/s41534-019-0239-7

[60] Earl Campbell. Random compiler for fast Hamiltonian Simulation. Physical review letters, 123(7):070503, 2019. doi:10.1103/PhysRevLett.123.070503.
https://doi.org/10.1103/PhysRevLett.123.070503

[61] Chi-Fang Chen, Hsin-Yuan Huang, Richard Kueng, and Joel A. Tropp. Concentration for random product formulas. PRX Quantum, 2:040305, Oct 2021. doi:10.1103/PRXQuantum.2.040305.
https://doi.org/10.1103/PRXQuantum.2.040305

[62] Guang Hao Low and Isaac L Chuang. Optimal Hamiltonian Simulation by Quantum Signal Processing. Physical Review Letters, 118(1):010501, 2017. doi:10.1103/PhysRevLett.118.010501.
https://doi.org/10.1103/PhysRevLett.118.010501

[63] Daniel S Abrams and Seth Lloyd. Quantum algorithm providing exponential speed increase for finding eigenvalues and eigenvectors. Physical Review Letters, 83(24):5162, 1999. doi:10.1103/PhysRevLett.83.5162.
https://doi.org/10.1103/PhysRevLett.83.5162

[64] Lin Lin and Yu Tong. Near-optimal ground state preparation. Quantum, 4:372, 2020. doi:10.22331/q-2020-12-14-372.
https://doi.org/10.22331/q-2020-12-14-372

[65] Aram W Harrow, Avinatan Hassidim, and Seth Lloyd. Quantum algorithm for linear systems of equations. Physical Review Letters, 103(15):150502, 2009. doi:10.1103/PhysRevLett.103.150502.
https://doi.org/10.1103/PhysRevLett.103.150502

[66] Leonard Wossnig, Zhikuan Zhao, and Anupam Prakash. Quantum linear system algorithm for dense matrices. Physical Review Letters, 120(5):050502, 2018. doi:10.1103/PhysRevLett.120.050502.
https://doi.org/10.1103/PhysRevLett.120.050502

[67] Shantanav Chakraborty, Aditya Morolia, and Anurudh Peduri. Quantum Regularized Least Squares. Quantum, 7:988, April 2023. doi:10.22331/q-2023-04-27-988.
https://doi.org/10.22331/q-2023-04-27-988

[68] Yiğit Subaşı, Rolando D Somma, and Davide Orsucci. Quantum algorithms for systems of linear equations inspired by adiabatic quantum computing. Physical Review Letters, 122(6):060504, 2019. doi:10.1103/PhysRevLett.122.060504.
https://doi.org/10.1103/PhysRevLett.122.060504

[69] Lin Lin and Yu Tong. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum, 4:361, 2020. doi:10.22331/q-2020-11-11-361.
https://doi.org/10.22331/q-2020-11-11-361

[70] Hsin-Yuan Huang, Kishor Bharti, and Patrick Rebentrost. Near-term quantum algorithms for linear systems of equations with regression loss functions. New Journal of Physics, 23(11):113021, 2021. doi:10.1088/1367-2630/ac325f.
https://doi.org/10.1088/1367-2630/ac325f

[71] Andrew M Childs, Enrico Deotto, Edward Farhi, Jeffrey Goldstone, Sam Gutmann, and Andrew J Landahl. Quantum search by measurement. Physical Review A, 66(3):032314, 2002. doi:10.1103/PhysRevA.66.032314.
https://doi.org/10.1103/PhysRevA.66.032314

[72] Shantanav Chakraborty, Kyle Luh, and Jérémie Roland. Analog quantum algorithms for the mixing of markov chains. Physical Review A, 102(2):022423, 2020. doi:10.1103/PhysRevA.102.022423.
https://doi.org/10.1103/PhysRevA.102.022423

[73] Shantanav Chakraborty, Leonardo Novo, Andris Ambainis, and Yasser Omar. Spatial search by quantum walk is optimal for almost all graphs. Physical review letters, 116(10):100501, 2016. doi:10.1103/PhysRevLett.116.100501.
https://doi.org/10.1103/PhysRevLett.116.100501

[74] Shantanav Chakraborty, Leonardo Novo, and Jérémie Roland. Optimality of spatial search via continuous-time quantum walks. Physical Review A, 102(3):032214, 2020. doi:10.1103/PhysRevA.102.032214.
https://doi.org/10.1103/PhysRevA.102.032214

[75] Shantanav Chakraborty, Leonardo Novo, and Jérémie Roland. Finding a marked node on any graph via continuous-time quantum walks. Physical Review A, 102(2):022227, 2020. doi:10.1103/PhysRevA.102.022227.
https://doi.org/10.1103/PhysRevA.102.022227

[76] Anirban N Chowdhury, Rolando D Somma, and Yiğit Subaşı. Computing partition functions in the one-clean-qubit model. Physical Review A, 103(3):032422, 2021. doi:10.1103/PhysRevA.103.032422.
https://doi.org/10.1103/PhysRevA.103.032422

[77] Mary Beth Ruskai. Inequalities for traces on von neumann algebras. Communications in Mathematical Physics, 26:280–289, 1972. doi:doi.org/10.1007/BF01645523.
https://doi.org/10.1007/BF01645523

[78] Patrick Rall. Quantum algorithms for estimating physical quantities using block encodings. Physical Review A, 102(2):022408, 2020. doi:10.1103/PhysRevA.102.022408.
https://doi.org/10.1103/PhysRevA.102.022408

[79] Dmitry Grinko, Julien Gacon, Christa Zoufal, and Stefan Woerner. Iterative quantum amplitude estimation. npj Quantum Information, 7(1):52, 2021. doi:10.1038/s41534-021-00379-1.
https://doi.org/10.1038/s41534-021-00379-1

[80] Kianna Wan, Mario Berta, and Earl T. Campbell. Randomized quantum algorithm for statistical phase estimation. Physical Review Letters, 129:030503, Jul 2022. doi:10.1103/PhysRevLett.129.030503.
https://doi.org/10.1103/PhysRevLett.129.030503

[81] Jeongwan Haah, Matthew B Hastings, Robin Kothari, and Guang Hao Low. Quantum algorithm for simulating real time evolution of lattice hamiltonians. SIAM Journal on Computing, 52(6):FOCS18–250–FOCS18–284, 2021. doi:10.1137/18M1231511.
https://doi.org/10.1137/18M1231511

[82] Julia Kempe, Alexei Kitaev, and Oded Regev. The complexity of the local hamiltonian problem. SIAM Journal on Computing, 35(5):1070–1097, 2006. doi:10.1137/S00975397044452.
https://doi.org/10.1137/S00975397044452

[83] Andris Ambainis. Variable time amplitude amplification and quantum algorithms for linear algebra problems. In Christoph Dürr and Thomas Wilke, editors, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012), volume 14 of Leibniz International Proceedings in Informatics (LIPIcs), pages 636–647, Dagstuhl, Germany, 2012. Schloss Dagstuhl – Leibniz-Zentrum für Informatik. doi:10.4230/LIPIcs.STACS.2012.636.
https://doi.org/10.4230/LIPIcs.STACS.2012.636

[84] Sushant Sachdeva, Nisheeth K Vishnoi, et al. Faster algorithms via approximation theory. Foundations and Trends in Theoretical Computer Science, 9(2):125–210, 2014. doi:10.1561/0400000065.
https://doi.org/10.1561/0400000065

[85] Yudong Cao, Jonathan Romero, Jonathan P. Olson, Matthias Degroote, Peter D. Johnson, Mária Kieferová, Ian D. Kivlichan, Tim Menke, Borja Peropadre, Nicolas P. D. Sawaya, Sukin Sim, Libor Veis, and Alán Aspuru-Guzik. Quantum chemistry in the age of quantum computing. Chemical Reviews, 119(19):10856–10915, 2019. PMID: 31469277. doi:10.1021/acs.chemrev.8b00803.
https://doi.org/10.1021/acs.chemrev.8b00803

[86] Yuan Su, Dominic W. Berry, Nathan Wiebe, Nicholas Rubin, and Ryan Babbush. Fault-tolerant quantum simulations of chemistry in first quantization. PRX Quantum, 2:040332, Nov 2021. doi:10.1103/PRXQuantum.2.040332.
https://doi.org/10.1103/PRXQuantum.2.040332

[87] Ryan Babbush, Nathan Wiebe, Jarrod McClean, James McClain, Hartmut Neven, and Garnet Kin-Lic Chan. Low-depth quantum simulation of materials. Phys. Rev. X, 8:011044, Mar 2018. doi:10.1103/PhysRevX.8.011044.
https://doi.org/10.1103/PhysRevX.8.011044

[88] Javier Argüello-Luengo, Alejandro González-Tudela, Tao Shi, Peter Zoller, and J Ignacio Cirac. Analogue quantum chemistry simulation. Nature, 574(7777):215–218, 2019. doi:10.1038/s41586-019-1614-4.
https://doi.org/10.1038/s41586-019-1614-4

[89] Rolando D Somma, Sergio Boixo, Howard Barnum, and Emanuel Knill. Quantum simulations of classical annealing processes. Physical Review Letters, 101(13):130504, 2008. doi:10.1103/PhysRevLett.101.130504.
https://doi.org/10.1103/PhysRevLett.101.130504

[90] Kristan Temme, Tobias J Osborne, Karl G Vollbrecht, David Poulin, and Frank Verstraete. Quantum metropolis sampling. Nature, 471(7336):87–90, 2011. doi:10.1038/nature09770.
https://doi.org/10.1038/nature09770

[91] Man-Hong Yung and Alán Aspuru-Guzik. A quantum–quantum metropolis algorithm. Proceedings of the National Academy of Sciences, 109(3):754–759, 2012. doi:10.1073/pnas.1111758109.
https://doi.org/10.1073/pnas.1111758109

[92] Samson Wang, Sam McArdle, and Mario Berta. Qubit-efficient randomized quantum algorithms for linear algebra. PRX Quantum, 5:020324, 2024. doi:10.1103/PRXQuantum.5.020324.
https://doi.org/10.1103/PRXQuantum.5.020324

[93] Nai-Hui Chia, András Gilyén, Han-Hsuan Lin, Seth Lloyd, Ewin Tang, and Chunhao Wang. Quantum-Inspired Algorithms for Solving Low-Rank Linear Equation Systems with Logarithmic Dependence on the Dimension. In Yixin Cao, Siu-Wing Cheng, and Minming Li, editors, 31st International Symposium on Algorithms and Computation (ISAAC 2020), volume 181 of Leibniz International Proceedings in Informatics (LIPIcs), pages 47:1–47:17, Dagstuhl, Germany, 2020. Schloss Dagstuhl–Leibniz-Zentrum für Informatik. doi:10.4230/LIPIcs.ISAAC.2020.47.
https://doi.org/10.4230/LIPIcs.ISAAC.2020.47

[94] Ewin Tang. Quantum principal component analysis only achieves an exponential speedup because of its state preparation assumptions. Physical Review Letters, 127:060503, 2021. doi:10.1103/PhysRevLett.127.060503.
https://doi.org/10.1103/PhysRevLett.127.060503

[95] András Gilyén, Zhao Song, and Ewin Tang. An improved quantum-inspired algorithm for linear regression. Quantum, 6:754, 2022. doi:10.22331/q-2022-06-30-754.
https://doi.org/10.22331/q-2022-06-30-754

[96] Changpeng Shao and Ashley Montanaro. Faster quantum-inspired algorithms for solving linear systems. ACM Transactions on Quantum Computing, 3(4):1–23, 2022. doi:10.1145/3520141.
https://doi.org/10.1145/3520141

Cited by

[1] Norifumi Matsumoto, Shoichiro Tsutsui, Yuya O. Nakagawa, Yuichiro Hidaka, Shota Kanasugi, Kazunori Maruyama, Hirotaka Oshima, and Shintaro Sato, "Quantum many-body simulation of finite-temperature systems with sampling a series expansion of a quantum imaginary-time evolution", Physical Review Research 7 1, 013254 (2025).

[2] Amir Kalev and Itay Hen, "A simple quantum simulation algorithm with near-optimal precision scaling", Quantum Science and Technology 10 4, 045052 (2025).

[3] Yuya Kawamata, Kosuke Mitarai, and Keisuke Fujii, "Quasi Monte Carlo method for linear combination unitaries via classical postprocessing", Physical Review Research 8 2, 023161 (2026).

[4] Océane Koska, Marc Baboulin, and Arnaud Gazda, ISC High Performance 2024 Research Paper Proceedings (39th International Conference) 1 (2024) ISBN:978-3-9826336-0-2.

[5] Arthur Braida, Shantanav Chakraborty, Alapan Chaudhuri, Joseph Cunningham, Rutvij Menavlikar, Leonardo Novo, and Jérémie Roland, "Unstructured Adiabatic Quantum Optimization: Optimality with Limitations", Quantum 9, 1790 (2025).

[6] Paul K. Faehrmann, Jens Eisert, Mária Kieferová, and Richard Kueng, "Short-time simulation of quantum dynamics by Pauli measurements", Physical Review A 112 1, 012602 (2025).

[7] Giuliana Siddi Moreau, Lorenzo Pisani, Manuela Profir, Carlo Podda, Lidia Leoni, and Giacomo Cao, "Quantum Artificial Intelligence Scalability in the NISQ Era: Pathways to Quantum Utility", Advanced Quantum Technologies 8 10, 2400716 (2025).

[8] Yuan Liu, Shraddha Singh, Kevin C. Smith, Eleanor Crane, John M. Martyn, Alec Eickbusch, Alexander Schuckert, Richard D. Li, Jasmine Sinanan-Singh, Micheline B. Soley, Takahiro Tsunoda, Isaac L. Chuang, Nathan Wiebe, and Steven M. Girvin, "Hybrid Oscillator-Qubit Quantum Processors: Instruction Set Architectures, Abstract Machine Models, and Applications", PRX Quantum 7 1, 010201 (2026).

[9] Samuel Godwood, Dog̃a Murat Kürkçüog̃lu, Gabriel N. Perdue, Marina Maneyro, and Alessandro Roggero, "Fault-tolerant resource comparison of qudit and qubit encodings for diagonal quadratic operators", Physical Review A 114 2, 022427 (2026).

[10] Jim Furches, Sarah Chehade, Kathleen Hamilton, Nathan Wiebe, and Carlos Ortiz Marrero, "Application-level benchmarking of quantum computers using nonlocal game strategies", Quantum Science and Technology 10 4, 045002 (2025).

[11] Jinzhao Sun, Pei Zeng, Tom Gur, and M. S. Kim, "High-precision and low-depth quantum algorithm design for eigenstate problems", Science Advances 12 3, eaeb1622 (2026).

[12] Lennart Binkowski, Gereon Koßmann, Tobias J. Osborne, René Schwonnek, and Timo Ziegler, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 111 (2025) ISBN:979-8-3315-5736-2.

[13] Vladimir Vargas-Calderón, "Quantum deep sets and sequences", Quantum Machine Intelligence 7 2, 65 (2025).

[14] Ibsal Assi, Michael Vogl, Meenu Kumari, and J. P. F. LeBlanc, "Beyond trotterization: Variational product formulas for quantum simulation", Physical Review B 113 21, 214314 (2026).

[15] Christopher F. Kane, Siddharth Hariprakash, Neel S. Modi, Michael Kreshchuk, and Christian W Bauer, "Block encoding bosons by signal processing", Quantum 9, 1747 (2025).

[16] Shantanav Chakraborty, Siddhartha Das, Arnab Ghorui, Soumyabrata Hazra, and Uttam Singh, "Sample complexity of black box work extraction", Quantum Science and Technology 10 4, 045070 (2025).

[17] Chukwudubem Umeano, Stefano Scali, and Oleksandr Kyriienko, "Quantum community detection via deterministic elimination", Physical Review A 112 5, 052422 (2025).

[18] Michelle Wynne Sze, Yao Tang, Silas Dilkes, David Muñoz Ramo, Ross Duncan, and Nathan Fitzpatrick, "Hamiltonian dynamics simulation using linear combination of unitaries on an ion trap quantum computer", Quantum Science and Technology 11 1, 015023 (2026).

[19] Vinay Kumar, Claudio Cicconetti, Riccardo Bassoli, Marco Conti, and Andrea Passarella, 2026 International Conference on Quantum Communications, Networking, and Computing (QCNC) 1038 (2026) ISBN:979-8-3315-6110-9.

[20] Dimitar Trenev, Pauline J Ollitrault, Stuart M. Harwood, Tanvi P. Gujarati, Sumathy Raman, Antonio Mezzacapo, and Sarah Mostame, "Refining resource estimation for the quantum computation of vibrational molecular spectra through Trotter error analysis", Quantum 9, 1630 (2025).

[21] Yudai Suzuki, Bi Hong Tiang, Jeongrak Son, Nelly H. Y. Ng, Zoe Holmes, and Marek Gluza, "Double-bracket algorithm for quantum signal processing without post-selection", Quantum 9, 1954 (2025).

[22] Fanxu Meng, Zhiguo Huang, Yuan Long, Tian Luan, Xianchao Zhang, Xutao Yu, and Zaichen Zhang, "Multiagent reinforcement learning for efficient variational fast-forwarding quantum circuits", Physical Review A 113 1, 012618 (2026).

[23] Joseph Peetz, Scott E. Smart, and Prineha Narang, "Quantum simulation via stochastic combination of unitaries", npj Quantum Information 12 1, 52 (2026).

[24] Dong An, Andrew M. Childs, and Lin Lin, "Quantum Algorithm for Linear Non-unitary Dynamics with Near-Optimal Dependence on All Parameters", Communications in Mathematical Physics 407 1, 19 (2026).

[25] Fu Chen, Li Feng, Zhengdong Hu, and Yangbiao Ren, "External Quantum Self-Attention Model", IEEE Access 13, 179556 (2025).

[26] D. Faílde, A. Gómez, and J. Fernández-Rossier, "Preparation of the single-spinon wave function on a quantum computer", Physical Review Research 8 3, 033152 (2026).

[27] Kushagra Garg, Zeeshan Ahmed, Subhadip Mitra, and Shantanav Chakraborty, "Simulating quantum collision models with Hamiltonian simulations using early fault-tolerant quantum computers", Physical Review A 112 2, 022425 (2025).

[28] Pavel P. Popov, Kevin T. Geier, Valentin Kasper, Maciej Lewenstein, and Philipp Hauke, "Qudit-native measurement protocol for dynamical correlations using Hadamard tests", Physical Review A 111 4, 042604 (2025).

[29] Salahuddin Abdul Rahman, Özkan Karabacak, and Rafal Wisniewski, "Feedback-based quantum strategies for constrained combinatorial optimization problems", Future Generation Computer Systems 174, 107979 (2026).

[30] Dong An, Andrew M. Childs, Lin Lin, and Lexing Ying, "Laplace Transform–Based Quantum Eigenvalue Transformation via Linear Combination of Hamiltonian Simulation", SIAM Journal on Computing 55 2, 376 (2026).

[31] Kasra Rajabzadeh Dizaji, Ariq Haqq, Alicia B Magann, and Christian Arenz, "Hamiltonian simulation in Zeno subspaces", Physica Scripta 101 25, 255106 (2026).

[32] Paul K. Faehrmann, Jens Eisert, and Richard Kueng, "In the Shadow of the Hadamard Test: Using the Garbage State for Good and Further Modifications", Physical Review Letters 135 15, 150603 (2025).

[33] Suguru Endo, Keitaro Anai, Yuichiro Matsuzaki, Yuuki Tokunaga, and Yasunari Suzuki, "Projective squeezing for translation symmetric bosonic codes", arXiv:2403.14218, (2024).

[34] Kaoru Yamamoto, Yuichiro Matsuzaki, Yasunari Suzuki, Yuuki Tokunaga, and Suguru Endo, "$N$-Party Hadamard Test for Distributed Quantum Computation", arXiv:2411.10024, (2024).

[35] Dong An, Jin-Peng Liu, and Lin Lin, "Linear Combination of Hamiltonian Simulation for Nonunitary Dynamics with Optimal State Preparation Cost", Physical Review Letters 131 15, 150603 (2023).

[36] Jacob M. Leamer, Alicia B. Magann, Denys I. Bondar, and Gerard McCaul, "Quantum Dynamical Emulation of Imaginary Time Evolution", arXiv:2403.03350, (2024).

[37] Samson Wang, Sam McArdle, and Mario Berta, "Qubit-Efficient Randomized Quantum Algorithms for Linear Algebra", PRX Quantum 5 2, 020324 (2024).

[38] Océane Koska, Marc Baboulin, and Arnaud Gazda, "A tree-approach Pauli decomposition algorithm with application to quantum computing", arXiv:2403.11644, (2024).

[39] Shantanav Chakraborty, Soumyabrata Hazra, Tongyang Li, Changpeng Shao, Xinzhao Wang, and Yuxin Zhang, "Quantum singular value transformation without block encodings: Near-optimal complexity with minimal ancilla", arXiv:2504.02385, (2025).

[40] Dhrumil Patel and Mark M. Wilde, "Wave Matrix Lindbladization II: General Lindbladians, Linear Combinations, and Polynomials", Open Systems and Information Dynamics 30 3, 2350014-242 (2023).

[41] Junaid Aftab, Dong An, and Konstantina Trivisa, "Multi-product Hamiltonian simulation with explicit commutator scaling", arXiv:2403.08922, (2024).

[42] Eric R. Anschuetz, "A Unified Theory of Quantum Neural Network Loss Landscapes", arXiv:2408.11901, (2024).

[43] Lennart Binkowski, Tobias J. Osborne, Marvin Schwiering, René Schwonnek, and Timo Ziegler, "One for All: Universal Quantum Conic Programming Framework for Hard-Constrained Combinatorial Optimization Problems", arXiv:2411.00435, (2024).

[44] Junaid Aftab and Haizhao Yang, "Approximating Korobov Functions via Quantum Circuits", arXiv:2404.14570, (2024).

[45] Songqinghao Yang and Jin-Peng Liu, "Circuit-Efficient Randomized Quantum Simulation of Non-Unitary Dynamics with Observable-Driven and Symmetry-Aware Designs", arXiv:2509.08030, (2025).

[46] Gekko Budiutama, Shunsuke Daimon, Hirofumi Nishi, and Yu-ichiro Matsushita, "General Transform: A Unified Framework for Adaptive Transform to Enhance Representations", arXiv:2505.04969, (2025).

[47] Siddharth Hariprakash, Roel Van Beeumen, Katherine Klymko, and Daan Camps, "The Practicality of Randomized Quantum Linear Systems Solvers", arXiv:2510.13766, (2025).

[48] Yizhi Shen, Niel Van Buggenhout, Daan Camps, Katherine Klymko, and Roel Van Beeumen, "Quantum Rational Transformation Using Linear Combinations of Hamiltonian Simulations", arXiv:2408.07742, (2024).

[49] Vladimir Vargas-Calderón, "Quantum Deep Sets and Sequences", arXiv:2504.02241, (2025).

[50] Giulio Crognaletti, Giovanni Di Bartolomeo, Michele Vischi, and Luciano Loris Viteritti, "Equivariant Variational Quantum Eigensolver to detect phase transitions through energy level crossings", Quantum Science and Technology 10 1, 015048 (2025).

[51] Giulio Crognaletti, Giovanni Di Bartolomeo, Michele Vischi, and Luciano Loris Viteritti, "Equivariant Variational Quantum Eigensolver to detect Phase Transitions through Energy Level Crossings", arXiv:2403.07100, (2024).

[52] Fan Yang, Dafa Zhao, Chao Wei, Xinyu Chen, Shijie Wei, Hefeng Wang, Guilu Long, and Tao Xin, "A parallel quantum eigensolver for quantum machine learning", New Journal of Physics 26 4, 043011 (2024).

[53] Fabio Cumbo, Rui-Hao Li, Bryan Raubenolt, Jayadev Joshi, Abu Kaisar Mohammad Masum, Sercan Aygun, and Daniel Blankenberg, "Quantum Hyperdimensional Computing: a foundational paradigm for quantum neuromorphic architectures", arXiv:2511.12664, (2025).

[54] Zexian Li, Guofeng Zhang, and Xiao-Ming Zhang, "Reducing C-NOT Counts for State Preparation and Block Encoding via Diagonal Matrix Migration", arXiv:2603.16492, (2026).

[55] Yacine Haddad, Kaidi Xu, Vincent Croft, Jad C. Halimeh, and Michele Grossi, "Coherent Quantum Evaluation of Collider Amplitudes for Effective Field Theory Constraints", arXiv:2602.21311, (2026).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-13 23:41:21) and SAO/NASA ADS (last updated successfully 2026-08-13 23:41:22). The list may be incomplete as not all publishers provide suitable and complete citation data.

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.