Entanglement Genesis by Ancilla-Based Parity Measurement in 2D Circuit QED

We present an indirect two-qubit parity meter in planar circuit quantum electrodynamics, realized by discrete interaction with an ancilla and a subsequent projective ancilla measurement with a dedicated, dispersively coupled resonator. Quantum process tomography and successful entanglement by measurement demonstrate that the meter is intrinsically quantum nondemolition. Separate interaction and measurement steps allow the execution of subsequent data-qubit operations in parallel with ancilla measurement, offering time savings over continuous schemes.

Figure 1

(a) cQED processor with four transmons ($A$, $D_1$, $D_2$, and an unused one at top right) coupled to a bus resonator. Each transmon has a dedicated readout resonator that is addressed through the shared feedline (ports 1–8–4–5) and a local flux-bias line (ports 2, 3, 6, 7) that allows tuning of the transition frequencies with $\sim 1\mathrm{ns}$ resolution [5]. A coaxial cable connects ports 4 and 8 off the chip. (b) Gate sequence for coherent swapping of excitations between $D_1$ and the bus by nonadiabatic qubit tuning. (c),(d) Measured average qubit populations at the end of the sequence for the one- and two-excitation manifolds, respectively. An excitation can be swapped from $D_1$ to the bus (or vice versa) in 13.1 ns when the $g_1\text{−}{e}_1$ transition is resonant with $B$. (d) In the two-excitation manifold, population transfer occurs when either the $g_1\text{−}{e}_1$ or the $e_1\text{−}{f}_1$ transition is resonant with $B$ with a half-period of 9.3 ns. The ratio of the measured periods agrees with the $\sqrt{2}$ enhancement of the effective coupling predicted by theory.

Figure 2

(a) Histograms of $V_{H,A}$ (${\tau }_{\mathrm{MA}}=450\mathrm{ns}$) for computational states of $A$. The dashed line is the fidelity maximizing digitizing threshold. (b) Single-shot ancilla readout fidelity ${\mathcal{F}}_A$ as a function of the measurement pulse duration ${\tau }_{\mathrm{MA}}$. Maximal ${\mathcal{F}}_A=89\%$ is attained at ${\tau }_{\mathrm{MA}}=450\mathrm{ns}$ (arrow). Inset: Corresponding calibrated readout error model. (c) Gate sequence used to study qubit dephasing induced by $A$ measurement. A readout pulse of duration ${\tau }_{\mathrm{MA}}$ and power $P_{\text{meas}}$ is embedded in a fixed-length echo sequence performed on qubit (d) $A$, (e) $D_1$, and (f) $D_2$. The azimuthal angle $\varphi$ of the final $\pi /2$ rotation is swept from 0 to $8\pi$ jointly with ${\tau }_{\mathrm{MA}}$ to facilitate discerning deterministic phase shifts and dephasing. The plots show averaged $\widehat{Z}$ measurements normalized to compensate for the fixed loss of contrast due to intrinsic decoherence. In (d), the dashed line is a theoretical prediction for 98% loss in phase contrast. Dashed lines in (e) are equal-phase contours accounting for ac Stark shift on $D_1$. All theory curves were calculated using measured parameters [28]. (f) No effect on $D_2$ is observed. Arrows indicate the power used for $A$ readout in Figs. 3 and 4. The incident power corresponding to the one-photon average population in the $A$ readout resonator is $-133\mathrm{dBm}$.

Figure 3

(a) Gate sequence realizing an indirect parity measurement of $D_1$ and $D_2$ using coherent interactions with $A$ and a subsequent projective measurement of $A$. (b) Histograms of the integrated voltage $V_{H,A}$ (${\tau }_{\mathrm{MA}}=450\mathrm{ns}$) for computational basis-state inputs prepared using initial rotations $R_{\alpha }^{\theta },R_{\beta }^{{\theta }^{\prime }}\in \{I,R_x^{\pi }\}$. The digitizing threshold indicated by the dashed line produces the maximum parity fidelity ${\mathcal{F}}_P=69\%$. (c)–(e) Manhattan-style plots of data-qubit density matrices after the parity measurement for the separable superposition input state prepared with $R_{\alpha }^{\theta }=R_{\beta }^{{\theta }^{\prime }}=R_y^{\pi /2}$. (c) In the unconditioned tomogram, the density matrix elements indicative of coherences between the parity subspaces have been suppressed by 90% in magnitude. The spurious residual probability amplitudes can be attributed almost entirely to the infidelity of the two $i$swap operations. (d),(e) The conditioned tomograms demonstrate probabilistic entanglement by measurement, reaching 87% (81%) fidelity to the even (odd) Bell state and 0.74 (0.63) concurrence for the even (odd) outcome. The higher fidelity of the even projection is in accordance with the error model of ancilla readout, where the dominating error mechanism is relaxation of $A$ during readout. For (c)–(e), the timing of tomographic prerotations and measurements corresponds to the parallel variation with $t_{\mathrm{MD}}=0$, as illustrated in Fig. 4.

Figure 4

Genesis and decay of two-qubit entanglement as a function of the time $t_{\mathrm{MD}}$ between the end of the interaction step and the beginning of the readout pulses on the data qubits. (a) Timing diagram for the serial variation, in which the ancilla measurement is completed strictly before operations on data qubits continue. (b) Timing diagram for the parallel variation, in which the duration ${\tau }_{\mathrm{MA}}$ of the ancilla readout pulse is constant at $1\mu \mathrm{s}$. The homodyne voltage is integrated only during the first ${\tau }_{\mathrm{MA}}^i=450\mathrm{ns}$. The tomographic prerotations and the readout pulses for the data qubits can overlap with the ancilla readout. For $t_{\mathrm{MD}}=0$, the prerotations are done simultaneously with the ancilla $R_y^{\pi /2}$ gate at the end of the interaction step. (a),(b) Elements drawn in green move together in time. (c) Wootters’s $\mathrm{\Lambda }$ [33] and concurrence $C=\max (\mathrm{\Lambda },0)$ as a function of $t_{\mathrm{MD}}$ for the parallel and serial timings (filled and open markers, respectively) extracted from the two-qubit density matrix that is unconditioned (black diamonds), conditioned on the even outcome ($M_P=+1$, blue circles), and conditioned on the odd outcome ($M_P=-1$, red squares) of the ancilla measurement. Solid curves are based on a model that includes single-qubit relaxation and dephasing processes and uses the experimental density matrix at $t=0$ as the initial value. For the dashed curves, mixing of the parity subspaces according to calibrated readout errors is explicitly included.