Introduction to Haar Measure Tools in Quantum Information: A Beginner’s Tutorial

Antonio Anna Mele

Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany

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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.

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.

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[13] Robbie King, Kianna Wan, and Jarrod McClean, "Exponential learning advantages with conjugate states and minimal quantum memory", arXiv:2403.03469, (2024).

[14] 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).

[15] Jason Saied, Jeffrey Marshall, Namit Anand, Shon Grabbe, and Eleanor G. Rieffel, "Advancing Quantum Networking: Some Tools and Protocols for Ideal and Noisy Photonic Systems", arXiv:2403.02515, (2024).

[16] Hela Mhiri, Leo Monbroussou, Mario Herrero-Gonzalez, Slimane Thabet, Elham Kashefi, and Jonas Landman, "Constrained and Vanishing Expressivity of Quantum Fourier Models", arXiv:2403.09417, (2024).

[17] Y. S. Teo, "Robustness of optimized numerical estimation schemes for noisy variational quantum algorithms", Physical Review A 109 1, 012620 (2024).

[18] Chirag Wadhwa and Mina Doosti, "Learning Quantum Processes with Quantum Statistical Queries", arXiv:2310.02075, (2023).

[19] 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).

[20] Eric Kubischta and Ian Teixeira, "Free Quantum Codes from Twisted Unitary $t$-groups", arXiv:2402.01638, (2024).

[21] Oliver DeWolfe and Kenneth Higginbotham, "Bulk reconstruction and non-isometry in the backwards-forwards holographic black hole map", arXiv:2311.12921, (2023).

[22] Marco Schumann, Frank K. Wilhelm, and Alessandro Ciani, "Emergence of noise-induced barren plateaus in arbitrary layered noise models", arXiv:2310.08405, (2023).

[23] Armando Angrisani, "Learning unitaries with quantum statistical queries", arXiv:2310.02254, (2023).

[24] Prabhanjan Ananth, Aditya Gulati, Fatih Kaleoglu, and Yao-Ting Lin, "Pseudorandom Isometries", arXiv:2311.02901, (2023).

[25] Vahid Asadi, Richard Cleve, Eric Culf, and Alex May, "Linear gate bounds against natural functions for position-verification", arXiv:2402.18648, (2024).

[26] Andreas Bluhm, Matthias C. Caro, and Aadil Oufkir, "Hamiltonian Property Testing", arXiv:2403.02968, (2024).

[27] Zhong-Xia Shang, Zi-Han Chen, and Cai-Sheng Cheng, "Unconditionally decoherence-free quantum error mitigation by density matrix vectorization", arXiv:2405.07592, (2024).

[28] A. E. Teretenkov, "Superoperator master equations for depolarizing dynamics", arXiv:2404.06595, (2024).

[29] Daniel Grier, Hakop Pashayan, and Luke Schaeffer, "Principal eigenstate classical shadows", arXiv:2405.13939, (2024).

[30] Neil Dowling, Maxwell T. West, Angus Southwell, Azar C. Nakhl, Martin Sevior, Muhammad Usman, and Kavan Modi, "Adversarial Robustness Guarantees for Quantum Classifiers", arXiv:2405.10360, (2024).

The above citations are from Crossref's cited-by service (last updated successfully 2024-05-26 10:22:49) and SAO/NASA ADS (last updated successfully 2024-05-26 10:22:49). The list may be incomplete as not all publishers provide suitable and complete citation data.