Operation and intrinsic error budget of a two-qubit cross-resonance gate

We analyze analytically, semianalytically, and numerically the operation of a cross-resonance (CR) gate for superconducting qubits (transmons). We find that a relatively simple semianalytical method gives accurate results for the controlled-not (cnot) -equivalent gate duration and compensating single-qubit rotations. It also allows us to minimize the cnot gate duration over the amplitude of the applied microwave drive and find the dependence on the detuning between the qubits. However, full numerical simulations are needed to calculate the intrinsic fidelity of the CR gate. We decompose numerical infidelity into contributions from various physical mechanisms, thus finding the intrinsic error budget. In particular, at small drive amplitudes, the CR gate fidelity is limited by imperfections of the target-qubit rotations, while at large amplitudes it is limited by leakage. The gate duration and fidelity are analyzed numerically as functions of the detuning between qubits, their coupling, drive frequency, relative duration of pulse ramps, and microwave crosstalk. The effect of the echo sequence is also analyzed numerically. Our results show that the CR gate can provide intrinsic infidelity of less than ${10}^{-3}$ when a simple pulse shape is used.

Figure 1

Schematic of the CR gate. Detuned control and target qubits (transmons with frequencies ${\omega }_{\mathrm{c}}$ and ${\omega }_{\mathrm{t}}$) have coupling $g$ and the control qubit is microwave driven at the frequency of the target qubit, ${\omega }_{\mathrm{d}}\approx {\omega }_{\mathrm{t}}$. The microwave drive amplitude is $\varepsilon$.

Figure 2

Classical CR gate counterpart: two coupled nonlinear oscillators, with one oscillator driven by a periodic force $F$ on-resonance with the other oscillator.

Figure 3

Diagram of bare energy levels for the CR gate. Each vertical ladder is for all control-qubit states, with a fixed state of the target qubit (we use the notation $|\mathrm{control},\mathrm{target}\rangle$). Slanted blue lines illustrate coupling between the bare levels due to the qubit-qubit coupling $H_g$; orange lines are due to the drive Hamiltonian $H_{\varepsilon }$. On this diagram we assumed $\delta =0$ (a nonzero $\delta$ would produce an energy shift between the ladders; also, in this case, $\mathrm{\Delta }$ should be replaced with $\mathrm{\Delta }+\delta$). Control-qubit states above $|3\rangle$ and target-qubit states above $|2\rangle$ are not shown.

Figure 4

Effective drive amplitudes ${\tilde{\varepsilon }}_0$ (blue lines) and ${\tilde{\varepsilon }}_1$ (orange lines) as functions of the drive amplitude $\varepsilon$, calculated using the ideal-case approximation (15) and (16) (dashed straight lines), the third-order formulas (38) and (41) (solid lines without symbols), and numerically (solid lines with symbols). We used the qubit-qubit coupling $g/2\pi =3$ MHz, qubit anharmonicity ${\eta }_{\mathrm{c}}/2\pi ={\eta }_{\mathrm{t}}/2\pi =300$ MHz, and detuning $\mathrm{\Delta }/2\pi =130$ MHz.

Figure 5

The CR gate speed ${\tilde{\varepsilon }}_1-{\tilde{\varepsilon }}_0$ as a function of the drive amplitude $\varepsilon$, calculated using the linear-order approximation (19) (dashed straight line), third-order approximation (38) and (41) (solid green line), and numerically (solid blue line with symbols). The parameters are the same as in Fig. 4.

Figure 6

Effective drive amplitudes ${\tilde{\varepsilon }}_0$ and ${\tilde{\varepsilon }}_1$ as functions of $\varepsilon$, calculated numerically (solid lines with symbols) and using the semianalytical approach (26, 27, 28, 29) (dashed lines, practically coinciding with the solid lines). We used $g/2\pi =3$ MHz, ${\eta }_{\mathrm{c}}/2\pi ={\eta }_{\mathrm{t}}/2\pi =300$ MHz, and two values for the detuning: $\mathrm{\Delta }/2\pi =130$ and 190 MHz.

Figure 7

Dimensionless CR gate speed $({\tilde{\varepsilon }}_1-{\tilde{\varepsilon }}_0)/g$ as a function of the dimensionless drive amplitude $\varepsilon /{\eta }_{\mathrm{c}}$ for several values of the dimensionless detuning $\mathrm{\Delta }/{\eta }_{\mathrm{c}}$. The lines are calculated using the semianalytical method (26, 27, 28, 29). At large $\varepsilon$, the lines group into bands. Group I is for $\mathrm{\Delta }/{\eta }_{\mathrm{c}}<0$ and group II is for $\mathrm{\Delta }/{\eta }_{\mathrm{c}}$ in the interval $(0,\frac{1}{2})$. Similarly, groups III, IV, and V are for $\mathrm{\Delta }/{\eta }_{\mathrm{c}}$ in the intervals $(\frac{1}{2},1), (1,\frac{3}{2})$, and $(\frac{3}{2},2)$, respectively.

Figure 8

Maximized (minimized for negative values) dimensionless speed ${({\tilde{\varepsilon }}_1-{\tilde{\varepsilon }}_0)}_{\max }/g$ (solid blue line) and corresponding dimensionless drive amplitude $\varepsilon /{\eta }_{\mathrm{c}}$ (dashed orange line), as functions of the dimensionless detuning $\mathrm{\Delta }/{\eta }_{\mathrm{c}}$. The lines are calculated using the semianalytical method (26, 27, 28, 29).

Figure 9

Pulse shape $\varepsilon \left(t\right)$ used in numerical simulations. The total pulse duration is ${\tau }_{\mathrm{p}}$, each cosine-shaped ramp has duration ${\tau }_{\mathrm{r}}$, and the drive amplitude in the middle flat part is ${\varepsilon }_{\mathrm{m}}$.

Figure 10

cnot gate duration ${\tau }_{\mathrm{p}}^{\mathrm{CNOT}}$ (neglecting single-qubit rotations) as a function of the midpulse drive amplitude ${\varepsilon }_{\text{m}}$ for several detunings: $\mathrm{\Delta }/2\pi =-70$, 70, 130, and 190 MHz, while $g/2\pi =3$ MHz, ${\eta }_{\mathrm{c}}/2\pi ={\eta }_{\mathrm{t}}/2\pi =300$ MHz, ${\tau }_{\mathrm{r}}/{\tau }_{\mathrm{p}}=0.3$, and ${\omega }_{\mathrm{d}}={\omega }_{\mathrm{t}}^{\mathrm{c}0}$. Solid lines are calculated numerically; dashed lines (almost coinciding with the solid lines) are calculated using the semianalytical method.

Figure 11

Solid lines show the dependence of the cnot gate duration ${\tau }_{\mathrm{p}}^{\mathrm{CNOT}}$ on ${\varepsilon }_{\mathrm{m}}$ for three drive frequencies: ${\omega }_{\mathrm{d}}={\omega }_{\mathrm{t}}^{\mathrm{c}0}$ (exact resonance for the control-qubit state $|0\rangle$, blue line), ${\omega }_{\mathrm{d}}={\omega }_{\mathrm{t}}^{\mathrm{c}1}$ (resonance for the control-qubit state $|1\rangle$, orange line), and ${\omega }_{\mathrm{d}}=({\omega }_{\mathrm{t}}^{\mathrm{c}0}+{\omega }_{\mathrm{t}}^{\mathrm{c}1})/2$ (exactly in between, green line). The dashed line shows the semianalytical results and the straight dotted line corresponds to Eq. (52). We use $\mathrm{\Delta }/2\pi =130$ MHz; the other parameters are the same as in Fig. 10.

Figure 12

Angle $-{\phi }_0$ of the compensating $x$ rotation of the target qubit to produce the cnot gate, as a function of the midpulse drive amplitude ${\varepsilon }_{\mathrm{m}}$. Numerical results are shown by solid lines, semianalytics are represented by dashed lines, and dotted lines show the ideal result $-{\phi }_0/\pi =({\eta }_{\mathrm{c}}-\mathrm{\Delta })/2{\eta }_{\mathrm{c}}$. Here we use $\mathrm{\Delta }/2\pi =130$ and 190 MHz, ${\omega }_{\mathrm{d}}={\omega }_{\mathrm{t}}^{\mathrm{c}0}, g/2\pi =3$ MHz, ${\eta }_{\mathrm{c}}/2\pi ={\eta }_{\mathrm{t}}/2\pi =300$ MHz, and ${\tau }_{\mathrm{r}}/{\tau }_{\mathrm{p}}=0.3$.

Figure 13

The upper (blue) solid line shows the numerical result for the angle ${\theta }_0^{\prime }-{\theta }_1^{\prime }+\pi /2$ of $z$ rotation of the control qubit, needed to produce the cnot gate. The horizontal (brown) line shows the ideal value $\pi /2$. Dashed and dotted red lines show the contribution ${\theta }_{\mathrm{rep}}$ due to $\varepsilon$-induced level repulsion (54), calculated either semianalytically (dashed line) or via Eqs. (37) and (40) (dotted line). The lower (green) solid line shows the contribution ${\theta }_{zz}$ given by Eq. (55). The upper solid and dashed black lines (very close to the solid blue line) show the sum ${\theta }_{\mathrm{rep}}+{\theta }_{zz}+\pi /2$. We use $\mathrm{\Delta }/2\pi =130$ MHz and the parameters from Fig. 12.

Figure 14

Infidelity $1-F_{MU}$ of the cnot-equivalent gate as a function of midpulse drive amplitude ${\varepsilon }_{\text{m}}$ for several values of detuning: $\mathrm{\Delta }/2\pi =-70$ MHz (red line), 70 MHz (brown line), 130 MHz (blue line), and 190 MHz (green line). The other parameters are $g/2\pi =3$ MHz, ${\eta }_{\mathrm{c}}/2\pi ={\eta }_{\mathrm{t}}/2\pi =300$ MHz, ${\omega }_{\mathrm{d}}={\omega }_{\mathrm{t}}^{\mathrm{c}0}$, and ${\tau }_{\mathrm{r}}/{\tau }_{\mathrm{p}}=0.3$.

Figure 15

Decomposition of the cnot gate infidelity $1-F_{MU}$ (solid blue line) into the leakage contribution $1-F_{M\tilde{M}}$ (orange line) and contribution $1-F_{\tilde{M}U}$ due to imperfect unitaries (green line). The sum $(1-F_{M\tilde{M}})+(1-F_{\tilde{M}U})$ (dashed blue line) is practically indistinguishable from the solid blue line. We use $\mathrm{\Delta }/2\pi =130$ MHz and the parameters from Fig. 14.

Figure 16

Further decomposition of the imperfect-unitary infidelity contribution $1-F_{\tilde{M}U}$ (green line) into contributions $\mathrm{\Delta }{F}_{\tilde{U},\mathrm{c}0}$ (orange line) and $\mathrm{\Delta }{F}_{\tilde{U},\mathrm{c}1}$ (solid blue line) for the control-qubit states $|0\rangle$ and $|1\rangle$, respectively. The dotted blue line shows the analytical approximation for $\mathrm{\Delta }{F}_{\tilde{U},\mathrm{c}1}$ by Eq. (59) with ideal values for ${\phi }_1$ and ${\tilde{\varepsilon }}_1$, while for the dashed blue line we use semianalytical values for ${\phi }_1$ and ${\tilde{\varepsilon }}_1$ in Eq. (59). The parameters are the same as in Fig. 15.

Figure 17

Numerical results for the leakage. The thick orange line shows the infidelity contribution $1-F_{M\tilde{M}}$. Four thin solid lines (labeled 1, 2, 3, and 4) show multiplied by the factor $\frac{1}{4}$ probabilities of the main leakage channels: $\overline{|0,0\rangle }\to \overline{|2,0\rangle }, \overline{|0,1\rangle }\to \overline{|2,1\rangle }, \overline{|0,0\rangle }\to \overline{|2,1\rangle }$, and $\overline{|0,1\rangle }\to \overline{|2,0\rangle }$. The sum of these four lines is shown by the dotted brown line; its closeness to the thick orange line verifies that the infidelity $1-F_{M\tilde{M}}$ is mainly due to these leakage channels. The dashed black line is the leakage probability estimate given by Eq. (61). The parameters are the same as in Fig. 15.

Figure 18

Parametric plot for the cnot gate infidelity $1-F_{MU}$ versus the cnot gate duration ${\tau }_{\mathrm{p}}^{\mathrm{CNOT}}$ (the running parameter is ${\varepsilon }_{\mathrm{m}}$) for the detunings $\mathrm{\Delta }/2\pi =-70$, 70, 130, and 190 MHz. We assume $g/2\pi =3$ MHz, ${\eta }_{\mathrm{c}}/2\pi ={\eta }_{\mathrm{t}}/2\pi =300$ MHz, ${\omega }_{\mathrm{d}}={\omega }_{\mathrm{t}}^{\mathrm{c}0}$, and ${\tau }_{\mathrm{r}}/{\tau }_{\mathrm{p}}=0.3$.

Figure 19

cnot gate infidelity $1-F_{MU}$ versus duration ${\tau }_{\mathrm{p}}^{\mathrm{CNOT}}$ (both are functions of ${\varepsilon }_{\mathrm{m}}$) for $\mathrm{\Delta }/2\pi =70$ and 190 MHz and the drive frequency on-resonance with the target qubit for the control qubit either in the state $|0\rangle$ (solid lines, ${\omega }_{\mathrm{d}}={\omega }_{\mathrm{t}}^{\mathrm{c}0}$) or in the state $|1\rangle$ (dotted lines, ${\omega }_{\mathrm{d}}={\omega }_{\mathrm{t}}^{\mathrm{c}1}$) or exactly in between [dashed lines, ${\omega }_{\mathrm{d}}=({\omega }_{\mathrm{t}}^{\mathrm{c}0}+{\omega }_{\mathrm{t}}^{\mathrm{c}1})/2]$. The other parameters are the same as in Fig. 18.

Figure 20

cnot gate infidelity $1-F_{MU}$ versus duration ${\tau }_{\mathrm{p}}^{\mathrm{CNOT}}$ for $\mathrm{\Delta }/2\pi =190$ MHz and several values of the qubit-qubit coupling: $g/2\pi =1.5$, 3, and 6 MHz. The other parameters are the same as in Fig. 18.

Figure 21

cnot gate infidelity $1-F_{MU}$ versus duration ${\tau }_{\mathrm{p}}^{\mathrm{CNOT}}$ for $\mathrm{\Delta }/2\pi =190$ MHz and several values of the relative duration of the pulse ramps: ${\tau }_{\mathrm{r}}/{\tau }_{\mathrm{p}}=0.1$ (dotted line), 0.2 (dashed line), 0.3 (solid line), 0.4 (dash-dotted line), and 0.5 (long-dashed line). The other parameters are the same as in Fig. 18. The lines are cut at the left for clarity.

Figure 22

cnot gate infidelity $1-F_{MU}$ versus duration ${\tau }_{\mathrm{p}}^{\mathrm{CNOT}}$ for $\mathrm{\Delta }/2\pi =190$ MHz and several values of the microwave crosstalk coefficient: $c_{\mathrm{ct}}=0$ (solid line), 0.05 (dotted line), 0.1 (dashed line), and 0.2 (dash-dotted line). The other parameters are the same as in Fig. 18. The lines are cut at the left for clarity.

Figure 23

cnot gate infidelity $1-F_{MU}$ versus duration ${\tau }_{\mathrm{p}}^{\mathrm{CNOT}}$ for the echo-CR gate (solid lines) and the basic CR gate (dashed lines). The detunings $\mathrm{\Delta }/2\pi$ are 190 MHz (green lines), 130 MHz (blue lines), 70 MHz (brown lines), and $-70$ MHz (magenta lines). The parameters are the same as in Fig. 18, except we use ${\omega }_{\mathrm{d}}=({\omega }_{\mathrm{t}}^{\mathrm{c}0}+{\omega }_{\mathrm{t}}^{\mathrm{c}1})/2$. The lines are cut at the left for clarity.

Figure 24

Symbols show the numerical results for the cnot gate duration ${\tau }_{\mathrm{p}}^{\mathrm{CNOT}}$ for the infidelity levels of 0.003 (crosses), 0.01 (triangles), and 0.03 (circles), for several values of the detuning $\mathrm{\Delta }$ (horizontal axis): $-200, -170, -130, -70, -40$, 40, 70, 100, 130, 170, 190, 210, 230, and 250 MHz. Green (thicker) symbols are for the echo-CR gate; orange (thinner) symbols are for the basic CR gate. We use the parameters $g/2\pi =3$ MHz, ${\eta }_{\mathrm{c}}/2\pi ={\eta }_{\mathrm{t}}/2\pi =300$ MHz, and ${\tau }_{\mathrm{r}}/{\tau }_{\mathrm{p}}=0.3$ (results on the dimensionless scales should not depend much on these parameters, except for ${\tau }_{\mathrm{r}}/{\tau }_{\mathrm{p}}$). The solid line shows the results of the semianalytical theory (optimized over the drive amplitude) for the same pulse shape. The dashed line shows the semianalytical results for a rectangular pulse (the same as in Fig. 8, but the vertical scale is inverted). Both dimensionless and dimensional scales are given for the axes.