Quantum Event Learning and Gentle Random Measurements

Authors Adam Bene Watts , John Bostanci

Thumbnail PDF


  • Filesize: 0.8 MB
  • 22 pages

Document Identifiers

Author Details

Adam Bene Watts
  • Institute for Quantum Computing, University of Waterloo, Canada
John Bostanci
  • Computer Science Department, Columbia University, New York, NY, USA


The authors would like to that Aram Harrow, Scott Aaronson and Luke Schaeffer for helpful discussions. Particular thanks is due to Luke Schaeffer for finding the counterexample discussed in Appendix B of this paper. The authors would also like to thank Matthias Caro for spotting a typo in an earlier version of this paper. Additionally, ABW would like to thank Nilin Abrahamsen and Juspreet Singh Sandhu for a particularly motivating discussion over ice cream during the early days of this work. JB would like to thank Ashwin Nayak and Angus Lowe for the many useful discussions and motivation.

Cite AsGet BibTex

Adam Bene Watts and John Bostanci. Quantum Event Learning and Gentle Random Measurements. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 97:1-97:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


We prove the expected disturbance caused to a quantum system by a sequence of randomly ordered two-outcome projective measurements is upper bounded by the square root of the probability that at least one measurement in the sequence accepts. We call this bound the Gentle Random Measurement Lemma. We then extend the techniques used to prove this lemma to develop protocols for problems in which we are given sample access to an unknown state ρ and asked to estimate properties of the accepting probabilities Tr[M_i ρ] of a set of measurements {M₁, M₂, … , M_m}. We call these types of problems Quantum Event Learning Problems. In particular, we show randomly ordering projective measurements solves the Quantum OR problem, answering an open question of Aaronson. We also give a Quantum OR protocol which works on non-projective measurements and which outperforms both the random measurement protocol analyzed in this paper and the protocol of Harrow, Lin, and Montanaro. However, this protocol requires a more complicated type of measurement, which we call a Blended Measurement. Given additional guarantees on the set of measurements {M₁, …, M_m}, we show the random and blended measurement Quantum OR protocols developed in this paper can also be used to find a measurement M_i such that Tr[M_i ρ] is large. We call the problem of finding such a measurement Quantum Event Finding. We also show Blended Measurements give a sample-efficient protocol for Quantum Mean Estimation: a problem in which the goal is to estimate the average accepting probability of a set of measurements on an unknown state. Finally we consider the Threshold Search Problem described by O'Donnell and Bădescu where, given given a set of measurements {M₁, …, M_m} along with sample access to an unknown state ρ satisfying Tr[M_i ρ] ≥ 1/2 for some M_i, the goal is to find a measurement M_j such that Tr[M_j ρ] ≥ 1/2 - ε. By building on our Quantum Event Finding result we show that randomly ordered (or blended) measurements can be used to solve this problem using O(log²(m) / ε²) copies of ρ. This matches the performance of the algorithm given by O'Donnell and Bădescu, but does not require injected noise in the measurements. Consequently, we obtain an algorithm for Shadow Tomography which matches the current best known sample complexity (i.e. requires Õ(log²(m)log(d)/ε⁴) samples). This algorithm does not require injected noise in the quantum measurements, but does require measurements to be made in a random order, and so is no longer online.

Subject Classification

ACM Subject Classification
  • Theory of computation
  • Theory of computation → Quantum computation theory
  • Theory of computation → Quantum information theory
  • Event learning
  • gentle measurments
  • random measurements
  • quantum or
  • threshold search
  • shadow tomography


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Scott Aaronson. Limitations of quantum advice and one-way communication. In Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004., pages 320-332. IEEE, 2004. Google Scholar
  2. Scott Aaronson. Qma/qpoly/spl sube/pspace/poly: de-merlinizing quantum protocols. In 21st Annual IEEE Conference on Computational Complexity (CCC'06), pages 13-pp. IEEE, 2006. Google Scholar
  3. Scott Aaronson. Shadow tomography of quantum states. SIAM Journal on Computing, 49(5):STOC18-368, 2019. Google Scholar
  4. Scott Aaronson and Guy N Rothblum. Gentle measurement of quantum states and differential privacy. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 322-333, 2019. Google Scholar
  5. Costin Bădescu and Ryan O'Donnell. Improved quantum data analysis. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021, pages 1398-1411, New York, NY, USA, 2021. Association for Computing Machinery. URL: https://doi.org/10.1145/3406325.3451109.
  6. Jingliang Gao. Quantum union bounds for sequential projective measurements. Physical Review A, 92(5):052331, 2015. Google Scholar
  7. Aram W Harrow, Cedric Yen-Yu Lin, and Ashley Montanaro. Sequential measurements, disturbance and property testing. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1598-1611. SIAM, 2017. Google Scholar
  8. Hsin-Yuan Huang, Richard Kueng, and John Preskill. Predicting many properties of a quantum system from very few measurements. arXiv preprint, 2020. URL: https://arxiv.org/abs/2002.08953.
  9. B Kaulakys and V Gontis. Quantum anti-zeno effect. Physical Review A, 56(2):1131, 1997. Google Scholar
  10. Samad Khabbazi Oskouei, Stefano Mancini, and Mark M Wilde. Union bound for quantum information processing. Proceedings of the Royal Society A, 475(2221):20180612, 2019. Google Scholar
  11. A. Uhlmann. The “transition probability” in the state space of a *-algebra. Reports on Mathematical Physics, 9(2):273-279, 1976. URL: https://doi.org/10.1016/0034-4877(76)90060-4.
  12. Mark M Wilde. Quantum information theory. Cambridge University Press, 2013. Google Scholar
  13. Andreas Winter. Coding theorem and strong converse for quantum channels. IEEE Transactions on Information Theory, 45(7):2481-2485, 1999. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail