[Paper Review] An Optimal Separation of Randomized and Quantum Query Complexity

Alexander A. Sherstov, Andrey A. Storozhenko+1 more|arXiv (Cornell University)|Aug 24, 2020

Quantum Computing Algorithms and Architecture28 references5 citations

TL;DR

This paper establishes an optimal separation between randomized and quantum query complexity by proving tight bounds on the Fourier coefficients of decision trees, which enables the construction of a partial Boolean function with bounded-error quantum query complexity $k$ and randomized query complexity $\tilde{\Omega}(n^{1-1/(2k)})$. The result resolves a long-standing open problem by achieving the best possible separation, matching the theoretical limits established by Aaronson and Ambainis and Bravyi et al.

ABSTRACT

We prove that for every decision tree, the absolute values of the Fourier coefficients of a given order $\ell\geq1$ sum to at most $c^{\ell}\sqrt{\binom{d}{\ell}(1+\log n)^{\ell-1}},$ where $n$ is the number of variables, $d$ is the tree depth, and $c>0$ is an absolute constant. This bound is essentially tight and settles a conjecture due to Tal (arxiv 2019; FOCS 2020). The bounds prior to our work degraded rapidly with $\ell,$ becoming trivial already at $\ell=\sqrt{d}.$ As an application, we obtain, for every integer $k\geq1,$ a partial Boolean function on $n$ bits that has bounded-error quantum query complexity at most $k$ and randomized query complexity $ ildeΩ(n^{1-\frac{1}{2k}}).$ This separation of bounded-error quantum versus randomized query complexity is best possible, by the results of Aaronson and Ambainis (STOC 2015) and Bravyi, Gosset, Grier, and Schaeffer (2021). Prior to our work, the best known separation was polynomially weaker: $O(1)$ versus $Ω(n^{2/3-ε})$ for any $ε>0$ (Tal, FOCS 2020). As another application, we obtain an essentially optimal separation of $O(\log n)$ versus $Ω(n^{1-ε})$ for bounded-error quantum versus randomized communication complexity, for any $ε>0.$ The best previous separation was polynomially weaker: $O(\log n)$ versus $Ω(n^{2/3-ε})$ (implicit in Tal, FOCS 2020).

Motivation & Objective

• To close the gap in understanding the maximum possible separation between bounded-error quantum and randomized query complexity for Boolean functions.
• To resolve a conjecture by Tal (2019, FOCS 2020) on the Fourier weight of decision trees for higher-order coefficients.
• To construct a partial Boolean function that achieves the optimal trade-off between quantum and randomized query complexity.
• To extend the result to communication complexity, achieving an optimal separation between quantum and randomized communication complexity.

Proposed method

• Prove a tight upper bound on the $\ell$-th order Fourier coefficients of decision trees: $\sum |\hat{f}(S)| \leq c^\ell \sqrt{\binom{d}{\ell} (1+\log n)^{\ell-1}}$ for $\ell \geq 1$, where $d$ is the tree depth and $c$ is an absolute constant.
• Use this Fourier bound to analyze the forrelation problem and its generalizations, particularly the $k$-fold forrelation function $f_{n,k,U}$.
• Apply error reduction techniques and concentration arguments to relate the acceptance probabilities under uniform and structured distributions to the Fourier bias of decision trees.
• Derive a lower bound on the randomized query complexity by comparing the expected outputs of decision trees under different input distributions, leveraging the Fourier coefficient bounds.
• Use the resulting complexity gap to construct a partial Boolean function with quantum complexity $k$ and randomized complexity $\tilde{\Omega}(n^{1-1/(2k)})$.
• Extend the result to communication complexity by applying the same function to a two-party setting, achieving $O(\log n)$ quantum vs. $\Omega(n^{1-\varepsilon})$ randomized communication complexity.

Experimental results

Research questions

• RQ1 What is the maximum possible separation between bounded-error quantum and randomized query complexity for a Boolean function on $n$ bits?
• RQ2 Can the Fourier coefficients of decision trees be bounded in a way that captures their influence on query complexity for higher-order interactions?
• RQ3 Is the $k$-fold forrelation problem the optimal candidate for achieving the largest known quantum speedup in the query model?
• RQ4 Can the separation between quantum and randomized query complexity be extended to communication complexity with optimal parameters?

Key findings

• The paper establishes a tight bound on the $\ell$-th order Fourier coefficients of decision trees: $\sum |\hat{f}(S)| \leq c^\ell \sqrt{\binom{d}{\ell} (1+\log n)^{\ell-1}}$, which resolves a conjecture by Tal (2019, FOCS 2020).
• It constructs a partial Boolean function with bounded-error quantum query complexity $k$ and randomized query complexity $\tilde{\Omega}(n^{1-1/(2k)})$, achieving the optimal separation up to logarithmic factors.
• The separation is optimal in the sense that no larger gap is possible, as confirmed by prior results of Aaronson and Ambainis (STOC 2015) and Bravyi et al. (2021).
• An essentially optimal separation of $O(\log n)$ vs. $\Omega(n^{1-\varepsilon})$ is achieved for bounded-error quantum versus randomized communication complexity, for any $\varepsilon > 0$.
• The result improves upon the previous best separation, which was polynomially weaker: $O(1)$ vs. $\Omega(n^{2/3-\varepsilon})$ for any $\varepsilon > 0$, as shown by Tal (FOCS 2020).
• The analysis relies on a novel application of Fourier analysis to decision trees, with precise control over the influence of input distributions and error propagation in randomized algorithms.