Quantum-information processing using strongly dipolar coupled nuclear spins

Dipolar coupled homonuclear spins present challenging, yet useful systems for quantum-information processing. In such systems, the eigenbasis of the system Hamiltonian is the appropriate computational basis and coherent control can be achieved by specially designed strongly modulating pulses. In this paper we describe the first experimental implementation of the quantum algorithm for numerical gradient estimation by nuclear magnetic resonance, using the eigenbasis of a four spin system.

Figure 1

Circuit diagram for QNGE.

Figure 2

(Color online) The SMP for QNGE: (Top) Amplitude vs time, (middle) phase vs time, and (bottom) fidelity vs rf inhomogeneity (RFI, 0.9 to 1.05 of the ideal field) and static field inhomogeneity (SFI, $-5\mathrm{Hz}\text{to}5\mathrm{Hz}$). The 30-segment SMP has an average fidelity $F_{\mathrm{avg}}^{\prime }=0.981$.

Figure 3

(Color online) Molecular structure of 1-chloro-2-iodobenzene (top left), the elements of the diagonalized Hamiltonian (top right), and the energy level diagram with prominent transitions (bottom). The eigenstates are labeled as shown.

Figure 4

${}^1\mathrm{H}$ spectra of CIB obtained by applying a small angle pulse $(\theta =3°)$ to the equilibrium state (bottom), $\mid 0100⟩⟨0100\mid -\mid 0000⟩⟨0000\mid$ POPS (middle), and QNGE (top). The transitions numbers correspond to those in Fig. 3. The pulse sequences for POPS and QNGE are shown in insets (by applying a transition-selective Gaussian $\pi$ pulse of $10\mathrm{ms}$ duration). Pulsed field gradients $\left(G_z\right)$ are used to destroy coherence after POPS and after the SMP. In the QNGE experiment, the residual zero-quantum coherence is dephased by a 32-step average over a random delay $\tau$ (between 0 and $10\mathrm{ms}$).

Figure 5

(Color online) Bar plots showing the diagonal elements of the density matrix in the eigenbasis: Equilibrium (top left), simulated POPS (middle left), experimental POPS (bottom left), theoretical output of QNGE (top right), simulated output (with the SMP shown in Fig. 2, middle right), and experimental output of the quantum algorithm (bottom right).