# Tight adversary bounds for composite functions

@article{Hyer2005TightAB, title={Tight adversary bounds for composite functions}, author={Peter H{\o}yer and Troy Lee and Robert Spalek}, journal={arXiv: Quantum Physics}, year={2005} }

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…

## 16 Citations

### Negative weights make adversaries stronger

- Computer Science, MathematicsSTOC '07
- 2007

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

- Computer ScienceSIAM J. Comput.
- 2014

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

- Computer ScienceBull. EATCS
- 2005

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

- Computer ScienceElectron. 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

- Computer Science
- 2010

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

- Computer Science, MathematicsIEICE Trans. Inf. Syst.
- 2010

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

- Computer ScienceTQC
- 2011

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

- Computer ScienceTheory Comput.
- 2012

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

- Computer ScienceTheory 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

- Computer Science, MathematicsChicago 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.

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