41 research outputs found
Pattern Matching for Superpositional Graphs
Käesolev magistritöö esitab algoritmi mustrite leidmiseks superpositsioonigraafides. Algoritm leiab mustrite arvu ajaga O(kn), kus n on teksti pikkus ja k mustri pikkus. Samasugune ajaline keerukus kehtib ka mustrite leidmisel lahutatavates permutatsioonides.This master's thesis presents a pattern matching algorithm for superpositional graphs which counts a number of matches with time complexity O(kn), where n is a length of a text and k is a length of a pattern. Consequently, the same time complexity is achieved for the case, when both text and pattern are separable permutations
Thermalization, Error-Correction, and Memory Lifetime for Ising Anyon Systems
We consider two-dimensional lattice models that support Ising anyonic
excitations and are coupled to a thermal bath. We propose a phenomenological
model for the resulting short-time dynamics that includes pair-creation,
hopping, braiding, and fusion of anyons. By explicitly constructing topological
quantum error-correcting codes for this class of system, we use our
thermalization model to estimate the lifetime of the quantum information stored
in the encoded spaces. To decode and correct errors in these codes, we adapt
several existing topological decoders to the non-Abelian setting. We perform
large-scale numerical simulations of these two-dimensional Ising anyon systems
and find that the thresholds of these models range between 13% to 25%. To our
knowledge, these are the first numerical threshold estimates for quantum codes
without explicit additive structure.Comment: 34 pages, 9 figures; v2 matches the journal version and corrects a
misstatement about the detailed balance condition of our Metropolis
simulations. All conclusions from v1 are unaffected by this correctio
Edge-coloured graphs with only monochromatic perfect matchings and their connection to quantum physics
Krenn, Gu and Zeilinger initiated the study of PMValid edge-colourings
because of its connection to a problem from quantum physics. A graph is defined
to have a PMValid -edge-colouring if it admits a -edge-colouring (i.e. an
edge colouring with -colours) with the property that all perfect matchings
are monochromatic and each of the colour classes contain at least one
perfect matching.
The matching index of a graph , is defined as the maximum value
of for which admits a PMValid -edge-colouring. It is easy to see
that if and only if has a perfect matching (due to the
trivial -edge-colouring which is PMValid). Bogdanov observed that for all
graphs non-isomorphic to , and . However, the
characterisation of graphs for which and is not known. In
this work, we answer this question. Using this characterisation, we also give a
fast algorithm to compute of a graph . In view of our work, the
structure of PMValid -edge-colourable graphs is now fully understood for all
. Our characterisation, also has an implication to the aforementioned
quantum physics problem. In particular, it settles a conjecture of Krenn and Gu
for a sub-class of graphs.Comment: 18 pages and 7 figure