Foundations and Trends® in Theoretical Computer Science

Approximate Degree in Classical and Quantum Computing

Mark Bun, Justin Thaler

Published Dec 30, 2022• 2 citations

Abstract

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”.

DOI: 10.1561/0400000107

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Approximate Degree in Classical and Quantum Computing

Mark Bun, Justin Thaler

Jan 1, 20220 citations

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Mark Bun, Justin Thaler

The approximate degree of a Boolean function f captures how well f can be approximated pointwise by low-degree polynomials. This article surveys what we know about approximate degree and illustrates some of its applications in theoretical computer science.

ACM SIGACT NewsDec 14, 20206 citations

The polynomial method strikes back: tight quantum query bounds via dual polynomials

Mark Bun, Robin Kothari, Justin Thaler

The approximate degree of a Boolean function f is the least degree of a real polynomial that approximates f pointwise to error at most 1/3. The approximate degree of f is known to be a lower bound on the quantum query complexity of f (Beals et al., FOCS 1998 and J. ACM 2001).

arXiv:1710.09079ACM Symposium on Theory of Computing (STOC)Jun 20, 201839 citations

Dual Lower Bounds for Approximate Degree and Markov-Bernstein Inequalities

Mark Bun, Justin Thaler

arXiv:1302.6191Feb 25, 20131 citation

Lower Bounds for the Approximate Degree of Block-Composed Functions

Justin Thaler

Leibniz international proceedings in informaticsJan 1, 201615 citations

A General Composition Theorem for Approximate Degree

Samruddhi Pednekar, Supartha Podder

cs.CCCross-listedarXiv:2609.30139Sep 25, 20260 citations

Foundations and Trends® in Theoretical Computer Science

Approximate Degree in Classical and Quantum Computing

Mark Bun, Justin Thaler

Published Dec 30, 2022• 2 citations

Publisher

Abstract

DOI: 10.1561/0400000107

Related papers

Approximate Degree in Classical and Quantum Computing

Mark Bun, Justin Thaler

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 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.

Jan 1, 20220 citations

Guest Column

Mark Bun, Justin Thaler

ACM SIGACT NewsDec 14, 20206 citations

The polynomial method strikes back: tight quantum query bounds via dual polynomials

Mark Bun, Robin Kothari, Justin Thaler

arXiv:1710.09079ACM Symposium on Theory of Computing (STOC)Jun 20, 201839 citations

Dual Lower Bounds for Approximate Degree and Markov-Bernstein Inequalities

Mark Bun, Justin Thaler

The $ε$ -approximate degree of a Boolean function $f: \{-1, 1\}^n \to \{-1, 1\}$ is the minimum degree of a real polynomial that approximates $f$ to within $ε$ in the $\ell_\infty$ norm. We prove several lower bounds on this important complexity measure by explicitly constructing solutions to the dual of an appropriate linear program. Our first result resolves the $ε$ -approximate degree of the two-level AND-OR tree for any constant $ε> 0$ . We show that this quantity is $Θ(\sqrt{n})$ , closing a line of incrementally larger lower bounds. The same lower bound was recently obtained independently by Sherstov using related techniques. Our second result gives an explicit dual polynomial that witnesses a tight lower bound for the approximate degree of any symmetric Boolean function, addressing a question of Špalek. Our final contribution is to reprove several Markov-type inequalities from approximation theory by constructing explicit dual solutions to natural linear programs. These inequalities underly the proofs of many of the best-known approximate degree lower bounds, and have important uses throughout theoretical computer science.The $ε$-approximate degree of a Boolean function $f: \{-1, 1\}^n \to \{-1, 1\}$ is the minimum degree of a real polynomial that approximates $f$ to within $ε$ in the $\ell_\infty$ norm. We prove several lower bounds on this important complexity measure by explicitly constructing solutions to the dual of an appropriate linear program. Our first result resolves the $ε$-approximate degree of the two-level AND-OR tree for any constant $ε> 0$. We show that this quantity is $Θ(\sqrt{n})$, closing a line of incrementally larger lower bounds. The same lower bound was recently obtained independently by Sherstov using related techniques. Our second result gives an explicit dual polynomial that witnesses a tight lower bound for the approximate degree of any symmetric Boolean function, addressing a question of Špalek. Our final contribution is to reprove several Markov-type inequalities from approximation theory by constructing explicit dual solutions to natural linear programs. These inequalities underly the proofs of many of the best-known approximate degree lower bounds, and have important uses throughout theoretical computer science.

arXiv:1302.6191Feb 25, 20131 citation

Lower Bounds for the Approximate Degree of Block-Composed Functions

Justin Thaler

We describe a new hardness amplification result for point-wise approximation of Boolean functions by low-degree polynomials. Specifically, for any function f on N bits, define F(x_1,...,x_M) = OMB(f(x_1),...,f(x_M)) to be the function on M*N bits obtained by block-composing f with a function known as ODD-MAX-BIT. We show that, if f requires large degree to approximate to error 2/3 in a certain one-sided sense (captured by a complexity measure known as positive one-sided approximate degree), then F requires large degree to approximate even to error 1-2^{-M}. This generalizes a result of Beigel (Computational Complexity, 1994), who proved an identical result for the special case f=OR. Unlike related prior work, our result implies strong approximate degree lower bounds even for many functions F that have low threshold degree. Our proof is constructive: we exhibit a solution to the dual of an appropriate linear program capturing the approximate degree of any function. We describe several applications, including improved separations between the complexity classes P^{NP} and PP in both the query and communication complexity settings. Our separations improve on work of Beigel (1994) and Buhrman, Vereshchagin, and de Wolf (CCC, 2007).We describe a new hardness amplification result for point-wise approximation of Boolean functions by low-degree polynomials. Specifically, for any function f on N bits, define F(x_1,...,x_M) = OMB(f(x_1),...,f(x_M)) to be the function on M*N bits obtained by block-composing f with a function known as ODD-MAX-BIT. We show that, if f requires large degree to approximate to error 2/3 in a certain one-sided sense (captured by a complexity measure known as positive one-sided approximate degree), then F requires large degree to approximate even to error 1-2^{-M}. This generalizes a result of Beigel (Computational Complexity, 1994), who proved an identical result for the special case f=OR. Unlike related prior work, our result implies strong approximate degree lower bounds even for many functions F that have low threshold degree. Our proof is constructive: we exhibit a solution to the dual of an appropriate linear program capturing the approximate degree of any function. We describe several applications, including improved separations between the complexity classes P^{NP} and PP in both the query and communication complexity settings. Our separations improve on work of Beigel (1994) and Buhrman, Vereshchagin, and de Wolf (CCC, 2007).

Leibniz international proceedings in informaticsJan 1, 201615 citations

A General Composition Theorem for Approximate Degree

Samruddhi Pednekar, Supartha Podder

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 $f:\{0,1\}^n\to\{0,1\}$ and $g:\{0,1\}^m\to\{0,1\}$ , $\widetilde{deg}(f\circ g) = Θ\!\left( \widetilde{deg}(f)\,\widetilde{deg}(g) \right),$ where $\widetilde{deg}$ 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.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 $f:\{0,1\}^n\to\{0,1\}$ and $g:\{0,1\}^m\to\{0,1\}$, \[ \widetilde{deg}(f\circ g) = Θ\!\left( \widetilde{deg}(f)\,\widetilde{deg}(g) \right), \] where $\widetilde{deg}$ 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.

cs.CCCross-listedarXiv:2609.30139Sep 25, 20260 citations

The

ε

-approximate degree of a Boolean function

f: \{-1, 1\}^n \to \{-1, 1\}

is the minimum degree of a real polynomial that approximates

f

to within

ε

in the

\ell_\infty

norm. We prove several lower bounds on this important complexity measure by explicitly constructing solutions to the dual of an appropriate linear program. Our first result resolves the

ε

-approximate degree of the two-level AND-OR tree for any constant

ε> 0

. We show that this quantity is

Θ(\sqrt{n})

, closing a line of incrementally larger lower bounds. The same lower bound was recently obtained independently by Sherstov using related techniques. Our second result gives an explicit dual polynomial that witnesses a tight lower bound for the approximate degree of any symmetric Boolean function, addressing a question of Špalek. Our final contribution is to reprove several Markov-type inequalities from approximation theory by constructing explicit dual solutions to natural linear programs. These inequalities underly the proofs of many of the best-known approximate degree lower bounds, and have important uses throughout theoretical computer science.

f:\{0,1\}^n\to\{0,1\}

and

g:\{0,1\}^m\to\{0,1\}

\widetilde{deg}(f\circ g) = Θ\!\left( \widetilde{deg}(f)\,\widetilde{deg}(g) \right),

where

\widetilde{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.