Generalized product formulas and quantum control

Abstract

We study the quantum evolution under the combined action of the exponentials of two not necessarily commuting operators. We consider the limit in which the two evolutions alternate at infinite frequency. This case appears in a plethora of situations, both in physics (Feynman integral) and mathematics (product formulas). We focus on the case in which the two evolution times are scaled differently in the limit and generalize standard techniques and results. arXiv:1906.04498v1 [quant-ph]

Quantum Zeno Dynamics from General Quantum Operations

“…where H Z = [H Z , q ] with H Z again given by Eq. (2.3) with the spectral projections {P k } of K [19,20,39].…”

One bound to rule them all: from Adiabatic to Zeno

“…• Theorem 7 solves a numerical conjecture of Ref. [41] by another generalization of Trotter's formula.…”

“…Another motivation for studying product formulas was recently highlighted in Ref. [41] in the attempt to establish a bridge between different control techniques such as the strong-coupling and bang-bang controls, which both yield quantum Zeno dynamics. To this aim, a "rescaled" version of the Trotter product formula was proved in Ref.…”

“…To this aim, a "rescaled" version of the Trotter product formula was proved in Ref. [41], with an analytical bound on the error which depends on the scaling parameter, although numerical evidence suggested an error uniform in the scaling parameter. Using the main theorem introduced in this paper, we will prove that the error in the rescaled product formula is indeed uniform.…”

Continuous and Pulsed Quantum Control

“…The second protocol consists in the application of periodic pulses to the system, implemented by an instantaneous unitary transformation U kick applied to the evolving state at time intervals t/n, as shown in Figure 2a. The idea at the basis of this procedure-and of the proof of Theorem 2-can be understood by looking at each step as an effective "rotation" of the Hamiltonian (see Figure 2b), so that the global effect over the whole time interval (0, t) is to average out of the Hamiltonian the off-diagonal part with respect to the eigenprojections of the unitary kick [3,6]. Such result is expressed formally in Theorem 2.…”