Use of non-adiabatic geometric phase for quantum computing by NMR

Abstract

Geometric phases have stimulated researchers for its potential applications in many areas of science. One of them is fault-tolerant quantum computation. A preliminary requisite of quantum computation is the implementation of controlled logic gates by controlled dynamics of qubits. In controlled dynamics, one qubit undergoes coherent evolution and acquires appropriate phase, depending on the state of other qubits. If the evolution is geometric, then the phase acquired depend only on the geometry of the path executed, and is robust against certain types of errors. This phenomenon leads to an inherently fault-tolerant quantum computation. Here we suggest a technique of using non-adiabatic geometric phase for quantum computation, using selective excitation. In a two-qubit system, we selectively evolve a suitable subsystem where the control qubit is in state |1 , through a closed circuit. By this evolution, the target qubit gains a phase controlled by the state of the control qubit. Using these geometric phase gates we demonstrate implementation of Deutsch-Jozsa algorithm and Grover's search algorithm in a two-qubit system.

Dynamic evolution for liquid-state nuclear spins and Berry phase of mixed state in a magnetic resonance

“…The definition of the geometric phase for the open system has still been a controversial issue up to now. So it is important to investigate the correlation between the physical phenomenon and the geometric phase in order to understand all aspects of the geometric phase [19][20][21][22][23].…”

Geometric quantum gates in liquid-state NMR based on a cancellation of dynamical phases

“…A promising way to achieve this is to employ geometric phases (or, more generally, nonAbelian holonomies) [1,2], because geometric phases are expected to be robust against noise and decoherence under a proper condition [3,4]. A large number of studies for applying their potential robustness to quantum computing have been done, e,g., phase-shift gates with Berry phases [5], nonadiabatic geometric quantum gates [6,7,8,9,10,11,12,13], holonomic quantum computing [14,15,16,17,18,19,20,21], quantum gates with noncyclic geometric phases [22], and so on.…”

“…Although several experimental techniques for the application of geometric phases to quantum computation are available [11,12,13], explicit implementations of ge-ometric phase gates have not been extensively studied so far. Without explicit implementations, the often-cited advantage of the holonomic quantum gates is nothing more than a desk plan.…”

“…In this paper, we combine Zhu and Wang's approach with Jones et al's one, employing an Aharonov-Anandan phase for fast gate operation and a spin echo technique for dynamical phase cancellation, and demonstrate one-qubit gates with a commercial liquid-state nuclear magnetic resonance (NMR) system. In many experiments of nonadiabatic geometric quantum gates [11,12,13], the gate operations in which the dynamical phase is arranged to vanish [4,28] have been adopted. In the present paper, we show that we may have another option for physical realization of geometric quantum gates.…”

Nonadiabatic dynamics and geometric phase of an ultrafast rotating electron spin

“…The adiabatic geometric can be used for measuring rotating frequency in the adiabatic limit [3][4][5]. Our analysis shows that this method still works in the nonadiabatic regime ω ∼ D, where the nonadiabatic geometric phase can be measured with spectroscopic or interference [39][40][41]. Moreover, as the rotating frequency can be determined with high precision by measuring the scattering photon of the nano-diamond [30,31], the angle θ could be measured through the Floquet quasi-energy spectrum [42][43][44].…”