Quantum Event Learning and Gentle Random Measurements
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.
Event learning
gentle measurments
random measurements
quantum or
threshold search
shadow tomography
Theory of computation
Theory of computation~Quantum computation theory
Theory of computation~Quantum information theory
97:1-97:22
Regular Paper
https://arxiv.org/abs/2210.09155
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.
Adam Bene
Watts
Adam Bene Watts
Institute for Quantum Computing, University of Waterloo, Canada
https://orcid.org/0000-0002-3289-3339
John
Bostanci
John Bostanci
Computer Science Department, Columbia University, New York, NY, USA
https://orcid.org/0000-0001-9666-7114
JB is partially supported by NSF CAREER award CCF-2144219.
10.4230/LIPIcs.ITCS.2024.97
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Adam Bene Watts and John Bostanci
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