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.

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 On Share Conversions for Private Information Retrieval

Entropy ◽  
2019 ◽  
Vol 21 (9) ◽  
pp. 826
Author(s):  
Anat Paskin-Cherniavsky ◽  
Leora Schmerler
Keyword(s):  
Information Retrieval ◽  
Private Information ◽  
Communication Complexity ◽  
Infinite Family ◽  
Log P ◽  
Triple P ◽  
Universal Relation ◽  
Private Information Retrieval ◽  
Full Characterization ◽  
A Share

Beimel et al. in CCC 12’ put forward a paradigm for constructing Private Information Retrieval (PIR) schemes, capturing several previous constructions for k ≥ 3 servers. A key component in the paradigm, applicable to three-server PIR, is a share conversion scheme from corresponding linear three-party secret sharing schemes with respect to a certain type of “modified universal” relation. In a useful particular instantiation of the paradigm, they used a share conversion from ( 2 , 3 ) -CNF over Z m to three-additive sharing over Z p β for primes p 1 , p 2 , p where p 1 ≠ p 2 and m = p 1 · p 2 . The share conversion is with respect to the modified universal relation C S m . They reduced the question of whether a suitable share conversion exists for a triple ( p 1 , p 2 , p ) to the (in)solvability of a certain linear system over Z p . Assuming a solution exists, they also provided a efficient (in m , log p ) construction of such a sharing scheme. They proved a suitable conversion exists for several triples of small numbers using a computer program; in particular, p = p 1 = 2 , p 2 = 3 yielded the three-server PIR with the best communication complexity at the time. This approach quickly becomes infeasible as the resulting matrix is of size Θ ( m 4 ) . In this work, we prove that the solvability condition holds for an infinite family of ( p 1 , p 2 , p ) ’s, answering an open question of Beimel et al. Concretely, we prove that if p 1 , p 2 > 2 and p = p 1 , then a conversion of the required form exists. We leave the full characterization of such triples, with potential applications to PIR complexity, to future work. Although larger (particularly with m a x ( p 1 , p 2 ) > 3 ) triples do not yield improved three-server PIR communication complexity via BIKO’s construction, a richer family of PIR protocols we obtain by plugging in our share conversions might have useful properties for other applications. Moreover, we hope that the analytic techniques for understanding the relevant matrices we developed would help to understand whether share conversion as above for C S m , where m is a product of more than two (say three) distinct primes, exists. The general BIKO paradigm generalizes to work for such Z m ’s. Furthermore, the linear condition in Beimel et al. generalizes to m’s, which are products of more than two primes, so our hope is somewhat justified. In case such a conversion does exist, plugging it into BIKO’s construction would lead to major improvement to the state of the art of three-server PIR communication complexity (reducing Communication Complexity (CC) in correspondence with certain matching vector families).

scholarly journals Symmetrized persistency of Bell correlations for Dicke states and GHZ-based mixtures

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Marcin Wieśniak
Keyword(s):  
Communication Complexity ◽  
Quantum Communication ◽  
Bell Inequality ◽  
Bell Inequalities ◽  
Quantum Correlations ◽  
Complexity Reduction ◽  
Main Difficulty ◽  
Quantum Communication Complexity ◽  
Number Of Particles ◽  
Dicke States

AbstractQuantum correlations, in particular those, which enable to violate a Bell inequality, open a way to advantage in certain communication tasks. However, the main difficulty in harnessing quantumness is its fragility to, e.g, noise or loss of particles. We study the persistency of Bell correlations of GHZ based mixtures and Dicke states. For the former, we consider quantum communication complexity reduction (QCCR) scheme, and propose new Bell inequalities (BIs), which can be used in that scheme for higher persistency in the limit of large number of particles N. In case of Dicke states, we show that persistency can reach 0.482N, significantly more than reported in previous studies.

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