• Corpus ID: 15027971

Tight adversary bounds for composite functions

  title={Tight adversary bounds for composite functions},
  author={Peter H{\o}yer and Troy Lee and Robert Spalek},
  journal={arXiv: Quantum Physics},
The quantum adversary method is a very versatile method for proving lower bounds on quantum algorithms. It has many equivalent formulations, yields tight bounds for many computational problems, and has natural connections to classical lower bounds. One of its formulations is in terms of the spectral norm of some matrices. We define a weighted version of this spectral method and show that it possesses useful composition properties. The results generalize and unify previously known composition… 

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.

Span Programs are Equivalent to Quantum Query Algorithms

It is proved that for any boolean function, the optimal “witness size” of a span program equals the general adversary bound, therefore, span program witness size and quantum query complexity are equivalent measures.

Lower Bounds on Quantum Query Complexity

This paper discusses here what quantum computers cannot do, and specifically how to prove limits on their computational power.

Least span program witness size equals the general adversary lower bound on quantum query complexity

  • B. Reichardt
  • Computer Science
    Electron. Colloquium Comput. Complex.
  • 2010
This work proves that for any boolean function, the optimal “witness size” of a span program for that function coincides exactly with the general adversary bound, which is an optimal quantum algorithm for evaluating “balanced,” read-once formulas over any finite boolean gate set.

Reflections for quantum query complexity : The general adversary bound is tight for every boolean function

We show that any boolean function can be evaluated optimally by a bounded-error quantum query algorithm that alternates a certain fixed, input-independent reflection with coherent queries to the

Lower Bounds on Quantum Query Complexity for Read-Once Formulas with XOR and MUX Operators

It is shown that for any Boolean formula F in the class F, r(F) is a lower bound on the quantum query complexity of the Boolean function that F represents, which gives further evidence for the conjecture that there is an only quadratic gap for all functions.

Span-Program-Based Quantum Algorithm for Evaluating Unbalanced Formulas

A quantum algorithm is given to evaluate formulas over any finite boolean gate set, Provided that the complexities of the input subformulas to any gate differ by at most a constant factor, the algorithm has optimal query complexity.

Span-program-based quantum algorithm for evaluating formulas

A quantum algorithm for evaluating formulas over an extended gate set, including all two- and three-bit binary gates (e.g., NAND, 3-majority), generalizes the optimal quantum AND-OR formula evaluation algorithm and is optimal for evaluating the balanced ternary majority formula.

Quantum Search with Variable Times

  • A. Ambainis
  • Computer Science
    Theory of Computing Systems
  • 2009
A new variant of the usual search problem, in which evaluating xi for different i may take a different number of time steps, is considered, which is optimal and shows a matching lower bound.

A composition theorem for decision tree complexity

  • A. Montanaro
  • Computer Science, Mathematics
    Chicago J. Theor. Comput. Sci.
  • 2015
It is shown that the complexity in the decision tree model of computing composite relations of the form h = g ◦ (f, . . . , f), where each relation f i is boolean-valued, is completely characterised.



Quantum lower bounds by quantum arguments

Two new Ω(√N) lower bounds on computing AND of ORs and inverting a permutation and more uniform proofs for several known lower bounds which have been previously proven via a variety of different techniques are proved.

Polynomial degree vs. quantum query complexity

  • A. Ambainis
  • Computer Science
    44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings.
  • 2003
The degree of a polynomial representing (or approximating) a function f is a lower bound for the quantum query complexity of f and this lower bound is shown by a new, more general version of quantum adversary method.


Two new complexity measures for Boolean functions are introduced, which are named sumPI and maxPI, and the main result is proven via a combinatorial lemma which relates the square of the spectral norm of a matrix to the squares ofthe spectral norms of its submatrices.

Lower bounds for randomized and quantum query complexity using Kolmogorov arguments

  • Sophie LaplanteF. Magniez
  • Computer Science, Mathematics
    Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004.
  • 2004
A very general lower bound technique for quantum and randomized query complexity, that is easy to prove as well as to apply, and derives a general form of the ad hoc weighted method used by Hoyer, Neerbek and Shi to give a quantum lower bound on ordered search and sorting.

Quantum Decision Trees and Semidefinite Programming.

The output condition on quantum algorithms used by Ambainis and others is not sufficient for an algorithm to compute a function with {var_epsilon}-bounded error: the existence of algorithms whose final entanglement matrix satisfies the condition, but for which the value of f cannot be determined from a quantum measurement on the accessible part of the computer is shown.

A lower bound on the quantum query complexity of read-once functions

  • H. BarnumM. Saks
  • Computer Science, Mathematics
    Electron. Colloquium Comput. Complex.
  • 2002

All Quantum Adversary Methods are Equivalent

Disclosed is a method of developing an electrostatic image by bringing an electrostatic image-carrying surface of a substrate into sliding contact with a magnetic brush of a developer formed on a

The spectral norm of a nonnegative matrix