arXiv subjects

Rebekah Herrman

Publications and source records attributed to Rebekah Herrman.

41 records · Page 3Linked to original sources

arXiv2020

Lower Bounds on Circuit Depth of the Quantum Approximate Optimization Algorithm

The quantum approximate optimization algorithm (QAOA) is a method of approximately solving combinatorial optimization problems. While QAOA is developed to solve a broad class of combinatorial optimization problems, it is not clear which classes of problems are best suited for it. One factor in demonstrating quantum advantage is the relationship between a problem instance and the circuit depth required to implement the QAOA method. As errors in NISQ devices increases exponentially with circuit depth, identifying lower bounds on circuit depth can provide insights into when quantum advantage could be feasible. Here, we identify how the structure of problem instances can be used to identify lower bounds for circuit depth for each iteration of QAOA and examine the relationship between problem structure and the circuit depth for a variety of combinatorial optimization problems including MaxCut and MaxIndSet. Specifically, we show how to derive a graph, $G$, that describes a general combinatorial optimization problem and show that the depth of circuit is at least the chromatic index of $G$. By looking at the scaling of circuit depth, we argue that MaxCut, MaxIndSet, and some instances of Vertex Covering and Boolean satisifiability problems are suitable for QAOA approaches while Knapsack and Traveling Sales Person problems are not.

quant-ph↗

arXiv2020

The Eternal Game Chromatic Number of Random Graphs

The eternal graph colouring problem, recently introduced by Klostermeyer and Mendoza, is a version of the graph colouring game, where two players take turns properly colouring a graph. In this note, we study the eternal game chromatic number of random graphs. We show that with high probability $χ_{g}^{\infty}(G_{n,p}) = (\frac{p}{2} + o(1))n$ for odd $n$, and also for even $n$ when $p=\frac{1}{k}$ for some $k \in \mathbb{N}$. The upper bound applies for even $n$ and any other value of $p$ as well, but we conjecture in this case this upper bound is not sharp. Finally, we answer a question posed by Klostermeyer and Mendoza.

math.CO↗

arXiv2019

(t,r) broadcast domination in the infinite grid

The $(t,r)$ broadcast domination number of a graph $G$, $γ_{t,r}(G)$, is a generalization of the domination number of a graph. $γ_{t,r}(G)$ is the minimal number of towers needed, placed on vertices of $G$, each transmitting a signal of strength $t$ which decays linearly, such that every vertex receives a total amount of at least $r$ signal. In this paper we prove a conjecture by Drews, Harris, and Randolph about the minimal density of towers in $\mathbb{Z}^2$ that provide a $(t,3)$ domination broadcast for $t>17$ and explore generalizations. Additionally, we determine the $(t,r)$ broadcast domination number of powers of paths, $P_n^{(k)}$ and powers of cycles, $C_n^{(k)}$.

math.CO↗

arXiv2019

Continuous-Time Quantum Walks on Dynamic Graphs

Continuous-time quantum walks (CTQWs) on static graphs provide efficient methods for search and sampling as well as a model for universal quantum computation. We consider an extension of CTQWs to the case of dynamic graphs, in which an ordered sequence of graphs governs free evolution of the quantum walk. We then consider how perfect state transfer during the quantum walk can be used to design dynamic graphs that implement a universal set of quantum logic gates. We give explicit examples for a complete logical basis, and we validate implementations using numerical simulations for quantum teleportation and addition circuits. Finally, we discuss the potential for realizing CTQWs on dynamic graphs using actively controlled quantum optical waveguides.

quant-ph↗

arXiv2018

Uniform Bounds for Non-negativity of the Diffusion Game

We study a variant of the chip-firing game called the diffusion game. In the diffusion game, we begin with some integer labelling of the vertices of a graph, interpreted as a number of chips on each vertex, and then for each subsequent step every vertex simultaneously fires a chip to each neighbour with fewer chips. In general, this could result in negative vertex labels. Long and Narayanan asked whether there exists an $f(n)$ for each $n$, such that whenever we have a graph on $n$ vertices and an initial allocation with at least $f(n)$ chips on each vertex, then the number of chips on each vertex will remain non-negative. We answer their question in the affirmative, showing further that $f(n)=n-2$ is the best possible bound. We also consider the existence of a similar bound $g(d)$ for each $d$, where $d$ is the maximum degree of the graph.

math.CO↗