Theoretical analysis of measurement crosstalk for coupled Josephson phase qubits

We analyze the crosstalk error mechanism in measurement of two capacitively coupled superconducting flux-biased phase qubits. The damped oscillations of the superconducting phase after the measurement of the first qubit may significantly excite the second qubit, leading to its measurement error. The first qubit, which is highly excited after the measurement, is described classically. The second qubit is treated both classically and quantum mechanically. The results of the analysis are used to find the upper limit for the coupling capacitance (thus limiting the frequency of two-qubit operations) for a given tolerable value of the measurement error probability.

Figure 1

The circuit schematic of a flux-biased phase qubit and the corresponding potential profile (as a function of the phase difference $\delta$ across the Josephson junction). During the measurement the state ∣1⟩ escapes from the “left” well through the barrier, which is followed by oscillations in the “right” well. This dissipative evolution leads to the two-qubit crosstalk.

Figure 2

The circuit schematic for two capacitively coupled flux-biased phase qubits.

Figure 3

The qubit potential $U\left(\delta \right)$ [Eq. (2)] for $N_l=10$ (dotted line), $N_l=5$ (solid line), and $N_l=1.355$ (dashed line).

Figure 4

The first-qubit oscillation frequency $f_d$ as a function of time $t$ (normalized by the energy relaxation time $T_1$) for $C_x=0$ (solid line) and $C_x=6\mathrm{fF}$ (dashed line), assuming $N_{l1}=1.355$ and parameters of Eq. (3). Dash-dotted horizontal line, ${\omega }_{r1}∕2\pi =15.3\mathrm{GHz}$, shows the long-time limit of $f_d\left(t\right)$. Two dotted horizontal lines show the plasma frequency for the second qubit: ${\omega }_{l2}∕2\pi =10.2\mathrm{GHz}$ for $N_{l2}=10$ and ${\omega }_{l2}∕2\pi =8.91\mathrm{GHz}$ for $N_{l2}=5$. The arrow shows the moment $t_c$ of exact resonance in the case $N_{l2}=5$.

Figure 5

The second qubit energy $E_2$ (in units of $\hslash {\omega }_{l2}$) in the oscillator model as a function of time $t$ (in ns) for (a) $C_x=5\mathrm{fF}$ and $T_1=25\mathrm{ns}$ and (b) $C_x=2.5\mathrm{fF}$ and $5\mathrm{fF}$ and $T_1=500\mathrm{ns}$, while $N_{l2}=5$. Dashed line in (a) shows analytical approximation using Eq. (D3). The arrows show the moment $t_c$ when the driving frequency $f_d$ (see Fig. 4) is in resonance with ${\omega }_{l2}∕2\pi =8.91\mathrm{GHz}$.

Figure 6

The second-qubit energy $E_2$ (in units of $\hslash {\omega }_{l2}$) as a function of time $t$ (in ns) for $N_{l2}=5$, $T_1=25\mathrm{ns}$, and (a) $C_x=5\mathrm{fF}$, (b) $C_x=6\mathrm{fF}$, and (c) $C_x=9\mathrm{fF}$. The classical model with actual qubit potential is used; energy relaxation in the second qubit is neglected, $T_1^{\prime }=\infty$. In (b) and (c) the qubit switches (goes over the barrier) at $44\mathrm{ns}$ and $2.1\mathrm{ns}$, respectively.

Figure 7

Same as in Fig. 6 for $N_{l2}=10$, $T_1=25\mathrm{ns}$, and $C_x=6\mathrm{fF}$.

Figure 8

The log-log plot of the threshold coupling capacitance $C_{x,T}$ vs $T_1$ in the classical model assuming $T_1^{\prime }=\infty$ (solid lines) or $T_1^{\prime }=T_1$ (dashed line) for $N_{l2}=5$ (dashed and lower solid line) and 10 (upper solid line). The crosstalk excitation does not switch the second qubit if $C_x<C_{x,T}$. The numerical data are shown by the dots; the lines are just guides for the eye.

Figure 9

The second-qubit energy $E_2\left(t\right)$ in the classical model taking into account energy dissipation in the second qubit, for $N_{l2}=5$, $C_x=6\mathrm{fF}$, and $T_1^{\prime }=T_1=25\mathrm{ns}$. [Compare with Fig. 6b.]

Figure 10

(Color online) The second qubit potential $U\left(\delta \right)$ (thick line) and eigenfunctions ${\mid {\psi }_n\left(\delta \right)\mid }^2(145⩽n⩽171)$, shifted vertically by the energy eigenvalues $E_n$ (in units of $\hslash {\omega }_{l2}$) for $N_{l2}=10$ and $C_x=6\mathrm{fF}$ $({\omega }_{l2}∕2\pi =10.2\mathrm{GHz})$. The energy origin is chosen at the minimum of the left well. An arbitrary scale for ${\mid {\psi }_n\mid }^2$ is chosen to be $1∕8$ of the $E_n∕\hslash {\omega }_{l2}$ scale. The arrow illustrates the dominating transition responsible for the escape from the level $k=9$ at $t\simeq 16\mathrm{ns}$.

Figure 11

The probability $P_l=1-P_s$ for the second qubit to remain in the left well as a function of time $t$ for $N_{l2}=10$, $C_x=6\mathrm{fF}$, and $T_1=25\mathrm{ns}$.

Figure 12

The occupation probabilities $Q_k\left(t\right)$ for the left-well levels $k$ as functions of time $t$ (in ns) for $N_{l2}=10$, $C_x=6\mathrm{fF}$, and $T_1=25\mathrm{ns}$.

Figure 13

Time dependence of the mean energy $⟨E⟩$ of the second qubit [Eq. (23)] in units of $\hslash {\omega }_{l2}$ for $N_{l2}=10$, $C_x=6\mathrm{fF}$, and $T_1=25\mathrm{ns}$.

Figure 14

Dots: Rabi frequencies $R_{k,k-1}∕2\pi$ for the left-well transitions at $t=t_c$, for $N_{l2}=10$, $C_x=6\mathrm{fF}$, and $T_1=25\mathrm{ns}$. Dashed line shows analytical dependence $1.1\sqrt{k}\mathrm{GHz}$.

Figure 15

Time dependence of the frequency detunings $[{\omega }_{k,k-1}-{\omega }_d\left(t\right)]∕2\pi$ of left-well transitions $(k-1)\leftrightarrow k$ for $N_{l2}=10$, $C_x=6\mathrm{fF}$, and $T_1=25\mathrm{ns}$. Curves from top to bottom correspond to $k=1,2\cdots 11$.

Figure 16

The qubit switching (error) probability $P_s$ as a function of coupling capacitance $C_x$ (in fF) for $T_1=25$, 50, 100, 200, and $500\mathrm{ns}$ for (a) $N_{l2}=5$ and (b) $N_{l2}=10$. The numerical data are represented by points, connected by solid lines as guides for the eye. The dashed straight lines are results of the least-squares fit (notice the logarithmic scale).

Figure 17

Solid lines: log-log contour plots for the values of the error (switching) probability $P_s=0.01$, 0.1, and 0.3 on the plane of relaxation time $T_1$ (in ns) and coupling capacitance $C_x$ (in fF) in the quantum model for (a) $N_{l2}=5$ and (b) $N_{l2}=10$. The corresponding results for $C_{x,T}\left(T_1\right)$ in the classical models are shown by the dashed lines (actual potential model) and the dotted lines [oscillator model, Eq. (15)]. The numerical data are represented by the points, connected by lines as guides for the eye. The scale at the right corresponds to the operation frequency of the two-qubit imaginary-swap[9] quantum gate.

Figure 18

The graphical solution of Eq. (A1); see text.