We show a connection between the deMorgan formula size of a Boolean function and the noise stability of the function. Using this connection, we show that the Fourier spectrum of any balanced Boolean function computed by a deMorgan formula of size s is concentrated on coefficients of degree up to O( √ s). These results have several applications that apply to any function f that can be computed by a deMorgan formula of size s. First, we get that f can be approximated (in L2-norm) with constant error by a polynomial of degree O( √ s). Second, we show an upper bound of O( √ s) on the average sensitivity of f . Our main result stems from a generalization of Khrapchenko’s bound [Khr71], that might be of independent interest, and some Fourier analysis on the Boolean cube. Previous works prove that any function f : {0, 1}n → {0, 1} that can be computed by a deMorgan formula of size s, can be approximated point-wise by a polynomial of degree O(s/2+o(1)) with constant point-wise error. We note that this result can be easily extended to have a polynomial of degree O(t ·s/2+o(1)) that approximates f with point-wise error 2−t, for any t > 0. This was shown in a long line of results in quantum complexity, including [BBC+01] and [FGG08, ACR+07, RS08, Rei09].We show a connection between the deMorgan formula size of a Boolean function and the noise stability of the function. Using this connection, we show that the Fourier spectrum of any balanced Boolean function computed by a deMorgan formula of size s is concentrated on coefficients of degree up to O( √ s). These results have several applications that apply to any function f that can be computed by a deMorgan formula of size s. First, we get that f can be approximated (in L2-norm) with constant error by a polynomial of degree O( √ s). Second, we show an upper bound of O( √ s) on the average sensitivity of f . Our main result stems from a generalization of Khrapchenko’s bound [Khr71], that might be of independent interest, and some Fourier analysis on the Boolean cube. Previous works prove that any function f : {0, 1}n → {0, 1} that can be computed by a deMorgan formula of size s, can be approximated point-wise by a polynomial of degree O(s/2+o(1)) with constant point-wise error. We note that this result can be easily extended to have a polynomial of degree O(t ·s/2+o(1)) that approximates f with point-wise error 2−t, for any t > 0. This was shown in a long line of results in quantum complexity, including [BBC+01] and [FGG08, ACR+07, RS08, Rei09].

Electronic colloquium on computational complexityJan 1, 20124 citations

Polynomial Approximations

Osama M. Raisuddin, Suvranu De

Springer eBooksNov 14, 20250 citations

Three lectures on Fourier analysis and learning theory

H. M. Zhang

Fourier analysis on the discrete hypercubes $\{-1,1\}^n$ has found numerous applications in learning theory. A recent breakthrough involves the use of a classical result from Fourier analysis, the Bohnenblust--Hille inequality, in the context of learning low-degree Boolean functions. In these lecture notes, we explore this line of research and discuss recent progress in discrete quantum systems and classical Fourier analysis.Fourier analysis on the discrete hypercubes $\{-1,1\}^n$ has found numerous applications in learning theory. A recent breakthrough involves the use of a classical result from Fourier analysis, the Bohnenblust--Hille inequality, in the context of learning low-degree Boolean functions. In these lecture notes, we explore this line of research and discuss recent progress in discrete quantum systems and classical Fourier analysis.

arXiv:2409.10886Sep 17, 20240 citations

Constrained minimax approximation for quantum signal processing

Yulong Dong, James B. Larsen, Lin Lin +1 more

Quantum signal processing (QSP) provides a simple and efficient framework for implementing polynomial transformations using quantum circuits. Its classical design stage leads to a constrained minimax approximation problem: find a polynomial of prescribed parity that approximates a target function uniformly on a fitting set while remaining bounded in magnitude by one on the domain $[0,1]$ , which can be viewed as a semi-infinite constraint. Discretization converts the problem into a linear program, but feasibility at a set of finitely many sampled points does not ensure feasibility on the whole domain, especially when an optimal approximant reaches the boundary of the feasible set. We investigate two approaches to address this difficulty. A Remez exchange method combined with active-set constraint enforcement is efficient on many tested instances, but its stability depends on the target and problem geometry. We then introduce nonlinear Fourier retraction, which uses QSP completion and phase synthesis to turn a nearly feasible polynomial into phase factors for a feasible QSP polynomial without increasing the degree. Across representative problems, retraction largely preserves approximation accuracy and remains effective on instances where the Remez heuristic is unstable. The resulting workflow connects classical minimax approximation and semi-infinite optimization with nonlinear Fourier analysis, and is implemented in the qsppack software package.Quantum signal processing (QSP) provides a simple and efficient framework for implementing polynomial transformations using quantum circuits. Its classical design stage leads to a constrained minimax approximation problem: find a polynomial of prescribed parity that approximates a target function uniformly on a fitting set while remaining bounded in magnitude by one on the domain $[0,1]$, which can be viewed as a semi-infinite constraint. Discretization converts the problem into a linear program, but feasibility at a set of finitely many sampled points does not ensure feasibility on the whole domain, especially when an optimal approximant reaches the boundary of the feasible set. We investigate two approaches to address this difficulty. A Remez exchange method combined with active-set constraint enforcement is efficient on many tested instances, but its stability depends on the target and problem geometry. We then introduce nonlinear Fourier retraction, which uses QSP completion and phase synthesis to turn a nearly feasible polynomial into phase factors for a feasible QSP polynomial without increasing the degree. Across representative problems, retraction largely preserves approximation accuracy and remains effective on instances where the Remez heuristic is unstable. The resulting workflow connects classical minimax approximation and semi-infinite optimization with nonlinear Fourier analysis, and is implemented in the qsppack software package.

quant-pharXiv:2608.30937Sep 1, 20260 citations

Information-Theoretic Lower Bounds for Approximating Monomials via Optimal Quantum Tsallis Entropy Estimation

Qisheng Wang

This paper reveals a conceptually new connection from information theory to approximation theory via quantum algorithms for entropy estimation. Specifically, we provide an information-theoretic lower bound $Ω(\sqrt{n})$ on the approximate degree of the monomial $x^n$ , compared to the analytic lower bounds shown in Newman and Rivlin (Aequ. Math. 1976) via Fourier analysis and in Sachdeva and Vishnoi (Found. Trends Theor. Comput. Sci. 2014) via the Markov brothers' inequality. This is done by relating the polynomial approximation of monomials to quantum Tsallis entropy estimation. This further implies a quantum algorithm that estimates to within additive error $\varepsilon$ the Tsallis entropy of integer order $q \geq 2$ of an unknown probability distribution $p$ or an unknown quantum state $ρ$ , using $\widetilde Θ(\frac{1}{\sqrt{q}\varepsilon})$ queries to the quantum oracle that produces a sample from $p$ or prepares a copy of $ρ$ , improving the prior best $O(\frac{1}{\varepsilon})$ via the Shift test due to Ekert, Alves, Oi, Horodecki, Horodecki and Kwek (Phys. Rev. Lett. 2002). To the best of our knowledge, this is the first quantum entropy estimator with optimal query complexity (up to polylogarithmic factors) for all parameters simultaneously.This paper reveals a conceptually new connection from information theory to approximation theory via quantum algorithms for entropy estimation. Specifically, we provide an information-theoretic lower bound $Ω(\sqrt{n})$ on the approximate degree of the monomial $x^n$, compared to the analytic lower bounds shown in Newman and Rivlin (Aequ. Math. 1976) via Fourier analysis and in Sachdeva and Vishnoi (Found. Trends Theor. Comput. Sci. 2014) via the Markov brothers' inequality. This is done by relating the polynomial approximation of monomials to quantum Tsallis entropy estimation. This further implies a quantum algorithm that estimates to within additive error $\varepsilon$ the Tsallis entropy of integer order $q \geq 2$ of an unknown probability distribution $p$ or an unknown quantum state $ρ$, using $\widetilde Θ(\frac{1}{\sqrt{q}\varepsilon})$ queries to the quantum oracle that produces a sample from $p$ or prepares a copy of $ρ$, improving the prior best $O(\frac{1}{\varepsilon})$ via the Shift test due to Ekert, Alves, Oi, Horodecki, Horodecki and Kwek (Phys. Rev. Lett. 2002). To the best of our knowledge, this is the first quantum entropy estimator with optimal query complexity (up to polylogarithmic factors) for all parameters simultaneously.

arXiv:2509.03496Sep 3, 20250 citations

Fourier analysis on the discrete hypercubes

\{-1,1\}^n

has found numerous applications in learning theory. A recent breakthrough involves the use of a classical result from Fourier analysis, the Bohnenblust--Hille inequality, in the context of learning low-degree Boolean functions. In these lecture notes, we explore this line of research and discuss recent progress in discrete quantum systems and classical Fourier analysis.

[0,1]

, which can be viewed as a semi-infinite constraint. Discretization converts the problem into a linear program, but feasibility at a set of finitely many sampled points does not ensure feasibility on the whole domain, especially when an optimal approximant reaches the boundary of the feasible set. We investigate two approaches to address this difficulty. A Remez exchange method combined with active-set constraint enforcement is efficient on many tested instances, but its stability depends on the target and problem geometry. We then introduce nonlinear Fourier retraction, which uses QSP completion and phase synthesis to turn a nearly feasible polynomial into phase factors for a feasible QSP polynomial without increasing the degree. Across representative problems, retraction largely preserves approximation accuracy and remains effective on instances where the Remez heuristic is unstable. The resulting workflow connects classical minimax approximation and semi-infinite optimization with nonlinear Fourier analysis, and is implemented in the qsppack software package.

Ω(\sqrt{n})

on the approximate degree of the monomial

x^n

, compared to the analytic lower bounds shown in Newman and Rivlin (Aequ. Math. 1976) via Fourier analysis and in Sachdeva and Vishnoi (Found. Trends Theor. Comput. Sci. 2014) via the Markov brothers' inequality. This is done by relating the polynomial approximation of monomials to quantum Tsallis entropy estimation. This further implies a quantum algorithm that estimates to within additive error

\varepsilon

the Tsallis entropy of integer order

q \geq 2

of an unknown probability distribution

p

or an unknown quantum state

ρ

, using

\widetilde Θ(\frac{1}{\sqrt{q}\varepsilon})

queries to the quantum oracle that produces a sample from

p

or prepares a copy of

ρ

, improving the prior best

O(\frac{1}{\varepsilon})

via the Shift test due to Ekert, Alves, Oi, Horodecki, Horodecki and Kwek (Phys. Rev. Lett. 2002). To the best of our knowledge, this is the first quantum entropy estimator with optimal query complexity (up to polylogarithmic factors) for all parameters simultaneously.