Continuous-time quantum search and time-dependent two-level quantum systems

Abstract

It was recently emphasized by Byrnes, Forster, and Tessler [Phys. Rev. Lett. 120, 060501 (2018)] that the continuous-time formulation of Grover's quantum search algorithm can be intuitively understood in terms of Rabi oscillations between the source and the target subspaces.In this work, motivated by this insightful remark and starting from the consideration of a timeindependent generalized quantum search Hamiltonian as originally introduced by Bae and Kwon [Phys. Rev. A 66, 012314 (2002)], we present a detailed investigation concerning the physical connection between quantum search Hamiltonians and exactly solvable time-dependent two-level quantum systems. Specifically, we compute in an exact analytical manner the transition probabilities from a source state to a target state in a number of physical scenarios specified by a spin-1/2 particle immersed in an external time-dependent magnetic field. In particular, we analyze both the periodic oscillatory as well as the monotonic temporal behaviors of such transition probabilities and, moreover, explore their analogy with characteristic features of Grover-like and fixed-point quantum search algorithms, respectively. Finally, we discuss from a physics standpoint the connection between the schedule of a search algorithm, in both adiabatic and nonadiabatic quantum mechanical evolutions, and the control fields in a time-dependent driving Hamiltonian.Recently, Dolzell, Yoder, and Chuang investigated in detail the possibility for adiabatic quantum search algorithms to exhibit the fixed-point property in addition to quadratic quantum speedup [14]. From their theoretical analysis, supported by a number of illustrative examples, they concluded that depending on the choice of the interpolation schedule of the algorithm, it is possible to construct quantum search Hamiltonians with a variety of features: Grover-

Information geometry aspects of minimum entropy production paths from quantum mechanical evolutions

“…Following our previous work presented in Ref. [16] and exploiting the results in Refs. [38,39], we consider here four quantum mechanical scenarios in which the transition probability P |w ⊥ →|w (t) from an initial state |w ⊥ to a final state |w , where w ⊥ |w = δ w ⊥ , w and with σ z |w = + |w and σ z |w ⊥ = − |w ⊥ , can be expressed in an exact analytical manner.…”

“…Therefore, by minimizing the entropy production, we shall find the optimum paths on the manifold of state space parametrized by θ along which one drives the system. To be more specific, in view of the connection between analog quantum search and two-level quantum systems [16,17], we assume that the output of a continuous-time quantum search algorithm where the input is the normalized N = 2 n -dimensional n-qubit source state |s def = |ψ (θ 0 ) can be described as,…”

Information Geometric Perspective on Off-Resonance Effects in Driven Two-Level Quantum Systems

“…In the framework of analog quantum searching, we have recently presented a detailed investigation concerning the physical connection between quantum search Hamiltonians and exactly solvable time-dependent two-level quantum systems in Ref. [7]. In Refs.…”

“…The resonance phenomenon happens to play a key role in problems where one seeks to drive an initial source state towards a final target state. For instance, typical problems of this kind can be found when controlling population transfer in quantum systems [3][4][5] or when searching for a target state [6,7]. In the framework of quantum control theory, it is commonly believed that an off-resonant driving scheme cannot help achieving population transfer with high fidelity between two quantum states.…”

Transition probabilities in generalized quantum search Hamiltonian evolutions

“…In conclusion, our proposed analysis was partially inspired by some of our previous findings reported in Refs. [32,33] and can be improved in a number of ways in the immediate short term. For instance, it would be interesting to address the following question: How large should the nearly optimal fidelity value be chosen, given the finite precision of quantum measurements and the unavoidable presence of decoherence effects in physical implementations of quantum information processing tasks?…”