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Register a new userQuantum versus Randomized Communication Complexity, with Efficient Players
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We study a new type of separation between quantum and classical communication complexity which is obtained using quantum protocols where all parties are efficient, in the sense that they can be implemented by small quantum circuits with oracle access to their inputs. More precisely, we give an explicit partial Boolean function that can be computed in the quantum-simultaneous-with-entanglement model of communication, however, every interactive randomized protocol is of exponentially larger cost. Furthermore, all the parties in the quantum protocol can be implemented by quantum circuits of small size with blackbox access to the inputs. Our result qualitatively matches the strongest known separation between quantum and classical communication complexity and is obtained using a quantum protocol where all parties are efficient.
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We investigate the randomized and quantum communication complexities of the well-studied Equality function with small error probability $epsilon$, getting the optimal constant factors in the leading terms in a number of different models. In the randomized model, 1) we give a general technique to convert public-coin protocols to private-coin protocols by incurring a small multiplicative error, at a small additive cost. This is an improvement over Newmans theorem [Inf. Proc. Let.91] in the dependence on the error parameter. 2) Using this we obtain a $(log(n/epsilon^2)+4)$-cost private-coin communication protocol that computes the $n$-bit Equality function, to error $epsilon$. This improves upon the $log(n/epsilon^3)+O(1)$ upper bound implied by Newmans theorem, and matches the best known lower bound, which follows from Alon [Comb. Prob. Comput.09], up to an additive $loglog(1/epsilon)+O(1)$. In the quantum model, 1) we exhibit a one-way protocol of cost $log(n/epsilon)+4$, that uses only pure states and computes the $n$-bit Equality function to error $epsilon$. This bound was implicitly already shown by Nayak [PhD thesis99]. 2) We show that any $epsilon$-error one-way protocol for $n$-bit Equality that uses only pure states communicates at least $log(n/epsilon)-loglog(1/epsilon)-O(1)$ qubits. 3) We exhibit a one-way protocol of cost $log(sqrt{n}/epsilon)+3$, that uses mixed states and computes the $n$-bit Equality function to error $epsilon$. This is also tight up to an additive $loglog(1/epsilon)+O(1)$, which follows from Alons result. Our upper bounds also yield upper bounds on the approximate rank and related measures of the Identity matrix. This also implies improved upper bounds on these measures for the distributed SINK function, which was recently used to refute the randomized and quant
We give a direct product theorem for the entanglement-assisted interactive quantum communication complexity of an $l$-player predicate $mathsf{V}$. In particular we show that for a distribution $p$ that is product across the input sets of the $l$ players, the success probability of any entanglement-assisted quantum communication protocol for computing $n$ copies of $mathsf{V}$, whose communication is $o(log(mathrm{eff}^*(mathsf{V},p))cdot n)$, goes down exponentially in $n$. Here $mathrm{eff}^*(mathsf{V}, p)$ is a distributional version of the quantum efficiency or partition bound introduced by Laplante, Lerays and Roland (2014), which is a lower bound on the distributional quantum communication complexity of computing a single copy of $mathsf{V}$ with respect to $p$. As an application of our result, we show that it is possible to do device-independent quantum key distribution (DIQKD) without the assumption that devices do not leak any information after inputs are provided to them. We analyze the DIQKD protocol given by Jain, Miller and Shi (2017), and show that when the protocol is carried out with devices that are compatible with $n$ copies of the Magic Square game, it is possible to extract $Omega(n)$ bits of key from it, even in the presence of $O(n)$ bits of leakage. Our security proof is parallel, i.e., the honest parties can enter all their inputs into their devices at once, and works for a leakage model that is arbitrarily interactive, i.e., the devices of the honest parties Alice and Bob can exchange information with each other and with the eavesdropper Eve in any number of rounds, as long as the total number of bits or qubits communicated is bounded.
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.
This work addresses two problems in the context of two-party communication complexity of functions. First, it concludes the line of research, which can be viewed as demonstrating qualitative advantage of quantum communication in the three most common communication layouts: two-way interactive communication; one-way communication; simultaneous message passing (SMP). We demonstrate a functional problem, whose communication complexity is $O((log n)^2)$ in the quantum version of SMP and $tildeOmega(sqrt n)$ in the classical (randomised) version of SMP. Second, this work contributes to understanding the power of the weakest commonly studied regime of quantum communication $-$ SMP with quantum messages and without shared randomness (the latter restriction can be viewed as a somewhat artificial way of making the quantum model as weak as possible). Our function has an efficient solution in this regime as well, which means that even lacking shared randomness, quantum SMP can be exponentially stronger than its classical counterpart with shared randomness.
We study quantum algorithms that learn properties of a matrix using queries that return its action on an input vector. We show that for various problems, including computing the trace, determinant, or rank of a matrix or solving a linear system that it specifies, quantum computers do not provide an asymptotic speedup over classical computation. On the other hand, we show that for some problems, such as computing the parities of rows or columns or deciding if there are two identical rows or columns, quantum computers provide exponential speedup. We demonstrate this by showing equivalence between models that provide matrix-vector products, vector-matrix products, and vector-matrix-vector products, whereas the power of these models can vary significantly for classical computation.
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