Time-optimal synthesis of unitary transformations in a coupled fast and slow qubit system

In this paper, we study time-optimal control problems related to a system of two coupled qubits where the time scales involved in performing unitary transformations on each qubit are significantly different. In particular, we address the case where unitary transformations produced by evolutions of the coupling take a much longer time as compared to the time required to produce unitary transformations on the first qubit, but a much shorter time as compared to the time to produce unitary transformations on the second qubit. We present a canonical decomposition of SU(4) in terms of the subgroup $\text{SU}\left(2\right)\times \text{SU}\left(2\right)\times \text{U}\left(1\right)$, which is natural in understanding the time-optimal control problem of such a coupled qubit system with significantly different time scales. A typical setting involves dynamics of a coupled electron-nuclear spin system in pulsed electron paramagnetic resonance experiments at high fields. Using the proposed canonical decomposition, we give time-optimal control algorithms to synthesize various unitary transformations of interest in coherent spectroscopy and quantum-information processing.

Figure 1

The eigenstates of the Hamiltonian $H_0^{\text{lab}}$ are shown. The $\alpha$ and $\beta$ states of the spins denote their orientation along and opposite to the static magnetic field, respectively. The first and second index refer to the orientation of the electron and nuclear spin, respectively. The transitions $\alpha \alpha \leftrightarrow \alpha \beta$ and $\beta \alpha \leftrightarrow \beta \beta$ can be induced by $H^{\alpha }\left({\varphi }_I\right)$ and $H^{\beta }\left({\varphi }_I\right)$, respectively. Refer to the text for details.

Figure 2

The figure shows a canonical pulse sequence for synthesizing unitary transformations in the coupled qubit system. Let ${\tilde{R}}_2=R_2\text{exp}\left(i\pi S_x\right)$ and ${\tilde{R}}_0=\text{exp}\left(-i{v}_0{S}_z\right)\text{exp}\left(-i\pi S_x\right)$. Since $1/J⪡1/{\Omega }^I$, the length of the time intervals $t_j/{\Omega }^I$ is larger as depicted. Refer to the text for details.

Figure 3

The figure shows the pulse sequences for synthesizing the unitary transformations (a) $\text{exp}\left(i\pi /4\right)\text{CNOT}\left[1,2\right]$ and (b) $\text{exp}\left(i\pi /4\right)\text{SWAP}$, where ${\tilde{R}}_5=\text{exp}\left(i\pi S_z/2\right)\text{exp}\left(-i\pi S_x/2\right)$. Since $1/J⪡1/{\Omega }^I$, the length of the time intervals $\pi /{\Omega }^I$ is larger as depicted. Refer to the text for details.