Experimental Implementation of Optimal Linear-Optical Controlled-Unitary Gates

We show that it is possible to reduce the number of two-qubit gates needed for the construction of an arbitrary controlled-unitary transformation by up to 2 times using a tunable controlled-phase gate. On the platform of linear optics, where two-qubit gates can only be achieved probabilistically, our method significantly reduces the amount of components and increases success probability of a two-qubit gate. The experimental implementation of our technique presented in this Letter for a controlled single-qubit unitary gate demonstrates that only one tunable controlled-phase gate is needed instead of two standard controlled-not gates. Thus, not only do we increase the success probability by about 1 order of magnitude (with the same resources), but also avoid the need for conducting quantum nondemolition measurement otherwise required to join two probabilistic gates. Subsequently, we generalize our method to a higher order, showing that $n$-times controlled gates can be optimized by replacing blocks of controlled-not gates with tunable controlled-phase gates.

Figure 1

Quantum computation circuit implementing an arbitrary single-qubit controlled-unitary operation $W$ [see Eq. (4)] by means of one tunable controlled-phase gate and several unconditional single-qubit operations.

Figure 2

Schematic drawing of the experimental setup. The components are labeled as follows: MT—motorized translation, HWP—half-wave plate, QWP—quarter-wave plate, PBS—polarizing beam splitter, BDA—beam divider assembly, BD—beam divider, F—neutral density filter, D—detector. For more details on the setup function, see the Supplemental Material [32].

Figure 3

Estimated process matrices for $\phi =(3\pi /4)$, $\theta =(\pi /2)$ and $\alpha =0$ (a) with control qubit $|0⟩$ and (b) with control qubit $|1⟩$. Moduli of matrix elements are visualized by bar heights and their phase by arrow directions. The experimentally acquired density matrices are compared to their theoretical counterparts in the Supplemental Material [32].

Figure 4

2-times and 3-times cphase [$Z(\phi )$] gates decomposed with a method of Lanyon et al. [22] based on extending the Hilbert space of the target qubit (the two top rows show the idea from Ref. [22] to demonstrate the benefits of our approach). The control lines are denoted as $C_n$ for $n=1,2,3$ and $T$ denotes the target qubit line. For the 2(3)-times controlled gate the dimension of the target qubit is extended by 1(2). The additional dimensions are visualized as additional dashed channels. For technical details on the operation of the $X_a$ and $X_b$ gates see Ref. [22]. By iterating the depicted procedure for an $n$-times controlled gate we conclude that $2(n-1)$ cnots and a single cphase are needed for its implementation. An arbitrary controlled-$U$ gate, where $U=A^{†}Z(\phi )A$ can be easily implemented using the same circuits by applying $A$ and $A^{†}$ single-qubit operations.