scholarly journals Span Programs and Quantum Query Complexity: The General Adversary Bound Is Nearly Tight for Every Boolean Function

2009 ◽  
Author(s):  
Ben W. Reichardt
Keyword(s):  
Boolean Function ◽  
Query Complexity ◽  
Quantum Query Complexity ◽  
Span Programs ◽  
Quantum Query

scholarly journals Exact quantum algorithms have advantage for almost all Boolean functions

2015 ◽  
pp. 435-452
Author(s):  
Andris Ambainis ◽  
Jozef Gruska ◽  
Shenggen Zheng
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Query Complexity ◽  
Exact Quantum ◽  
Quantum Query Complexity ◽  
Almost All ◽  
Quantum Query

It has been proved that almost all n-bit Boolean functions have exact classical query complexity n. However, the situation seemed to be very different when we deal with exact quantum query complexity. In this paper, we prove that almost all n-bit Boolean functions can be computed by an exact quantum algorithm with less than n queries. More exactly, we prove that ANDn is the only n-bit Boolean function, up to isomorphism, that requires n queries.

scholarly journals Span program for non-binary functions

2019 ◽  
Vol 19 (9&10) ◽  
Author(s):  
Salman Beigi ◽  
Leila Taghavi
Keyword(s):  
Constant Factor ◽  
Query Complexity ◽  
Input Output ◽  
Quantum Query Complexity ◽  
Span Programs ◽  
Binary Input ◽  
Binary Functions ◽  
Quantum Query ◽  
Query Algorithms ◽  
Program Show

Span programs characterize the quantum query complexity of binary functions f:\{0,\ldots,\ell\}^n \to \{0,1\} up to a constant factor. In this paper we generalize the notion of span programs for functions with non-binary input/output alphabets f: [\ell]^n \to [m]. We show that non-binary span program characterizes the quantum query complexity of any such function up to a constant factor. We argue that this non-binary span program is indeed the generalization of its binary counterpart. We also generalize the notion of span programs for a special class of relations. Learning graphs provide another tool for designing quantum query algorithms for binary functions. In this paper, we also generalize this tool for non-binary functions, and as an application of our non-binary span program show that any non-binary learning graph gives an upper bound on the quantum query complexity.

scholarly journals Partial Boolean Functions With Exact Quantum Query Complexity One

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 189
Author(s):  
Guoliang Xu ◽  
Daowen Qiu
Keyword(s):  
Boolean Function ◽  
Upper Bound ◽  
Boolean Functions ◽  
Necessary Conditions ◽  
Quantum Algorithm ◽  
Query Complexity ◽  
Exact Quantum ◽  
Quantum Query Complexity ◽  
Sufficient And Necessary Conditions ◽  
Quantum Query

We provide two sufficient and necessary conditions to characterize any n-bit partial Boolean function with exact quantum query complexity 1. Using the first characterization, we present all n-bit partial Boolean functions that depend on n bits and can be computed exactly by a 1-query quantum algorithm. Due to the second characterization, we construct a function F that maps any n-bit partial Boolean function to some integer, and if an n-bit partial Boolean function f depends on k bits and can be computed exactly by a 1-query quantum algorithm, then F(f) is non-positive. In addition, we show that the number of all n-bit partial Boolean functions that depend on k bits and can be computed exactly by a 1-query quantum algorithm is not bigger than an upper bound depending on n and k. Most importantly, the upper bound is far less than the number of all n-bit partial Boolean functions for all efficiently big n.

Open Problems Related to Quantum Query Complexity

2021 ◽  
Vol 2 (4) ◽  
pp. 1-9
Author(s):  
Scott Aaronson
Keyword(s):  
Synthesis Problem ◽  
Query Complexity ◽  
Partial Functions ◽  
Open Problems ◽  
Polynomial Degree ◽  
Quantum Query Complexity ◽  
Time Space ◽  
Quantum Query

I offer a case that quantum query complexity still has loads of enticing and fundamental open problems—from relativized QMA versus QCMA and BQP versus IP , to time/space tradeoffs for collision and element distinctness, to polynomial degree versus quantum query complexity for partial functions, to the Unitary Synthesis Problem and more.

Quantum Query Complexity for Some Graph Problems

2004 ◽  
pp. 140-150 ◽  
Author(s):  
Aija Berzina ◽  
Andrej Dubrovsky ◽  
Rusins Freivalds ◽  
Lelde Lace ◽  
Oksana Scegulnaja
Keyword(s):  
Query Complexity ◽  
Graph Problems ◽  
Quantum Query Complexity ◽  
Quantum Query

scholarly journals Improving Quantum Query Complexity of Boolean Matrix Multiplication Using Graph Collision

Algorithmica ◽  
2015 ◽  
Vol 76 (1) ◽  
pp. 1-16 ◽  
Author(s):  
Stacey Jeffery ◽  
Robin Kothari ◽  
François Le Gall ◽  
Frédéric Magniez
Keyword(s):  
Matrix Multiplication ◽  
Boolean Matrix ◽  
Query Complexity ◽  
Quantum Query Complexity ◽  
Boolean Matrix Multiplication ◽  
Quantum Query
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