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

Abstract

A proposal for applying nonadiabatic geometric phases to quantum computing, called double-loop method [S.-L. Zhu and Z. D. Wang, Phys. Rev. A 67, 022319 (2003)], is demonstrated in a liquid state nuclear magnetic resonance quantum computer. Using a spin-echo technique, the original method is modified so that quantum gates are implemented in a standard high-precision nuclear magnetic resonance system for chemical analysis. We show that a dynamical phase is successfully eliminated and a one-qubit quantum gate is realized, although the gate fidelity is not high.

Interplay between geometric and dynamic phases in a single-spin system

“…Attention then turned to nonadiabatic geometric phases, where the requirement for slow evolution is relaxed, but the presence of dynamic phases that accompany the spin manipulations used to drive geometric gates reintroduces a conduit for noise to enter the system. Several schemes have been investigated to remove the dynamic phase, which typically cannot be eliminated in a time-reversal measurement such as spin-echo without also removing the geometric phase [24][25][26].…”

Suppressing the geometric dephasing of Berry phase by using modified dynamical decoupling sequences

“…Here, the phase difference g g g = --+ 1 1 is the physical observable, which contains the contributions from the dynamic phase, Berry phase and the other corrections. To observe the phase difference caused by the Berry phase, ones usually take advantage of the SE technique [2,4,5,7,9] to eliminate the part of the dynamic phase (here, we omit the correction term). For instance, to obtain the phase difference of the Berry phase for the close anti-clockwise loop + C (m= 2), ones first adiabatically drives the Hamiltonian H to vary along the anti-clockwise loop + C (m = 1), next, a π pulse is added to swap the eigenstates of the Hamiltonian H, then, an adiabatic evolution along the clockwise loop -C ( =m 1) is applied, finally, a π pulse is used again to swap the eigenstates of the Hamiltonian H. In this process, by virtue of the swapping of eigenstates, the dynamic phase is offset.…”

“…By using spin echo (SE) methods [2] one can eliminate the dynamic phase included in the total phase [3] to realize the pure geometric quantum gate [4]. Up to now, nuclear magnetic-resonance (NMR) [5,6], ultracold neutrons [7], graphene [8], superconducting circuit-QED [9,10], etc have been used to study the Berry phase. The concept of geometric phase has been also generalized to the Aharonov-Anandan phase [11], non-unitary evolution [12], and mixed state [13].…”

Geometric aspects of composite pulses

“…For example, Zanardi & Rasetti [24] proposed to use the Wilczek-Zee holonomy to implement unitary gates. It is also possible to implement unitary gates by using the Aharonov-Anandan phase [10,25,[36][37][38]. To see this, let {|j a } 1≤a≤n be the eigenvectors of a Hamiltonian H (l(0)) and suppose their dynamical evolution is cyclic, that is, When there is no degeneracy, the spectral decomposition of U (T ) is written as…”