[Paper Review] Vanishing-Error Approximate Degree and QMA Complexity

Alexander A. Sherstov, Justin Thaler|arXiv (Cornell University)|Sep 16, 2019

Complexity and Algorithms in Graphs20 references4 citations

TL;DR

This paper establishes tight bounds on the vanishing-error approximate degree of three fundamental Boolean functions—element distinctness, surjectivity, and permutation testing—showing they scale as $\Theta(n^{2/3}\log^{1/3}(1/\epsilon))$, $\tilde{\Theta}(n^{3/4}\log^{1/4}(1/\epsilon))$, and $\Theta(n^{1/3}\log^{2/3}(1/\epsilon))$ respectively. It further links this approximate degree to quantum Merlin-Arthur (QMA) query complexity, proving a $\Omega(n^{1/4})$ lower bound for permutation testing, significantly improving upon the prior $\Omega(n^{1/6})$ bound.

ABSTRACT

The $ε$-approximate degree of a function $f\colon X o \{0, 1\}$ is the least degree of a multivariate real polynomial $p$ such that $|p(x)-f(x)| \leq ε$ for all $x \in X$. We determine the $ε$-approximate degree of the element distinctness function, the surjectivity function, and the permutation testing problem, showing they are $Θ(n^{2/3} \log^{1/3}(1/ε))$, $ ildeΘ(n^{3/4} \log^{1/4}(1/ε))$, and $Θ(n^{1/3} \log^{2/3}(1/ε))$, respectively. Previously, these bounds were known only for constant $ε.$ We also derive a connection between vanishing-error approximate degree and quantum Merlin--Arthur (QMA) query complexity. We use this connection to show that the QMA complexity of permutation testing is $Ω(n^{1/4})$. This improves on the previous best lower bound of $Ω(n^{1/6})$ due to Aaronson (Quantum Information & Computation, 2012), and comes somewhat close to matching a known upper bound of $O(n^{1/3})$.

Motivation & Objective

• To determine the vanishing-error approximate degree of element distinctness, surjectivity, and permutation testing functions for arbitrarily small $\epsilon > 0$, extending prior results that were only known for constant $\epsilon$.
• To establish a novel connection between vanishing-error approximate degree and quantum Merlin-Arthur (QMA) query complexity.
• To improve the lower bound on the QMA complexity of permutation testing from $\Omega(n^{1/6})$ to $\Omega(n^{1/4})$, approaching the known upper bound of $O(n^{1/3})$.
• To develop a general technique for lifting bounded-error approximate degree lower bounds to vanishing-error regimes for functions with a certain subfunction structure.

Proposed method

• Introduce a new technique to lift bounded-error approximate degree lower bounds to vanishing-error regimes by analyzing functions that contain $\text{AND}_k \circ f_{\lfloor n/k \rfloor}$ as a subfunction for all $k \leq n$.
• Use symmetrization and polynomial averaging over permutations to construct low-degree approximating polynomials that preserve error bounds under input symmetry.
• Leverage the connection between QMA query protocols and low-degree polynomials, using a variant of the Marriott-Watrous amplification technique to derive polynomial representations with controlled error and degree.
• Apply the approximate degree lower bounds to derive QMA query complexity lower bounds via a comparison of one-sided approximate degree and query cost.
• Use the fact that the acceptance probability of a $T$-query quantum algorithm is a degree-$2T$ polynomial, as established by Beals et al., to relate QMA protocols to low-degree polynomials.
• Prove optimality of bounds using quantum query lower bounds and interpolation arguments for $\epsilon < 1/3^n$.

Experimental results

Research questions

• RQ1 What is the exact vanishing-error approximate degree of the element distinctness function for arbitrarily small $\epsilon$?
• RQ2 Can the vanishing-error approximate degree of surjectivity and permutation testing be tightly characterized beyond the constant-$\epsilon$ regime?
• RQ3 Is there a general method to lift bounded-error approximate degree lower bounds to the vanishing-error regime for structured functions?
• RQ4 How does the vanishing-error approximate degree relate to quantum query complexity, particularly QMA complexity?
• RQ5 Can the gap between the $\Omega(n^{1/4})$ QMA lower bound and the $O(n^{1/3})$ upper bound for permutation testing be closed?

Key findings

• The $\epsilon$-approximate degree of element distinctness is $\Theta(n^{2/3}\log^{1/3}(1/\epsilon))$, matching the best-known constructions and proving optimality.
• The $\epsilon$-approximate degree of surjectivity is $\tilde{\Theta}(n^{3/4}\log^{1/4}(1/\epsilon))$, extending prior results to vanishing error.
• The $\epsilon$-approximate degree of permutation testing is $\Theta(n^{1/3}\log^{2/3}(1/\epsilon))$, which is tight by a quantum query argument.
• The QMA query complexity of permutation testing is $\Omega(n^{1/4})$, improving upon the previous $\Omega(n^{1/6})$ bound.
• For $k$-element distinctness, the $\epsilon$-approximate degree is $\tilde{\Omega}(n^{3/4 - 1/(2k)} \log^{1/4 + 1/(2k)}(1/\epsilon))$, approaching the known upper bound.
• The paper shows that the standard error-reduction method for approximate degree is not known to be tight for any function with sublinear approximate degree, highlighting an open problem in the field.