Abstract
We prove two results about the relationship between quantum and classical messages. Our first contribution is to show how to replace a quantum message in a one-way communication protocol by a deterministic message, establishing that for all partial Boolean functions f:{0,1}n×{0,1}m → {0,1} we have D A → B(f) ≤ O(Q A → B,*(f)·m). This bound was previously known for total functions, while for partial functions this improves on results by Aaronson [1,2], in which either a log-factor on the right hand is present, or the left hand side is R A → B(f), and in which also no entanglement is allowed.
In our second contribution we investigate the power of quantum proofs over classical proofs. We give the first example of a scenario in which quantum proofs lead to exponential savings in computing a Boolean function, for quantum verifiers. The previously only known separation between the power of quantum and classical proofs is in a setting where the input is also quantum [3].
We exhibit a partial Boolean function f, such that there is a one-way quantum communication protocol receiving a quantum proof (i.e., a protocol of type QMA) that has cost O(logn) for f, whereas every one-way quantum protocol for f receiving a classical proof (protocol of type QCMA) requires communication \(\Omega(\sqrt n/\log n)\).
This work is funded by the Singapore Ministry of Education (partly through the Academic Research Fund Tier 3 MOE2012-T3-1-009) and by the Singapore National Research Foundation.
Supported by a CQT Graduate Scholarship.
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Klauck, H., Podder, S. (2014). Two Results about Quantum Messages. In: Csuhaj-Varjú, E., Dietzfelbinger, M., Ésik, Z. (eds) Mathematical Foundations of Computer Science 2014. MFCS 2014. Lecture Notes in Computer Science, vol 8635. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44465-8_38
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