arXiv subjects

Hartmut Klauck

Publications and source records attributed to Hartmut Klauck.

31 records · Page 2Linked to original sources

arXiv2009

Depth-Independent Lower bounds on the Communication Complexity of Read-Once Boolean Formulas

We show lower bounds of $Ω(\sqrt{n})$ and $Ω(n^{1/4})$ on the randomized and quantum communication complexity, respectively, of all $n$-variable read-once Boolean formulas. Our results complement the recent lower bound of $Ω(n/8^d)$ by Leonardos and Saks and $Ω(n/2^{Ω(d\log d)})$ by Jayram, Kopparty and Raghavendra for randomized communication complexity of read-once Boolean formulas with depth $d$. We obtain our result by "embedding" either the Disjointness problem or its complement in any given read-once Boolean formula.

cs.CC↗

arXiv2009

New Results in the Simultaneous Message Passing Model

Consider the following Simultaneous Message Passing (SMP) model for computing a relation f subset of X x Y x Z. In this model Alice, on input x in X and Bob, on input y in Y, send one message each to a third party Referee who then outputs a z in Z such that (x,y,z) in f. We first show optimal 'Direct sum' results for all relations f in this model, both in the quantum and classical settings, in the situation where we allow shared resources (shared entanglement in quantum protocols and public coins in classical protocols) between Alice and Referee and Bob and Referee and no shared resource between Alice and Bob. This implies that, in this model, the communication required to compute k simultaneous instances of f, with constant success overall, is at least k-times the communication required to compute one instance with constant success. This in particular implies an earlier Direct sum result, shown by Chakrabarti, Shi, Wirth and Yao, 2001, for the Equality function (and a class of other so-called robust functions), in the classical smp model with no shared resources between any parties. Furthermore we investigate the gap between the smp model and the one-way model in communication complexity and exhibit a partial function that is exponentially more expensive in the former if quantum communication with entanglement is allowed, compared to the latter even in the deterministic case.

cs.DC↗

arXiv2006

Interaction in Quantum Communication

In some scenarios there are ways of conveying information with many fewer, even exponentially fewer, qubits than possible classically. Moreover, some of these methods have a very simple structure--they involve only few message exchanges between the communicating parties. It is therefore natural to ask whether every classical protocol may be transformed to a ``simpler'' quantum protocol--one that has similar efficiency, but uses fewer message exchanges. We show that for any constant k, there is a problem such that its k+1 message classical communication complexity is exponentially smaller than its k message quantum communication complexity. This, in particular, proves a round hierarchy theorem for quantum communication complexity, and implies, via a simple reduction, an Omega(N^{1/k}) lower bound for k message quantum protocols for Set Disjointness for constant k. Enroute, we prove information-theoretic lemmas, and define a related measure of correlation, the informational distance, that we believe may be of significance in other contexts as well.

quant-ph↗

arXiv2004

Quantum and Classical Communication-Space Tradeoffs from Rectangle Bounds

We derive lower bounds for tradeoffs between the communication C and space S for communicating circuits. The first such bound applies to quantum circuits. If for any function f with image Z the multicolor discrepancy of the communication matrix of f is 1/2^d, then any bounded error quantum protocol with space S, in which Alice receives some l inputs, Bob r inputs, and they compute f(x_i,y_j) for the lr pairs of inputs (x_i,y_j) needs communication C=Ω(lrd \log |Z|/S). In particular, n\times n-matrix multiplication over a finite field F requires C=Θ(n^3\log^2 |F|/S). We then turn to randomized bounded error protocols, and derive the bound C=Ω(n^3/S^2) for Boolean matrix multiplication, utilizing a new direct product result for the one-sided rectangle lower bound on randomized communication complexity. This implies a separation between quantum and randomized protocols.

quant-ph↗

arXiv2004

Quantum and Classical Strong Direct Product Theorems and Optimal Time-Space Tradeoffs

A strong direct product theorem says that if we want to compute k independent instances of a function, using less than k times the resources needed for one instance, then our overall success probability will be exponentially small in k. We establish such theorems for the classical as well as quantum query complexity of the OR function. This implies slightly weaker direct product results for all total functions. We prove a similar result for quantum communication protocols computing k instances of the Disjointness function. Our direct product theorems imply a time-space tradeoff T^2*S=Omega(N^3) for sorting N items on a quantum computer, which is optimal up to polylog factors. They also give several tight time-space and communication-space tradeoffs for the problems of Boolean matrix-vector multiplication and matrix multiplication.

quant-ph↗

arXiv2004

One-way communication complexity and the Neciporuk lower bound on formula size

In this paper the Neciporuk method for proving lower bounds on the size of Boolean formulae is reformulated in terms of one-way communication complexity. We investigate the scenarios of probabilistic formulae, nondeterministic formulae, and quantum formulae. In all cases we can use results about one-way communication complexity to prove lower bounds on formula size. In the latter two cases we newly develop the employed communication complexity bounds. The main results regarding formula size are as follows: A polynomial size gap between probabilistic/quantum and deterministic formulae. A near-quadratic size gap for nondeterministic formulae with limited access to nondeterministic bits. A near quadratic lower bound on quantum formula size, as well as a polynomial separation between the sizes of quantum formulae with and without multiple read random inputs. The methods for quantum and probabilistic formulae employ a variant of the Neciporuk bound in terms of the VC-dimension. Regarding communication complexity we give optimal separations between one-way and two-way protocols in the cases of limited nondeterministic and quantum communication, and we show that zero-error quantum one-way communication complexity asymptotically equals deterministic one-way communication complexity for total functions.

cs.CC↗

arXiv2003

On quantum and approximate privacy

This paper studies privacy and secure function evaluation in communication complexity. The focus is on quantum versions of the model and on protocols with only approximate privacy against honest players. We show that the privacy loss (the minimum divulged information) in computing a function can be decreased exponentially by using quantum protocols, while the class of privately computable functions (i.e., those with privacy loss 0) is not enlarged by quantum protocols. Quantum communication combined with small information leakage on the other hand makes certain functions computable (almost) privately which are not computable using either quantum communication without leakage or classical communication with leakage. We also give an example of an exponential reduction of the communication complexity of a function by allowing a privacy loss of $o(1)$ instead of privacy loss 0.

quant-ph↗

arXiv2003

Quantum Time-Space Tradeoffs for Sorting

We investigate the complexity of sorting in the model of sequential quantum circuits. While it is known that in general a quantum algorithm based on comparisons alone cannot outperform classical sorting algorithms by more than a constant factor in time complexity, this is wrong in a space bounded setting. We observe that for all storage bounds n/\log n\ge S\ge \log^3 n, one can devise a quantum algorithm that sorts n numbers (using comparisons only) in time T=O(n^{3/2}\log^{3/2} n/\sqrt S). We then show the following lower bound on the time-space tradeoff for sorting $n$ numbers from a polynomial size range in a general sorting algorithm (not necessarily based on comparisons): TS=Ω(n^{3/2}). Hence for small values of S the upper bound is almost tight. Classically the time-space tradeoff for sorting is TS=Θ(n^2).

quant-ph↗

arXiv2003

Individual Communication Complexity

We initiate the theory of communication complexity of individual inputs held by the agents, rather than worst-case or average-case. We consider total, partial, and partially correct protocols, one-way versus two-way, with and without help bits. The results are expressed in trems of Kolmogorov complexity.

cs.CC↗

arXiv2002

Rectangle Size Bounds and Threshold Covers in Communication Complexity

We investigate the power of the most important lower bound technique in randomized communication complexity, which is based on an evaluation of the maximal size of approximately monochromatic rectangles, minimized over all distributions on the inputs. While it is known that the 0-error version of this bound is polynomially tight for deterministic communication, nothing in this direction is known for constant error and randomized communication complexity. We first study a one-sided version of this bound and obtain that its value lies between the MA- and AM-complexities of the considered function. Hence the lower bound actually works for a (communication complexity) class between MA cap co-MA and AM cap co-AM. We also show that the MA-complexity of the disjointness problem is Omega(sqrt(n)). Following this we consider the conjecture that the lower bound method is polynomially tight for randomized communication complexity. First we disprove a distributional version of this conjecture. Then we give a combinatorial characterization of the value of the lower bound method, in which the optimization over all distributions is absent. This characterization is done by what we call a uniform threshold cover. We also study relaxations of this notion, namely approximate majority covers and majority covers, and compare these three notions in power, exhibiting exponential separations. Each of these covers captures a lower bound method previously used for randomized communication complexity.

cs.CC↗

arXiv2002

Lower bounds for quantum communication complexity

We prove new lower bounds for bounded error quantum communication complexity. Our methods are based on the Fourier transform of the considered functions. First we generalize a method for proving classical communication complexity lower bounds developed by Raz to the quantum case. Applying this method we give an exponential separation between bounded error quantum communication complexity and nondeterministic quantum communication complexity. We develop several other lower bound methods based on the Fourier transform, notably showing that \sqrt{\bar{s}(f)/\log n}, for the average sensitivity \bar{s}(f) of a function f, yields a lower bound on the bounded error quantum communication complexity of f(x AND y XOR z), where x is a Boolean word held by Alice and y,z are Boolean words held by Bob. We then prove the first large lower bounds on the bounded error quantum communication complexity of functions, for which a polynomial quantum speedup is possible. For all the functions we investigate, the only previously applied general lower bound method based on discrepancy yields bounds that are O(\log n).

quant-ph↗

arXiv2001

On rounds in quantum communication

We investigate the power of interaction in two player quantum communication protocols. Our main result is a rounds-communication hierarchy for the pointer jumping function $f_k$. We show that $f_k$ needs quantum communication $Ω(n)$ if Bob starts the communication and the number of rounds is limited to $k$ (for any constant $k$). Trivially, if Alice starts, $O(k\log n)$ communication in $k$ rounds suffices. The lower bound employs a result relating the relative von Neumann entropy between density matrices to their trace distance and uses a new measure of information. We also describe a classical probabilistic $k$ round protocol with communication $O(n/k\cdot(\log^{(k/2)}n+\log k)+k\log n)$ in which Bob starts the communication. Furthermore as a consequence of the lower bound for pointer jumping we show that any $k$ round quantum protocol for the disjointness problem needs communication $Ω(n^{1/k})$ for $k=O(1)$.

quant-ph↗

arXiv2000

Quantum Communication Complexity

This paper surveys the field of quantum communication complexity. Some interesting recent results are collected concerning relations to classical communication, lower bound methods, one-way communication, and applications of quantum communication complexity.

quant-ph↗