Time-optimal generation of cluster states

The definition of a cluster state naturally suggests an implementation scheme: find a physical system with an Ising coupling topology identical to that of the target state and evolve freely for a time of $\frac{1}{2J}$. Using the tools of optimal control theory, we address the question of whether or not this implementation is time-optimal. We present some examples where it is not and provide an explanation in terms of geodesics on the Bloch sphere.

Figure 1

$K_3$ coupling graph.

Figure 2

(a) Maximum achievable fidelity as a function of the transfer time. (b) A sample solution at the minimal time for initial controls $u_x^j\left(t\right)=u_y^j\left(t\right)=0$. The $y$ controls are omitted as they remain zero, and the $x$ controls are identical on each qubit due to the permutation symmetry of the drift Hamiltonian and the initial and target states.

Figure 3

(Color online) Transfer of Eq. (5) on the Bloch sphere. The $u=0$ solution (blue upper line) transfers $a$ to $c$ in a time of $\frac{1}{2J}$. A time-optimal solution (red lower line) transfers $a$ to $b$ in $\frac{2}{3\sqrt{3}J}$, followed by a hard pulse (dashed red lower line) from $b$ to $c$ in negligible time.

Figure 4

Geometric constructions used. All points shown here lie in the $I_x\text{−}{I}_z$ plane except for $b$, which lies above it on the upper surface of the sphere.

Figure 5

(Color online) (a) An arbitrary path can be decomposed into rotation about $H_c^{\prime }$ and rotation about the orthogonal axis $H^{\perp }$ via the Trotter decomposition. (b) Rotating our viewpoint by $90°$, we see that the optimal trajectory from $a_1\to a_2$ minimizes the angle rotated through in each segment (red) when compared to a generic trajectory (blue), i.e., ${\theta }_1<{\theta }_2$.

Figure 6

Some of the graphs considered: (a) $C_4$, (b) $K_4$, and (c) $G_{2,3}$.

Figure 7

Numerically optimized sequences for (a) $K_5$ and (b) $K_7$.