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.
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, which we name 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 (Ambainis 2002, 2003; Barnum et al. 2003; Laplante & Magniez 2004; Zhang 2005), culminating in Špalek & Szegedy (2005) 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 (Khrapchenko 1971; Koutsoupias 1993), including a key lemma of Håstad (1998), 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. Our main result is proven via a combinatorial lemma which relates the square of the spectral norm of a matrix to the squares of the spectral norms of its submatrices. The generality of this lemma implies that our methods can also be used to lower-bound the communication complexity of relations, and a related combinatorial quantity, the rectangle partition number. To exhibit the strengths and weaknesses of our methods, we look at the sumPI and maxPI complexity of a few examples, including the recursive majority of three function, a function defined by Ambainis (2003), and the collision problem.
We introduce two new complexity measures for Boolean functions, which we name 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, culminating 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/sup 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 [Khrapchenko, 1971, Koutsoupias, 1993], including a key lemma of [Hastad, 1998], 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/sup 2/(f) remains a lower bound on formula size. Our main result is proven via a combinatorial lemma which relates the square of the spectral norm of a matrix to the squares of the spectral norms of its submatrices. The generality of this lemma gives that our methods can also be used to lower bound the communication complexity of relations, and a related combinatorial quantity, the rectangle partition number. To exhibit the strengths and weaknesses of our methods, we look at the sumPI and maxPI complexity of a few examples, including the recursive majority of three function, a function defined by Ambainis [2003], and the collision problem.
Span programs form a linear-algebraic model of computation, with span program “size” used in proving classical lower bounds. Quantum query complexity is a coherent generalization, for quantum algorithms, of classical decision-tree complexity. It is bounded below by a semi-definite program (SDP) known as the general adversary bound. We connect these classical and quantum models by proving that for any boolean function, the optimal “witness size” of a span program for that function coincides exactly with the general adversary bound. A consequence is an optimal quantum algorithm for evaluating “balanced,” read-once formulas over any finite boolean gate set. For example, the gate set may be taken to be all functions {0, 1} → {0, 1} with k ≤ 1000. A formula is a tree whose nodes are associated to functions from the gate set. The notion of balance is technical, but it includes layered formulas. A previous quantum algorithm optimally evaluates formulas for which an optimal span program is given for each constant-size gate. However, span programs have been found only by hand. The SDP automates this procedure, and its value surprisingly always matches the lower bound. Other implications of the SDP include an exact composition rule for the general adversary bound, and that the general adversary bound upper-bounds the sign degree. The connection can also be seen as half of a universality result for span programs. For any boolean function, there exists a span program with witness size at most the function’s quantum query complexity. Conversely, solutions to the SDP give span programs, and therefore also new quantum algorithms—beyond evaluating formulas. Subsequent work has bounded the query complexity by the witness size, thus implying that the general adversary bound is tight.
The general adversary bound is a semi-definite program that lower-bounds the number of input queries needed by a quantum algorithm to evaluate a function. It is known to be tight up to constant factors for functions (total or partial) with boolean output and binary input alphabet. We show that the general adversary bound is tight for any function whatsoever, with potentially non-boolean input or output alphabets. We also show that quantum query complexity exhibits a remarkable composition property: Q ( f(g(x), . . . , g(x)) ) = O ( Q(f)Q(g) ) for any compatible functions f, g. This was previously known only in the boolean case. Both of these results are obtained by defining a new, but closely related, semi-definite program that we call the witness size. The minimization formulation of the witness size adds constraints compared with the general adversary bound to enable dual solutions to correspond to eigenvalue-zero eigenvectors of certain graphs. While the witness size can be strictly larger than the adversary bound, we show that it can be at most a factor of two larger.
One of the main reasons for query model's prominence in quantum complexity is the presence of concrete lower bounding techniques: polynomial method and adversary method. There have been considerable efforts to not just give lower bounds using these methods but even to compare and relate them. We explore the value of these bounds on quantum query complexity for the class of symmetric functions, arguably one of the most natural and basic set of Boolean functions. We show (using the recently introduced spectral sensitivity) that both these bounds (positive adversary and approximate degree) give the same value for every total symmetric Boolean function. We also look at the quantum query complexity of Gap Majority, a partial symmetric function. It has gained importance recently in regard to understanding the composition of randomized query complexity. We characterize the quantum query complexity of Gap Majority and show a lower bound on noisy randomized query complexity (Ben-David and Blais, FOCS 2020) in terms of quantum query complexity. In addition, we study how large certificate complexity and block sensitivity can be as compared to sensitivity (even up to constant factors) for symmetric functions. We show tight separations, i.e., give upper bound on possible separations and construct functions achieving the same.