scholarly journals How Low Can Approximate Degree and Quantum Query Complexity Be for Total Boolean Functions?

2013 ◽  
Author(s):  
Andris Ambainis ◽  
Ronald de Wolf
Keyword(s):  
Boolean Functions ◽  
Query Complexity ◽  
Quantum Query Complexity ◽  
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 Explicit relation between all lower bound techniques for quantum query complexity

2015 ◽  
Vol 13 (04) ◽  
pp. 1350059
Author(s):  
Loïck Magnin ◽  
Jérémie Roland
Keyword(s):  
Lower Bound ◽  
Boolean Functions ◽  
Query Complexity ◽  
Polynomial Method ◽  
Quantum Query Complexity ◽  
State Generation ◽  
Adversary Method ◽  
Explicit Relation ◽  
Special Case ◽  
Quantum Query

The polynomial method and the adversary method are the two main techniques to prove lower bounds on quantum query complexity, and they have so far been considered as unrelated approaches. Here, we show an explicit reduction from the polynomial method to the multiplicative adversary method. The proof goes by extending the polynomial method from Boolean functions to quantum state generation problems. In the process, the bound is even strengthened. We then show that this extended polynomial method is a special case of the multiplicative adversary method with an adversary matrix that is independent of the function. This new result therefore provides insight on the reason why in some cases the adversary method is stronger than the polynomial method. It also reveals a clear picture of the relation between the different lower bound techniques, as it implies that all known techniques reduce to the multiplicative adversary method.

Quantum Query Complexity of Boolean Functions with Small On-Sets

2008 ◽  
pp. 907-918 ◽  
Author(s):  
Andris Ambainis ◽  
Kazuo Iwama ◽  
Masaki Nakanishi ◽  
Harumichi Nishimura ◽  
Rudy Raymond ◽  
...  
Keyword(s):  
Boolean Functions ◽  
Query Complexity ◽  
Quantum Query Complexity ◽  
Quantum Query

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
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