Lower bounds for one-way probabilistic communication complexity

Jürgen Forster

Shengyu Zhang

Hartmut Klauck

In this paper the Nečiporuk method for proving lower bounds on the size of Boolean formulas is reformulated in terms of one-way communication complexity. We investigate the settings of probabilistic formulas, nondeterministic formulas, and quantum formulas. In all cases we can use results about one-way communication complexity to prove lower bounds on formula size. The main results regarding formula size are as follows: We show a polynomial size gap between probabilistic/quantum and deterministic formulas, a near-quadratic gap between the sizes of nondeterministic formulas with limited access to nondeterministic bits and nondeterministic formulas with access to slightly more such bits, and a near-quadratic lower bound on quantum formula size. Furthermore we give a polynomial separation between the sizes of quantum formulas with and without multiple read random inputs. The lower bound methods for quantum and probabilistic formulas employ a variant of the Nečiporuk bound in terms of the Vapnik–Chervonenkis dimension. To establish our lower bounds we show optimal separations between one-way and two-way protocols for limited nondeterministic and quantum communication complexity, and we show that zero-error quantum one-way communication complexity asymptotically equals deterministic one-way communication complexity for total functions.

Ran Raz

Hartmut Klauck

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.

Hartmut Klauck

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 that the VC-dimension lower bound on one-way probabilistic communication of [26] holds for quantum protocols, too.Then we prove that for oneway protocols computing total functions quantum Las Vegas communication is asymptotically as efficient as exact quantum communication, which is exactly as efficient as deterministic communication.We describe applications of the lower bounds for one-way communication complexity to quantum finite automata and quantum formulae.