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Motivated by the fact that in a space where shortest paths are unique, no two shortest paths meet twice, we study a question posed by Greg Bodwin: Given a geodetic graph G, i.e., an unweighted graph in which the shortest path between any pair of vertices is unique, is there a philogeodetic drawing of G, i.e., a drawing of G in which the curves of any two shortest paths meet at most once? We answer this question in the negative by showing the existence of geodetic graphs that require some pair of shortest paths to cross at least four times. The bound on the number of crossings is tight for the class of graphs we construct. Furthermore, we exhibit geodetic graphs of diameter two that do not admit a philogeodetic drawing. 

An important measure of the development of quantum computing platforms has been the simulation of increasingly complex physical systems. Before fault-tolerant quantum computing, robust error-mitigation strategies were necessary to continue this growth. Here, we validate recently introduced error-mitigation strategies that exploit the expectation that the ideal output of a quantum algorithm would be a pure state. We consider the task of simulating electron systems in the seniority-zero subspace where all electrons are paired with their opposite spin. This affords a computational stepping stone to a fully correlated model. We compare the performance of error mitigations on the basis of doubling quantum resources in time or in space on up to 20 qubits of a superconducting qubit quantum processor. We observe a reduction of error by one to two orders of magnitude below less sophisticated techniques such as postselection. We study how the gain from error mitigation scales with the system size and observe a polynomial suppression of error with increased resources. Extrapolation of our results indicates that substantial hardware improvements will be required for classically intractable variational chemistry simulations.

 

Abstract Indistinguishability of particles is a fundamental principle of quantum mechanics 1 . For all elementary and quasiparticles observed to date—including fermions, bosons and Abelian anyons—this principle guarantees that the braiding of identical particles leaves the system unchanged 2,3 . However, in two spatial dimensions, an intriguing possibility exists: braiding of non-Abelian anyons causes rotations in a space of topologically degenerate wavefunctions 4–8 . Hence, it can change the observables of the system without violating the principle of indistinguishability. Despite the well-developed mathematical description of non-Abelian anyons and numerous theoretical proposals 9–22 , the experimental observation of their exchange statistics has remained elusive for decades. Controllable many-body quantum states generated on quantum processors offer another path for exploring these fundamental phenomena. Whereas efforts on conventional solid-state platforms typically involve Hamiltonian dynamics of quasiparticles, superconducting quantum processors allow for directly manipulating the many-body wavefunction by means of unitary gates. Building on predictions that stabilizer codes can host projective non-Abelian Ising anyons 9,10 , we implement a generalized stabilizer code and unitary protocol 23 to create and braid them. This allows us to experimentally verify the fusion rules of the anyons and braid them to realize their statistics. We then study the prospect of using the anyons for quantum computation and use braiding to create an entangled state of anyons encoding three logical qubits. Our work provides new insights about non-Abelian braiding and, through the future inclusion of error correction to achieve topological protection, could open a path towards fault-tolerant quantum computing. 

We initiate the study of Simultaneous Graph Embedding with Fixed Edges in the beyond planarity framework. In the QSEFE problem, we allow edge crossings, as long as each graph individually is drawn quasiplanar, that is, no three edges pairwise cross. %We call this problem the QSEFE problem. We show that a triple consisting of two planar graphs and a tree admit a QSEFE. This result also implies that a pair consisting of a 1-planar graph and a planar graph admits a QSEFE. We show several other positive results for triples of planar graphs, in which certain structural properties for their common subgraphs are fulfilled. For the case in which simplicity is also required, we give a triple consisting of two quasiplanar graphs and a star that does not admit a QSEFE. Moreover, in contrast to the planar SEFE problem, we show that it is not always possible to obtain a QSEFE for two matchings if the quasiplanar drawing of one matching is fixed.