# 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

### Quantum and classical strong direct product theorems and optimal time-space tradeoffs

- Computer Science, Mathematics45th Annual IEEE Symposium on Foundations of Computer Science
- 2004

Theorems for the classical as well as quantum query complexity of the OR function are established, which imply a time-space tradeoff T/sup 2/S = /spl Omega/(N/sup 3/) for sorting N items on a quantum computer, which is optimal up to polylog factors.

### Space-bounded communication complexity

- Computer Science, MathematicsITCS '13
- 2013

This work introduces memory models for 2-party communication complexity, obtaining memory hierarchy theorems, and showing super-linear lower bounds for some explicit (non-boolean) functions.

### The Power of One Clean Qubit in Communication Complexity

- Computer ScienceMFCS
- 2021

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 $\Omega(n)$.

### Strong direct product conjecture holds for all relations in public coin randomized one-way communication complexity

- MathematicsArXiv
- 2010

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.

### Direct product theorems for classical communication complexity via subdistribution bounds: extended abstract

- Computer ScienceSTOC
- 2008

The subdistribution bound is introduced, which is a relaxation of the well-studied rectangle or corruption bound in communication complexity, and it is shown that for the communication complexity of Boolean functions with constant error, the subdist distribution bound is the same as the latter measure, up to a constant factor.

### Strong direct product theorems for quantum communication and query complexity

- Computer Science, MathematicsSTOC '11
- 2011

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.

### On Arthur Merlin Games in Communication Complexity

- Computer Science2011 IEEE 26th Annual Conference on Computational Complexity
- 2011

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.

### New Bounds for the Garden-Hose Model

- Computer Science, MathematicsFSTTCS
- 2014

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).

### Communication memento: Memoryless communication complexity

- Computer ScienceITCS
- 2021

It is shown that the memoryless communication complexity of F characterizes the logarithm of the size of the smallest bipartite branching program computing $F$ (up to a factor 2); exponential separations between the classical variants of memoryless Communication models are given; and exponential quantum-classical separations in the four variants of the memorylessness communication model are exhibited.

### New Strong Direct Product Results in Communication Complexity

- Computer Science, MathematicsElectron. Colloquium Comput. Complex.
- 2011

It is proved that the new complexity measure gives a tight lower bound of Ω(n) for the set-disjointness problem on n-bit inputs (this strengthens the linear lower bound on the rectangle/corruption bound for set- Disjoints shown by Razborov [1992]).

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