Quantum process tomography of two-qubit controlled-Z and controlled-NOT gates using superconducting phase qubits

We experimentally demonstrate quantum process tomography of controlled-Z and controlled-NOT gates using capacitively coupled superconducting phase qubits. These gates are realized by using the $|2⟩$ state of the phase qubit. We obtain a process fidelity of 0.70 for the controlled phase and 0.56 for the controlled-NOT gate, with the loss of fidelity mostly due to single-qubit decoherence. The controlled-Z gate is also used to demonstrate a two-qubit Deutsch-Jozsa algorithm with a single function query.

Figure 1

Circuit diagram for the experimental device, showing two flux-biased phase qubits coupled by a fixed capacitance $C_c$. A bias coil and readout SQUID are coupled to each qubit. The design parameters of the circuit are $I_0^{\text{A}}=I_0^{\text{B}}=2\mu \text{A}$, $C_{\text{A}}=C_{\text{B}}=1\text{pF}$, $L_{\text{A}}=L_{\text{B}}=720\text{pH}$, and $C_{\text{c}}=2\text{fF}$.

Figure 2

(Color online) High-power spectroscopy for qubit B. The escape probability (gray scale) is plotted versus microwave frequency and the Z-pulse amplitude $\Delta i$. The single-photon $|0⟩\to |1⟩$ and two-photon $|0⟩\to |2⟩$ transitions are visible, along with two avoided-level crossings. The upper inset shows the calculated states and eigenenergies with the thick (thin) lines representing single (two) photon excitations. The lower inset illustrates the Z-pulse amplitude $i_0$ for the $\text{CZ}=-{\left(\text{SWAP}\right)}^2$ operation.

Figure 3

(Color online) (a) Operation sequence for (b), the state evolution between $|11⟩$ and $|02⟩$ states. (b) Plot of $|2⟩$ state probability of qubit B versus Z-pulse time $\Delta t$ and Z-pulse amplitude $\Delta i$. The dashed lines correspond to the optimal setting for the CZ gate. (c) Operation sequence for (d), demonstration of the CZ gate. (d) Plot of $|1⟩$ state probability of qubit B as a function of $i_{\text{cmp}}^{\text{B}}$ ($i_{\text{cmp}}^{\text{A}}$ is fixed as $3\times {10}^{-4}$). The (red) solid and (blue) dashed curves are for qubit A initialized to the $|0⟩$ and $|1⟩$ states, respectively. The vertical dotted-dashed line indicates the value of $i_{\text{cmp}}^{\text{B}}$ for the CZ gate.

Figure 4

(Color online) $\chi$ matrices of CZ and CNOT gates. (a) Left panel: the real part of the experimentally obtained $\chi$ matrix $\left({\chi }_{\text{p}}\right)$ for the CZ gate with $F_{\text{p}}=0.70$. Right panel: the real part of the simulated $\chi$ matrix for the CZ gate with $F_{\text{p}}=0.67$. (b) Left panel: the real part of ${\chi }_{\text{p}}$ for the CNOT gate with $F_{\text{p}}=0.56$. Right panel: the real part of the simulated $\chi$ matrix for the CNOT gate with $F_{\text{p}}=0.52$. The open boxes in the figure represent the ideal $\chi$ matrix.

Figure 5

(Color online) (a) Pulse sequence for the Deutsch-Jozsa algorithm. [(b)–(d)] Real part of the density matrix of the final state for four Deutsch-Jozsa functions. Open boxes represent the ideal density matrix.

Figure 6

(Color online) Histogram of the differences in the peak height of each of the 256 matrix elements in the real part of $\chi$ matrix. Data is for (a) CZ and (b) CNOT. The solid curves are a Gaussian fit to the data.

Figure 7

(Color online) ${\chi }_{\text{p}}$ of (a) ${\text{CZ}}_{00}$, (b) ${\text{CZ}}_{01}$, (c) ${\text{CZ}}_{10}$, (d) $\text{CZ}={\text{CZ}}_{11}$, and (e) CNOT. Process fidelity $F_{\text{p}}$ are 0.68, 0.69, 0.70, 0.70, and 0.56, respectively.

Figure 8

(Color online) Simulated $\chi$ matrix of (a) ${\text{CZ}}_{00}$, (b) ${\text{CZ}}_{01}$, (c) ${\text{CZ}}_{10}$, (d) CZ, and (e) CNOT. Process fidelity $F_{\text{p}}$ are 0.67, 0.67, 0.66, 0.67, and 0.52, respectively.