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}
}
  • H. Klauck
  • Published in FSTTCS 11 December 2004
  • Computer Science, Mathematics
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… 

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References

SHOWING 1-10 OF 26 REFERENCES

Quantum vs. classical communication and computation

TLDR
A simple and general simulation technique is presented that transforms any black-box quantum algorithm to a quantum communication protocol for a related problem, in a way that fully exploits the quantum parallelism, to obtain new positive and negative results.

Improved Quantum Communication Complexity Bounds for Disjointness and Equality

TLDR
It is shown that the complexities of the disjointness and equality problems of exact and non-deterministic protocols are all equal to n+1, the previous best lower bound being n/2.

Limitations of quantum advice and one-way communication

  • S. Aaronson
  • Computer Science
    Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004.
  • 2004
TLDR
It is shown in three settings that quantum messages have only limited advantages over classical ones, and the polynomial method is used to give the first correct proof of a direct product theorem for quantum search.

Rectangle size bounds and threshold covers in communication complexity

  • H. Klauck
  • Mathematics, Computer Science
    18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings.
  • 2003
TLDR
This work investigates the power of the most important lower bound technique in randomized communication complexity, which is based on an evaluation of the maximal size of approximately monochromatic rectangles, with respect to arbitrary distributions on the inputs, and disproves the conjecture that the lower bound method is polynomially tight for randomized communications complexity.

The Cost of the Missing Bit: Communication Complexity with Help

TLDR
The multiparty communication model of Chandra, Furst, and Lipton (1983) is generalized to functions with b-bit output, and families of explicit functions for which bits of communication are required to find the "missing bit" are constructed.

Direct product results and the GCD problem, in old and new communication models

TLDR
The improved parallel repetition theorem of 2-prover games is applied to derive, for the first time, a direct product theorem for communication complexity, based on interaction between the two models.

Quantum search of spatial regions

  • S. AaronsonA. Ambainis
  • Computer Science
    44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings.
  • 2003
TLDR
An 0(/spl radic/n)-qubit communication protocol for the disjointness problem is given, which improves an upper bound of Hoyer and de Wolf and matches a lower bound of Razborov.

Quantum time-space tradeoffs for sorting

TLDR
The complexity of sorting in the model of sequential quantum circuits is investigated and the following lower bound on the time-space tradeoff for sorting numbers from a polynomial size range is shown: <i>TS</i>=Ω(<i>n</i><sup>3/2</sup>).

The computational complexity of universal hashing

TLDR
Lower bounds on the complexity of any implementation of Carter-Wegman universal hashing are given: quadratic AT/sup 2/ bound for VLSI implementation; Omega (log n) parallel time bound on a CREW PRAM; and exponential size for constant depth circuits.