We investigate the power of randomness in two-party communication complexity. In particular, we study the model where the parties can make a constant number of queries to a function with an efficient one-sided-error randomized protocol. The complexity classes defined by this model comprise the Randomized Boolean Hierarchy, which is analogous to the Boolean Hierarchy but defined with one-sided-error randomness instead of nondeterminism. Our techniques connect the Nondeterministic and Randomized Boolean Hierarchies, and we provide a complete picture of the relationships among complexity classes within and across these two hierarchies. In particular, we prove that the Randomized Boolean Hierarchy does not collapse, and we prove a query-to-communication lifting theorem for all levels of the Nondeterministic Boolean Hierarchy and use it to resolve an open problem stated in the paper by Halstenberg and Reischuk (CCC 1988) which initiated the study of this hierarchy.
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This content will become publicly available on June 30, 2025
The role of randomness in quantum state certification with unentangled measurements
We study quantum state certification using unentangled quantum measurements, namely measurements which operate only on one copy of the state at a time. When there is a common source of randomness available and the unentangled measurements are chosen based on this randomness, prior work has shown that copies are necessary and sufficient. We show a separation between algorithms with and without randomness. We develop a lower bound framework for both fixed and randomized measurements that relates the hardness of testing to the well-established Lüders rule. More precisely, we obtain lower bounds for randomized and fixed schemes as a function of the eigenvalues of the Lüders channel which characterizes one possible post-measurement state transformation.
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- Award ID(s):
- 1846300
- PAR ID:
- 10541062
- Editor(s):
- Agrawal, Shipra; Roth, Aaron
- Publisher / Repository:
- Proceedings of Machine Learning Research
- Date Published:
- Format(s):
- Medium: X
- Location:
- Edmonton, Canada
- Sponsoring Org:
- National Science Foundation
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