[Paper Review] A note on quantum algorithms and the minimal degree of epsilon-error polynomials for symmetric functions

Ronald de Wolf|ArXiv.org|Feb 13, 2008

Quantum Computing Algorithms and Architecture28 references4 citations

TL;DR

This paper provides a tighter characterization of the minimal degree of ε-error polynomials for symmetric Boolean functions by leveraging quantum algorithms. It proves that degε(f) = Θ(deg₁/₃(f) + √(n log(1/ε))) without hidden logarithmic factors, using a quantum query-based construction that achieves optimal approximation via Grover's algorithm and symmetrization techniques.

ABSTRACT

The degrees of polynomials representing or approximating Boolean functions are a prominent tool in various branches of complexity theory. Sherstov recently characterized the minimal degree deg_{\eps}(f) among all polynomials (over the reals) that approximate a symmetric function f:{0,1}^n-->{0,1} up to worst-case error \eps: deg_{\eps}(f) = ~Θ(deg_{1/3}(f) + \sqrt{n\log(1/\eps)}). In this note we show how a tighter version (without the log-factors hidden in the ~Θ-notation), can be derived quite easily using the close connection between polynomials and quantum algorithms.

Motivation & Objective

• To refine Sherstov's [She08a] Ω̃(deg₁/₃(f) + √(n log(1/ε))) bound on ε-error polynomial degree for symmetric functions by removing hidden logarithmic factors.
• To establish a tight Θ-bound for degε(f) that holds for all ε ∈ [2⁻ⁿ, 1/3] and non-constant symmetric f.
• To demonstrate that quantum query algorithms can be used to construct optimal polynomial approximations for symmetric functions.
• To extend the result beyond symmetric functions to any function constant on inputs of Hamming weight in {t, ..., n−t}, showing degε(f) = O(√(tn) + √(n log(1/ε))).

Proposed method

• Uses the known correspondence between quantum query algorithms and multilinear polynomials: a T-query quantum algorithm computes a polynomial of degree at most 2T.
• Constructs a 4-step quantum algorithm using repeated Grover search to estimate Hamming weight and detect whether |x| < t, |x| ∈ {t, ..., n−t}, or |x| > n−t.
• Employs exact Grover search (O(√(tn)) queries) to find up to t solutions (1s) when |x| ≤ t, and ε/2-error Grover search (O(√(n log(1/ε))) queries) to detect an additional solution.
• Applies the same logic to 0s to determine if |x| > n−t, ensuring full knowledge of x in extreme cases.
• Uses symmetrization to reduce the analysis of n-variate multilinear polynomials to univariate polynomials over Hamming weight.
• Combines Paturi’s tight bound on deg₁/₃(f) = Θ(√(tn)) with the known exact degree of OR function: degε(ORₙ) = Θ(√(n log(1/ε))).

Experimental results

Research questions

• RQ1 Can the hidden logarithmic factors in Sherstov’s Ω̃(deg₁/₃(f) + √(n log(1/ε))) bound be removed to yield a tight Θ-bound?
• RQ2 To what extent can quantum query algorithms be used to construct optimal polynomial approximations for symmetric functions?
• RQ3 Does the degree bound degε(f) = Θ(deg₁/₃(f) + √(n log(1/ε))) extend to non-symmetric functions that are constant on a central Hamming weight interval?
• RQ4 What is the minimal degree of ε-error polynomial approximations for symmetric functions when ε is sub-exponentially small?
• RQ5 How does the structure of the function (e.g., constancy on {t, ..., n−t}) affect the minimal degree of ε-approximating polynomials?

Key findings

• The paper establishes a tight Θ-bound: degε(f) = Θ(deg₁/₃(f) + √(n log(1/ε))) for all non-constant symmetric f and ε ∈ [2⁻ⁿ, 1/3], removing hidden logarithmic factors from prior work.
• The upper bound is achieved via a quantum algorithm using O(√(tn) + √(n log(1/ε))) queries, where t is the smallest value such that f is constant on inputs of Hamming weight in {t, ..., n−t}.
• The algorithm computes f with error probability at most ε by combining exact and ε/2-error Grover searches on both 1s and 0s, ensuring full knowledge of x when |x| < t or |x| > n−t.
• The lower bound follows from embedding the OR function into f when t < n/4, leveraging the known exact degree degε(ORₙ) = Θ(√(n log(1/ε))).
• The result extends to non-symmetric functions that are constant on {t, ..., n−t}, with degε(f) = O(√(tn) + √(n log(1/ε))) for such functions.
• The construction provides an explicit method to build ε-approximating polynomials via quantum algorithms, in contrast to Sherstov’s use of Chebyshev polynomials.