Kenneth W. Regan

We study multi-structural games, played on two sets $\mathcal{A}$ and $\mathcal{B}$ of structures. These games generalize Ehrenfeucht-Fra\"{i}ss\'{e} games. Whereas Ehrenfeucht-Fra\"{i}ss\'{e} games capture the quantifier rank of a first-order sentence, multi-structural games capture the number of quantifiers, in the sense that Spoiler wins the $r$-round game if and only if there is a first-order sentence $\phi$ with at most $r$ quantifiers, where every structure in $\mathcal{A}$ satisfies $\phi$ and no structure in $\mathcal{B}$ satisfies $\phi$. We use these games to give a complete characterization of the number of quantifiers required to distinguish linear orders of different sizes, and develop machinery for analyzing structures beyond linear orders.

We study multi-structural games, played on two sets $\mathcal{A}$ and $\mathcal{B}$ of structures. These games generalize Ehrenfeucht-Fra\"{i}ss\'{e} games. Whereas Ehrenfeucht-Fra\"{i}ss\'{e} games capture the quantifier rank of a first-order sentence, multi-structural games capture the number of quantifiers, in the sense that Spoiler wins the $r$-round game if and only if there is a first-order sentence $\phi$ with at most $r$ quantifiers, where every structure in $\mathcal{A}$ satisfies $\phi$ and no structure in $\mathcal{B}$ satisfies $\phi$. We use these games to give a complete characterization of the number of quantifiers required to distinguish linear orders of different sizes, and develop machinery for analyzing structures beyond linear orders.

We study multi-structural games, played on two sets ${\mathcal{A}}$ and ${\mathcal{B}}$ of structures. These games generalize Ehrenfeucht-Fraïssé games. Whereas Ehrenfeucht-Fraïssé games capture the quantifier rank of a first-order sentence, multi-structural games capture the number of quantifiers, in the sense that Spoiler wins the r-round game if and only if there is a first-order sentence ϕ with at most r quantifiers, where every structure in ${\mathcal{A}}$ satisfies ϕ and no structure in ${\mathcal{B}}$ satisfies ϕ. We use these games to give a complete characterization of the number of quantifiers required to distinguish linear orders of different sizes, and develop machinery for analyzing structures beyond linear orders.

We show that a form of strong simulation for $n$-qubit quantum stabilizer circuits $C$ is computable in $O(s + n^\omega)$ time, where $\omega$ is the exponent of matrix multiplication. Solution counting for quadratic forms over $\mathbb{F}_2$ is also placed into $O(n^\omega)$ time. This improves previous $O(n^3)$ bounds. Our methods in fact show an $O(n^2)$-time reduction from matrix rank over $\mathbb{F}_2$ to computing $p = |\langle \; 0^n \;|\; C \;|\; 0^n \;\rangle|^2$ (hence also to solution counting) and a converse reduction that is $O(s + n^2)$ except for matrix multiplications used to decide whether $p > 0$. The current best-known worst-case time for matrix rank is $O(n^{\omega})$ over $\mathbb{F}_2$, indeed over any field, while $\omega$ is currently upper-bounded by $2.3728\dots$ Our methods draw on properties of classical quadratic forms over $\mathbb{Z}_4$. We study possible distributions of Feynman paths in the circuits and prove that the differences in $+1$ vs. $-1$ counts and $+i$ vs. $-i$ counts are always $0$ or a power of $2$. Further properties of quantum graph states and connections to graph theory are discussed.