Quantum Adiabatic Brachistochrone

We formulate a time-optimal approach to adiabatic quantum computation (AQC). A corresponding natural Riemannian metric is also derived, through which AQC can be understood as the problem of finding a geodesic on the manifold of control parameters. This geometrization of AQC is demonstrated through two examples, where we show that it leads to improved performance of AQC, and sheds light on the roles of entanglement and curvature of the control manifold in algorithmic performance.

Figure 1

Left: Final-time error $\delta (T)$ for the RC and 2D geodesic paths for the Grover search problem, for $n=6$ qubits. Squares (cyan) indicate where the 2D geodesic path outperforms the RC path (${\delta }_{\mathrm{geodesic}}\le {\delta }_{\mathrm{RC}}$); circles (red) correspond to the opposite case. Oscillations are due to $\delta (T)$ itself being highly oscillatory, with an envelope $\propto 1/T$. Middle: $\delta (T)$ for the 2D and 4D geodesic paths, for $n=1$. Cyan squares (red circles) indicate where the 4D (2D) geodesic path results in a smaller error. Right: Component $R_{1212}(x^1,x^2)$ of the curvature tensor for $n=3$. The curves on the curvature surface show the critical line (vanishing gap as $n\to \infty$), the RC interpolation, and the 2D geodesic (QAB). Here $R_{1212}$ is the only independent component of the curvature tensor.

Figure 2

Red diamonds or solid line (blue circles or dashed line) represent the 2D QAB (RC) path. Left: Average curvature component $R_{1212}$ vs $n$. Inset: instantaneous curvature for $n=4$. Results for other values of $n$ are qualitatively similar. Right: Same for negativity $N_{12}$ [21]—between qubits 1 and 2. Inset: instantaneous gap (lower left) and negativity (upper right) for $n=4$. Here, ${\mathrm{Ave}}_{\gamma }[X]\equiv {\int }_{\gamma }X[\mathbf{x}(s)]ds/{\int }_{\gamma }ds$ for given path $\gamma$; $X\in \{N_{12},R_{1212}\}$ and $\gamma \in \{\mathrm{RC},\mathrm{QAB}\}$.