We show that Nechiporuk's method for proving lower bound for Boolean formulas can be extended to the quantum case. This leads to an n^2 / log^2 n lower bound for quantum formulas computing an explicit function. The only known previous explicit lower bound for quantum formulas (by Yao) states that the majority function does not have a linear-size quantum formula.
We show that Nechiporuk's method for proving lower bound for Boolean formulas can be extended to the quantum case. This leads to an n^2 / log^2 n lower bound for quantum formulas computing an explicit function. The only known previous explicit lower bound for quantum formulas (by Yao) states that the majority function does not have a linear-size quantum formula.
We show that Nechiporuk's method [I. Wegener, The Complexity of Boolean Functions, Teubner-Wiley, New York, 1987] for proving lower bounds for Boolean formulas can be extended to the quantum case. This leads to an Ω(n2/log2n) lower bound for quantum formulas computing an explicit function. The only known previous explicit lower bound for quantum formulas [A. Yao, Proceedings of 34th IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1993, pp. 352--361] states that the majority function does not have a linear-size quantum formula. We also show that quantum formulas can be simulated by Boolean circuits of almost the same size.
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 introduce two new complexity measures for Boolean functions, or more generally for functions of the form f:S->T. We call these measures sumPI and maxPI. The quantity sumPI has been emerging through a line of research on quantum query complexity lower bounds via the so-called quantum adversary method [Amb02, Amb03, BSS03, Zha04, LM04], culminating in [SS04] with the realization that these many different formulations are in fact equivalent. Given that sumPI turns out to be such a robust invariant of a function, we begin to investigate this quantity in its own right and see that it also has applications to classical complexity theory. As a surprising application we show that sumPI^2(f) is a lower bound on the formula size, and even, up to a constant multiplicative factor, the probabilistic formula size of f. We show that several formula size lower bounds in the literature, specifically Khrapchenko and its extensions [Khr71, Kou93], including a key lemma of [Has98], are in fact special cases of our method. The second quantity we introduce, maxPI(f), is always at least as large as sumPI(f), and is derived from sumPI in such a way that maxPI^2(f) remains a lower bound on formula size. While sumPI(f) is always a lower bound on the quantum query complexity of f, this is not the case in general for maxPI(f). A strong advantage of sumPI(f) is that it has both primal and dual characterizations, and thus it is relatively easy to give both upper and lower bounds on the sumPI complexity of functions. To demonstrate this, we look at a few concrete examples, for three functions: recursive majority of three, a function defined by Ambainis, and the collision problem.
We introduce two new complexity measures for Boolean functions, or more generally for functions of the form f:S->T. We call these measures sumPI and maxPI. The quantity sumPI has been emerging through a line of research on quantum query complexity lower bounds via the so-called quantum adversary method [Amb02, Amb03, BSS03, Zha04, LM04], culminating in [SS04] with the realization that these many different formulations are in fact equivalent. Given that sumPI turns out to be such a robust invariant of a function, we begin to investigate this quantity in its own right and see that it also has applications to classical complexity theory. As a surprising application we show that sumPI^2(f) is a lower bound on the formula size, and even, up to a constant multiplicative factor, the probabilistic formula size of f. We show that several formula size lower bounds in the literature, specifically Khrapchenko and its extensions [Khr71, Kou93], including a key lemma of [Has98], are in fact special cases of our method. The second quantity we introduce, maxPI(f), is always at least as large as sumPI(f), and is derived from sumPI in such a way that maxPI^2(f) remains a lower bound on formula size. While sumPI(f) is always a lower bound on the quantum query complexity of f, this is not the case in general for maxPI(f). A strong advantage of sumPI(f) is that it has both primal and dual characterizations, and thus it is relatively easy to give both upper and lower bounds on the sumPI complexity of functions. To demonstrate this, we look at a few concrete examples, for three functions: recursive majority of three, a function defined by Ambainis, and the collision problem.
A de Morgan formula over Boolean variables x1,…,xn is a binary tree whose internal nodes are marked with AND or OR gates and whose leaves are marked with variables or their negation. We define the size of the formula as the number of leaves in it. Proving that some explicit function (in P or NP) requires a large formula is a central open question in computational complexity. While we believe that some explicit functions require exponential formula size, currently the best lower bound for an explicit function is the Ω(n3) lower bound for Andreev's function.