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 Quantum Speedup Based on Classical Decision Trees

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 241
Author(s):  
Salman Beigi ◽  
Leila Taghavi
Keyword(s):  
Decision Tree ◽  
Main Tool ◽  
Upper Bounds ◽  
Query Complexity ◽  
Bipartite Matching ◽  
Quantum Query Complexity ◽  
Maximum Bipartite Matching ◽  
Query Algorithm ◽  
Binary Functions ◽  
Quantum Query

Lin and Lin \cite{LL16} have recently shown how starting with a classical query algorithm (decision tree) for a function, we may find upper bounds on its quantum query complexity. More precisely, they have shown that given a decision tree for a function f:{0,1}n→[m] whose input can be accessed via queries to its bits, and a guessing algorithm that predicts answers to the queries, there is a quantum query algorithm for f which makes at most O(GT) quantum queries where T is the depth of the decision tree and G is the maximum number of mistakes of the guessing algorithm. In this paper we give a simple proof of and generalize this result for functions f:[ℓ]n→[m] with non-binary input as well as output alphabets. Our main tool for this generalization is non-binary span program which has recently been developed for non-binary functions, and the dual adversary bound. As applications of our main result we present several quantum query upper bounds, some of which are new. In particular, we show that topological sorting of vertices of a directed graph G can be done with O(n3/2) quantum queries in the adjacency matrix model. Also, we show that the quantum query complexity of the maximum bipartite matching is upper bounded by O(n3/4m+n) in the adjacency list model.

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.

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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