Quantum lower bounds by quantum arguments

  title={Quantum lower bounds by quantum arguments},
  author={Andris Ambainis},
  journal={J. Comput. Syst. Sci.},
  • 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… 

Quantum Adversary (Upper) Bound

  • S. Kimmel
  • Computer Science, Mathematics
    Chic. J. Theor. Comput. Sci.
  • 2013
A method for upper bounding the quantum query complexity of certain boolean formula evaluation problems, using fundamental theorems about the general adversary bound is described, which gives an upper bound on query complexity without producing an algorithm.

An optimal adiabatic quantum query algorithm

This work revisits the quantum adversary bound result by providing a direct proof in the continuous-time model, and draws new connections between the adversary bound, a modern theoretical computer science technique, and early theorems of quantum mechanics.

Explicit relation between all lower bound techniques for quantum query complexity

An explicit reduction from the polynomial method to the multiplicative adversary method is shown, which reveals a clear picture of the relation between the different lower bound techniques, as it implies that all known techniques reduce to the multiplier method.

The Multiplicative Quantum Adversary

  • R. Spalek
  • Computer Science, Mathematics
    2008 23rd Annual IEEE Conference on Computational Complexity
  • 2008
A new variant of the quantum adversary method, a method for proving lower bounds on the quantum query complexity of a function, is presented, rooted in the quantum lower-bound method by Ambainis, based on the analysis of eigenspaces of the density matrix.

Applications of Adversary Method in Quantum Query Algorithms

A recently developed tight characterisation of quantum query complexity, the adversary bound, is used to develop new quantum algorithms and lower bounds, and a generalisation ofquantum walks that connects electrical properties of a graph and its quantum hitting time is developed.

A Universal Adiabatic Quantum Query Algorithm

This work revisits the result that quantum adversary bound holds for continuous-time quantum computation, and uses for the first time in the context of quantum computation a version of the adiabatic theorem that does not require a spectral gap.

Negative weights make adversaries stronger

A stronger version of the adversary method which goes beyond this principle to make explicit use of the stronger condition that the algorithm actually computes the function, and which is a lower bound on bounded-error quantum query complexity.

Efficient algorithms in quantum query complexity

These algorithms are a novel application of the quantum walk search framework and give improved upper bounds for several subgraph-finding problems and study the quantum query complexity of matrix multiplication and related problems over rings, semirings, and the Boolean semiring in particular.

A quantum lower bound for distinguishing random functions from random permutations

  • H. Yuen
  • Computer Science, Mathematics
    Quantum Inf. Comput.
  • 2014
The quantum query complexity of this problem is studied, and it is shown that any quantum algorithm that solves this problem with bounded error must make $\Omega(N^{1/5}/\log N)$ queries to the input function.

A New Quantum Lower Bound Method, with an Application to a Strong Direct Product Theorem for Quantum Search

  • A. Ambainis
  • Mathematics, Computer Science
    Theory Comput.
  • 2010
A new method for proving lower bounds on quantum query al- gorithms is presented, an extension of the adversary method, by analyzing the eigenspace structure of the problem by proving a strong direct product theorem for quantum search.



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.