Introduction to Haar Measure Tools in Quantum Information: A Beginner’s Tutorial
Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
| Published: | 2024-05-08, volume 8, page 1340 |
| Eprint: | arXiv:2307.08956v4 |
| Doi: | https://doi.org/10.22331/q-2024-05-08-1340 |
| Citation: | Quantum 8, 1340 (2024). |
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Abstract
The Haar measure plays a vital role in quantum information, but its study often requires a deep understanding of representation theory, posing a challenge for beginners. This tutorial aims to provide a basic introduction to Haar measure tools in quantum information, utilizing only basic knowledge of linear algebra and thus aiming to make this topic more accessible. The tutorial begins by introducing the Haar measure with a specific emphasis on characterizing the moment operator, an essential element for computing integrals over the Haar measure. It also covers properties of the symmetric subspace and introduces helpful tools like tensor network diagrammatic notation, which aid in visualizing and simplifying calculations. Next, the tutorial explores the concept of unitary designs, providing equivalent definitions, and subsequently explores approximate notions of unitary designs, shedding light on the relationships between these different notions. Practical examples of Haar measure calculations are illustrated, including the derivation of well-known formulas such as the twirling of a quantum channel. Lastly, the tutorial showcases the applications of Haar measure calculations in quantum machine learning and classical shadow tomography.
Featured image: Diagrammatic representation of second moment integral over the Haar measure
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[1] Hsin-Yuan Huang, Richard Kueng, and John Preskill, ``Predicting many properties of a quantum system from very few measurements'' Nature Physics 16, 1050-1057 (2020).
https://doi.org/10.1038/s41567-020-0932-7
[2] Jeongwan Haah, Aram W. Harrow, Zhengfeng Ji, Xiaodi Wu, and Nengkun Yu, ``Sample-optimal tomography of quantum states'' IEEE Transactions on Information Theory 1–1 (2017).
https://doi.org/10.1109/tit.2017.2719044
[3] Andreas Elben, Steven T. Flammia, Hsin-Yuan Huang, Richard Kueng, John Preskill, Benoı̂t Vermersch, and Peter Zoller, ``The randomized measurement toolbox'' Nature Reviews Physics 5, 9–24 (2022).
https://doi.org/10.1038/s42254-022-00535-2
[4] Ryan O'Donnelland John Wright ``Efficient quantum tomography'' (2015).
https://doi.org/10.48550/arXiv.1508.01907
arXiv:1508.01907
[5] Richard Kueng, Holger Rauhut, and Ulrich Terstiege, ``Low rank matrix recovery from rank one measurements'' Applied and Computational Harmonic Analysis 42, 88–116 (2017).
https://doi.org/10.1016/j.acha.2015.07.007
https://www.sciencedirect.com/science/article/pii/S1063520315001037
[6] Andreas Elben, Richard Kueng, Hsin-Yuan (Robert) Huang, Rick van Bijnen, Christian Kokail, Marcello Dalmonte, Pasquale Calabrese, Barbara Kraus, John Preskill, Peter Zoller, and Benoı̂t Vermersch, ``Mixed-State Entanglement from Local Randomized Measurements'' Phys. Rev. Lett. 125, 200501 (2020).
https://doi.org/10.1103/PhysRevLett.125.200501
[7] M GuÅ£Ä, J Kahn, R Kueng, and J A Tropp, ``Fast state tomography with optimal error bounds'' Journal of Physics A: Mathematical and Theoretical 53, 204001 (2020).
https://doi.org/10.1088/1751-8121/ab8111
[8] PaweÅ CieÅliÅski, Satoya Imai, Jan Dziewior, Otfried Gühne, Lukas Knips, WiesÅaw Laskowski, Jasmin Meinecke, Tomasz Paterek, and Tamás Vértesi, ``Analysing quantum systems with randomised measurements'' (2023).
https://doi.org/10.48550/arXiv.2307.01251
arXiv:2307.01251
[9] Aram W. Harrowand Saeed Mehraban ``Approximate Unitary t-Designs by Short Random Quantum Circuits Using Nearest-Neighbor and Long-Range Gates'' Communications in Mathematical Physics 401, 1531â1626 (2023).
https://doi.org/10.1007/s00220-023-04675-z
[10] Adam Bouland, Bill Fefferman, Chinmay Nirkhe, and Umesh Vazirani, ``On the complexity and verification of quantum random circuit sampling'' Nature Physics 15, 159–163 (2018).
https://doi.org/10.1038/s41567-018-0318-2
[11] Ramis Movassagh ``Quantum supremacy and random circuits'' (2020).
https://doi.org/10.48550/arXiv.1909.06210
arXiv:1909.06210
[12] Alexander M. Dalzell, Nicholas Hunter-Jones, and Fernando G. S. L. Brandão, ``Random quantum circuits transform local noise into global white noise'' (2021).
https://doi.org/10.48550/arXiv.2111.14907
arXiv:2111.14907
[13] Dominik Hangleiter, Juan Bermejo-Vega, Martin Schwarz, and Jens Eisert, ``Anticoncentration theorems for schemes showing a quantum speedup'' Quantum 2, 65 (2018).
https://doi.org/10.22331/q-2018-05-22-65
[14] Adam Bouland, Joseph F. Fitzsimons, and Dax Enshan Koh, ``Complexity Classification of Conjugated Clifford Circuits'' 33rd Computational Complexity Conference (CCC 2018) 102, 21:1–21:25 (2018).
https://doi.org/10.4230/LIPIcs.CCC.2018.21
https://drops.dagstuhl.de/opus/volltexte/2018/8867
[15] Hakop Pashayan, Stephen D. Bartlett, and David Gross, ``From estimation of quantum probabilities to simulation of quantum circuits'' Quantum 4, 223 (2020).
https://doi.org/10.22331/q-2020-01-13-223
[16] Dominik Hangleiterand Jens Eisert ``Computational advantage of quantum random sampling'' Reviews of Modern Physics 95 (2023).
https://doi.org/10.1103/revmodphys.95.035001
[17] Sandu Popescu, Anthony J. Short, and Andreas Winter, ``Entanglement and the foundations of statistical mechanics'' Nature Physics 2, 754–758 (2006).
https://doi.org/10.1038/nphys444
[18] Joseph Emerson, Robert Alicki, and Karol Ż yczkowski, ``Scalable noise estimation with random unitary operators'' Journal of Optics B: Quantum and Semiclassical Optics 7, S347–S352 (2005).
https://doi.org/10.1088/1464-4266/7/10/021
[19] J. Helsen, I. Roth, E. Onorati, A.H. Werner, and J. Eisert, ``General Framework for Randomized Benchmarking'' PRX Quantum 3 (2022).
https://doi.org/10.1103/prxquantum.3.020357
[20] Easwar Magesan, J. M. Gambetta, and Joseph Emerson, ``Scalable and Robust Randomized Benchmarking of Quantum Processes'' Phys. Rev. Lett. 106, 180504 (2011).
https://doi.org/10.1103/PhysRevLett.106.180504
[21] D.P. DiVincenzo, D.W. Leung, and B.M. Terhal, ``Quantum data hiding'' IEEE Transactions on Information Theory 48, 580–598 (2002).
https://doi.org/10.1109/18.985948
[22] Aram Harrow, Patrick Hayden, and Debbie Leung, ``Superdense Coding of Quantum States'' Physical Review Letters 92 (2004).
https://doi.org/10.1103/physrevlett.92.187901
[23] Andris Ambainis, Jan Bouda, and Andreas Winter, ``Nonmalleable encryption of quantum information'' Journal of Mathematical Physics 50, 042106 (2009).
https://doi.org/10.1063/1.3094756
[24] Michał Horodecki, Paweł Horodecki, and Ryszard Horodecki, ``General teleportation channel, singlet fraction, and quasidistillation'' Phys. Rev. A 60, 1888–1898 (1999).
https://doi.org/10.1103/PhysRevA.60.1888
[25] Anura Abeyesinghe, Igor Devetak, Patrick Hayden, and Andreas Winter, ``The mother of all protocols: restructuring quantum information's family tree'' Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, 2537–2563 (2009).
https://doi.org/10.1098/rspa.2009.0202
[26] Michał Horodeckiand Paweł Horodecki ``Reduction criterion of separability and limits for a class of distillation protocols'' Phys. Rev. A 59, 4206–4216 (1999).
https://doi.org/10.1103/PhysRevA.59.4206
[27] Josep Batle-Vallespir ``Characterization of Quantum Entangled States and Information Measures'' (2006).
https://doi.org/10.48550/arXiv.quant-ph/0603124
[28] Karol Å»yczkowski, PaweÅ Horodecki, Anna Sanpera, and Maciej Lewenstein, ``Volume of the set of separable states'' Physical Review A 58, 883â892 (1998).
https://doi.org/10.1103/physreva.58.883
[29] Karol Zyczkowskiand Hans-Jürgen Sommers ``Induced measures in the space of mixed quantum states'' Journal of Physics A: Mathematical and General 34, 7111â7125 (2001).
https://doi.org/10.1088/0305-4470/34/35/335
[30] Jarrod R. McClean, Sergio Boixo, Vadim N. Smelyanskiy, Ryan Babbush, and Hartmut Neven, ``Barren plateaus in quantum neural network training landscapes'' Nature Communications 9 (2018).
https://doi.org/10.1038/s41467-018-07090-4
[31] Zoë Holmes, Kunal Sharma, M. Cerezo, and Patrick J. Coles, ``Connecting Ansatz Expressibility to Gradient Magnitudes and Barren Plateaus'' PRX Quantum 3 (2022).
https://doi.org/10.1103/prxquantum.3.010313
[32] M. Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C. Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R. McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, and Patrick J. Coles, ``Variational quantum algorithms'' Nature Reviews Physics 3, 625–644 (2021).
https://doi.org/10.1038/s42254-021-00348-9
[33] John Napp ``Quantifying the barren plateau phenomenon for a model of unstructured variational ansätze'' (2022).
https://doi.org/10.48550/arXiv.2203.06174
arXiv:2203.06174
[34] Brian Skinner, Jonathan Ruhman, and Adam Nahum, ``Measurement-Induced Phase Transitions in the Dynamics of Entanglement'' Phys. Rev. X 9, 031009 (2019).
https://doi.org/10.1103/PhysRevX.9.031009
[35] Matthew P.A. Fisher, Vedika Khemani, Adam Nahum, and Sagar Vijay, ``Random Quantum Circuits'' Annual Review of Condensed Matter Physics 14, 335–379 (2023).
https://doi.org/10.1146/annurev-conmatphys-031720-030658
[36] Amos Chan, Andrea De Luca, and J. T. Chalker, ``Solution of a Minimal Model for Many-Body Quantum Chaos'' Phys. Rev. X 8, 041019 (2018).
https://doi.org/10.1103/PhysRevX.8.041019
[37] Patrick Haydenand John Preskill ``Black holes as mirrors: quantum information in random subsystems'' Journal of High Energy Physics 2007, 120–120 (2007).
https://doi.org/10.1088/1126-6708/2007/09/120
[38] Jonas Haferkamp, Philippe Faist, Naga B. T. Kothakonda, Jens Eisert, and Nicole Yunger Halpern, ``Linear growth of quantum circuit complexity'' Nature Physics 18, 528–532 (2022).
https://doi.org/10.1038/s41567-022-01539-6
[39] Fernando G.S.L. Brandão, Wissam Chemissany, Nicholas Hunter-Jones, Richard Kueng, and John Preskill, ``Models of Quantum Complexity Growth'' PRX Quantum 2 (2021).
https://doi.org/10.1103/prxquantum.2.030316
[40] Daniel A. Robertsand Beni Yoshida ``Chaos and complexity by design'' Journal of High Energy Physics 2017 (2017).
https://doi.org/10.1007/jhep04(2017)121
[41] Zi-Wen Liu, Seth Lloyd, Elton Zhu, and Huangjun Zhu, ``Entanglement, quantum randomness, and complexity beyond scrambling'' Journal of High Energy Physics 2018 (2018).
https://doi.org/10.1007/jhep07(2018)041
[42] E. Onorati, O. Buerschaper, M. Kliesch, W. Brown, A. H. Werner, and J. Eisert, ``Mixing Properties of Stochastic Quantum Hamiltonians'' Communications in Mathematical Physics 355, 905–947 (2017).
https://doi.org/10.1007/s00220-017-2950-6
[43] Oles Shtankoand Ramis Movassagh ``Stability of Periodically Driven Topological Phases against Disorder'' Phys. Rev. Lett. 121, 126803 (2018).
https://doi.org/10.1103/PhysRevLett.121.126803
[44] Ramis Movassaghand Alan Edelman ``Density of States of Quantum Spin Systems from Isotropic Entanglement'' Phys. Rev. Lett. 107, 097205 (2011).
https://doi.org/10.1103/PhysRevLett.107.097205
[45] Richard J. Kuengand Joel Tropp ``Quantum and Classical Information Processing with Tensors'' (2019).
https://doi.org/10.7907/my9r-p178
https://resolver.caltech.edu/CaltechAUTHORS:20220412-220843410
[46] Matthias Christandl ``The Structure of Bipartite Quantum States - Insights from Group Theory and Cryptography'' (2006).
https://doi.org/10.48550/arXiv.quant-ph/0604183
[47] Martin Klieschand Ingo Roth ``Theory of Quantum System Certification'' PRX Quantum 2 (2021).
https://doi.org/10.1103/prxquantum.2.010201
[48] Jonas Haferkamp ``Randomness and complexity in random complex quantum systems'' thesis (2022).
https://doi.org/10.17169/refubium-38286
[49] Richard A. Low ``Pseudo-randomness and Learning in Quantum Computation'' (2010).
https://doi.org/10.48550/arXiv.1006.5227
arXiv:1006.5227
[50] John Watrous ``The Theory of Quantum Information'' Cambridge University Press (2018).
https://doi.org/10.1017/9781316848142
[51] Benoı̂t Collinsand Piotr Śniady ``Integration with Respect to the Haar Measure on Unitary, Orthogonal and Symplectic Group'' Communications in Mathematical Physics 264, 773–795 (2006).
https://doi.org/10.1007/s00220-006-1554-3
[52] Barry Simon ``Representations of finite and compact groups'' (1995).
https://doi.org/10.1090/gsm/010
https://api.semanticscholar.org/CorpusID:117744597
[53] Roe Goodmanand Nolan Wallach ``Symmetry, Representations, and Invariants'' (2009).
https://doi.org/10.1007/978-0-387-79852-3
https://api.semanticscholar.org/CorpusID:18376245
[54] Lin Zhang ``Matrix integrals over unitary groups: An application of Schur-Weyl duality'' (2015).
https://doi.org/10.48550/arXiv.1408.3782
arXiv:1408.3782
[55] Benoit Collins ``Moments and Cumulants of Polynomial random variables on unitary groups, the Itzykson-Zuber integral and free probability'' (2002).
https://doi.org/10.48550/arXiv.math-ph/0205010
[56] Don Weingarten ``Asymptotic Behavior of Group Integrals in the Limit of Infinite Rank'' J. Math. Phys. 19, 999 (1978).
https://doi.org/10.1063/1.523807
[57] Benoit Collins, Sho Matsumoto, and Jonathan Novak, ``The Weingarten Calculus'' Notices of the American Mathematical Society 69, 1 (2022).
https://doi.org/10.1090/noti2474
[58] Georg Köstenberger ``Weingarten Calculus'' (2021).
https://doi.org/10.48550/arXiv.2101.00921
arXiv:2101.00921
[59] Fernando G. S. L. Brandão, Aram W. Harrow, and Michał Horodecki, ``Local Random Quantum Circuits are Approximate Polynomial-Designs'' Communications in Mathematical Physics 346, 397–434 (2016).
https://doi.org/10.1007/s00220-016-2706-8
[60] Diego GarcÃa-MartÃn, Martin Larocca, and M. Cerezo, ``Deep quantum neural networks form Gaussian processes'' (2023).
https://doi.org/10.48550/arXiv.2305.09957
arXiv:2305.09957
[61] Aram W. Harrow ``The Church of the Symmetric Subspace'' (2013).
https://doi.org/10.48550/arXiv.1308.6595
arXiv:1308.6595
[62] Jacob C Bridgemanand Christopher T Chubb ``Hand-waving and interpretive dance: an introductory course on tensor networks'' Journal of Physics A: Mathematical and Theoretical 50, 223001 (2017).
https://doi.org/10.1088/1751-8121/aa6dc3
[63] Michael A. Nielsenand Isaac L. Chuang ``Quantum Computation and Quantum Information: 10th Anniversary Edition'' Cambridge University Press (2010).
https://doi.org/10.1017/CBO9780511976667
[64] Christoph Dankert, Richard Cleve, Joseph Emerson, and Etera Livine, ``Exact and approximate unitary 2-designs and their application to fidelity estimation'' Physical Review A 80 (2009).
https://doi.org/10.1103/physreva.80.012304
[65] D. Gross, K. Audenaert, and J. Eisert, ``Evenly distributed unitaries: On the structure of unitary designs'' Journal of Mathematical Physics 48 (2007).
https://doi.org/10.1063/1.2716992
[66] Aidan Royand A. J. Scott ``Unitary designs and codes'' Designs, Codes and Cryptography 53, 13–31 (2009).
https://doi.org/10.1007/s10623-009-9290-2
[67] Andris Ambainisand Joseph Emerson ``Quantum t-designs: t-wise independence in the quantum world'' (2007).
https://doi.org/10.48550/arXiv.quant-ph/0701126
[68] Daniel Gottesman ``The Heisenberg Representation of Quantum Computers'' (1998).
https://doi.org/10.48550/arXiv.quant-ph/9807006
[69] Zak Webb ``The Clifford group forms a unitary 3-design'' (2016).
https://doi.org/10.48550/arXiv.1510.02769
arXiv:1510.02769
[70] Huangjun Zhu ``Multiqubit Clifford groups are unitary 3-designs'' Physical Review A 96 (2017).
https://doi.org/10.1103/physreva.96.062336
[71] Huangjun Zhu, Richard Kueng, Markus Grassl, and David Gross, ``The Clifford group fails gracefully to be a unitary 4-design'' (2016).
https://doi.org/10.48550/arXiv.1609.08172
arXiv:1609.08172
[72] Scott Aaronsonand Daniel Gottesman ``Improved simulation of stabilizer circuits'' Physical Review A 70 (2004).
https://doi.org/10.1103/physreva.70.052328
[73] Ewout Van Den Berg ``A simple method for sampling random Clifford operators'' 2021 IEEE International Conference on Quantum Computing and Engineering (QCE) 54–59 (2021).
https://doi.org/10.1109/QCE52317.2021.00021
[74] Robert Koenigand John A. Smolin ``How to efficiently select an arbitrary Clifford group element'' Journal of Mathematical Physics 55, 122202 (2014).
https://doi.org/10.1063/1.4903507
[75] M. Wilde ``Quantum Information Theory'' Cambridge University Press (2016).
https://doi.org/10.1017/9781316809976.001
[76] Jonas Haferkamp ``Random quantum circuits are approximate unitary $t$-designs in depth $O\left(nt^{5+o(1)}\right)$'' Quantum 6, 795 (2022).
https://doi.org/10.22331/q-2022-09-08-795
[77] Aram W. Harrowand Richard A. Low ``Random Quantum Circuits are Approximate 2-designs'' Communications in Mathematical Physics 291, 257–302 (2009).
https://doi.org/10.1007/s00220-009-0873-6
[78] Aram W. Harrowand Saeed Mehraban ``Approximate Unitary t-Designs by Short Random Quantum Circuits Using Nearest-Neighbor and Long-Range Gates'' Communications in Mathematical Physics 401, 1531–1626 (2023).
https://doi.org/10.1007/s00220-023-04675-z
[79] 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'' Physical Review A 77 (2008).
https://doi.org/10.1103/physreva.77.012307
[80] Michael A Nielsen ``A simple formula for the average gate fidelity of a quantum dynamical operation'' Physics Letters A 303, 249–252 (2002).
https://doi.org/10.1016/s0375-9601(02)01272-0
[81] Elihu Lubkinand Thelma Lubkin ``Average quantal behavior and thermodynamic isolation'' International Journal of Theoretical Physics 32, 933–943 (1993).
https://doi.org/10.1007/BF01215300
[82] Eshed Ramand Igal Sason ``On Renyi Entropy Power Inequalities'' (2016).
https://doi.org/10.48550/arXiv.1601.06555
arXiv:1601.06555
[83] Don N. Page ``Average entropy of a subsystem'' Physical Review Letters 71, 1291–1294 (1993).
https://doi.org/10.1103/physrevlett.71.1291
[84] Patrick Hayden, Debbie W. Leung, and Andreas Winter, ``Aspects of Generic Entanglement'' Communications in Mathematical Physics 265, 95–117 (2006).
https://doi.org/10.1007/s00220-006-1535-6
[85] M. Ledoux ``The Concentration of Measure Phenomenon'' AMS Surveys and Monographs 89 (2001).
https://doi.org/10.1090/surv/089
[86] Richard A. Low ``Large deviation bounds for $k$-designs'' Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, 3289–3308 (2009).
https://doi.org/10.1098/rspa.2009.0232
[87] Martin Larocca, Piotr Czarnik, Kunal Sharma, Gopikrishnan Muraleedharan, Patrick J. Coles, and M. Cerezo, ``Diagnosing Barren Plateaus with Tools from Quantum Optimal Control'' Quantum 6, 824 (2022).
https://doi.org/10.22331/q-2022-09-29-824
[88] Enrico Fontana, Dylan Herman, Shouvanik Chakrabarti, Niraj Kumar, Romina Yalovetzky, Jamie Heredge, Shree Hari Sureshbabu, and Marco Pistoia, ``The Adjoint Is All You Need: Characterizing Barren Plateaus in Quantum Ansätze'' (2023).
https://doi.org/10.48550/arXiv.2309.07902
arXiv:2309.07902
[89] Michael Ragone, Bojko N. Bakalov, Frédéric Sauvage, Alexander F. Kemper, Carlos Ortiz Marrero, Martin Larocca, and M. Cerezo, ``A Unified Theory of Barren Plateaus for Deep Parametrized Quantum Circuits'' (2023).
https://doi.org/10.48550/arXiv.2309.09342
arXiv:2309.09342
[90] MartÃn Larocca, Frédéric Sauvage, Faris M. Sbahi, Guillaume Verdon, Patrick J. Coles, and M. Cerezo, ``Group-Invariant Quantum Machine Learning'' PRX Quantum 3 (2022).
https://doi.org/10.1103/prxquantum.3.030341
[91] Johannes Jakob Meyer, Marian Mularski, Elies Gil-Fuster, Antonio Anna Mele, Francesco Arzani, Alissa Wilms, and Jens Eisert, ``Exploiting Symmetry in Variational Quantum Machine Learning'' PRX Quantum 4 (2023).
https://doi.org/10.1103/prxquantum.4.010328
[92] Samson Wang, Enrico Fontana, M. Cerezo, Kunal Sharma, Akira Sone, Lukasz Cincio, and Patrick J. Coles, ``Noise-induced barren plateaus in variational quantum algorithms'' Nature Communications 12 (2021).
https://doi.org/10.1038/s41467-021-27045-6
[93] P. Singkanipaand D. A. Lidar ``Beyond unital noise in variational quantum algorithms: noise-induced barren plateaus and fixed points'' (2024).
https://doi.org/10.48550/arXiv.2402.08721
arXiv:2402.08721
[94] Antonio Anna Mele, Armando Angrisani, Soumik Ghosh, Sumeet Khatri, Jens Eisert, Daniel Stilck França, and Yihui Quek, ``Noise-induced shallow circuits and absence of barren plateaus'' (2024).
https://doi.org/10.48550/arXiv.2403.13927
arXiv:2403.13927
[95] Dax Enshan Kohand Sabee Grewal ``Classical Shadows With Noise'' Quantum 6, 776 (2022).
https://doi.org/10.22331/q-2022-08-16-776
[96] Kianna Wan, William J. Huggins, Joonho Lee, and Ryan Babbush, ``Matchgate Shadows for Fermionic Quantum Simulation'' Communications in Mathematical Physics 404, 629â700 (2023).
https://doi.org/10.1007/s00220-023-04844-0
[97] Hong-Ye Hu, Soonwon Choi, and Yi-Zhuang You, ``Classical shadow tomography with locally scrambled quantum dynamics'' Phys. Rev. Res. 5, 023027 (2023).
https://doi.org/10.1103/PhysRevResearch.5.023027
[98] Christian Bertoni, Jonas Haferkamp, Marcel Hinsche, Marios Ioannou, Jens Eisert, and Hakop Pashayan, ``Shallow shadows: Expectation estimation using low-depth random Clifford circuits'' (2023).
https://doi.org/10.48550/arXiv.2209.12924
arXiv:2209.12924
[99] Kaifeng Bu, Dax Enshan Koh, Roy J. Garcia, and Arthur Jaffe, ``Classical shadows with Pauli-invariant unitary ensembles'' npj Quantum Information 10 (2024).
https://doi.org/10.1038/s41534-023-00801-w
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[14] Tony Metger, Alexander Poremba, Makrand Sinha, and Henry Yuen, "Simple constructions of linear-depth t-designs and pseudorandom unitaries", arXiv:2404.12647, (2024).
[15] Chirag Wadhwa and Mina Doosti, "Learning Quantum Processes with Quantum Statistical Queries", arXiv:2310.02075, (2023).
[16] Y. S. Teo, "Robustness of optimized numerical estimation schemes for noisy variational quantum algorithms", Physical Review A 109 1, 012620 (2024).
[17] Rahul Arvind, Kishor Bharti, Jun Yong Khoo, Dax Enshan Koh, and Jian Feng Kong, "A quantum tug of war between randomness and symmetries on homogeneous spaces", arXiv:2309.05253, (2023).
[18] Eric Kubischta and Ian Teixeira, "Free Quantum Codes from Twisted Unitary $t$-groups", arXiv:2402.01638, (2024).
[19] Oliver DeWolfe and Kenneth Higginbotham, "Bulk reconstruction and non-isometry in the backwards-forwards holographic black hole map", arXiv:2311.12921, (2023).
[20] Prabhanjan Ananth, Aditya Gulati, Fatih Kaleoglu, and Yao-Ting Lin, "Pseudorandom Isometries", arXiv:2311.02901, (2023).
[21] Marco Schumann, Frank K. Wilhelm, and Alessandro Ciani, "Emergence of noise-induced barren plateaus in arbitrary layered noise models", arXiv:2310.08405, (2023).
[22] Armando Angrisani, "Learning unitaries with quantum statistical queries", arXiv:2310.02254, (2023).
[23] Vahid Asadi, Richard Cleve, Eric Culf, and Alex May, "Linear gate bounds against natural functions for position-verification", arXiv:2402.18648, (2024).
[24] Andreas Bluhm, Matthias C. Caro, and Aadil Oufkir, "Hamiltonian Property Testing", arXiv:2403.02968, (2024).
[25] Zhong-Xia Shang, Zi-Han Chen, and Cai-Sheng Cheng, "Unconditionally decoherence-free quantum error mitigation by density matrix vectorization", arXiv:2405.07592, (2024).
[26] A. E. Teretenkov, "Superoperator master equations for depolarizing dynamics", arXiv:2404.06595, (2024).
[27] Alessio Paviglianiti, Guglielmo Lami, Mario Collura, and Alessandro Silva, "Estimating Non-Stabilizerness Dynamics Without Simulating It", arXiv:2405.06054, (2024).
[28] Kento Tsubouchi, Yosuke Mitsuhashi, Kunal Sharma, and Nobuyuki Yoshioka, "Symmetric Clifford twirling for cost-optimal quantum error mitigation in early FTQC regime", arXiv:2405.07720, (2024).
The above citations are from Crossref's cited-by service (last updated successfully 2024-05-19 22:10:05) and SAO/NASA ADS (last updated successfully 2024-05-19 22:10:07). The list may be incomplete as not all publishers provide suitable and complete citation data.
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