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SIAM Journal on Computing

One-Way Communication Complexity and the Nečiporuk Lower Bound on Formula Size

Hartmut Klauck

Published Jan 1, 2007• 31 citations
Publisher

Abstract

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.

DOI
10.1137/s009753970140004x

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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.
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.
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.
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 sˉ(f)/log⁡n\sqrt{\bar{s}(f)/\log n}sˉ(f)/logn​, for the average sensitivity sˉ(f)\bar{s}(f)sˉ(f) of a function f, yields a lower bound on the bounded error quantum communication complexity of f((x∧y)⊕z)f((x \wedge y)\oplus z)f((x∧y)⊕z), where x is a Boolean word held by Alice and y,zy,zy,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)O(\log n)O(logn).
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).
A major open question in communication complexity is if randomized and quantum communication are polynomially related for all total functions. So far, no gap larger than a power of two is known, despite significant efforts. We examine this question in the number-on-the-forehead model of multiparty communication complexity. We show that essentially all lower bounds known on randomized complexity in this model also hold for quantum communication. This includes bounds of size Omega(n/2k) for the k-party complexity of explicit functions, bounds for the generalized inner product function, and recent work on the multiparty complexity of disjointness. To the best of our knowledge, these are the first lower bounds of any kind on quantum communication in the general number-on-the-forehead model. We show this result in the following way. In the two-party case, there is a lower bound on quantum communication complexity in terms of a norm gamma2, which is known to subsume nearly all other techniques in the literature. For randomized complexity there is another natural bound in terms of a different norm mu which is also one of the strongest techniques available. A deep theorem in functional analysis, Grothendieck's inequality, implies that gamma2and mu are equivalent up to a constant factor. This connection is one of the major obstacles to showing a larger gap between randomized and quantum communication complexity in the two-party case. The lower bound technique in terms of the norm mu was recently extended to the multiparty number-on-the-forehead model. Here we show how the gamma2norm can be also extended to lower bound quantum multiparty complexity. Surprisingly, even in this general setting the two lower bounds, on quantum and classical communication, are still very closely related. This implies that separating quantum and classical communication in this setting will require the development of new techniques. The relation between these extensions of mu and gamma2is proved by a multi-dimensional version of Grothendieck's inequality.
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).