This article addresses the formulation for implementing a single source, single-destination shortest path algorithm on a quantum annealing computer. Three distinct approaches are presented. In all the three cases, the shortest path problem is formulated as a quadratic unconstrained binary optimization problem amenable to quantum annealing. The first implementation builds on existing quantum annealing solutions to the traveling salesman problem, and requires the anticipated maximum number of vertices on the solution path |P| to be provided as an input. For a graph with |V | vertices, |E| edges, and no self-loops, it encodes the problem instance using |V ||P| qubits. The second implementation adapts the linear programming formulation of the shortest path problem, and encodes the problem instance using |E| qubits for directed graphs or 2|E| qubits for undirected graphs. The third implementation, designed exclusively for undirected graphs, encodes the problem in |E| + |V | qubits. Scaling factors for penalty terms, complexity of coupling matrix construction, and numerical estimates of the annealing time required to find the shortest path are made explicit in the article. INDEX TERMSQuantum annealing, quantum computing, shortest path problem, simulated annealing. Engineering uantum Transactions on IEEE Krauss and McCollum: SOLVING THE NETWORK SHORTEST PATH PROBLEM ON A QUANTUM ANNEALER FIGURE 1. Example undirected graph with specified edge costs.

This article addresses the question of implementing a maximum flow algorithm on directed graphs in a formulation suitable for a quantum annealing computer. Three distinct approaches are presented. In all three cases, the flow problem is formulated as a quadratic unconstrained binary optimization (QUBO) problem amenable to quantum annealing. The first implementation augments a graph with integral edge capacities into a multigraph with unit-capacity edges and encodes the fundamental objective and constraints of the maximum flow problem using a number of qubits equal to the total capacity of the graph i c i . The second implementation, which encodes flows through edges using a binary representation, reduces the required number of qubits to O(|E| log C max ), where |E| and C max denote the number of edges and maximum edge capacity of the graph, respectively. The third implementation adapts the dual minimum cut formulation and encodes the problem instance using |V | qubits, where |V | is the number of vertices in the graph. Scaling factors for penalty terms and coupling matrix construction times are made explicit in this article.INDEX TERMS Maximum flow problem, minimum cut problem, quantum annealing, quantum computing, simulated annealing.

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Textual scholars have been using phylogenetics to analyze manuscript traditions since the early 1990s (Robinson & O'Hara, 1992). Many standard phylogenetic software packages accept as input the NEXUS file format (Maddison et al., 1997). The teiphy program takes a collation of texts encoded in TEI XML format and can convert it to any of the following formats amenable to phylogenetic analysis: NEXUS (with support for ambiguous states and clock model calibration data blocks for MrBayes or BEAST2), Hennig86, PHYLIP (relaxed for use with RAxML), FASTA (relaxed for use with RAxML), and STEMMA (designed for Stephen C. Carlson's stemmatic software tailored for textual data). For machine learning-based analyses, teiphy can also convert a TEI XML collation to a collation matrix in NumPy, Pandas DataFrame, CSV, TSV, or Excel format.

Bayesian phylogenetic methods offer various models that would be especially suitable in the reconstruction of textual traditions, but text-critical applications of phylogenetics to date have generally not taken advantage of these features. In this paper, we offer a way forward for text-critical phylogenetics. On the side of theory, we highlight multiple Bayesian phylogenetic models and discuss their relevance to textual criticism. More practically, we show how TEI XML collations of textual traditions can be encoded to facilitate robust analyses using these models in BEAST 2, with the teiphy Python package mediating the conversion from TEI XML to BEAST XML. Finally, we give a proof of concept for this approach, showing that the results of BEAST 2 analyses of a sample collation of the Epistle to the Ephesians under different clock models cohere with established findings on the textual tradition of this work.