# Quantum and Classical Communication-Space Tradeoffs from Rectangle Bounds

@inproceedings{Klauck2004QuantumAC, title={Quantum and Classical Communication-Space Tradeoffs from Rectangle Bounds}, author={Hartmut Klauck}, booktitle={FSTTCS}, year={2004} }

We derive lower bounds for tradeoffs between the communication C and space S for communicating circuits. The first such bound applies to quantum circuits. If for any problem f: X x Y → Z the multicolor discrepancy of the communication matrix of f is 1/2 d , then any bounded error quantum protocol with space S, in which Alice receives some I inputs, Bob r inputs, and they compute f(x i ,y j ) for the l . r pairs of inputs (x i ,y i ) needs communication C = ω(lrd log |Z|/S). In particular, n x n…

## 18 Citations

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There is a quantum protocol using one clean qubit only and using $O(\log n)$ qubits of communication, such that any classical protocol simulating the acceptance behaviour of the quantum protocol within additive error needs communication $\Omega(n)$.

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It is shown that if for computing f^k (k independent copies of f), o(k R 1/3(f) communication is provided, then the success is exponentially small in k, which settles the strong direct product conjecture for all relations in public coin one-way communication complexity.

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A new measure, the subdistribution bound, is defined, which is a generalization of the well-studied rectangle or corruption bound in communication complexity, and it is proved that the one-way version of this bound tightly captures theOne-way public-coin randomized communication complexity of any relation.

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This work proves that quantum communication complexity obeys an SDPT whenever the communication lower bound for a single instance is proved by the generalized discrepancy method, the strongest technique in that model.

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Lower bounds for the QMA-communication complexity of the functions Inner Product and Disjointness are shown, and how one can 'transfer' hardness under an analogous measure in the query complexity model to the communication model using Sherstov's pattern matrix method is described.

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A strong direct product theorem is established that if the authors want to compute k independent instances of a function, using less than k times the resources needed for one instance, then the overall success probability will be exponentially small in k, which solves an open problem of [KSW07, LSS08].

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Improved lower bounds based on non-deterministic communication complexity (leading to the previously unknown $\Theta(n)$ bounds for Inner Product mod 2 and Disjointness), as well as an upper bound for the Distributed Majority function (previously conjectured to have quadratic complexity).

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