Span Programs and Quantum Space Complexity

While quantum computers hold the promise of significant computational speedups, the limited size of early quantum machines motivates the study of space-bounded quantum computation. We relate the quantum space complexity of computing a function f with one-sided error to the logarithm of its span program size, a classical quantity that is well-studied in attempts to prove formula size lower bounds. In the more natural bounded error model, we show that the amount of space needed for a unitary quantum algorithm to compute f with bounded (two-sided) error is lower bounded by the logarithm of its approximate span program size. Approximate span programs were introduced in the field of quantum algorithms but not studied classically. However, the approximate span program size of a function is a natural generalization of its span program size. While no non-trivial lower bound is known on the span program size (or approximate span program size) of any concrete function, a number of lower bounds are known on the monotone span program size. We show that the approximate monotone span program size of f is a lower bound on the space needed by quantum algorithms of a particular form, called monotone phase estimation algorithms, to compute f. We then give the first non-trivial lower bound on the approximate span program size of an explicit function

Approximate Degree and the Complexity of Depth Three Circuits

Threshold weight, margin complexity, and Majority-of-Threshold circuit size are basic complexity measures of Boolean functions that arise in learning theory, communication complexity, and circuit complexity. Each of these measures might exhibit a chasm at depth three: namely, all polynomial size Boolean circuits of depth two have polynomial complexity under the measure, but there may exist Boolean circuits of depth three that have essentially maximal complexity exp(Theta(n)). However, existing techniques are far from showing this: for all three measures, the best lower bound for depth three circuits is exp(Omega(n^{2/5})). Moreover, prior methods exclusively study block-composed functions. Such methods appear intrinsically unable to prove lower bounds better than exp(Omega(sqrt{n})) even for depth four circuits, and have yet to prove lower bounds better than exp(Omega(sqrt{n})) for circuits of any constant depth. We take a step toward showing that all of these complexity measures indeed exhibit a chasm at depth three. Specifically, for any arbitrarily small constant delta > 0, we exhibit a depth three circuit of polynomial size (in fact, an O(log n)-decision list) of complexity exp(Omega(n^{1/2-delta})) under each of these measures. Our methods go beyond the block-composed functions studied in prior work, and hence may not be subject to the same barriers. Accordingly, we suggest natural candidate functions that may exhibit stronger bounds

A Nearly Optimal Lower Bound on the Approximate Degree of AC0^0

The approximate degree of a Boolean function $f \colon \{-1, 1\}^n \rightarrow \{-1, 1\}$ is the least degree of a real polynomial that approximates $f$ pointwise to error at most $1/3$. We introduce a generic method for increasing the approximate degree of a given function, while preserving its computability by constant-depth circuits. Specifically, we show how to transform any Boolean function $f$ with approximate degree $d$ into a function $F$ on $O(n \cdot \operatorname{polylog}(n))$ variables with approximate degree at least $D = \Omega(n^{1/3} \cdot d^{2/3})$. In particular, if $d= n^{1-\Omega(1)}$, then $D$ is polynomially larger than $d$. Moreover, if $f$ is computed by a polynomial-size Boolean circuit of constant depth, then so is $F$. By recursively applying our transformation, for any constant $\delta > 0$ we exhibit an AC$^0$ function of approximate degree $\Omega(n^{1-\delta})$. This improves over the best previous lower bound of $\Omega(n^{2/3})$ due to Aaronson and Shi (J. ACM 2004), and nearly matches the trivial upper bound of $n$ that holds for any function. Our lower bounds also apply to (quasipolynomial-size) DNFs of polylogarithmic width. We describe several applications of these results. We give: * For any constant $\delta > 0$, an $\Omega(n^{1-\delta})$ lower bound on the quantum communication complexity of a function in AC$^0$. * A Boolean function $f$ with approximate degree at least $C(f)^{2-o(1)}$, where $C(f)$ is the certificate complexity of $f$. This separation is optimal up to the $o(1)$ term in the exponent. * Improved secret sharing schemes with reconstruction procedures in AC$^0$.Comment: 40 pages, 1 figur

Algorithmic Polynomials

The approximate degree of a Boolean function $f(x_{1},x_{2},\ldots,x_{n})$ is the minimum degree of a real polynomial that approximates $f$ pointwise within $1/3$. Upper bounds on approximate degree have a variety of applications in learning theory, differential privacy, and algorithm design in general. Nearly all known upper bounds on approximate degree arise in an existential manner from bounds on quantum query complexity. We develop a first-principles, classical approach to the polynomial approximation of Boolean functions. We use it to give the first constructive upper bounds on the approximate degree of several fundamental problems: - $O\bigl(n^{\frac{3}{4}-\frac{1}{4(2^{k}-1)}}\bigr)$ for the $k$-element distinctness problem; - $O(n^{1-\frac{1}{k+1}})$ for the $k$-subset sum problem; - $O(n^{1-\frac{1}{k+1}})$ for any $k$-DNF or $k$-CNF formula; - $O(n^{3/4})$ for the surjectivity problem. In all cases, we obtain explicit, closed-form approximating polynomials that are unrelated to the quantum arguments from previous work. Our first three results match the bounds from quantum query complexity. Our fourth result improves polynomially on the $\Theta(n)$ quantum query complexity of the problem and refutes the conjecture by several experts that surjectivity has approximate degree $\Omega(n)$. In particular, we exhibit the first natural problem with a polynomial gap between approximate degree and quantum query complexity

A short list of Equalities induces large sign-rank

We exhibit a natural function Fn on n variables that can be computed by just a linear-size decision list of "Equalities," but whose sign-rank is 2Ω (n1/4). This yields the following two new unconditional complexity class separations. 1. Boolean circuit complexity. The function Fn can be computed by linear-size depth-two threshold formulas when the weights of the threshold gates are unrestricted (THR ∘ THR), but any THR ∘ MAJ circuit (the weights of the bottom threshold gates are polynomially bounded in n) computing Fn requires size 2Ω (n1/4). This provides the first separation between the Boolean circuit complexity classes THR ∘ MAJ and THR ∘ THR. While Amano and Maruoka [Proceedings of the 30th International Symposium on Mathematical Foundations of Computer Science, 2005, pp. 107-118] and Hansen and Podolskii [Proceedings of the 25th Annual IEEE Conference on Computational Complexity, 2010, pp. 270-279] emphasized that superpolynomial separations between the two classes remained a basic open problem, our separation is in fact exponential. In contrast, Goldmann, Håstad, and Razborov [Comput. Complexity, 2 (1992), pp. 277-300] showed more than twenty-five years ago that functions efficiently computable by MAJ ∘ THR circuits can also be efficiently computed by MAJ ∘ MAJ circuits. In view of this, it was not even clear if THR ∘ THR was significantly more powerful than THR ∘ MAJ until our work, and there was no candidate function identified for the potential separation. 2. Communication complexity. The function Fn (under the natural partition of the inputs) lies in the communication complexity class PMA. Since Fn has large sign-rank, this implies PMA ⊈ UPP, strongly resolving a recent open problem posed by Göös, Pitassi, and Watson [Comput. Complexity, 27 (2018), pp. 245-304]. In order to prove our main result, we view Fn as an XOR function and develop a technique to lower bound the sign-rank of such functions. This requires novel approximation-theoretic arguments against polynomials of unrestricted degree. Further, our work highlights for the first time the class "decision lists of exact thresholds" as a common frontier for making progress on longstanding open problems in threshold circuits and communication complexity

Near-Optimal Lower Bounds on the Threshold Degree and Sign-Rank of AC^0

The threshold degree of a Boolean function $f\colon\{0,1\}^n\to\{0,1\}$ is the minimum degree of a real polynomial $p$ that represents $f$ in sign: $\mathrm{sgn}\; p(x)=(-1)^{f(x)}.$ A related notion is sign-rank, defined for a Boolean matrix $F=[F_{ij}]$ as the minimum rank of a real matrix $M$ with $\mathrm{sgn}\; M_{ij}=(-1)^{F_{ij}}$. Determining the maximum threshold degree and sign-rank achievable by constant-depth circuits ($\text{AC}^{0}$) is a well-known and extensively studied open problem, with complexity-theoretic and algorithmic applications. We give an essentially optimal solution to this problem. For any $\epsilon>0,$ we construct an $\text{AC}^{0}$ circuit in $n$ variables that has threshold degree $\Omega(n^{1-\epsilon})$ and sign-rank $\exp(\Omega(n^{1-\epsilon})),$ improving on the previous best lower bounds of $\Omega(\sqrt{n})$ and $\exp(\tilde{\Omega}(\sqrt{n}))$, respectively. Our results subsume all previous lower bounds on the threshold degree and sign-rank of $\text{AC}^{0}$ circuits of any given depth, with a strict improvement starting at depth $4$. As a corollary, we also obtain near-optimal bounds on the discrepancy, threshold weight, and threshold density of $\text{AC}^{0}$, strictly subsuming previous work on these quantities. Our work gives some of the strongest lower bounds to date on the communication complexity of $\text{AC}^{0}$.Comment: 99 page