General Methods and Properties for Evaluation of Continuum Limits of Discrete Time Quantum Walks in One and Two Dimensions
This event is part of the PhD Final Oral Exams.
Models of quantum walks which admit continuous time and continuous spacetime limits have recently led to quantum simulation schemes for simulating fermions in relativistic and non relativistic regimes. This work continues the study of relationships between discrete time quantum walks (DTQW) and their ostensive continuum counterparts by developing a more general framework than was done in Di Molfetta’s 2019 work to evaluate the continuous time limit of these discrete quantum systems. Under this framework, we prove two constructive theorems concerning which internal discrete transitions (“coins”) admit nontrivial continuum limits in 1D + 1. We additionally prove that the continuous space limit of the continuous time limit of the DTQW can only yield massless states which obey the Dirac equation. We also demonstrate that for general coins the continuous time limit of the DTQW can be identified with the canonical continuous time quantum walk (CTQW) when the coin is allowed to transition through the continuum limit process. Finally, we introduce the Plastic Quantum Walk, or a quantum walk which admits both continuous time and continuous spacetime limits and, as a novel result, we use our 1D + 1 results to obtain necessary and sufficient conditions concerning which DTQWs admit plasticity in 2D + 1, showing the resulting Hamiltonians. We consider coin operators as general 4 parameter unitary matrices, with parameters which are functions of the lattice step size ε. This dependence on ε encapsulates all functions of ε for which a Taylor series expansion in ε is well defined, making our results very general.
Meeting ID: 920 6269 7327