Experimental replication of single-qubit quantum phase gates

We experimentally demonstrate the underlying physical mechanism of the recently proposed protocol for superreplication of quantum phase gates [W. Dür, P. Sekatski, and M. Skotiniotis, Phys. Rev. Lett. 114, 120503 (2015)], which allows producing up to $N^2$ high-fidelity replicas from $N$ input copies in the limit of large $N$. Our implementation of $1\to 2$ replication of the single-qubit phase gates is based on linear optics and qubits encoded into states of single photons. We employ the quantum Toffoli gate to imprint information about the structure of an input two-qubit state onto an auxiliary qubit, apply the replicated operation to the auxiliary qubit, and then disentangle the auxiliary qubit from the other qubits by a suitable quantum measurement. We characterize the replication protocol by full quantum process tomography and observe good agreement of the experimental results with theory.

Figure 1

(a) Quantum circuit for superreplication of single-qubit quantum phase gates $U\left(\varphi \right)$ [7]. (b) $1\to 2$ quantum-gate replication circuit involving two quantum Toffoli gates and one ancilla qubit. (c) The same circuit as in panel (b), but the second Toffoli gate is replaced with measurement on ancilla qubit and feedforward. (d) The circuits in panels (b), (c) convert a single-qubit phase gate $U\left(\varphi \right)$ into a two-qubit controlled-phase gate. For details, see text.

Figure 2

Quantum circuit for optimal $1\to 2$ replication of single-qubit unitary phase gates $U\left(\varphi \right)$ [6].

Figure 3

Experimental setup. HWP: half-wave plate; QWP: quarter-wave plate; PPBS: partially polarizing beam splitter with reflectances $R_V=2/3$ and $R_H=0$ for vertical and horizontal polarizations, respectively; PBS: polarizing beam splitter; BD: calcite beam displacer; APD: single-photon detector. Inset (a) shows the implemented quantum circuit and inset (b) depicts the single-photon detection block DB.

Figure 4

Real and imaginary parts of the experimentally determined quantum process matrices $\chi$ representing the two-qubit operations $\mathcal{R}$ implemented on the two signal qubits are plotted for four values of the phase shift: $\varphi =0$ (a), $\varphi =\pi /2$ (b), $\varphi =\pi$ (c), and $\varphi =3\pi /2$ (d). All process matrices are normalized such that $\mathrm{Tr}\left(\chi \right)=1$.

Figure 5

Experimentally determined quantum gate fidelities $F_{CU}\left(\varphi \right)$ (a) and $F_{UU}\left(\varphi \right)$ (b) are plotted for the eight values of $\varphi$ that were probed experimentally. The solid line in panel (b) represents the ideal theoretical dependence (9) and the dashed line is the least-square fit of the form $A+Bcos\varphi$, with $A=0.543$ and $B=0.322$. Statistical error bars are smaller than the markers.

Figure 6

Real and imaginary parts of the experimentally determined quantum process matrix $\chi$ representing the quantum operation implemented on the two signal qubits (first and second column), and real and imaginary parts of the quantum process matrix ${\chi }_{\mathrm{th}}$ of the corresponding ideal unitary operation $CU\left(\varphi \right)$ (third and fourth column), are plotted for four values of the phase shift: $\varphi =0$ (a), $\varphi =\pi /4$ (b), $\varphi =\pi /2$ (c), and $\varphi =3\pi /4$ (d). All process matrices are normalized such that $\mathrm{Tr}\left(\chi \right)=1$.

Figure 7

Same as Fig. 6: $\varphi =\pi$ (a), $\varphi =5\pi /4$ (b), $\varphi =3\pi /2$ (c), and $\varphi =7\pi /4$ (d).