scholarly journals Quantum communication complexity of distribution testing

2021 ◽  
Vol 21 (15&16) ◽  
pp. 1261-1273
Author(s):  
Aleksandrs Belovs ◽  
Arturo Castellanos ◽  
Francois Le Gall ◽  
Guillaume Malod ◽  
Alexander A. Sherstov
Keyword(s):  
Lower Bound ◽  
Communication Complexity ◽  
Quantum Communication ◽  
Discrete Distributions ◽  
Classical Communication ◽  
Testing Problem ◽  
Quantum Protocols ◽  
Quantum Communication Complexity ◽  
Pattern Matrix ◽  
Distribution Testing

The classical communication complexity of testing closeness of discrete distributions has recently been studied by Andoni, Malkin and Nosatzki (ICALP'19). In this problem, two players each receive $t$ samples from one distribution over $[n]$, and the goal is to decide whether their two distributions are equal, or are $\epsilon$-far apart in the $l_1$-distance. In the present paper we show that the quantum communication complexity of this problem is $\tilde{O}(n/(t\epsilon^2))$ qubits when the distributions have low $l_2$-norm, which gives a quadratic improvement over the classical communication complexity obtained by Andoni, Malkin and Nosatzki. We also obtain a matching lower bound by using the pattern matrix method. Let us stress that the samples received by each of the parties are classical, and it is only communication between them that is quantum. Our results thus give one setting where quantum protocols overcome classical protocols for a testing problem with purely classical samples.

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.

On the orthogonal rank and impossibility of quantum round elimination

2017 ◽  
Vol 17 (1&2) ◽  
pp. 106-116
Author(s):  
Jop Briet ◽  
Jeroen Zuiddam
Keyword(s):  
Lower Bound ◽  
Communication Complexity ◽  
Quantum Communication ◽  
Hamming Distance ◽  
Cayley Graphs ◽  
Affirmative Answer ◽  
Complex Vector ◽  
Total Communication ◽  
Quantum Communication Complexity ◽  
The Cost

After Bob sends Alice a bit, she responds with a lengthy reply. At the cost of a factor of two in the total communication, Alice could just as well have given Bob her two possible replies at once without listening to him at all, and have him select which one applies. Motivated by a conjecture stating that this form of “round elimination” is impossible in exact quantum communication complexity, we study the orthogonal rank and a symmetric variant thereof for a certain family of Cayley graphs. The orthogonal rank of a graph is the smallest number d for which one can label each vertex with a nonzero d-dimensional complex vector such that adjacent vertices receive orthogonal vectors. We show an exp(n) lower bound on the orthogonal rank of the graph on {0, 1} n in which two strings are adjacent if they have Hamming distance at least n/2. In combination with previous work, this implies an affirmative answer to the above conjecture.

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.

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 Communication complexity and the reality of the wave function

2015 ◽  
Vol 30 (01) ◽  
pp. 1530001 ◽  
Author(s):  
Alberto Montina
Keyword(s):  
Quantum State ◽  
Communication Complexity ◽  
Quantum Communication ◽  
Communication Process ◽  
Minimal Amount ◽  
Quantum Processes ◽  
Independence Property ◽  
Classical Communication ◽  
Asymptotic Equipartition Property ◽  
Quantum Communication Complexity

In this review, we discuss a relation between quantum communication complexity and a long-standing debate in quantum foundation concerning the interpretation of the quantum state. Is the quantum state a physical element of reality as originally interpreted by Schrödinger? Or is it an abstract mathematical object containing statistical information about the outcome of measurements as interpreted by Born? Although these questions sound philosophical and pointless, they can be made precise in the framework of what we call classical theories of quantum processes, which are a reword of quantum phenomena in the language of classical probability theory. In 2012, Pusey, Barrett and Rudolph (PBR) proved, under an assumption of preparation independence, a theorem supporting the original interpretation of Schrödinger in the classical framework. The PBR theorem has attracted considerable interest revitalizing the debate and motivating other proofs with alternative hypotheses. Recently, we showed that these questions are related to a practical problem in quantum communication complexity, namely, quantifying the minimal amount of classical communication required in the classical simulation of a two-party quantum communication process. In particular, we argued that the statement of the PBR theorem can be proved if the classical communication cost of simulating the communication of n qubits grows more than exponentially in n. Our argument is based on an assumption that we call probability equipartition property. This property is somehow weaker than the preparation independence property used in the PBR theorem, as the former can be justified by the latter and the asymptotic equipartition property of independent stochastic sources. The probability equipartition property is a general and natural hypothesis that can be assumed even if the preparation independence hypothesis is dropped. In this review, we further develop our argument into the form of a theorem.

scholarly journals Robust Bell inequalities from communication complexity

Quantum ◽  
2018 ◽  
Vol 2 ◽  
pp. 72 ◽  
Author(s):  
Sophie Laplante ◽  
Mathieu Laurière ◽  
Alexandre Nolin ◽  
Jérémie Roland ◽  
Gabriel Senno
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Communication Complexity ◽  
Bell Inequality ◽  
Bell Inequalities ◽  
Local Distribution ◽  
Classical Communication ◽  
Quantum Communication Complexity ◽  
Complexity Lower Bounds ◽  
Quantum Distributions

The question of how large Bell inequality violations can be, for quantum distributions, has been the object of much work in the past several years. We say that a Bell inequality is normalized if its absolute value does not exceed 1 for any classical (i.e. local) distribution. Upper and (almost) tight lower bounds have been given for the quantum violation of these Bell inequalities in terms of number of outputs of the distribution, number of inputs, and the dimension of the shared quantum states. In this work, we revisit normalized Bell inequalities together with another family: inefficiency-resistant Bell inequalities. To be inefficiency-resistant, the Bell value must not exceed 1 for any local distribution, including those that can abort. This makes the Bell inequality resistant to the detection loophole, while a normalized Bell inequality is resistant to general local noise. Both these families of Bell inequalities are closely related to communication complexity lower bounds. We show how to derive large violations from any gap between classical and quantum communication complexity, provided the lower bound on classical communication is proven using these lower bound techniques. This leads to inefficiency-resistant violations that can be exponential in the size of the inputs. Finally, we study resistance to noise and inefficiency for these Bell inequalities.

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