Hybrid quantum computation

We present a hybrid model of the unitary-evolution-based quantum computation model and the measurement-based quantum computation model. In the hybrid model, part of a quantum circuit is simulated by unitary evolution and the rest by measurements on star graph states, thereby combining the advantages of the two standard quantum computation models. In the hybrid model, a complicated unitary gate under simulation is decomposed in terms of a sequence of single-qubit operations, the controlled-z gates, and multiqubit rotations around the $z$ axis. Every single-qubit and the controlled-z gate are realized by a respective unitary evolution, and every multiqubit rotation is executed by a single measurement on a required star graph state. The classical information processing in our model requires only an information flow vector and propagation matrices. We provide the implementation of multicontrol gates in the hybrid model. They are very useful for implementing Grover’s search algorithm, which is studied as an illustrative example.

Figure 1

(i) This is called a star graph because of its appearance, and the associated graph state $|\varphi \rangle {}_{(1+n)}$ is given by Eq. (8). The (blue) circles represent the logical qubits which carry the input information; the bonds represent the $\text{CZ}$ operations given by Eq. (1), and the ancilla qubit $a$ is represented by the (black) diamond. (ii) The effect on the input state $|{\psi }_{\mathrm{in}}(n)\rangle$ when the qubit $a$ of the graph state $|\varphi \rangle {}_{(1+n)}$ is measured in an appropriately chosen basis.

Figure 2

(i) The quantum circuit merely represents the temporal order of rotations for ${\Lambda }^1{U}_{z...z}^{2...n}(-2\theta )$. Horizontal lines represent $n$ logical qubits. The left (green) rectangular box symbolizes the $n$-qubit rotation $U_{\mathit{zz}...z}^{12...n}(\theta )$, and the right box symbolizes the $(n-1)$-qubit rotation $U_{z...z}^{2...n}(-\theta )$. Both of them are executed under the scheme described in Sec. 2a. (ii) This circuit represents ${\Lambda }^1{U}_{z...z}^{2...n}(-2\theta )$, where qubit 1 is the control qubit and the other qubits are targets. The dashed (red) vertical lines in the input and the output sections represent the by-product operators $U_{B,\mathrm{in}}$ and $U_{B,\mathrm{out}}$, respectively. Circuits (i) and (ii) are equivalent.

Figure 3

Horizontal lines represent $n$ logical qubits. (i) The four (green) rectangular boxes (from left to right) represent ${\Lambda }^1{U}_{z...z}^{2...n}(-2\theta )$, $U_{z...z}^{2...n}(-\theta )$, $U_{z...z}^{13...n}(-\theta )$, and $U_{z...z}^{3...n}(\theta )$, respectively. Each rotation is realized under the scheme described in Sec. 2a. (ii) The left (green) rectangular box symbolizes the $n$-qubit operation ${\Lambda }^1{U}_{z...z}^{2...n}(-2\theta )$, and the right box symbolizes the $(n-1)$-qubit operation ${\Lambda }^1{U}_{z...z}^{3...n}(2\theta )$. Both of them have qubit 1 as the control. (iii) The diagram represents ${\Lambda }^{12}{U}_{z...z}^{3...n}(4\theta )$, where qubits 1 and 2 are the control qubits. In (i) and (iii), the dashed (red) vertical lines in the input and output sections represent the by-product operators $U_{B,\mathrm{in}}$ and $U_{B,\mathrm{out}}$, respectively. Circuit (ii) merely depicts the intermediate stage of circuits (i) and (iii), and they all are mutually equivalent.

Figure 4

The quantum circuit for implementing the four-qubit gate ${\Lambda }^{123}{Z}^6$. From top to bottom, the three black horizontal lines represent control qubits 1, 2, and 3, and the next two, gray horizontal lines represent work qubits 4 and 5, which are prepared in state $|+\rangle {}^{\otimes 2}$. The black horizontal line at the bottom represents target qubit 6. Every three-qubit gate is the special case of ${\Lambda }^{12}{U}_{z...z}^{3...n}(4\theta )$ (for $n=3$ and $\theta =\pm \pi /4$), and they are realized by the four rotations according to Fig. 3. The Hadamard gate $H$ and $\text{CZ}(5,6)$ are executed by the unitary evolution. Computation steps ($\tau$) are expressed by dashed (red) vertical lines for the classical information processing according to Table .

Figure 5

(i) Graph associated with graph state $|\varphi \rangle {}_{(1+1)}$ given by Eq. (A1). The (blue) circle, the bond, and the (black) diamond represent the logical qubit, which is in input state $|{\psi }_{\mathrm{in}}(1)\rangle$, the $\text{CZ}$ operations given by Eq. (1), and the ancilla qubit $a$, respectively. (ii) The quantum circuit illustrates the effect on the input state when qubit $a$ is measured in an appropriately chosen basis.