Quantum lower bounds by quantum arguments

@article{Ambainis2000QuantumLB,
  title={Quantum lower bounds by quantum arguments},
  author={Andris Ambainis},
  journal={J. Comput. Syst. Sci.},
  year={2000},
  volume={64},
  pages={750-767}
}
  • A. Ambainis
  • Published 23 February 2000
  • Computer Science
  • J. Comput. Syst. Sci.
We propose a new method for proving lower bounds on quantum query algorithms. Instead of a classical adversary that runs the algorithm with on input and then modifies the input, we use a quantum adversary that runs the algorithm with a superposition of inputs. If the algorithm works correctly, its state becomes entangled with the superposition over inputs. We bound the number of queries needed to achieve a sufficient entanglement and this implies a lower bound on the number of queries for the… 

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...

References

SHOWING 1-10 OF 25 REFERENCES

Quantum lower bounds by polynomials

This work examines the number T of queries that a quantum network requires to compute several Boolean functions on {0,1}/sup N/ in the black-box model and gives asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings.

Communication complexity lower bounds by polynomials

  • H. BuhrmanR. D. Wolf
  • Computer Science
    Proceedings 16th Annual IEEE Conference on Computational Complexity
  • 2001
The "log rank" lower bound extends to the strongest variant of quantum communication complexity (qubit communication+unlimited prior entanglement) and the polynomial equivalence of quantum and classical communication complexity for various classes of functions is proved.

Quantum vs. classical communication and computation

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.

Bounds for small-error and zero-error quantum algorithms

We present a number of results related to quantum algorithms with small error probability and quantum algorithms that are zero-error. First, we give a tight analysis of the trade-offs between the

On the power of quantum computation

  • Daniel R. Simon
  • Computer Science
    Proceedings 35th Annual Symposium on Foundations of Computer Science
  • 1994
This work presents here a problem of distinguishing between two fairly natural classes of function, which can provably be solved exponentially faster in the quantum model than in the classical probabilistic one, when the function is given as an oracle drawn equiprobably from the uniform distribution on either class.

Quantum cryptanalysis of hash and claw-free functions

A quantum algorithm that finds collisions in arbitrary functions after only O(3√N/τ) expected evaluations of the function, more efficient than the best possible classical algorithm, even allowing probabilism.

Quantum algorithms for element distinctness

We present several applications of quantum amplitude amplification to finding claws and collisions in ordered or unordered functions. Our algorithms generalize those of Brassard, Hoyer, and Tapp

Limit on the Speed of Quantum Computation in Determining Parity

It is shown that any quantum algorithm capable of determining the parity of f contains at least N/2 applications of the unitary operator which evaluates f and quantum computers cannot outperform classical computers.

Communication capacity of quantum computation.

By considering quantum computation as a communication process, this formalism enables the link the mixedness of a quantum computer to its efficiency and also allows the critical level of mixedness beyond which there is no quantum advantage in computation to be derived.