Emma O'Neil | Searcharxiv

arXiv2025

Visualizing the state space and transformations of higher order quantum logics via toric geometry

We propose some new uses of toric variety structures in the study of quantum computation for small radices. In particular, we observe the concurrence of the equivalence classes of quantum states under quantum measurement and the orbits of the toric geometric structure of the state space. Visualizations of these state spaces and of certain fundamental unitary transformations in binary and ternary quantum logic and a method to develop new transformations based on these visualization techniques are presented. Transformations discussed included minimal universal sets for permutative ternary quantum circuits. In addition, general structures and synthesis methods based on quantum multiplexers are presented. A general framework for the design of optimal ternary quantum transformations and circuits is additionally presented. Finally, a number of open research areas that are extensions of the work presented herein are given.

quant-ph↗

arXiv2025

The Origami flip graph of the $2\times n$ Miura-ori

Given an origami crease pattern $C=(V,E)$, a straight-line planar graph embedded in a region of $\mathbb{R}^2$, we assign each crease to be either a mountain crease (which bends convexly) or a valley crease (which bends concavely), creating a mountain-valley (MV) assignment $\mu:E\to\{-1,1\}$. An MV assignment $\mu$ is locally valid if the faces around each vertex in $C$ can be folded flat under $\mu$. In this paper, we investigate locally valid MV assignments of the Miura-ori, $M_{m,n}$, an $m\times n$ parallelogram tessellation used in numerous engineering applications. The origami flip graph $OFG(C)$ of $C$ is a graph whose vertices are locally valid MV assignments of $C$, and two vertices are adjacent if they differ by a face flip, an operation that swaps the MV-parity of every crease bordering a given face of $C$. We enumerate the number of vertices and edges in $OFG(M_{2,n})$ and prove several facts about the degrees of vertices in $OFG(M_{2,n})$. By finding recurrence relations, we show that the number of vertices of degree $d$ and $2n-a$ (for $0\leq a$) are both described by polynomials of particular degrees. We then prove that the diameter of $OFG(M_{2,n})$ is $\lceil \frac{n^2}{2}\rceil$ and find lower bounds on the diameter of $OFG(M_{m,n})$ using techniques from $3$-coloring reconfiguration graphs.

math.CO↗