On the Range of the Permanent of $(\pm1)$-Matrices
We establish a superpolynomial lower bound on the range of the permanent function on the set of $n\times n$ matrices with $\pm1$ entries.
math.CO↗
arXiv subjects
DeVon Ingram
Publications and source records attributed to DeVon Ingram.
The Acrobatics of BQP
One can fix the randomness used by a randomized algorithm, but there is no analogous notion of fixing the quantumness used by a quantum algorithm. Underscoring this fundamental difference, we show that, in the black-box setting, the behavior of quantum polynomial-time ($\mathsf{BQP}$) can be remarkably decoupled from that of classical complexity classes like $\mathsf{NP}$. Specifically: -There exists an oracle relative to which $\mathsf{NP^{BQP}}\not\subset\mathsf{BQP^{PH}}$, resolving a 2005 problem of Fortnow. As a corollary, there exists an oracle relative to which $\mathsf{P}=\mathsf{NP}$ but $\mathsf{BQP}\neq\mathsf{QCMA}$. -Conversely, there exists an oracle relative to which $\mathsf{BQP^{NP}}\not\subset\mathsf{PH^{BQP}}$. -Relative to a random oracle, $\mathsf{PP}=\mathsf{PostBQP}$ is not contained in the "$\mathsf{QMA}$ hierarchy" $\mathsf{QMA}^{\mathsf{QMA}^{\mathsf{QMA}^{\cdots}}}$. -Relative to a random oracle, $\mathsf{\Sigma}_{k+1}^\mathsf{P}\not\subset\mathsf{BQP}^{\mathsf{\Sigma}_{k}^\mathsf{P}}$ for every $k$. -There exists an oracle relative to which $\mathsf{BQP}=\mathsf{P^{\# P}}$ and yet $\mathsf{PH}$ is infinite. -There exists an oracle relative to which $\mathsf{P}=\mathsf{NP}\neq\mathsf{BQP}=\mathsf{P^{\# P}}$. To achieve these results, we build on the 2018 achievement by Raz and Tal of an oracle relative to which $\mathsf{BQP}\not \subset \mathsf{PH}$, and associated results about the Forrelation problem. We also introduce new tools that might be of independent interest. These include a "quantum-aware" version of the random restriction method, a concentration theorem for the block sensitivity of $\mathsf{AC^0}$ circuits, and a (provable) analogue of the Aaronson-Ambainis Conjecture for sparse oracles.
cs.CC↗
An Upper Bound on the GKS Game via Max Bipartite Matching
The sensitivity conjecture is a longstanding conjecture concerning the relationship between the degree and sensitivity of a Boolean function. In 2015, a communication game was formulated by Justin Gilmer, Michal Koucký, and Michael Saks to attempt to make progress on this conjecture. Andrew Drucker independently formulated this game. Shortly after the creation of the GKS game, Nisan Szegedy obtained a protocol for the game with a cost of $O(n^{.4732})$. We improve Szegedy's result to a cost of $O(n^{.4696})$ by providing a technique to identify whether a set of codewords can be used as a viable strategy in this game.
cs.CC↗