Scaling the robustness of the solutions for quantum controllable problems

The major task in quantum control theory is to find an external field that transforms the system from one state to another or executes a predetermined unitary transformation. We investigate the difficulty of computing the control field as the size of the Hilbert space is increased. In the models studied the controls form a small closed subalgebra of operators. Complete controllability is obtained by the commutators of the controls with the stationary Hamiltonian. We investigate the scaling of the computation effort required to converge a solution for the quantum control task with respect to the size of the Hilbert space. The models studied include the double-well Bose Hubbard model with the SU(2) control subalgebra and the Morse oscillator with the Heisenberg-Weil algebra. We find that for initial and target states that are classified as generalized coherent states (GCSs) of the control subalgebra the control field is easily found independent of the size of the Hilbert space. For such problems, a control field generated for a small system can serve as a pilot for finding the field for larger systems. Attempting to employ pilot fields that generate superpositions of GCSs or cat states failed. No relation was found between control solutions of different Hilbert space sizes. In addition the task of finding such a field scales unfavorably with Hilbert space sizes. We demonstrate the use of symmetry to obtain quantum transitions between states without phase information. Implications to quantum computing are discussed.

Figure 1

Fitness vs the Hilbert space size obtained by OCT: deviation from unity fitness vs the number of iterations, for different $j$ values. Initial GCS state: $\langle {\stackrel{\wedge }{\mathbf{J}}}_z\rangle =j$, all the particles in one of the wells. The target is the ground state of the system. Initial guess for all the optimization is taken as $\mu (t)=10\sin (25t\pi /T)$.

Figure 2

Fitness at the 200th iteration vs $j$. The control task is identical to that in Fig. 1.

Figure 3

Generalized purity vs time for $j=5$ (solid blue) and $j=320$ (dashed red), for the control task of Fig. 1, after 200 iterations. The generalized purity is defined with respect to projection on the SU(2) subalgebra. The period of the motion is $2\pi /\Delta =0.42.$

Figure 4

Temporal fitness values for different $j$ values; the pilot field was the optimal solution of $j=5$. Different curves correspond to different $j$ values.

Figure 5

Generalized purity vs time for different $j$ values, for a converged field starting from the pilot field generated from the optimal solution of $j=5$.

Figure 6

Expectation values $\langle {\stackrel{\wedge }{\mathbf{J}}}_x\rangle$, $\langle {\stackrel{\wedge }{\mathbf{J}}}_y\rangle$, and $\langle {\stackrel{\wedge }{\mathbf{J}}}_z\rangle$ vs time for $j=5$ and $j=160$, in solid and dashed lines, respectively. The close resemblance indicates that both states are GCSs.

Figure 7

Fitness vs time for local control. Optimization for $j=5,$ solid curved line. The fitness for $j=40$ and $160$, using the pilot $j=5$ field, are plotted in grey (red) and black lines, respectively.

Figure 8

Control field from local control, in logarithm scale. The OCT corrected field is very similar (not plotted).

Figure 9

Fitness vs time for local control with a field corrected by a single OCT iteration.

Figure 10

Fitness vs $j$ for a ground-state target state. Left: Initial GCS: all the particles in one of the wells $|\psi (0)\rangle =|j\rangle$. Right: Initial cat state: $|\psi (0)\rangle =\frac{1}{\sqrt{2}}(|j\rangle +|-j\rangle )$. The dashed line in the right panel corresponds to the inverse GCS to cat state transition. In all cases the field was optimized for $j=10$.

Figure 11

Conditional optimization. Fitness vs time for the conditional optimization of the half cat states $\frac{1}{\sqrt{2}}|j\rangle$ and $\frac{1}{\sqrt{2}}|-j\rangle$, in blue dashed-dotted and red dashed lines, respectively. (solid) Fitness/2 vs. time for the cat state. In all of the cases here $J=5$.

Figure 12

Relative phase of the cat state’s components vs the ratio of the amplitude between the components. In all of the cases here, $J=5$.

Figure 13

Fitness for a control using a pilot field generated for $\hslash =1$ for the Morse oscillator. Solid black line: fitness vs $\hslash$ for a coherent-state to ground-state transition. Target time $T\approx \tau$. At dashed blue line: the fitness for $T\approx 20\tau$. The dashed-dotted red and dotted green lines represent the fitness for transforming a superposition of two coherent states to the ground state, in the case of low and high initial energy of the superposition, respectively.