Quantum Process Tomography of a Controlled-Phase Gate for Time-Bin Qubits

Time-bin qubits, where information is encoded in a single photon at different times, have been widely used in optical-fiber- and waveguide-based quantum communications. With the recent developments in distributed quantum computation, it is logical to ask whether time-bin encoded qubits may be useful in that context. We have recently realized a time-bin qubit controlled-phase (C-phase) gate using a $2\times 2$ optical switch based on a lithium-niobate waveguide, with which we demonstrated the generation of an entangled state. However, the experiment was performed with only a pair of input states and thus the functionality of the C-phase gate was not fully verified. In this research, we use quantum process tomography to establish a process fidelity of $97.1\mathrm{\%}$. Furthermore, we demonstrate the controlled-not gate operation with a process fidelity greater than $94\mathrm{\%}$. This study confirms that typical two-qubit logic gates used in quantum computational circuits can be implemented with time-bin qubits and thus it is a significant step forward for the realization of distributed quantum computation based on time-bin qubits.

Figure 1

The experimental setup for the implementation and QPT measurement of our time-bin qubit C-phase gate, with schematic illustrations of the time-dependent beam splitter (TDBS) and mode-dependent attenuator (MDA) operations using a $2\times 2$ optical switch. (a) Correlated photon-pair generation. (b) Four methods of time-bin qubit preparation for TDBS and quantum-state tomography measurement. cw laser, continuous-wave laser; IM, intensity modulator; EDFA, erbium-doped fiber amplifier; OVA, optical variable attenuator; PPLN, periodically poled lithium-niobate waveguide; SHG, second-harmonic generation; PPKTP, periodically poled potassium-titanyl-phosphate waveguide; PBS, polarization beam splitter; PLC, planar light-wave circuit; MZI, Mach-Zehnder interferometer; circles 1, 2, 3, and 4, preparation methods; PC, polarization controller; POL, polarizer; PM, electro-optic phase modulator; T, temperature controller for relative phase (${\varphi }_y$) adjusting; Tatt, temperature setting of MZI in short arm of PLC by independent temperature controller; SSPD, superconducting single-photon detector; TIA, time-interval analyzer; ATT, one-third amplitude attenuator; DL, delay line.

Figure 2

The density matrices of single time-bin qubits prepared in the states (a) $\left|t_1\right.⟩$, (b) $\left|t_2\right.⟩$, (c) $\left|+\right.⟩$, and (d) $\left|L\right.⟩$. The left and right parts are the real and imaginary parts of each experimental density matrix.

Figure 3

The process matrix of the C-phase gate for time-bin qubits. (a) The ideal process matrix ${\chi }_{\mathrm{th}}$. (b) The physical process matrix $\tilde{\chi }$ obtained by applying MLE to process matrix ${\chi }_{\exp }$. (c) The C-phase-gate process matrix ${\chi }_{\mathrm{cphase}}$ including compensation for the imperfect experimental input time-bin qubits. The imaginary components of the gate are approximately zero and are not shown.

Figure 4

The density matrices for the C-phase-gate entangling operation of inputs (a) $\left|++\right.⟩$, (b) $\left|+L\right.⟩$, (c) $\left|LL\right.⟩$, and (d) $\left|L+\right.⟩$ states. In each result, we show both the theoretical expectation on the left and the experimental result on the right. The upper and lower parts are the real and imaginary parts of each experimental density matrix.

Figure 5

A bar graph of our experimental cnot gate for time-bin qubits in the (a) computational $zz$ and (b) $xx$ bases.