Qubit measurement error from coupling with a detuned neighbor in circuit QED

In modern circuit QED architectures, superconducting transmon qubits are measured via the state-dependent phase and amplitude shift of a microwave field leaking from a coupled resonator. Determining this shift requires integrating the field quadratures for a nonzero duration, which can permit unwanted concurrent evolution. Here we investigate such dynamical degradation of the measurement fidelity caused by a detuned neighboring qubit. We find that in realistic parameter regimes, where the qubit ensemble-dephasing rate is slower than the qubit-qubit detuning, the joint qubit-qubit eigenstates are better discriminated by measurement than the bare states. Furthermore, we show that when the resonator leaks much more slowly than the qubit-qubit detuning, the measurement tracks the joint eigenstates nearly adiabatically. However, the measurement process also causes rare quantum jumps between the eigenstates. The rate of these jumps becomes significant if the resonator decay is comparable to or faster than the qubit-qubit detuning, thus significantly degrading the measurement fidelity in a manner reminiscent of energy relaxation processes.

Figure 1

Analyzed system. A measured (main) transmon qubit (blue) with frequency ${\omega }_{\mathrm{q}}$ has capacitive coupling $g$ to a detuned neighboring qubit (red) with a frequency ${\omega }_{\mathrm{n}}$ such that $g\ll \left|\mathrm{\Delta }\right|$, where $\mathrm{\Delta }\equiv {\omega }_{\mathrm{q}}-{\omega }_{\mathrm{n}}$ (${\omega }_{\mathrm{q}}$ includes the ac Stark shift). A readout resonator with frequency ${\omega }_{\mathrm{r}}$ is dispersively coupled to the main qubit, and thus is frequency shifted by $\pm \chi$ depending on the main qubit state. During measurement, this resonator is driven with a coherent microwave $\varepsilon$ at a frequency ${\omega }_{\mathrm{d}}$. The field leaks from the resonator (with the energy decay rate $\kappa$) to a transmission line, where it is amplified and mixed with a local oscillator to measure the quadrature $I\left(t\right)$ that is sensitive to the qubit-state-dependent phase and amplitude shift. The coupling of the main qubit with the neighboring qubit contributes to the qubit measurement error.

Figure 2

Blue lines: ensemble-averaged evolution of the population $P_{10}$ of the bare-basis state $|10\rangle$ when the evolution starts in this state (left panels) and the population $P_{\overline{10}}$ of the eigenstate $|\overline{10}\rangle$ when starting in this state (right panels). The dashed red lines show the initial value 1 of all blue lines for reference. The time is normalized by the ensemble-dephasing rate ${\mathrm{\Gamma }}_{\mathrm{m}}$ due to measurement; we assume fixed qubit-qubit coupling and detuning with $g/\mathrm{\Delta }=1/10$ for all regimes. (a) Textbook regime with $\mathrm{\Delta }\ll {\mathrm{\Gamma }}_{\mathrm{m}}\ll \kappa$, using directly applied qubit dephasing of ${\mathrm{\Gamma }}_{\mathrm{m}}/\mathrm{\Delta }=20$ for simplicity (i.e., assuming $\kappa \to \infty$). The bare state $|10\rangle$ is best preserved by the evolution, but slowly decays at the rate $2g^2/{\mathrm{\Gamma }}_{\mathrm{m}}$, while the eigenstate population $P_{\overline{10}}$ additionally drops by approximately $2{(g/\mathrm{\Delta })}^2$. (b) Adiabatic (experimental) regime with $({\mathrm{\Gamma }}_{\mathrm{m}},\kappa )\ll \mathrm{\Delta }$, using $\kappa /\mathrm{\Delta }={10}^{-1}$ and ${\mathrm{\Gamma }}_{\mathrm{m}}/\mathrm{\Delta }={10}^{-2}$, set by assuming a weak response $\chi /\kappa =3.5\times {10}^{-2}$ and a resonator drive ${\omega }_{\mathrm{d}}={\omega }_{\mathrm{r}}$ with power tuned to produce the steady-state photon number $\overline{n}=10$. The eigenstate $|\overline{10}\rangle$ is best preserved by the evolution, but slowly decays (analogously to the textbook regime for $P_{10}$), while the bare population $P_{10}$ additionally drops by $2{(g/\mathrm{\Delta })}^2$. (c) Frustrated regime with ${\mathrm{\Gamma }}_{\mathrm{m}}\ll \mathrm{\Delta }\ll \kappa$, using $\kappa /\mathrm{\Delta }=10$ and ${\mathrm{\Gamma }}_{\mathrm{m}}/\mathrm{\Delta }={10}^{-4}$, keeping the same $\chi /\mathrm{\Delta }$ and $\overline{n}$ as in the adiabatic regime. The bare state population $P_{10}$ drops by $2{(g/\mathrm{\Delta })}^2$ compared to the eigenstate population $P_{\overline{10}}$, and both populations show rapid decay. The decay rate seen in regimes (b) and (c) matches the analytical results for averaged incoherent quantum jumps between the eigenstates (see Fig. 4), an example of which is shown here in the bare (c) plot as the overlaid dashed yellow curve.

Figure 3

Qubit-qubit single-excitation ensemble-averaged evolution, using bare Bloch sphere coordinates defined by $x=|10\rangle \langle 01|+|01\rangle \langle 10|,y=-i\left(\right|10\rangle \langle 01|-|01\rangle \langle 10\left|\right)$ and $z=|10\rangle \langle 10|-|01\rangle \langle 01|$, and parameters $g/\mathrm{\Delta }=1/10,\kappa /\mathrm{\Delta }=1$, and ${\mathrm{\Gamma }}_{\mathrm{m}}/\mathrm{\Delta }=1/100$ (with $\overline{n}=2$ and correspondingly $\chi /\kappa =1/40$). An initially bare state $|10\rangle \equiv (0,0,1)$ oscillates rapidly around the tilted axis corresponding to the eigenbasis $\left\{\right|\overline{10}\rangle ,|\overline{01}\rangle \}$, and approaches this axis, indicating the gradual collapse, which produces an incoherent mixture of the eigenstates. At longer times, the ensemble-averaged state continues moving along the eigenstate axis at a slow rate, indicating an additional classical mixing process. Top: 3D plot of $(x,y,z)$ evolution, showing the spiraling evolution to the eigenstate axis and then along the axis, simulated for ${\mathrm{\Gamma }}_{\mathrm{m}}t\in [0,40]$. Bottom: Slice of the $x-z$ plane, with the sphere surface shown as the dashed gray curve and the eigenstate axis shown as the dashed red line tilted from the bare $z$ axis by the angle $2\theta =arctan(2g/\mathrm{\Delta })$. Black dots show time intervals of ${\mathrm{\Gamma }}_{\mathrm{m}}t=5$.

Figure 4

Switching rate vs cavity decay rate. Blue solid line: switching rate analytics ${\mathrm{\Gamma }}_{\text{sw}}/{\mathrm{\Gamma }}_{\mathrm{m}}=2{(g/\mathrm{\Omega })}^2/[1+{(2\mathrm{\Omega }/\kappa )}^2]$ as a function of the ratio $\kappa /\mathrm{\Delta }$. Red boxes: numerical switching rate obtained from solving the master equation and extracting the decay rate ${\mathrm{\Gamma }}_{\text{sw}}$ from fitting $P_{\overline{10}}\left(t\right)$ [as shown in the right panels of Figs. 2 and 2] with Eq. (20), assuming ${\mathrm{\Gamma }}_{\text{sw}}^{+}={\mathrm{\Gamma }}_{\text{sw}}^{-}$. A typical relative difference between the numerical and analytical results is about ${10}^{-3}$, which is comparable to an inaccuracy from the fitting procedure. The used parameters are $g/\mathrm{\Delta }=1/10$ and $1/20,\overline{n}=10,{\mathrm{\Delta }}_{\mathrm{r}}=0$, and $\chi /\mathrm{\Delta }={10}^{-3}$.

Figure 5

An example of quantum jump (switching event) between eigenstates $|\overline{10}\rangle$ and $|\overline{01}\rangle$, obtained in the quantum trajectory simulation for $g/\mathrm{\Delta }=1/20,{\mathrm{\Gamma }}_{\mathrm{m}}/\mathrm{\Delta }=2.5\times {10}^{-4}$, and $\kappa \gg \mathrm{\Delta }$ (bad cavity regime). The eigenstate Bloch coordinate $z_{\mathrm{e}}=P_{\overline{10}}-P_{\overline{01}}$ noisily hovers near $\pm 1$, except when it rapidly jumps between the eigenstates on the time scale of ${\mathrm{\Gamma }}_{\mathrm{m}}^{-1}$. Averaging these random jumps produces the decay observed in Figs. 2 and 2. The physical picture of these jumps is used later to calculate the measurement error.

Figure 6

Signal histograms $P\left(\overline{I}\right|00)$ and $P\left(\overline{I}\right|\overline{10})$ for the integrated quadrature $\overline{I}$, given the initial logical eigenstates $|00\rangle$ and $|\overline{10}\rangle$. Shown are the binned readouts for $100000$ trajectories simulated as in Fig. 5 for duration $t/\tau =7$, with $g/\mathrm{\Delta }=1/10,{\mathrm{\Gamma }}_{\mathrm{m}}/\mathrm{\Delta }={10}^{-3}$, and $\eta =1$. Note that $P\left(\overline{I}\right|00)$ is a Gaussian centered at $I_0=-1$, but $P\left(\overline{I}\right|\overline{10})$ is a slightly shifted Gaussian [centered at $1-2{(g/\mathrm{\Delta })}^2$ instead of $I_1=1$], with a significant extended “tail” (red shaded region) caused by quantum jumps. The overlap of the two Gaussians decreases with integration time $t$, but the histogram overlap due to the tail increases with $t$, thus preventing perfect discrimination.

Figure 7

Simple analytics for the measurement error $P_{\text{err}}$ as a function of an integration duration $t$, normalized by the distinguishability time $\tau$. The orange dashed line shows the monotonically decreasing separation error from integrated white noise. The green dashed line shows the linearly increasing error ${\mathrm{\Gamma }}_{\text{sw}}^{-}t/4$ from switching events, for which we choose ${\mathrm{\Gamma }}_{\text{sw}}^{-}\tau ={10}^{-3}$. The blue solid line shows the total measurement error, which has a minimum of $P_{\text{err}}^{\min }\approx ({\mathrm{\Gamma }}_{\text{sw}}^{-}\tau /2)ln(0.6/{\mathrm{\Gamma }}_{\text{sw}}^{-}\tau )$ at the optimum time $t_{\mathrm{opt}}/\tau \approx 2ln(1/4{\mathrm{\Gamma }}_{\text{sw}}^{-}\tau )$.

Figure 8

Simulated measurement error $P_{\text{err}}^{\left(1\right)}$ for misidentifying the initial state $|\overline{10}\rangle$, using the discrimination threshold of $I_{\text{th}}=0$. Red solid line: measurement error obtained by binning the integrated readouts for $M=1500000$ individual quantum trajectories, with $g/\mathrm{\Delta }=1/20,\mathrm{\Delta }/{\mathrm{\Gamma }}_{\mathrm{m}}=2000$, and $\eta =1$, in the bad cavity regime $\left(\kappa \gg \mathrm{\Delta }\right)$. Error bars show the standard deviation of ${[P_{\mathrm{err}}^{\left(1\right)}(1-P_{\mathrm{err}}^{\left(1\right)})/M]}^{1/2}$. Green dashed line: simple analytics that includes only single quantum jumps, Eq. (38). Blue dot-dashed line: refined analytics that includes single and double quantum jump events (see the Appendix).

Figure 9

(a) Left panel: schematic evolution of the eigenbasis population difference $z_{\mathrm{e}}$ with a jump from 1 to $-1$ at $t_1$. Right panel: the histogram $P_z^{\left(1\right)}\left({\overline{z}}_{\mathrm{tot}}\right|\overline{10})$ for time-averaged bare-basis population difference in the one-jump scenario (solid line) and the corresponding histogram $P^{\left(1\right)}\left(\overline{I}\right|\overline{10})$, which includes Gaussian noise (dashed line). (b) Similar schematic and histograms for the two-jump scenario. We assume $t/\tau =10,g/\mathrm{\Delta }=1/10$, and ${\mathrm{\Gamma }}_{\mathrm{sw}}^{+}={\mathrm{\Gamma }}_{\mathrm{sw}}^{-}$.

Figure 10

Comparison between analytical and numerical results for the measurement histograms $P\left(\overline{I}\right|00)$ and $P\left(\overline{I}\right|\overline{10})$. The green and blue lines show the same numerical results as in Fig. 6, but on an enlarged scale. The solid black line shows the analytical result [Eq. (A7)] for $P\left(\overline{I}\right|\overline{10})$, taking into account up to two jumps. The almost coinciding red dashed line shows the result with up to one jump. The black dashed line shows Eq. (A6) for $P\left(\overline{I}\right|00)$.

Figure 11

Time evolution of the integrated signal histogram for an initially excited eigenstate $|\overline{10}\rangle$, shown on a semilogarithmic scale, for times $t/\tau =5,10,15,20$, assuming ${\mathrm{\Gamma }}_{\mathrm{sw}}^{\pm }\tau ={10}^{-3}$. The histogram width due to Gaussian noise decreases with increasing integration time, analogously to the ground-state histogram (which is not shown). Also, a tail due to quantum jumps between the eigenstates appears at the left side of the histogram; this tail grows in amplitude and flattens with increasing integration time.