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Approximate Degree in Classical and Quantum Computing

Mark Bun, Justin Thaler

Published Jan 1, 2022• 0 citations
Publisher

Abstract

The ability (or inability) to represent or approximate Boolean functions by polynomials is a central concept in complexity theory, underlying interactive and probabilistically checkable proof systems, circuit lower bounds, quantum complexity theory, and more. In this book, the authors survey what is known about a particularly natural notion of approximation by polynomials, capturing pointwise approximation over This book covers recent progress on proving approximate degree lower and upper bounds and describes some applications of the new bounds to oracle separations, quantum query and communication complexity, and circuit complexity. The authors explain how several of these advances have been unlocked by a particularly simple and elegant technique, called dual block composition, for constructing solutions to this dual linear program. They also provide concise coverage of even more recent lower bound techniques based on a new complexity measure called spectral sensitivity. Finally, they show how explicit constructions of approximating polynomials have been inspired by quantum query algorithms. This book provides a comprehensive review of the foundational and recent developments of an important topic in both classical and quantum computing. The reader has a considerable body of knowledge condensed in an accessible form to quickly understand the principles and further their own research.

DOI
10.1561/9781638281412

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The ability (or inability) to represent or approximate Boolean functions by polynomials is a central concept in complexity theory, underlying interactive and probabilistically checkable proof systems, circuit lower bounds, quantum complexity theory, and more. In this book, the authors survey what is known about a particularly natural notion of approximation by polynomials, capturing pointwise approximation over This book covers recent progress on proving approximate degree lower and upper bounds and describes some applications of the new bounds to oracle separations, quantum query and communication complexity, and circuit complexity. The authors explain how several of these advances have been unlocked by a particularly simple and elegant technique, called dual block composition, for constructing solutions to this dual linear program. They also provide concise coverage of even more recent lower bound techniques based on a new complexity measure called spectral sensitivity. Finally, they show how explicit constructions of approximating polynomials have been inspired by quantum query algorithms. This book provides a comprehensive review of the foundational and recent developments of an important topic in both classical and quantum computing. The reader has a considerable body of knowledge condensed in an accessible form to quickly understand the principles and further their own research.
The approximate degree of a Boolean function f captures how well f can be approximated pointwise by low-degree polynomials. This monograph surveys what is known about approximate degree and illustrates its applications in theoretical computer science. A particular focus of the survey is a method of proving lower bounds via objects called dual polynomials. These represent a reformulation of approximate degree using linear programming duality. We discuss in detail a recent, powerful technique for constructing dual polynomials, called “dual block composition”.
The approximate degree of a Boolean function f(x1,x2,…,xn)f(x_{1},x_{2},\ldots,x_{n})f(x1​,x2​,…,xn​) is the minimum degree of a real polynomial that approximates fff pointwise within 1/31/31/3. 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 novel, first-principles 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(n34−14(2k−1))O(n^{\frac{3}{4}-\frac{1}{4(2^{k}-1)}})O(n43​−4(2k−1)1​) for the kkk-element distinctness problem; O(n1−1k+1)O(n^{1-\frac{1}{k+1}})O(n1−k+11​) for the kkk-subset sum problem; O(n1−1k+1)O(n^{1-\frac{1}{k+1}})O(n1−k+11​) for any kkk-DNF or kkk-CNF formula; and O(n3/4)O(n^{3/4})O(n3/4) 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 Θ(n)\Theta(n)Θ(n) quantum query complexity of the problem and refutes the conjecture by several experts that surjectivity has approximate degree Ω(n)\Omega(n)Ω(n). In particular, we exhibit the first natural problem with a polynomial gap between approximate degree and quantum query complexity.
Query complexity provides a basic framework to quantify the inherent computational cost of functions in the model where access to the input is limited to oracle queries. In this paper, we provide a detailed discussion of classical and quantum models of inquiry, emphasizing the structural differences and the common mathematical basis. We consider deterministic, randomized and quantum query complexities and highlight key techniques to prove lower bounds such as adversarial techniques, polynomial approximations and hybrid arguments . The main theme of this paper is the polynomial-spectral relation which underlies both the classical and the quantum models. In the classical setting, the decision tree complexity is closely related to the degree of real polynomials representing Boolean functions. In the quantum setting, the relation involves the spectral properties of the operators controlling quantum query algorithms, and gives deep insights into quantum accelerations. We study tight bounds for basic problems such as search and element distinctness, proving separations between classical and quantum query complexity. The present study uses methods from approximation theory, linear algebra and spectrum analysis to build a foundation for a complete understanding of the power and limitations of query-based computation.
We prove lower bounds on complexity measures, such as the approximate degree of a Boolean function and the approximate rank of a Boolean matrix, using quantum arguments. We prove these lower bounds using a quantum query algorithm for the combinatorial group testing problem. We show that for any function f, the approximate degree of computing the OR of n copies of f is Omega(sqrt n) times the approximate degree of f, which is optimal. No such general result was known prior to our work, and even the lower bound for the OR of ANDs function was only resolved in 2013. We then prove an analogous result in communication complexity, showing that the logarithm of the approximate rank (or more precisely, the approximate gamma-2 norm) of F: X x Y to 0,1 grows by a factor of Omega (sqrtn) when we take the OR of n copies of F, which is also essentially optimal. As a corollary, we give a new proof of Razborov's celebrated Omega(sqrtn) lower bound on the quantum communication complexity of the disjointness problem. Finally, we generalize both these results from composition with the OR function to composition with arbitrary symmetric functions, yielding nearly optimal lower bounds in this setting as well.
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.
A longstanding open question in Boolean function complexity asks whether approximate degree composes multiplicatively under block composition. Although a general multiplicative upper bound is known, matching lower bounds have previously been established only for restricted classes of functions. We resolve this question for all total Boolean functions by proving the matching lower bound. Together with Sherstov's upper bound, our result shows that, for every pair of total Boolean functions f:{0,1}n→{0,1}f:\{0,1\}^n\to\{0,1\}f:{0,1}n→{0,1} and g:{0,1}m→{0,1}g:\{0,1\}^m\to\{0,1\}g:{0,1}m→{0,1}, deg~(f∘g)=Θ ⁣(deg~(f) deg~(g)), \widetilde{deg}(f\circ g) = Θ\!\left( \widetilde{deg}(f)\,\widetilde{deg}(g) \right), deg​(f∘g)=Θ(deg​(f)deg​(g)), where deg~\widetilde{deg}deg​ denotes constant-error approximate degree. We also consider the analogous question for exact quantum query complexity and give a counterexample showing that multiplicative composition fails in this setting.