Local quantum control of Heisenberg spin chains

Motivated by some recent results of quantum control theory, we discuss the feasibility of local operator control in arrays of interacting qubits modeled as isotropic Heisenberg spin chains. Acting on one of the end spins, we aim at finding piecewise-constant control pulses that lead to optimal fidelities for a chosen set of quantum gates. We analyze the robustness of the obtained results for the gate fidelities to random errors in the control fields, finding that with faster switching between piecewise-constant controls the system is less susceptible to these errors. The observed behavior falls into a generic class of physical phenomena that are related to a competition between resonance- and relaxation-type behavior, exemplified by motional narrowing in NMR experiments. Finally, we discuss how the obtained optimal gate fidelities are altered when the corresponding rapidly varying piecewise-constant control fields are smoothened through spectral filtering.

Figure 1

Examples of optimal control fields for the $X_3$ gate, corresponding to (a) $N_t=70$ and $T=0.429$, with the resulting optimal fidelity of 1–${10}^{-8}$, and (b) $N_t=70$ and $T=0.243$ (minimal total time $t_f=17$), with the resulting optimal fidelity of 1–${10}^{-6}$.

Figure 2

Average fidelity versus half-width ($\delta$) for random noise affected optimal fields corresponding to (a) $X_3$ gate, and (b) ${\mathrm{CNOT}}_3$ gate, both implemented by alternate $x$ and $y$ controls with a total evolution time $t_f=30$.

Figure 3

Average fidelity versus half-width ($\delta$) for random noise affected optimal fields corresponding to the implementation of the $X_3$ gate by control of the $x$ field only: (a) fixed total evolution time ($t_f=25$) and (b) fixed number of pulses ($N_t=70$). The insets show the same curves for large values of the parameter $\delta$.

Figure 4

The power spectra corresponding to the optimal $x$ (a) and $y$ (b) control fields shown in Fig. 1a.

Figure 5

Optimal piecewise-constant control field which realizes the $X_3$ gate by alternate $x$ (a) and $y$ (b) pulses compared to the low-pass filtered counterpart (cut-off frequency ${\omega }_0/J=\pi /2$) with a fidelity of $0.9$. The optimal field corresponds to $N_t=70$ and $T=0.5$.

Figure 6

Optimal piecewise-constant control field which realizes the ${\text{CNOT}}_3$ gate by alternate $x$ (a) and $y$ (b) pulses, compared to the low-pass filtered counterpart (cut-off frequency ${\omega }_0/J=\pi /2$) with a fidelity of $0.9$. The optimal field corresponds to $N_t=70$ and $T=1.0$.

Figure 7

Optimal piecewise-constant control field which realizes the $X_3$ gate by alternate $x$ (a) and $y$ (b) pulses compared to the Gaussian-filtered counterpart ($\mathrm{FWHM}=4.3$) with a fidelity of $0.9$. The optimal control field corresponds to $N_t=70$ and $T=0.5$ as in the case of low-pass filtering shown in Fig. 5.

Figure 8

Fidelity ($F$) versus cutoff frequency (${\omega }_0$) for different low-pass filtered optimal control fields, all of them implementing the $X_3$ gate with $70$ pulses of durations $1.5$ alternately applied in the $x$ and $y$ directions. (a) The optimal control fields with average amplitudes indicated and variances ranging between $1.1$ and $1.3$. (b) The optimal control fields with average amplitudes between $1.9$ and $2.1$, and variances specified in the legend.