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arXiv:quant-ph/0502066 · Physical Review A

Experimental quantum communication complexity

Pavel Trojek, Christian Schmid, Mohamed Bourennane, Časlav Brukner, Marek Żukowski, Harald Weinfurter

Published Nov 28, 2005• 52 citations
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Abstract

We prove that by communicating $N\ensuremath{-}1$ times a single qubit, instead of $N\ensuremath{-}1$ classical bits, the success probability for solving some $N$ partner communication complexity tasks is strongly enhanced. The superiority, as measured by the quantum-to-classical fidelity ratio, of the quantum scheme grows exponentially with $N$. We report an experimental implementation of these tasks for $N=5$. Even without correcting for any inefficiencies of the state-of-the-art setup, our multiparty quantum protocol still outperforms the best classical protocols.

DOI
10.1103/physreva.72.050305

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We prove that by communicating N\ensuremath−1N\ensuremath{-}1N\ensuremath−1 times a single qubit, instead of N\ensuremath−1N\ensuremath{-}1N\ensuremath−1 classical bits, the success probability for solving some NNN partner communication complexity tasks is strongly enhanced. The superiority, as measured by the quantum-to-classical fidelity ratio, of the quantum scheme grows exponentially with NNN. We report an experimental implementation of these tasks for N=5N=5N=5. Even without correcting for any inefficiencies of the state-of-the-art setup, our multiparty quantum protocol still outperforms the best classical protocols.
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I show that a simple multiparty communication task can be performed more efficiently with quantum communication than with classical communication, even with low detection efficiency $\ensuremath{\eta}.$ The task is a communication complexity problem in which distant parties need to compute a function of the distributed inputs, while minimizing the amount of communication between them. A realistic quantum optical setup is suggested that can demonstrate a five-party quantum protocol with higher-than-classical performance, provided \ensuremath{\eta}>0.33.
I present a simple two-party quantum communication complexity protocol with higher success rate than the best possible classical protocol for the same task. The quantum protocol is shown to be equivalent to a quantum non-locality test, except that it is not necessary to close the locality loophole. I derive bounds for the detector efficiency and background count rates necessary for an experimental implementation and show that they are close to what can be currently achieved using ion trap technology. I also analyze the requirements for a three-party protocol and show that they are less demanding than those for the two-party protocol. The results can be interpreted as sufficient experimental conditions for quantum non-locality tests using two or three entangled qubits.
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Demonstrating a quantum advantage with currently available experimental systems is of utmost importance in quantum information science. While this remains elusive for quantum computation, the field of communication complexity offers the possibility to already explore and showcase this advantage for useful tasks. Here, we define such a task, the Sampling Matching problem, which is inspired by the Hidden Matching problem and features an exponential gap between quantum and classical protocols in the one-way communication model. Our problem allows by its conception a photonic implementation based on encoding in the phase of coherent states of light, the use of a fixed size linear optic circuit, and single-photon detection. This enables us to demonstrate in a proof-of-principle experiment an advantage in the transmitted information resource over the best known classical protocol, something impossible to reach for the original Hidden Matching problem. Our demonstration has implications in quantum verification and cryptographic settings.
We study a new type of separations between quantum and classical communication complexity, separations that are obtained using quantum protocols where all parties are efficient , in the sense that they can be implemented by small quantum circuits, with oracle access to their inputs. Our main result qualitatively matches the strongest known separation between quantum and classical communication complexity Gavinsky (2016) and is obtained using a quantum protocol where all parties are efficient. More precisely, we give an explicit partial Boolean function f over inputs of length N , such that: f can be computed by a simultaneous-message quantum protocol with communication complexity polylog( N ) (where at the beginning of the protocol Alice and Bob also have polylog( N ) entangled EPR pairs). Any classical randomized protocol for f , with any number of rounds, has communication complexity at least Ω~(N1/4)\tilde{\Omega}\left(N^{1/4}\right)Ω~(N1/4) . All parties in the quantum protocol of Item (1) (Alice, Bob and the referee) can be implemented by quantum circuits of size polylog( N ) (where Alice and Bob have oracle access to their inputs). Items (1), (2) qualitatively match the strongest known separation between quantum and classical communication complexity, proved by Gavinsky (2016). Item (3) is new. (Our result is incomparable to the one of Gavinsky. While he obtained a quantitatively better lower bound of Ω(N1/2)\Omega\left(N^{1/2}\right)Ω(N1/2) in the classical case, the referee in his quantum protocol is inefficient). Exponential separations of quantum and classical communication complexity have been studied in numerous previous works, but to the best of our knowledge the efficiency of the parties in the quantum protocol has not been addressed, and in most previous separations the quantum parties seem to be inefficient. The only separations that we know of that have efficient quantum parties are the recent separations that are based on lifting Göös et al. (2017), Chattopadhyay et al. (2019a). However, these separations seem to require quantum protocols with at least two rounds of communication, so they imply a separation of two-way quantum and classical communication complexity, but they do not give the stronger separations of simultaneous-message quantum communication complexity vs. two-way classical communication complexity (or even one-way quantum communication complexity vs. two-way classical communication complexity). Our proof technique is completely new, in the context of communication complexity, and is based on techniques from Raz & Tal (2019). Our function f is based on a lift of the forrelation problem, using xor as a gadget.