Effective Hamiltonian theory of the geometric evolution of quantum systems

Abstract

In this work we present an effective Hamiltonian description of the quantum dynamics of a generalized Lambda system undergoing adiabatic evolution. We assume the system to be initialized in the dark subspace and show that its holonomic evolution can be viewed as a conventional Hamiltonian dynamics in an appropriately chosen extended Hilbert space. In contrast to the existing approaches, our method does not require the calculation of the non-Abelian Berry connection and can be applied without any parametrization of the dark subspace, which becomes a challenging problem with increasing system size.

Algorithms for Quantum Control without Discontinuities; Application to the Simultaneous Control of two Qubits

“…Consider a three level quantum system where the controls couple level |1 to level |2 and level |1 to level |3 but not level |2 and |3 directly. Assuming that |1 is the highest energy level, the energy level diagram takes the so-called Λ configuration (see, e.g., [15]). The Schrödinger operator equation ( 2) is such that…”

“…The controls that drive system (15) to (±U f , ±V f ) drive system (2), (13) to the state U f ⊗ V f . Therefore we shall focus on the steering problem for system (15) which consists of steering one spin to U f and the other to V f , simultaneously. Since |γ| = 1, the dynamical Lie algebra associated with ( 15) is spanned by the pairs (σ 1 , σ 2 ) with σ 1 and σ 2 in su(2).…”

“…Such a Lie algebra can be written as K ⊕ P with K spanned by elements of the form (σ, σ) with σ ∈ su(2) and P spanned by elements of the form (σ, γσ) with σ ∈ su(2). At every p ∈ G = SU (2) × SU (2), the vector fields in (15) belong to R p * P.…”

“…Furthermore, we want u x (0) = u y (0) = u z (0) = 0 and u x (T 1 ) = u y (T 1 ) = u z (T 1 ) = 0 to avoid discontinuities at the initial time t = 0 and at the time of concatenation with the second portion of the control. We propose to prescribe a trajectory for U = U (t) in (15) and, from that trajectory, to derive the control to be used in the equation for V = V (t) in (15). We choose a smooth function δ := δ(t) such that δ(0) = 0 and δ(T 1 ) = δ 0 = 0, and δ(0) = δ(T 1 ) = 0.…”

Implementation of Geometric Quantum Gates on Microwave‐Driven Semiconductor Charge Qubits

“…[ 44 ] Since the non‐Abelian GQGs are using the state space more than two‐level structure, it is usually implemented in the three‐level system. [ 45–61 ] On the other hand, both the Abelian and non‐Abelian GQGs can be implemented by using the adiabatic [ 31,32,62–66 ] or non‐adiabatic [ 67,68 ] evolution of quantum states. The non‐adiabatic GQGs have attracted much more interest compared to the adiabatic approach, as the latter requires an overly long gating time that is impractical in experiments.…”