Efficient Toffoli gates using qudits

The simplest decomposition of a Toffoli gate acting on 3 qubits requires five 2-qubit gates. If we restrict ourselves to controlled-sign (or controlled-NOT) gates this number climbs to 6. We show that the number of controlled-sign gates required to implement a Toffoli gate can be reduced to just 3 if one of the three quantum systems has a third state that is accessible during the computation—i.e., is actually a qutrit. Such a requirement is not unreasonable or even atypical since we often artificially enforce a qubit structure on multilevel quantums systems (e.g., atoms, photonic polarization plus spatial modes). We explore the implementation of these techniques in optical quantum processing and show that linear optical circuits could operate with much higher probabilities of success.

Figure 1

(Color online) Realization of a TS gate using two qubits ($a$ and $b$) and a qutrit $\left(c\right)$. Controlled-NOT (CNOT) gates (first and last two-qubit gates) operate as normal on the qubit levels and implement the identity if the target is in the qutrit level (∣2⟩). Similarly for the CS gate (middle two-qubit gate). The $X_A$ gate flips the qutrit between the states 0 and 2. The sign change occurs on the ∣1,0,1⟩ component.

Figure 2

(Color online) Realization of an $(n=3)$ level TS gate using three qubits ($a$, $b$, and $c$) and a ququit $\left(d\right)$. CNOT gates operate as normal on the qubit levels and implement the identity if the target is in the ququit levels (2 or 3). Similarly for the CS gate. The $X_A$ gate flips the ququit between the states 0 and 2. The $X_B$ gate flips the ququit between the states 1 and 3. The sign change occurs on the ∣1,1,1,1⟩ component.

Figure 3

(Color online) Schematic of the implementation of an optical CS gate using a strong cross-Kerr nonlinearity $\chi$. PBS are polarizing beam splitters.

Figure 4

(Color online) Optical realization of a TS gate. Polarizing beam splitters (PBS) act as $X_A$ gates by accessing an additional spatial mode. The half-wave plates $(\lambda ∕2)$ are set at 22.5° so as to act as Hadamard gates and the CS gates are implemented as per Fig. 3.

Figure 5

(Color online) Nondeterministic, but heralded, optical realization of a TS gate. Polarizing beam splitters (PBS) act as $X_A$ gates by accessing an additional spatial mode. The half-wave plates $(\lambda ∕2)$ act as Hadamard gates and the CS gates are implemented using heralded nondeterministic gates. The sign change occurs on the ∣0,0,1⟩ component.

Figure 6

(Color online) Nondeterministic, post-selected, optical realization of a TS gate. In contrast to the other figures all optical modes are shown explicitly; thus, each input qubit is represented by two modes. Note that in the central part of the diagram there are a total of seven modes, due to the introduction of an additional target mode. Beam splitters are represented as black lines with their reflectivity indicated to the right. The beam splitters are assumed to be asymmetric; i.e., a phase flip is induced by reflection off one surface but all other components suffer no phase change. The surface for which the phase flip occurs is indicated by a dotted line. If we take occupation of the top mode of each qubit to represent logical 0 and occupation of the bottom mode to represent logical 1, then the circuit implements a TS gate in which a phase flip is only applied to the element ∣000⟩. The figure is represented, for clarity, as if all modes are spatial. In an experimental realization polarization modes would be utilized where ever possible.