量子斷層掃描（quantum state tomography）在量子資訊科學中扮演關鍵
角色，然而其所需的樣本數通常隨系統維度過度快速的增加，導致實務應
用上極其困難。此一挑戰促使研究者尋求更有效率的量子學習辦法，以
避免重建全部的量子態。其中，Scott Aaronson 提出的陰影斷層（shadow
tomography）即為一例[1]。陰影斷層的目標在於對一組給定的二元結果測
量（two-outcome measurements）之「陰影」進行估計，而非重建整個量子
態。因此當所需處理的陰影數目不多時，陰影斷層可在樣本複雜度上呈現
指數級的改善，相較於完整的量子斷層掃描可大幅降低樣本需求。受到陰
影斷層的啟發，我們提出量子事件識別（Quantum Event Identification, QEI）
問題，其目的在於以更少樣本數來辨識具高成功率的測量。不同於著重全
局精確度（對所有可觀察量均有良好估計）的標準陰影斷層，QEI 著重於每
個測量本身的成功率。透過利用可預期之接受機率落差，QEI 在理論上可
於樣本複雜度上優於陰影斷層。為評估QEI 的複雜度，我們進行各種測量
程序的模擬，靈感來自Adam Bene Watts 與John Bostanci 所提出的量子事件
搜尋（quantum event finding）方法[2]。在這些方法中，即便在低維度條件
下只利用單一未知態的複本，仍可穩定達成約60% 的成功率。我們亦將經
典陰影（classical shadow）及相關技術應用於QEI，探討其相對優勢，並驗
證1.0.4 中提出之事件搜尋界限。綜上，本研究結果顯示，QEI 極具潛力成
為高效利用資源的量子測量預測策略，同時也為進一步的理論精進及更可
擴張的計算方法研究，提供了未來發展方向。

Quantum state tomography is a central task in quantum information science, but it typically requires a prohibitively large number of samples that scale poorly with system dimension. This challenge has motivated the exploration of more efficient quantum learning problems that avoid full state reconstruction. One of such problems is shadow tomography introduced by Scott Aaronson [1], which requires the estimations of the shadow, a given set of two-outcome measurements. Consequently, when the number of shadows are not too large, shadow tomography was shown to be solvable in exponentially lower sample complexity compared to full quantum state tomography. Inspired by shadow tomography, we introduce the Quantum Event Identification (QEI) problem, which targets identifying measurements with high success rates while requiring fewer samples. Unlike standard shadow tomography, which focuses on obtaining global accuracy across all observables, QEI emphasizes per-measurement success. Exploiting a promised gap in acceptance probabilities, QEI can potentially achieve lower sample complexity compared to shadow tomography. We evaluate the complexity of QEI through simulations of various measurement procedures inspired by procedures of quantum event finding introduced by Adam Bene Watts and John Bostanci[2]. These procedures achieve stable success rates of around 60% with only a single copy of the unknown state in low-dimensional settings. We also adapt classical shadow and related techniques to QEI, examining their relative strengths and validating the event-finding bounds introduced in Section 1.0.4. Our results highlight the potential of QEI as a resource-efficient strategy for quantum measurement prediction, suggesting avenues for theoretical refinements and more scalable computational
methods.

[1] S. Aaronson, “Shadow tomography of quantum states,” in Proceedings of the 50th Annual
ACM SIGACT Symposium on Theory of Computing (STOC 2018), pp. 325–338, ACM,
2018.
[2] A. B. Watts and J. Bostanci, “Quantum event learning and gentle random measurements,”
in 15th Innovations in Theoretical Computer Science Conference (ITCS 2024), vol. 287
of Leibniz International Proceedings in Informatics (LIPIcs), pp. 97:1–97:22, Schloss
Dagstuhl–Leibniz-Zentrum für Informatik, 2024.
[3] E. Schrödinger, “Quantisierung als eigenwertproblem,” Annalen der Physik, vol. 79,
pp. 361–376, 1926.
[4] J. B. Altepeter, D. F. V. James, and P. G. Kwiat, “Quantum state tomography,” in Quantum
State Estimation (M. G. A. Paris and J. Řeháček, eds.), vol. 649 of Lecture Notes in
Physics, pp. 113–145, Springer, 2004.
[5] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting
an unknown quantum state via dual classical and einstein-podolsky-rosen channels,”
Physical Review Letters, vol. 70, pp. 1895–1899, March 1993.
[6] K. Vogel and H. Risken, “Determination of the density matrix via the Wigner function of
the light field,” Physical Review A, vol. 40, pp. 2847–2849, 1989.
[7] Z. Hradil, “Quantum-state estimation,” Physical Review A, vol. 55, p. R1561, March
1997.
[8] D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the wigner
distribution and the density matrix of a light mode using optical homodyne tomography:
Application to squeezed states and the vacuum,” Physical Review Letters, vol. 70, p. 1244,
March 1993.
[9] J. Haah, A. W. Harrow, Z. Ji, X. Wu, and N. Yu, “Sample-optimal tomography of quantum
states,” IEEE Transactions on Information Theory, vol. 63, no. 9, pp. 5628–5641, 2020.
[10] H.-Y. Huang, R. Kueng, and J. Preskill, “Predicting many properties of a quantum system
from very few measurements,” Nature Physics, vol. 16, pp. 1050–1057, 2020.
[11] C. Bădescu and R. O’Donnell, “Improved quantum data analysis,” in Proceedings of
the 53rd Annual ACM SIGACT Symposium on Theory of Computing (STOC 2021),
pp. 1398–1411, ACM, 2021.
[12] A. W. Harrow, C. Y.-Y. Lin, and A. Montanaro, “Sequential measurements, disturbance
and property testing,” in Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium
on Discrete Algorithms (SODA 2017), pp. 1598–1611, Society for Industrial and
Applied Mathematics, 2017.
[13] S. Aaronson, “QMA/qpoly ⊆ PSPACE/poly: de-merlinizing quantum protocols,” in Proceedings
of the 21st Annual IEEE Conference on Computational Complexity (CCC 2006),
(Prague, Czech Republic), pp. 261–273, IEEE, July 2006.
[14] S. Aaronson, “Quantum copy-protection and quantum money,” in 2009 24th Annual IEEE
Conference on Computational Complexity, (Paris, France), pp. 229–242, IEEE, July 2009.
[15] S. Aaronson, “The learnability of quantum states,” Proceedings of the Royal Society A:
Mathematical, Physical and Engineering Sciences, vol. 463, pp. 3089–3114, December
2007.
[16] V. Wei, W. A. Coish, P. Ronagh, and C. A. Muschik, “Neural-shadow quantum state tomography,”
Physical Review Research, vol. 6, p. 023250, June 2024.
[17] Y. Quek, S. Fort, and H. K. Ng, “Adaptive quantum state tomography with neural networks,”
npj Quantum Information, vol. 7, p. 105, June 2021.
[18] J. Lee, “Lecture 11: Sublinear algorithms for big data,” 2020. Available at:
https://cs.brown.edu/courses/csci1951-w/lec/lec