scholarly journals The Partition Bound for Classical Communication Complexity and Query Complexity

2010 ◽  
Author(s):  
Rahul Jain ◽  
Hartmut Klauck
Keyword(s):  
Communication Complexity ◽  
Query Complexity ◽  
Classical Communication

scholarly journals A new exponential separation between quantum and classical one-way communication complexity

2011 ◽  
Vol 11 (7&8) ◽  
pp. 574-591
Author(s):  
Ashley Montanaro
Keyword(s):  
Fourier Analysis ◽  
Boolean Function ◽  
Communication Complexity ◽  
Quantum Communication ◽  
Membership Problem ◽  
Classical Communication ◽  
Exponential Separation ◽  
Quantum Communication Complexity ◽  
Natural Generalisation ◽  
Subgroup Membership Problem

We present a new example of a partial boolean function whose one-way quantum communication complexity is exponentially lower than its one-way classical communication complexity. The problem is a natural generalisation of the previously studied Subgroup Membership problem: Alice receives a bit string $x$, Bob receives a permutation matrix $M$, and their task is to determine whether $Mx=x$ or $Mx$ is far from $x$. The proof uses Fourier analysis and an inequality of Kahn, Kalai and Linial.

scholarly journals Quantum communication complexity advantage implies violation of a Bell inequality

2016 ◽  
Vol 113 (12) ◽  
pp. 3191-3196 ◽  
Author(s):  
Harry Buhrman ◽  
Łukasz Czekaj ◽  
Andrzej Grudka ◽  
Michał Horodecki ◽  
Paweł Horodecki ◽  
...  
Keyword(s):  
Communication Complexity ◽  
Quantum Communication ◽  
Bell Inequality ◽  
Main Tool ◽  
Classical Communication ◽  
Quantum Communication Complexity ◽  
Bell Nonlocality ◽  
Large Quantum ◽  
General Connection ◽  
Inputs And Outputs

We obtain a general connection between a large quantum advantage in communication complexity and Bell nonlocality. We show that given any protocol offering a sufficiently large quantum advantage in communication complexity, there exists a way of obtaining measurement statistics that violate some Bell inequality. Our main tool is port-based teleportation. If the gap between quantum and classical communication complexity can grow arbitrarily large, the ratio of the quantum value to the classical value of the Bell quantity becomes unbounded with the increase in the number of inputs and outputs.

scholarly journals Quantum and Classical Log-Bounded Automata for the Online Disjointness Problem

Mathematics ◽  
2022 ◽  
Vol 10 (1) ◽  
pp. 143
Author(s):  
Kamil Khadiev ◽  
Aliya Khadieva
Keyword(s):  
Online Algorithms ◽  
Minimization Problem ◽  
Competitive Ratio ◽  
Communication Complexity ◽  
Point Of View ◽  
Online Version ◽  
Query Complexity

We consider online algorithms with respect to the competitive ratio. In this paper, we explore one-way automata as a model for online algorithms. We focus on quantum and classical online algorithms. For a specially constructed online minimization problem, we provide a quantum log-bounded automaton that is more effective (has less competitive ratio) than classical automata, even with advice, in the case of the logarithmic size of memory. We construct an online version of the well-known Disjointness problem as a problem. It was investigated by many researchers from a communication complexity and query complexity point of view.

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 Generalizations of the distributed Deutsch–Jozsa promise problem

2015 ◽  
Vol 27 (3) ◽  
pp. 311-331 ◽  
Author(s):  
JOZEF GRUSKA ◽  
DAOWEN QIU ◽  
SHENGGEN ZHENG
Keyword(s):  
Communication Complexity ◽  
Quantum Communication ◽  
Hamming Distance ◽  
Finite Automata ◽  
Classical Communication ◽  
Deterministic Finite Automata ◽  
Exact Quantum ◽  
Quantum Communication Complexity

In the distributed Deutsch–Jozsa promise problem, two parties are to determine whether their respective strings x, y ∈ {0,1}n are at the Hamming distanceH(x, y) = 0 or H(x, y) = $\frac{n}{2}$. Buhrman et al. (STOC' 98) proved that the exact quantum communication complexity of this problem is O(log n) while the deterministic communication complexity is Ω(n). This was the first impressive (exponential) gap between quantum and classical communication complexity. In this paper, we generalize the above distributed Deutsch–Jozsa promise problem to determine, for any fixed $\frac{n}{2}$ ⩽ k ⩽ n, whether H(x, y) = 0 or H(x, y) = k, and show that an exponential gap between exact quantum and deterministic communication complexity still holds if k is an even such that $\frac{1}{2}$n ⩽ k < (1 − λ)n, where 0 < λ < $\frac{1}{2}$ is given. We also deal with a promise version of the well-known disjointness problem and show also that for this promise problem there exists an exponential gap between quantum (and also probabilistic) communication complexity and deterministic communication complexity of the promise version of such a disjointness problem. Finally, some applications to quantum, probabilistic and deterministic finite automata of the results obtained are demonstrated.

scholarly journals The communication complexity of non-signaling distributions

2011 ◽  
Vol 11 (7&8) ◽  
pp. 649-676
Author(s):  
Julien Degorre ◽  
Marc Kaplan ◽  
Sophie Laplante ◽  
J\'er\'emie Roland
Keyword(s):  
Lower Bounds ◽  
Boolean Functions ◽  
Communication Complexity ◽  
Quantum Communication ◽  
Optimization Problems ◽  
Complexity Measures ◽  
Quantum Distribution ◽  
General Technique ◽  
Classical Communication ◽  
Information Theoretic

We study a model of communication complexity that encompasses many well-studied problems, including classical and quantum communication complexity, the complexity of simulating distributions arising from bipartite measurements of shared quantum states, and XOR games. In this model, Alice gets an input $x$, Bob gets an input $y$, and their goal is to each produce an output $a,b$ distributed according to some pre-specified joint distribution $p(a,b|x,y)$. Our results apply to any non-signaling distribution, that is, those where Alice's marginal distribution does not depend on Bob's input, and vice versa.~~~By taking a geometric view of the non-signaling distributions, we introduce a simple new technique based on affine combinations of lower-complexity distributions, and we give the first general technique to apply to all these settings, with elementary proofs and very intuitive interpretations. Specifically, we introduce two complexity measures, one which gives lower bounds on classical communication, and one for quantum communication. These measures can be expressed as convex optimization problems. We show that the dual formulations have a striking interpretation, since they coincide with maximum violations of Bell and Tsirelson inequalities. The dual expressions are closely related to the winning probability of XOR games. Despite their apparent simplicity, these lower bounds subsume many known communication complexity lower bound methods, most notably the recent lower bounds of Linial and Shraibman for the special case of Boolean functions. We show that as in the case of Boolean functions, the gap between the quantum and classical lower bounds is at most linear in the size of the support of the distribution, and does not depend on the size of the inputs. This translates into a bound on the gap between maximal Bell and Tsirelson inequality violations, which was previously known only for the case of distributions with Boolean outcomes and uniform marginals. It also allows us to show that for some distributions, information theoretic methods are necessary to prove strong lower bounds. ~~~ Finally, we give an exponential upper bound on quantum and classical communication complexity in the simultaneous messages model, for any non-signaling distribution. One consequence of this is a simple proof that any quantum distribution can be approximated with a constant number of bits of communication.

scholarly journals Quantum states cannot be transmitted efficiently classically

Quantum ◽  
2019 ◽  
Vol 3 ◽  
pp. 154 ◽  
Author(s):  
Ashley Montanaro
Keyword(s):  
Lower Bound ◽  
Classical Solution ◽  
Communication Complexity ◽  
Communication Protocol ◽  
Quantum States ◽  
Classical Communication ◽  
Fourier Sampling ◽  
Simple Protocol ◽  
Sampling Problem ◽  
Quantum Query

We show that any classical two-way communication protocol with shared randomness that can approximately simulate the result of applying an arbitrary measurement (held by one party) to a quantum state of n qubits (held by another), up to constant accuracy, must transmit at least Ω(2n) bits. This lower bound is optimal and matches the complexity of a simple protocol based on discretisation using an ϵ-net. The proof is based on a lower bound on the classical communication complexity of a distributed variant of the Fourier sampling problem. We obtain two optimal quantum-classical separations as easy corollaries. First, a sampling problem which can be solved with one quantum query to the input, but which requires Ω(N) classical queries for an input of size N. Second, a nonlocal task which can be solved using n Bell pairs, but for which any approximate classical solution must communicate Ω(2n) bits.

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