Robust controllability of two-qubit Hamiltonian dynamics

Physically, quantum gates (unitary gates) for quantum computation are implemented by controlling the Hamiltonian dynamics of quantum systems. When full descriptions of the Hamiltonians are given, the set of implementable quantum gates is easily characterized by quantum control theory. In many real systems, however, the Hamiltonians may include unknown parameters due to the difficulty of performing precise measurements or instability of the system. In this paper, we consider the situation that some parameters of the Hamiltonian are unknown, but we still want to perform a robust control of the Hamiltonian dynamics to implement a quantum gate irrespectively to the unknown parameters. The existence of the robust control was previously shown for single-qubit systems, and a constructive method was developed for two-qubit systems if a full control of each qubit is available. We analytically investigate the robust controllability of two-qubit systems, and apply Lie-algebraic approaches to handle the cases where only one of the two qubits is controllable. We also numerically analyze the robust controllability of the two-qubit systems where the analytical approach is not necessarily applicable and investigate the relationship between the robust controllability of systems with a discrete and continuous unknown parameter.

Figure 1

The error spectrum $\epsilon \left(\omega \right)$ for dimensionless $\omega \in [1,2]$ to implement a controlled-not (cnot) gate by the dynamics of $H\left(t\right)=H_d\left(\omega \right)+v\left(t\right)H_c$ for $H_d\left(\omega \right)=\omega X\otimes I+X\otimes X+Y\otimes Y+Z\otimes Z$ and $H_c=Z\otimes I$ (system A), where $v\left(t\right)$ is numerically obtained by the discretization approach for ${\mathrm{\Omega }}_{11}=\{1.0,1.1,\dots ,2.0\}$. The black line represents errors (a vertical axis) of the dynamics with $\omega \in [1,2]$ (a horizontal axis) from the ideal cnot gate evaluated by Eq. (17) where $d=4$ and $T=32$. The black dots represent errors for the points in ${\mathrm{\Omega }}_{11}$.

Figure 2

The control pulse $v\left(t\right)$ for the robust control realizing the error spectrum shown in Fig. 1. The total time of control $T$ is 32. We divide the total time by 128, and set $v\left(t\right)$ be a constant in $t\in \left[\right(n-1)T/128,nT/128]$ for any $n=1,2,\dots ,128$ where $t$ and $T$ are dimensionless parameters.

Figure 3

The error spectrum $\epsilon \left(\omega \right)$ for dimensionless $\omega \in [1,2]$ to implement a cnot gate by the dynamics of $H\left(t\right)=H_d\left(\omega \right)+v\left(t\right)H_c$ for $H_d\left(\omega \right)=X\otimes I+\omega (X\otimes X+Y\otimes Y+Z\otimes Z)$ and $H_c=Z\otimes I$ (system E), where $v\left(t\right)$ is numerically obtained by the discretization approach for ${\mathrm{\Omega }}_{11}=\{1.0,1.1,\dots ,2.0\}$. The black line represents errors (a vertical axis) of the dynamics with $\omega \in [1,2]$ (a horizontal axis) from the ideal cnot gate evaluated by Eq. (17) where $d=4$ and $T=32$. The black dots represent errors for the points in ${\mathrm{\Omega }}_{11}$.

Figure 4

(a) System A. (b) System E. The minimum control time $T_{\varepsilon }\left(N\right)$ (a vertical axis) to let all the $N$ (a horizontal axis) points in ${\mathrm{\Omega }}_N$ be under a given allowed error (line style) where ${\mathrm{\Omega }}_1:=\left\{1.0\right\}$ and we search the time in only integers. The black dots show the minimum control time $T_{\varepsilon }\left(N\right)$ achieving the cnot gate within each of allowed errors (${10}^{-2},{10}^{-3},{10}^{-4},{10}^{-5}$) for all $\omega \in {\mathrm{\Omega }}_N$, and we can find a robust control pulse to be under a given allowed error for all $\omega \in [1,2]$ on points with a gray square. $T_{\epsilon }\left(N\right)$ is a dimensionless quantity.