arXiv subjects

K. de Lacy

Publications and source records attributed to K. de Lacy.

3 recordsLinked to original sources

arXiv2026

A Variational Shape Optimisation Approach to Multi-region Relaxed Magnetohydrodynamic Equilibria

Let $\Lambda \subset\mathbb{R}^3$ be a region admitting a partition into $n$ compact, connected subregions $\Lambda_1,\dots,\Lambda_n$, each with smooth boundary. Consider a vector field $B$ on $\Lambda$ where $B|_{\Lambda_i}$ is smooth, divergence free, and tangent to $\partial \Lambda_i$ for all $i$. We show that the multi-region relaxed magnetohydrodynamics (MRxMHD) equilibrium equations are necessary and sufficient conditions for $ B $ and a metric to yield a stationary point of the magnetic energy under appropriate constraints. We constrain the pressure, relative helicity, and magnetic flux of $B$ through all smooth surfaces in $\Lambda_i$ whose boundary lies on $\partial \Lambda_i$. We identify a previously overlooked gauge condition. A definition for relative helicity is introduced, its gauge invariance is proved, and the existence of a gauge where relative helicity reduces to conventional helicity is demonstrated. In the case of a single region an additional condition is introduced that is sufficient to ensure a critical point of the magnetic energy is also a minimiser.

math-ph↗

arXiv2018

Graph comparison via nonlinear quantum search

In this paper we present an efficiently scaling quantum algorithm which finds the size of the maximum common edge subgraph for a pair of arbitrary graphs and thus provides a meaningful measure of graph similarity. The algorithm makes use of a two-part quantum dynamic process: in the first part we obtain information crucial for the comparison of two graphs through linear quantum computation. However, this information is hidden in the quantum system with vanishingly small amplitude that even quantum algorithms such as Grover's search are not fast enough to distill the information efficiently. In order to extract the information we call upon techniques in nonlinear quantum computing to provide the speed-up necessary for an efficient algorithm. The linear quantum circuit requires $\mathcal{O}(n^3 \log^3 (n) \log \log (n))$ elementary quantum gates and the nonlinear evolution under the Gross-Pitaevskii equation has a time scaling of $\mathcal{O}(\frac{1}{g} n^2 \log^3 (n) \log \log (n))$, where $n$ is the number of vertices in each graph and $g$ is the strength of the Gross-Pitaveskii non-linearity. Through this example, we demonstrate the power of nonlinear quantum search techniques to solve a subset of NP-hard problems.

quant-ph↗

arXiv2017

Controlled Quantum Search

Quantum searching for one of $N$ marked items in an unsorted database of $n$ items is solved in $\mathcal{O}(\sqrt{n/N})$ steps using Grover's algorithm. Using nonlinear quantum dynamics with a Gross-Pitaevskii type quadratic nonlinearity, Childs and Young discovered an unstructured quantum search algorithm with a complexity $\mathcal{O}( \min \{ 1/g \, \log (g n), \sqrt{n} \} ) $, which can be used to find a marked item after $o(\log(n))$ repetitions, where $g$ is the nonlinearity strength [PhysRevA.93.022314]. In this work we develop a structured search on a complete graph using a time dependent nonlinearity which obtains one of the $N$ marked items with certainty. The protocol has runtime $\mathcal{O}((N^{\perp} - N) / (G \sqrt{N N^{\perp}}) ) if N^{\perp} > N$, where $N^{\perp}$ denotes the number of unmarked items and $G$ is related to the time dependent nonlinearity. If $N^{\perp} \leq N$, we obtain a runtime $\mathcal{O}( 1 )$. We also extend the analysis to a quantum search on general symmetric graphs and can greatly simplify the resulting equations when the graph diameter is less than 5.

quant-ph↗