Optimizing for an arbitrary perfect entangler. I. Functionals

Abstract

Optimal control theory is a powerful tool for improving figures of merit in quantum information tasks. Finding the solution to any optimal control problem via numerical optimization depends crucially on the choice of the optimization functional. Here, we derive a functional that targets the full set of two-qubit perfect entanglers, gates capable of creating a maximally entangled state out of some initial product state. The functional depends on easily computable local invariants and unequivocally determines whether a gate is a perfect entangler. Optimization with our functional is most useful if the two-qubit dynamics allows for the implementation of more than one perfect entangler. We discuss the reachable set of perfect entanglers for a generic Hamiltonian that corresponds to several quantum information platforms of current interest.

Optimizing for an arbitrary perfect entangler. II. Application

“…The two variants of a perfect entanglers functional presented in [44] differ in the representation of the two-qubit nonlocality. In particular, one form utilizes the coefficients c 1 , c 2 , c 3 of σ x ⊗ σ x , σ y ⊗ σ y , σ z ⊗ σ z in the canonical parametrization of two-qubit gates, while the other form uses the so-called local invariants [41].…”

“…In the preceding paper [44], two variants of an optimization functional targeting all two-qubit perfect entanglers have been developed. Here, we apply these two functionals to two quantum information platforms that currently enjoy great popularity due to their promise of control and scalability, namely, nitrogen vacancy (NV) centers in diamond and superconducting Josephson junctions.…”

Controlling open quantum systems: tools, achievements, and limitations

“…While the additional computational effort for evaluating the control update is negligible, storage of allρ (i) j is necessary and may become a limiting factor when scaling up the system size. Such extensions of the optimization algorithm to control targets other than a specific state or unitary have been applied to coherent dynamics [29,[97][98][99]. For open quantum systems, they are still under exploration.…”

“…The corresponding figure of merit is based on the so-called local invariants [29,97]. Similarly, one can formulate a figure of merit for targeting an arbitrary perfect entangler [98,99]. Since these figures of merit are based on the local invariants which in turn are calculated from the unitary evolution, extension to non-unitary dynamics requires to first determine the unitary part of the overall evolution.…”

“…Moreover, weights may be introduced inσ j (t = T ) to speed up convergence by attaching different importance to different basis stateŝ ρ j (t = 0) [39]. Extension of the optimization algorithm to more flexible control targets, such as an arbitrary perfect entangler [98,99] instead of a specific unitary, requires two steps. First, a modified target functional results in a modification of Eq.…”