Hartmut Klauck

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 R. Raz (1995) 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 Fourier based lower bound methods, notably showing that /spl radic/(s~(f)/log n) n, for the average sensitivity s~(f) of a function f, yields a lower bound on the bounded error quantum communication complexity of f (x/spl and/y/spl oplus/yz), 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, only the previously applied general lower bound method based on discrepancy yields bounds that are O(log n).

On quantum and probabilistic communication: Las Vegas and one-way protocols

We investigate the power of quantum communication protocols compared to classical probabilistic protocols. In our first result we describe a total Boolean function that has a quantum Las Vegas protocol communicating at most O(N 1°/11+~) qubits for all e > 0, while any classical probabilistic protocol (with bounded error) needs ~(N/log N) bits. Then we investigate quantum one-way communication complexity. First we show tha t the VC-dimension lower bound on one-way probabilist ic communication of [26] holds for quantum protocols, too. Then we prove that for oneway protocols computing total functions quantum Las Vegas communication is asymptotical ly as efficient as exact quantum communication, which is exactly as efficient as determinist ic communication. We describe applications of the lower bounds for one-way communication complexity to quantum finite au tomata and quantum formulae.

Interaction in quantum communication and the complexity of set disjointness

One of the most intriguing facts about communication using quantum states is that these states cannot be used to transmit more classical bits than the number of qubits used, yet in some scenarios there are ways of conveying information with exponentially fewer qubits than possible classically [3, 26]. Moreover, these methods have a very simple structure---they involve only few message exchanges between the communicating parties. We consider the question as to 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 <italic>k</italic>, there is a problem such that its <italic>k+1</italic> message classical communication complexity is exponentially smaller than its <italic>k</italic> message quantum communication complexity, thus answering the above question in the negative. This in particular proves a round hierarchy theorem for quantum communication complexity, and implies via a simple reduction, an <italic>\Omega(N^{1/k})</italic> lower bound for <italic>k</italic> message protocols for Set Disjointness for constant~<italic>k</italic>. Our result builds on two primitives, <italic>local transitions in bi-partite states</italic> (based on previous work) and <italic>average encoding</italic> which may be of significance in other contexts as well.

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, with respect to arbitrary 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/spl cap/co - MA and AM/spl cap/co - AM, and allows to show that the MA-complexity of the disjointness problem is /spl Omega/(/spl radic/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 bounded error uniform threshold cover, and reduces showing tightness of the bound to the construction of an efficient protocol for a specific communication problem. We then study relaxations of bounded error uniform threshold covers, namely approximate majority covers and majority covers, and exhibit exponential separations between them. Each of these covers captures a lower bound method previously used for randomized communication complexity.

Communication Complexity Method for Measuring Nondeterminism in Finite Automata

While deterministic finite automata seem to be well understood, surprisingly many important problems concerning nondeterministic finite automata (nfas) remain open. One such problem area is the study of different measures of nondeterminism in finite automata and the estimation of the sizes of minimal nondeterministic finite automata. In this paper the concept of communication complexity is applied in order to achieve progress in this problem area. The main results are as follows:(1) Deterministic communication complexity provides lower bounds on the size of nfas with bounded unambiguity. Applying this fact, the proofs of several results about nfas with limited ambiguity can be simplified and presented in a uniform way. (2) There is a family of languages KONk2 with an exponential size gap between nfas with polynomial leaf number/ambiguity and nfas with ambiguity k. This partially provides an answer to the open problem posed by B. Ravikumar and O. Ibarra (1989, SIAM J. Comput. 18, 1263-1282) and H. Leung (1998, SIAM J. Comput. 27, 1073-1082).

A strong direct product theorem for disjointness

A strong direct product theorem states that if we want to compute k independent instances of a function, using less than k times the resources needed for one instance, then the overall success probability will be exponentially small in k. We establish such a theorem for the randomized communication complexity of the Disjointness problem, i.e., with communication const• kn the success probability of solving k instances of size n can only be exponentially small in k. This solves an open problem of [KSW07, LSS08]. We also show that this bound even holds for

The Partition Bound for Classical Communication Complexity and Query Complexity

AM

Lower Bounds for Quantum Communication Complexity

-communication protocols with limited ambiguity. The main result implies a new lower bound for Disjointness in a restricted 3-player NOF protocol, and optimal communication-space tradeoffs for Boolean matrix product. Our main result follows from a solution to the dual of a linear programming problem, whose feasibility comes from a so-called Intersection Sampling Lemma that generalizes a result by Razborov [Raz92].

Quantum and classical strong direct product theorems and optimal time-space tradeoffs

We describe new lower bounds for randomized communication complexity and query complexity which we call the partition bounds. They are expressed as the optimum value of linear programs. For communication complexity we show that the partition bound is stronger than both the rectangle/corruption bound and the γ2/generalized discrepancy bounds. In the model of query complexity we show that the partition bound is stronger than the approximate polynomial degree and classical adversary bounds. We also exhibit an example where the partition bound is quadratically larger than the approximate polynomial degree and adversary bounds.

Quantum Communication Complexity.

We prove lower bounds on the 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 [Comput. Complexity, 5 (1995), pp. 205-221] 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