Philip Grech - Academia.edu (original) (raw)

Uploads

Papers by Philip Grech

Research paper thumbnail of Optimal time schedule for adiabatic evolution

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.

Research paper thumbnail of Landau-Zener Tunneling for Dephasing Lindblad Evolutions

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.

Research paper thumbnail of Adiabatic Theorems for Generators of Contracting Evolutions

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.

Research paper thumbnail of The Excitation Spectrum for Weakly Interacting Bosons

Communications in Mathematical Physics, 2011

Research paper thumbnail of Divide the dollar and conquer more: sequential bargaining and risk aversion

International Journal of Game Theory

Research paper thumbnail of Divide the dollar and conquer more: sequential bargaining and risk aversion

International Journal of Game Theory

Research paper thumbnail of Optimal time schedule for adiabatic evolution

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.

Research paper thumbnail of Landau-Zener Tunneling for Dephasing Lindblad Evolutions

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.

Research paper thumbnail of Adiabatic Theorems for Generators of Contracting Evolutions

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.

Research paper thumbnail of The Excitation Spectrum for Weakly Interacting Bosons

Communications in Mathematical Physics, 2011

Research paper thumbnail of Divide the dollar and conquer more: sequential bargaining and risk aversion

International Journal of Game Theory

Research paper thumbnail of Divide the dollar and conquer more: sequential bargaining and risk aversion

International Journal of Game Theory

Log In