We establish upper bounds on the Shannon entropy $\mathbb H(\widehat f^{\,2})$ of the Fourier distribution of Boolean functions. We prove that $$ \mathbb H(\widehat f^{\,2}) =O\!\left(\log\bigl(1+\|\widehat f\|_{1,1/3}\bigr)\right), $$ where $\|\widehat f\|_{1,1/3}$ denotes the minimum spectral norm of a pointwise $1/3$-approximant to $f$. This implies $\mathbb H(\widehat f^{\,2})=O(\mathsf R^\oplus(f))$, where $\mathsf R^\oplus(f)$ is bounded-error randomized parity-query complexity. We also bound Fourier entropy by the smaller of the two one-sided average unambiguous parity-certificate complexities. More generally, for any exact representation of indicator of $f$ as $\sum_C w_C\mathbf1_C$, with nonnegative weights and affine subspaces $C$, we prove $$ \mathbb H(\widehat f^{\,2}) =O\!\left(\sum_C w_C\,2^{-k_C}k_C\right), \qquad k_C=\operatorname{codim}(C). $$ The quantity on the right is the average conical junta degree of the representation. These results strengthen earlier bounds of (Arunachalam et al. 2021) by extending approximation-based entropy estimates to Shannon entropy and allowing one-sided and overlapping exact representations. The proofs use an entropy bound from a sequence of increasingly accurate approximants of $f$ with \emph{small} spectral norm. Our techniques also find an application in quantum communication complexity, where it improves a lower bound for the bounded-error quantum communication complexity of XOR functions.
Chandrima Kayal, Rajat Mittal, Manaswi Paraashar, Nitin Saurabh
原文摘要
In the world of quantum computing, quantum Fourier transform (QFT) has been one of the most important subroutines for quantum algorithms, as exemplified by Shor's factoring algorithms and phase estimation (which in itself has been used as a primitive in various quantum algorithms). In fact most of the well known quantum algorithms can be divided into two classes, QFT based algorithms and search based algorithms. The usual circuit for QFT (covered in almost every basic quantum algorithms course) needs $O(q^2)$ gates (and $O(q)$ qubits) to perform the exact Fourier transform over $\mathbb{F}_{2^q}$. Cleve and Watrous (FOCS 2000) improved this gate count to $O(q (\log q)^2 (\log \log q))$ using the integer multiplication algorithm in $O(n (\log n) (\log \log n))$ time for two $n$-bit integers (best known at that time). Our first main result shows that this gate count can be improved to $O(q (\log q))$. In fact, if there is a multiplication circuit, to multiply two $n$-bit integers, with $O(n (\log n)^α)$ gates; we get a QFT circuit with $O(q (\log q)^α)$ gates (improving upon $O(q (\log q)^{α+ 1})$) gates from the construction of Cleve and Watrous). Given the recent claim by OpenAI about integer multiplication, this opens up the possibility of having a QFT circuit with $o \left( q (\log q)\right)$ gates. On approximating the QFT over $\ZZ/(2^q \ZZ)$, the result of Ahokas, Cleve and Hales (ERATO 2003) showed that an approximating circuit can be built with $O \left( q (\log \log(q/\varepsilon))^{α+1} \right)$ gates, where $ε$ is the approximating parameter. Our second main result again improves this to $O \left( q (\log \log(q/\varepsilon))^α \right)$ gates.