Characterization of Complex Quantum Dynamics with a Scalable NMR Information Processor

We present experimental results on the measurement of fidelity decay under contrasting system dynamics using a nuclear magnetic resonance quantum information processor. The measurements were performed by implementing a scalable circuit in the model of deterministic quantum computation with only one quantum bit. The results show measurable differences between regular and complex behavior and for complex dynamics are faithful to the expected theoretical decay rate. Moreover, we illustrate how the experimental method can be seen as an efficient way for either extracting coarse-grained information about the dynamics of a large system or measuring the decoherence rate from engineered environments.

Figure 1

Ideal quantum circuit for measuring fidelity decay after $n$ steps. The top qubit in a pure state can be considered either as a probe of the bottom system of $K$ qubits which starts in the maximally mixed state or as a toy system being decohered by the maximally mixed environment below. Making the final rotation about the $x$ ($y$) basis gives the real (imaginary) part of the trace in Eq. (2). Taken from Ref. 15.

Figure 2

The molecule transcrotonic acid and its natural Hamiltonian-diagonal terms are chemical shifts and off-diagonal elements coupling terms in Hz. The darkly shaded nuclei are hydrogen and the lightly shaded, carbon. The unlabeled nuclei are oxygen whose natural abundance have zero spin. Hence, we can ignore their coupling with the rest of the molecule. The three hydrogen nuclei of the methyl group are magnetically equivalent. They form a composite spin, from which we can select the spin $\frac{1}{2}$ subspace, giving us an additional qubit.

Figure 3

Comparison of fidelity decay curves for regular (a) versus chaotic (b) evolution under different forms of perturbation. The natural evolution of the molecule provided the regular system and a pseudorandom map was used to model chaotic evolution. The perturbation form was controlled by the presence or absence of conjugate rotations on either side of the coupling interaction between the probe and system. In the case of regular evolution, when the perturbation is a rotation about the $z$ axis (▲ curves), which commutes with a system coordinate, the fidelity decay shows substantial fluctuations away from the average. Changing the perturbation to a rotation about the $x$ axis (■ curves) substantially alters the form of the decay. In the chaotic case, the evolution looks random in the eigenbasis of either perturbation and the decays exhibit the universality of the fidelity decay for complex dynamics. The FGR result (solid curves) and standard deviations (dashed lines) calculated from numerical simulations assuming perfect control are also shown. The inset of (b) shows the experimental average fidelity decay from 20 different random maps. The dotted line shows the corresponding average from the numerical simulations, and we can see that deviation from the exponential decay is also present in the numerical results and is well explained by the finite size saturation level.