Quantum control with a multi-dimensional Gaussian quantum invariant

Abstract

The framework of quantum invariants is an elegant generalization of adiabatic quantum control to control fields that do not need to change slowly. Due to the unavailability of invariants for systems with more than one spatial dimension, the benefits of this framework have not yet been exploited in multi-dimensional systems. We construct a multi-dimensional Gaussian quantum invariant that permits the design of time-dependent potentials that let the ground state of an initial potential evolve towards the ground state of a final potential. The scope of this framework is demonstrated with the task of shuttling an ion around a corner which is a paradigmatic control problem in achieving scalability of trapped ion quantum information technology.

Quantum invariant-based control of interacting trapped ions

“…It is thus sufficient to resort to existing solutions [23] for single ions that ensure that M and thus also M is real symmetric.…”

“…( 26) to (28) determine M uniquely, since the anti-commutator as linear map is indeed invertible. The resulting matrix M is provably real and symmetric [23].…”

“…There exists a class of invariants that result in Hamiltonians with a given kinetic energy term that can be used to find time-dependent trapping potentials that realise ground state to ground state shuttling protocols for individual trapped ions [23]. Generalizing such approaches to interacting trapped ions, however, brings a crucial further difficulty: while the actual trapping potential of the trapped ions can be chosen at will, the Coulomb interaction between trapped ions cannot be modified in any way.…”

Inverse engineering of fast state transfer among coupled oscillators

“…We shall make use of recent work on two-dimensional invariants [1,22]. First we adapt to the rotation scenario a recently proposed two-dimensional quadratic invariant commuting with initial and final trap Hamiltonians [22] to define the time-dependent protocols for the harmonic trap.…”

“…We shall make use of recent work on two-dimensional invariants [1,22]. First we adapt to the rotation scenario a recently proposed two-dimensional quadratic invariant commuting with initial and final trap Hamiltonians [22] to define the time-dependent protocols for the harmonic trap. Inverse engineering will proceed from the 2D invariant to the evolution of the 2D harmonic potential, which will include both rotations and scaling of the instantaneous eigenfrequencies.…”