Characterization of a Two-Transmon Processor with Individual Single-Shot Qubit Readout

We report the characterization of a two-qubit processor implemented with two capacitively coupled tunable superconducting qubits of the transmon type, each qubit having its own nondestructive single-shot readout. The fixed capacitive coupling yields the $\sqrt{i\mathrm{SWAP}}$ two-qubit gate for a suitable interaction time. We reconstruct by state tomography the coherent dynamics of the two-bit register as a function of the interaction time, observe a violation of the Bell inequality by 22 standard deviations after correcting readout errors, and measure by quantum process tomography a gate fidelity of 90%.

Figure 1

(a) Circuit schematics of the experiment showing the qubits I and II in green, their readout devices in grayed blue, and the homodyne detection circuits with their digitizer (ACQ) in blue. (b) Left-hand panel: Spectroscopy of the sample showing the resonator frequencies ${\nu }_R^{\mathrm{I}}=6.84\mathrm{GHz}$ and ${\nu }_R^{\mathrm{II}}=6.70\mathrm{GHz}$ (horizontal lines), and the measured (disks, triangles) and fitted (lines) qubit frequencies ${\nu }_{\mathrm{I},\mathrm{II}}$ as a function of their flux bias ${\varphi }_{\mathrm{I},\mathrm{II}}$ when the other qubit is far detuned. Right-hand panel: Spectroscopic anticrossing of the two qubits revealed by the 2D plot of $p_{01}+p_{10}$ as a function of the probe frequency and of ${\varphi }_{\mathrm{I}}$, at ${\nu }_{\mathrm{II}}=5.124\mathrm{GHz}$. (c) Typical pulse sequence including $X$ or $Y$ rotations, a ${\sqrt{i\mathrm{SWAP}}}^{\alpha }$ gate, $Z$ rotations, and tomographic and readout pulses. Microwave pulses $a(t)$ for qubit (green) and for readout (blue) are drawn on top of the ${\nu }_{\mathrm{I},\mathrm{II}}(\varphi )$ dc pulses (red lines).

Figure 2

Coherent swapping of a single excitation between the qubits. (a) Experimental (solid lines) and fitted (dashed lines) occupation probabilities of the four computational states $|00⟩\dots |11⟩$ as a function of the coupling duration. No $Z$ or tomographic pulses are applied here. (b),(c) State tomography of the initial state (left) and of the state produced by the $\sqrt{i\mathrm{SWAP}}$ gate (right). (b) Ideal (empty bars) and experimental (color filling) expected values of the 15 Pauli operators $IX,\dots ,ZZ$. (c) Corresponding ideal (color-filled black circles with black arrow) and experimental (red circle and arrow) density matrices, as well as fidelity $F$ and concurrence C. Each complex matrix element is represented by a circle with an area proportional to its modulus (diameter equals cell size for unit modulus) and by an arrow giving its argument. See Sec. VI of the SM [14] for a real and imaginary part representation of the matrices.

Figure 3

Test of the CHSH-Bell inequality on a $|10⟩+e^{i\psi }|01⟩$ state by measuring the qubits along $X^{\mathrm{I}}$ or $Y^{\mathrm{I}}$ and $X_{\phi }^{\mathrm{II}}$ or $Y_{\phi }^{\mathrm{II}}$ (see top-left inset), respectively. Blue (red) error bars are the experimental CHSH entanglement witness determined from the raw (readout-error corrected) measurements as a function of the angle $\phi$ between the measuring basis, whereas solid line is a fit using $\psi$ as the only fitting parameter. Height of error bars is $\pm 1$ standard deviation $\sigma (N)$ (see bottom-right inset), with $N$ the number of sequences per point. Note that averaging beyond $N={10}^6$ does not improve the violation because of a slow drift of $\phi$.

Figure 4

Map of the implemented $\sqrt{i\mathrm{SWAP}}$ gate yielding a fidelity $F_g=90\%$. Superposition of the ideal (empty thick bars) and experimental (color-filled bars) upper part of the Hermitian process matrix $\chi$ (a) and lower part of the Hermitian error matrix $\tilde{\chi }$ (b), in the two-qubit Pauli operators basis $\{II,\dots ,ZZ\}$. Expected elements are marked with a star, and elements below 1% are not shown. Each complex matrix element is represented by a bar with height proportional to its modulus and by an arrow at the top of the bar (as well as a filling color for the experiment—see top inset) giving its argument. See also Sec. VI of the SM [14] for a real and imaginary part representation of these matrices and for additional information. Labeled arrows indicate the main visible contributions to errors, i.e., a too long swapping time (S), too small rotations ${\theta }_{\mathrm{I},\mathrm{II}}$ (Z), and relaxation ($T_1$)—see text.