Two Results about Quantum Messages

Abstract

We show 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. This bound was previously known for total functions, while for partial functions this improves on results by Aaronson [Aar05, Aar07], 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, where quantum proofs lead to exponential savings in computing a Boolean function. The previously only known separation between the power of quantum and classical proofs is in a setting where the input is also quantum [AK07].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(log n) for f , whereas every one-way quantum protocol for f receiving a classical proof (protocol of type QCMA) requires communication Ω( √ n/ log n).

Equality, Revisited

“…Note that we are closing the gap left by [9], as our results about non-deterministic SMP complexity of EQ imply that its quantum-classical complexity is also in Ω( √ n). This result has also recently been obtained by Klauck and Podder [12]. One of the key ingredients in our lower bound method is a tight analysis of a new communication primitive that we call "One out of two", which might be of independent interest:…”

“…In 2001, Buhrman, Cleve, Watrous and de Wolf [6] considered the version of SMP with quantum players and gave a very efficient and surprising protocol solving EQ at cost O(log n), and showed its optimality. In 2008, Gavinsky, Regev and de Wolf [9] studied the "quantumclassical" version of SMP, where only one of the players could send a quantum message, and they showed that the complexity of EQ in that model was Ω n/ log n (which was, tight up to the multiplicative factor √ log n by [16], and was improved to Ω( √ n) in [12]) . 1 The communication complexity of EQ becomes n in the most of deterministic models, but those results are usually trivial and we do not consider the deterministic setup in this work (except for one special case where a "semi-deterministic" protocol has complexity O( √ n) -that situation will be analysed in one of our lower bound proofs).…”

Practical position-based quantum cryptography

“…They showed in particular that finding an explicit Boolean function with polynomial circuit complexity (so that the honest prover can compute it) but exponential attack complexity in the garden-hose model is at least as difficult as separating the classes of languages P and L, corresponding respectively to decision problems decidable in polynomial time or logarithmic space. This result was recently extended by Klauck and Podder who showed that explicit Boolean functions on k variables with Garden-hose complexity Ω(k 2+ε ) will be hard to obtain [14]. These results give us little hope of finding an explicit position-verification based on the nonlocal computation of Boolean functions both practical and secure.…”

Distributed Quantum Proofs for Replicated Data

“…The present paper adds to this list another important task for which quantum distributed computing significantly outperforms classical distributed computing, namely, distributed certification. Note that while this paper is the first to study quantum Merlin-Arthur protocols in a distributed computing framework, there are a number of prior works studying them in communication complexity [38,26,27,9]. In particular, quantum Merlin-Arthur protocols are shown to improve some computational measure (say, the total length of the messages from the prover to Alice, and of the messages between Alice and Bob) exponentially compared to Merlin-Arthur protocols where the messages from the prover are classical [38,27].…”