Efficient universal quantum computation with auxiliary Hilbert space

We propose a scheme to construct the efficient universal quantum circuit for qubit systems with the assistance of possibly available auxiliary Hilbert spaces. An elementary two-ququart gate, termed the controlled-double-not gate, is proposed first in ququart (four-level) systems, and its physical implementation is illustrated in the four-dimensional Hilbert spaces built by the path and polarization states of photons. Then an efficient universal quantum circuit for ququart systems is constructed using the gate and the quantum Shannon decomposition method. By introducing auxiliary two-dimensional Hilbert spaces, the universal quantum circuit for qubit systems is finally achieved using the result obtained in ququart systems with the lowest complexity.

Figure 1

(a) Circuit representation for the controlled-double-not gate $▽\left(X\right)$. The control (target) ququart is denoted by the closed (open) triangle. (b) Circuit representation for the controlled-double-phase-flip gate $▽\left(P\right)$.

Figure 2

Optical realization of the exchange operator $E$, where the PBS transmits photons in the $\left|H\right.〉$ state while reflects photons in the $\left|V\right.〉$ state. The half-wave plates ($\lambda 2$) are set at 45${}^{\circ }$ so as to convert the polarization of photon in path $S$ from $H$ ($V$) to $V$ ($H$).

Figure 3

Schematic setup of implementing operation $▽\left(P\right)$ with the cavity-assisted interactions. The relevant energy level of the trapped atom [22] is shown in the inset. The atomic transitions $\left|1\right.〉\to \left|e_L\right.〉$ ($L=H,V$) are resonantly coupled to the cavity modes $a_L$.

Figure 4

(a) Recursive quantum circuits implementing the UQC in ququart systems, where the gates $V_p^1$ have the same decomposition form as gate $V_0^1$. The controlled gates $R_{0y}^g$ ($R_{qz}^g$) with half-closed circle in control ququarts represent the uniformly controlled gate $F_n^{n-1}\left(R_y^g\right)$ $(F_n^{n-1}\left(R_z^g\right))$ corresponding to $Y_0^g$ ($Z_q^g$) in Eq. (4). The line with the backslash symbol ($\setminus$) represents more than one wire in the quantum circuit. (b) Recursive quantum circuits of the gate $F_n^{n-1}\left(R_{\mathbf{a}}^1\right)$, where some gates ${\tilde{D}}_n^g$ have been canceled each other out.

Figure 5

(a) Quantum circuit for a general seven-qubit unitary operation assisted with auxiliary DOF, where $U^4=U$. The ququarts (qubits) are denoted by the bold red lines (blue lines). (b) Quantum circuit for realizing the transferring operator $V$. The left controlled gate applies operation $X$ to ququart $A$ if qubit $B$ is in state $\left|1\right.〉$, and the right controlled gate flips the state of qubit $B$ if ququart $A$ is in state $\left|2\right.〉$ or $\left|3\right.〉$. (c) Quantum circuit for realizing $U^4$. The controlled gates between the qubit and ququart in (b) and (c) can be implemented as in ququart systems after expanding the state space of the qubit (red line) to four levels.