scholarly journals Quantum entanglement and the communication complexity of the inner product function

2013 ◽  
Vol 486 ◽  
pp. 11-19 ◽  
Author(s):  
Richard Cleve ◽  
Wim van Dam ◽  
Michael Nielsen ◽  
Alain Tapp
Keyword(s):  
Quantum Entanglement ◽  
Communication Complexity ◽  
Inner Product ◽  
Product Function

Quantum communication complexity of block-composed functions

2009 ◽  
Vol 9 (5&6) ◽  
pp. 444-460
Author(s):  
Y.-Y. Shi ◽  
Y.-F. Zhu
Keyword(s):  
Boolean Function ◽  
Communication Complexity ◽  
Negative Answer ◽  
Inner Product ◽  
Classical Communication ◽  
Symmetric Boolean Function ◽  
Product Function ◽  
Quantum Protocols ◽  
Quantum Communication Complexity ◽  
Quantum Complexity

A major open problem in communication complexity is whether or not quantum protocols can be exponentially more efficient than classical ones for computing a {\em total} Boolean function in the two-party interactive model. Razborov's result ({\em Izvestiya: Mathematics}, 67(1):145--159, 2002) implies the conjectured negative answer for functions $F$ of the following form: $F(x, y)=f_n(x_1\cdot y_1, x_2\cdot y_2, ..., x_n\cdot y_n)$, where $f_n$ is a {\em symmetric} Boolean function on $n$ Boolean inputs, and $x_i$, $y_i$ are the $i$'th bit of $x$ and $y$, respectively. His proof critically depends on the symmetry of $f_n$. We develop a lower-bound method that does not require symmetry and prove the conjecture for a broader class of functions. Each of those functions $F$ is the ``block-composition'' of a ``building block'' $g_k : \{0, 1\}^k \times \{0, 1\}^k \rightarrow \{0, 1\}$, and an $f_n : \{0, 1\}^n \rightarrow \{0, 1\}$, such that $F(x, y) = f_n( g_k(x_1, y_1), g_k(x_2, y_2), ..., g_k(x_n, y_n) )$, where $x_i$ and $y_i$ are the $i$'th $k$-bit block of $x, y\in\{0, 1\}^{nk}$, respectively. We show that as long as g_k itself is "hard'' enough, its block-composition with an arbitrary f_n has polynomially related quantum and classical communication complexities. For example, when g_k is the Inner Product function with k=\Omega(\log n), the deterministic communication complexity of its block-composition with any f_n is asymptotically at most the quantum complexity to the power of 7.

scholarly journals Information cost of quantum communication protocols

2016 ◽  
Vol 16 (3&4) ◽  
pp. 181-196
Author(s):  
Iordanis Kerenidis ◽  
Mathieu Lauriere ◽  
Francois Le Gall ◽  
Mathys Rennela
Keyword(s):  
Information Retrieval ◽  
Quantum Information ◽  
Private Information ◽  
Communication Complexity ◽  
Quantum Communication ◽  
Inner Product ◽  
Private Information Retrieval ◽  
Information Cost ◽  
Product Function ◽  
Quantum Communication Complexity

In two-party quantum communication complexity, Alice and Bob receive some classical inputs and wish to compute some function that depends on both these inputs, while minimizing the communication. This model has found numerous applications in many areas of computer science. One notion that has received a lot of attention recently is the information cost of the protocol, namely how much information the players reveal about their inputs when they run the protocol. In the quantum world, it is not straightforward to define a notion of quantum information cost. We study two different notions and analyze their relation. We also provide new quantum protocols for the Inner Product function and for Private Information Retrieval, and show that protocols for Private Information Retrieval whose classical or quantum information cost for the user is zero can have exponentially different information cost for the server.

scholarly journals Algorithms and Lower Bounds for De Morgan Formulas of Low-Communication Leaf Gates

2021 ◽  
Vol 13 (4) ◽  
pp. 1-37
Author(s):  
Valentine Kabanets ◽  
Sajin Koroth ◽  
Zhenjian Lu ◽  
Dimitrios Myrisiotis ◽  
Igor C. Oliveira
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Boolean Functions ◽  
Communication Complexity ◽  
Inner Product ◽  
Threshold Function ◽  
Seed Length ◽  
Average Case ◽  
Linear Threshold ◽  
Product Function

The class FORMULA[s]∘G consists of Boolean functions computable by size- s De Morgan formulas whose leaves are any Boolean functions from a class G. We give lower bounds and (SAT, Learning, and pseudorandom generators ( PRG s )) algorithms for FORMULA[n 1.99 ]∘G, for classes G of functions with low communication complexity . Let R (k) G be the maximum k -party number-on-forehead randomized communication complexity of a function in G. Among other results, we show the following: • The Generalized Inner Product function GIP k n cannot be computed in FORMULA[s]° G on more than 1/2+ε fraction of inputs for s=o(n 2 /k⋅4 k ⋅R (k) (G)⋅log⁡(n/ε)⋅log⁡(1/ε)) 2 ). This significantly extends the lower bounds against bipartite formulas obtained by [62]. As a corollary, we get an average-case lower bound for GIP k n against FORMULA[n 1.99 ]∘PTF k −1 , i.e., sub-quadratic-size De Morgan formulas with degree-k-1) PTF ( polynomial threshold function ) gates at the bottom. Previously, it was open whether a super-linear lower bound holds for AND of PTFs. • There is a PRG of seed length n/2+O(s⋅R (2) (G)⋅log⁡(s/ε)⋅log⁡(1/ε)) that ε-fools FORMULA[s]∘G. For the special case of FORMULA[s]∘LTF, i.e., size- s formulas with LTF ( linear threshold function ) gates at the bottom, we get the better seed length O(n 1/2 ⋅s 1/4 ⋅log⁡(n)⋅log⁡(n/ε)). In particular, this provides the first non-trivial PRG (with seed length o(n)) for intersections of n halfspaces in the regime where ε≤1/n, complementing a recent result of [45]. • There exists a randomized 2 n-t #SAT algorithm for FORMULA[s]∘G, where t=Ω(n\√s⋅log 2 ⁡(s)⋅R (2) (G))/1/2. In particular, this implies a nontrivial #SAT algorithm for FORMULA[n 1.99 ]∘LTF. • The Minimum Circuit Size Problem is not in FORMULA[n 1.99 ]∘XOR; thereby making progress on hardness magnification, in connection with results from [14, 46]. On the algorithmic side, we show that the concept class FORMULA[n 1.99 ]∘XOR can be PAC-learned in time 2 O(n/log n) .

scholarly journals Fuzzy Inner Product Space: Literature Review and a New Approach

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 765
Author(s):  
Lorena Popa ◽  
Lavinia Sida
Keyword(s):  
Literature Review ◽  
Product Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Product Spaces ◽  
Comprehensive Overview ◽  
New Approach ◽  
Inner Product Spaces ◽  
Product Function ◽  
Suitable Definition

The aim of this paper is to provide a suitable definition for the concept of fuzzy inner product space. In order to achieve this, we firstly focused on various approaches from the already-existent literature. Due to the emergence of various studies on fuzzy inner product spaces, it is necessary to make a comprehensive overview of the published papers on the aforementioned subject in order to facilitate subsequent research. Then we considered another approach to the notion of fuzzy inner product starting from P. Majundar and S.K. Samanta’s definition. In fact, we changed their definition and we proved some new properties of the fuzzy inner product function. We also proved that this fuzzy inner product generates a fuzzy norm of the type Nădăban-Dzitac. Finally, some challenges are given.

scholarly journals Some properties of the superior and inferior semi inner product function associated to the 2-norm

2016 ◽  
Vol 12 (5) ◽  
pp. 6254-6260
Author(s):  
Stela Ceno
Keyword(s):  
Normed Space ◽  
Scalar Product ◽  
Inner Product ◽  
Normed Spaces ◽  
Close Link ◽  
Long Period ◽  
Product Function

Special properties that the scalar product enjoys and its close link with the norm function have raised the interest of researchers from a very long period of time. S.S. Dragomir  presents concrete generalizations of the scalar product functions  in a normed space and deals with the interesting properties of them. Based on S.S. Dragomirs idea in this paper we treat generalizations of superior and inferior scalar product functions in the case of semi-normed spaces and 2-normed spaces.

Reducing Three-party Communication Complexity with Quantum Entanglement

2016 ◽  
Vol 22 (1) ◽  
pp. 29-35
Author(s):  
牛瑞明 NIU Rui-ming ◽  
张融 ZHANG Rong ◽  
薛鹏 XUE Peng
Keyword(s):  
Quantum Entanglement ◽  
Communication Complexity

scholarly journals Quantum Entanglement and Communication Complexity

1997 ◽  
Vol 4 (40) ◽  
Author(s):  
Harry Buhrman ◽  
Richard Cleve ◽  
Wim Van Dam
Keyword(s):  
Quantum Entanglement ◽  
Quantum State ◽  
Communication Channel ◽  
Communication Complexity ◽  
Complexity Problem

We consider a variation of the multi-party communication complexity scenario where the parties are supplied with an extra resource: particles<br />in an entangled quantum state. We show that, although a prior<br />quantum entanglement cannot be used to simulate a communication channel, it can reduce the communication complexity of functions in<br />some cases. Specifically, we show that, for a particular function among three parties (each of which possesses part of the function's input), a prior quantum entanglement enables them to learn the value of the<br />function with only three bits of communication occurring among the parties, whereas, without quantum entanglement, four bits of communication are necessary. We also show that, for a particular two-party probabilistic communication complexity problem, quantum entanglement<br />results in less communication than is required with only classical<br />random correlations (instead of quantum entanglement). These results are a noteworthy contrast to the well-known fact that quantum entanglement cannot be used to actually simulate communication among<br />remote parties.

scholarly journals Quantum Entanglement and Communication Complexity

2001 ◽  
Vol 30 (6) ◽  
pp. 1829-1841 ◽  
Author(s):  
Harry Buhrman ◽  
Richard Cleve ◽  
Wim van Dam
Keyword(s):  
Quantum Entanglement ◽  
Communication Complexity

On axiomatic of inner product function

2013 ◽  
Vol 7 ◽  
pp. 3011-3018
Author(s):  
Silvana Liftaj ◽  
Eriola Sila
Keyword(s):  
Inner Product ◽  
Product Function
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