Philip Grech - Academia.edu (original) (raw)
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Papers by Philip Grech
Physical Review A, 2010
We show that, given a path of Hamiltonians and a fixed time to complete it, there are many parame... more We show that, given a path of Hamiltonians and a fixed time to complete it, there are many parametrizations, i.e. timetables for running along the path, such that the ground state of the initial Hamiltonian is mapped exactly on the final ground state. In contrast, we show that if dephasing is added to the dynamics then there is a unique parametrization which maximizes the fidelity of the final state with the target ground state. The optimizing parametrization solves a variational problem of Lagrangian type and has constant tunneling rate along the path irrespective of the gap. Application to quantum search algorithms recovers the Grover result for appropriate scaling of the dephasing with the size of the data base. Lindbladians that describe wide open systems require special care since they may mask hidden resources that enable beating the Grover bound.
Communications in Mathematical Physics, 2011
We consider a family of time dependent dephasing Lindblad generators which model the monitoring o... more We consider a family of time dependent dephasing Lindblad generators which model the monitoring of the instantaneous Hamiltonian of a system by a Markovian bath. In this family the time dependence of the dephasing operators is (essentially) governed by the instantaneous Hamiltonian. The evolution in the adiabatic limit admits a geometric interpretation and can be solved by quadrature. As an application we derive an analog of the Landau-Zener tunneling formula for this family.
Communications in Mathematical Physics, 2012
We develop an adiabatic theory for generators of contracting evolution on Banach spaces. This pro... more We develop an adiabatic theory for generators of contracting evolution on Banach spaces. This provides a uniform framework for a host of adiabatic theorems ranging from unitary quantum evolutions through quantum evolutions of open systems generated by Lindbladians all the way to classically driven stochastic systems. In all these cases the adiabatic evolution approximates, to lowest order, the natural notion of parallel transport in the manifold of instantaneous stationary states. The dynamics in the manifold of instantaneous stationary states and transversal to it have distinct characteristics: The former is irreversible and the latter is transient in a sense that we explain. Both the gapped and gapless cases are considered. Some applications are discussed.
Communications in Mathematical Physics, 2011
International Journal of Game Theory
International Journal of Game Theory
Physical Review A, 2010
We show that, given a path of Hamiltonians and a fixed time to complete it, there are many parame... more We show that, given a path of Hamiltonians and a fixed time to complete it, there are many parametrizations, i.e. timetables for running along the path, such that the ground state of the initial Hamiltonian is mapped exactly on the final ground state. In contrast, we show that if dephasing is added to the dynamics then there is a unique parametrization which maximizes the fidelity of the final state with the target ground state. The optimizing parametrization solves a variational problem of Lagrangian type and has constant tunneling rate along the path irrespective of the gap. Application to quantum search algorithms recovers the Grover result for appropriate scaling of the dephasing with the size of the data base. Lindbladians that describe wide open systems require special care since they may mask hidden resources that enable beating the Grover bound.
Communications in Mathematical Physics, 2011
We consider a family of time dependent dephasing Lindblad generators which model the monitoring o... more We consider a family of time dependent dephasing Lindblad generators which model the monitoring of the instantaneous Hamiltonian of a system by a Markovian bath. In this family the time dependence of the dephasing operators is (essentially) governed by the instantaneous Hamiltonian. The evolution in the adiabatic limit admits a geometric interpretation and can be solved by quadrature. As an application we derive an analog of the Landau-Zener tunneling formula for this family.
Communications in Mathematical Physics, 2012
We develop an adiabatic theory for generators of contracting evolution on Banach spaces. This pro... more We develop an adiabatic theory for generators of contracting evolution on Banach spaces. This provides a uniform framework for a host of adiabatic theorems ranging from unitary quantum evolutions through quantum evolutions of open systems generated by Lindbladians all the way to classically driven stochastic systems. In all these cases the adiabatic evolution approximates, to lowest order, the natural notion of parallel transport in the manifold of instantaneous stationary states. The dynamics in the manifold of instantaneous stationary states and transversal to it have distinct characteristics: The former is irreversible and the latter is transient in a sense that we explain. Both the gapped and gapless cases are considered. Some applications are discussed.
Communications in Mathematical Physics, 2011
International Journal of Game Theory
International Journal of Game Theory