Dynamical invariants for quantum control of four-level systems

Abstract

We present a Lie-algebraic classification and detailed construction of the dynamical invariants, also known as Lewis-Riesenfeld invariants, of the four-level systems including two-qubit systems which are most relevant and sufficiently general for quantum control and computation. These invariants not only solve the time-dependent Schrödinger equation of four-level systems exactly but also enable the control, and hence quantum computation based on which, of four-level systems fast and beyond adiabatic regimes.

Nonadiabatic quantum state engineering driven by fast quench dynamics

“…As also noticed in Ref. [22], this algebraic independent structure of the generators of I(τ ) allows for the decoupling of Eq. (6) into two independent subsets of differential equations for the coefficients {g i (τ )}, readinġ…”

“…Indeed, for sys-tems under decoherence, there is a competition between the time required for adiabaticity and the decoherence time scales [18], which typically limits the success of the adiabatic approach. Recently, as an alternative direction, several nonadiabatic protocols related to QSE have been proposed [19][20][21][22]. Berry [19] introduced an approach named transitionless quantum driving, in which Hamiltonians are designed in order to follow a path similar to the adiabatic one in an arbitrary time.…”

“…The dynamic invariant approach was also employed to gather a shortcut to adiabatic quantum algorithms, such as in the Grover and Deutsch-Jozsa problems [21]. In [22], a method based on Lie algebras is presented to build dynamic invariants for four-level-system Hamiltonians. Transitionless quantum driving was also studied for the scenario of open-system dynamics in Refs.…”

Non-Adiabatic Universal Holonomic Quantum Gates Based on Abelian Holonomies

“…Two-qubit Hamiltonians are elements of su (4) in general (the identity term can always be dropped as it only contributes a global phase, which can be restored at any time) which makes an analytic approach difficult. However, the problem can be systematically reduced when the Hamiltonian is restricted to a subalgebra of su(4) [29]. In this particular case, the generating set of H ′ spans the subalgebra su (2)⊕su (2)⊕u (1), which is…”

“…Using the corresponding dynamical invariants [29] Ω cos ωtG ′± x + Ω sin ωtG ′± y + (∆ ± − ω)G ′± z , (26) one can repeat the straightforward but tedious analysis (obtaining eigenvectors of the dynamical invariant, enforcing that dynamical phases vanish and that the gate is local unitarily equivalent to a non-trivial gate) and obtain a non-trivial two-qubit GQG.…”