Dmitry Gavinsky

Exponential Separation for One-Way Quantum Communication Complexity, with Applications to Cryptography

We give an exponential separation between one-way quantum and classical communication protocols for a partial Boolean function (a variant of the Boolean hidden matching problem of Bar-Yossef et al.). Previously, such an exponential separation was known only for a relational problem. The communication problem corresponds to a strong extractor that fails against a small amount of quantum information about its random source. Our proof uses the Fourier coefficients inequality of Kahn, Kalai, and Linial. We also give a number of applications of this separation. In particular, we show that there are privacy amplification schemes that are secure against classical adversaries but not against quantum adversaries; and we give the first example of a key-expansion scheme in the model of bounded-storage cryptography that is secure against classical memory-bounded adversaries but not against quantum ones.

A Separation of NP and coNP in Multiparty Communication Complexity

We prove that NP differs from coNP and coNP is not a subset of MA in the number-on-forehead model of multiparty communication complexity for up to k = (1-\epsilon)log(n) players, where \epsilon>0 is any constant. Specifically, we construct a function F with co-nondeterministic complexity O(log(n)) and Merlin-Arthur complexity n^{\Omega(1)}. The problem was open for k > 2.

Strengths and weaknesses of quantum fingerprinting

We study the power of quantum fingerprints in the simultaneous message passing (SMP) setting of communication complexity. Yao recently showed how to simulate, with exponential overhead, classical shared-randomness SMP protocols by means of quantum SMP protocols without shared randomness (Qpar-protocols). Our first result is to extend Yaos simulation to the strongest possible model: every many-round quantum protocol with unlimited shared entanglement can be simulated, with exponential overhead, by Qpar-protocols. We apply our technique to obtain an efficient Qpar-protocol for a function which cannot be efficiently solved through more restricted simulations. Second, we tightly characterize the power of the quantum fingerprinting technique by making a connection to arrangements of homogeneous halfspaces with maximal margin. These arrangements have been well studied in computational learning theory, and we use some strong results obtained in this area to exhibit weaknesses of quantum fingerprinting. In particular, this implies that for almost all functions, quantum fingerprinting protocols are exponentially worse than classical deterministic SMP protocols

Classical interaction cannot replace a quantum message

We demonstrate a two-player communication problem that can be solved in the one-way quantum model by a 0-error protocol of cost O(log n) but requires exponentially more communication in the classical interactive (bounded error) model.

En Route to the Log-Rank Conjecture: New Reductions and Equivalent Formulations

We prove that several measures in communication complexity are equivalent, up to polynomial factors in the logarithm of the rank of the associated matrix: deterministic communication complexity, randomized communication complexity, information cost and zero-communication cost. This shows that in order to prove the log-rank conjecture, it suffices to show that low-rank matrices have efficient protocols in any of the aforementioned measures.

Quantum Money with Classical Verification

We propose and construct a quantum money scheme that allows verification through classical communication with a bank. This is the first demonstration that a secure quantum money scheme exists that does not require quantum communication for coin verification. Our scheme is secure against adaptive adversaries - this property is not directly related to the possibility of classical verification, nevertheless none of the earlier quantum money constructions is known to possess it.

Quantum algorithm for the Boolean hidden shift problem

The hidden shift problem is a natural place to look for new separations between classical and quantum models of computation. One advantage of this problem is its flexibility, since it can be defined for a whole range of functions and a whole range of underlying groups. In a way, this distinguishes it from the hidden subgroup problem where more stringent requirements about the existence of a periodic subgroup have to be made. And yet, the hidden shift problem proves to be rich enough to capture interesting features of problems of algebraic, geometric, and combinatorial flavor. We present a quantum algorithm to identify the hidden shift for any Boolean function. Using Fourier analysis for Boolean functions we relate the time and query complexity of the algorithm to an intrinsic property of the function, namely its minimum influence. We show that for randomly chosen functions the time complexity of the algorithm is polynomial. Based on this we show an average case exponential separation between classical and quantum time complexity. A perhaps interesting aspect of this work is that, while the extremal case of the Boolean hidden shift problem over so-called bent functions can be reduced to a hidden subgroup problem over an abelian group, the more general case studied here does not seem to allow such a reduction.

Exponential Separation of Quantum and Classical Non-interactive Multi-party Communication Complexity

We give the first exponential separation between quantum and classical multi-party communication complexity in the (non-interactive) one-way and simultaneous message passing settings. For every k, we demonstrate a relational communication problem between k parties that can be solved exactly by a quantum simultaneous message passing protocol of cost O (log n) and requires protocols of cost nc/k2, where c > 0 is a constant, in the classical non-interactive one-way message passing model with shared randomness and bounded error. Thus our separation of corresponding communication classes is superpolynomial as long as k =0 (radic log n/ log log n ) and exponential for k = O(1).

On the Role of Shared Randomness in Simultaneous Communication

Two parties wish to carry out certain distributed computational tasks, and they are given access to a source of correlated random bits. It allows the parties to act in a correlated manner, which can be quite useful. But what happens if the shared randomness is not perfect? In this work, we initiate the study of the power of different sources of shared randomness in communication complexity. This is done in the setting of simultaneous message passing (SMP) model of communication complexity, which is one of the most suitable models for studying the resource of shared randomness. Toward characterising the power of various sources of shared randomness, we introduce a measure for the quality of a source - we call it collision complexity. Our results show that the collision complexity tightly characterises the power of a (shared) randomness resource in the SMP model. Of independent interest is our demonstration that even the weakest sources of shared randomness can in some cases increase the power of SMP substantially: the equality function can be solved very efficiently with virtually any nontrivial shared randomness.

Quantum Algorithms for Evaluating Min-Max Trees

We present a bounded-error quantum algorithm for evaluating Min - Max trees with