Robust quantum state transfer using tunable couplers

We analyze the transfer of a quantum state between two resonators connected by a superconducting transmission line. Nearly perfect state-transfer efficiency can be achieved by using adjustable couplers and destructive interference to cancel the back-reflection into the transmission line at the receiving coupler. We show that the transfer protocol is robust to parameter variations affecting the transmission amplitudes of the couplers. We also show that the effects of the Gaussian filtering, pulse-shape noise, and multiple reflections on the transfer efficiency are insignificant. However, the transfer protocol is very sensitive to frequency mismatch between the two resonators. Moreover, the tunable coupler we considered produces time-varying frequency detuning caused by the changing coupling. This detuning requires an active frequency compensation with an accuracy better than $90\%$ to yield the transfer efficiency above $99\%$.

Figure 1

(a) The state transfer setup. An initial microwave field amplitude $G\left(0\right)$ is transferred from the emitting resonator to the receiving resonator via a transmission line. This is done using variable couplers for both resonators, characterized by (effective) transmission amplitudes ${\mathbf{t}}_{\mathrm{e}}\left(t\right)$ and ${\mathbf{t}}_{\mathrm{r}}\left(t\right)$, and corresponding leakage rates ${\kappa }_{\mathrm{e}}\left(t\right)$ and ${\kappa }_{\mathrm{r}}\left(t\right)$. Almost perfect transfer can be achieved when the back-reflection of the propagating field $A\left(t\right)$ is canceled by arranging its destructive interference with the leaking part of the field $B\left(t\right)$ in the receiving resonator. (b) A variant of the setup that includes a circulator, which prevents multiple reflections of the small back-reflected field $F\left(t\right)$.

Figure 2

Time dependence (“pulse shapes”) of the absolute values of transmission amplitudes ${\mathbf{t}}_{\mathrm{e}}\left(t\right)$ for the emitting coupler (red dashed curve) and ${\mathbf{t}}_{\mathrm{r}}\left(t\right)$ for the receiving coupler (blue solid curve). The amplitude ${\mathbf{t}}_{\mathrm{e}}\left(t\right)$ is kept constant at the maximum level ${\mathbf{t}}_{\mathrm{e},\max }$ after the midtime $t_{\mathrm{m}}$, while ${\mathbf{t}}_{\mathrm{r}}\left(t\right)$ is kept at the maximum ${\mathbf{t}}_{\mathrm{r},\max }$ during the first part of the procedure, $t\le t_{\mathrm{m}}$. The propagating field $A\left(t\right)$ first increases exponentially and then decreases exponentially (black solid curve). In simulations we typically use $|{\mathbf{t}}_{\mathrm{e},\max }|=|{\mathbf{t}}_{\mathrm{r},\max }|=0.05$ for quarter-wavelength 6-GHz resonators (${\tau }_{\mathrm{e}}={\tau }_{\mathrm{r}}=33$ ns); then the transfer efficiency $\eta =0.999$ requires the procedure duration of $t_{\mathrm{f}}=460$ ns.

Figure 3

Inefficiency $1-\eta$ of the state transfer procedure as a function of relative variation of the maximum transmission amplitudes $\delta {\mathbf{t}}_{\max }/{\mathbf{t}}_{\max }=({\mathbf{t}}_{\mathrm{e}\left(\mathrm{r}\right),\max }^{\mathrm{a}}-{\mathbf{t}}_{\mathrm{e}\left(\mathrm{r}\right),\max })/{\mathbf{t}}_{\mathrm{e}\left(\mathrm{r}\right),\max }$ for design efficiencies ${\eta }_{\mathrm{d}}=0.99$ (blue curves) and $0.999$ (red curves). The maximum transmission amplitudes ${\mathbf{t}}_{\mathrm{e},\max }$ and ${\mathbf{t}}_{\mathrm{r},\max }$ are either varied simultaneously (dashed curves) or one of them is kept at the design value (solid curves). The superscript “a” indicates an “actual” value, different from the design value.

Figure 4

Dependence of the inefficiency $1-\eta$ on relative variation of the buildup/leakage time $\delta {\tau }_{\mathrm{e}\left(\mathrm{r}\right)}/\tau =({\tau }_{\mathrm{e}\left(\mathrm{r}\right)}^{\mathrm{a}}-\tau )/\tau$ for design efficiencies ${\eta }_{\mathrm{d}}=0.99$ (blue curves) and $0.999$ (red curves), assuming ${\tau }_{\mathrm{e}}={\tau }_{\mathrm{r}}=\tau$. The buildup/leakage times ${\tau }_{\mathrm{e}}^{\mathrm{a}}$ and ${\tau }_{\mathrm{r}}^{\mathrm{a}}$ are varied either simultaneously (dashed curves) or one of them is kept at the design value (solid curves).

Figure 5

Inefficiency $1-\eta$ as a function of the midtime shift $\delta t_{\mathrm{m}}^{\mathrm{r}}=t_{\mathrm{m}}^{\mathrm{a},\mathrm{r}}-t_{\mathrm{m}}$ normalized by the buildup/leakage time $\tau$. The midtime $t_{\mathrm{m}}^{\mathrm{a},\mathrm{e}}$ is either varied equally (dashed curves) or kept constant (solid curves). The results for varying only $t_{\mathrm{m}}^{\mathrm{a},\mathrm{e}}$ are the same as the solid curves up to the sign change, $\delta t_{\mathrm{m}}^{\mathrm{r}}\leftrightarrow -\delta t_{\mathrm{m}}^{\mathrm{e}}$.

Figure 6

Dependence of the inefficiency $1-\eta$ on the warping parameters ${\alpha }_{\mathrm{e}}$ and ${\alpha }_{\mathrm{r}}$, introduced in Eq. (37) to describe the pulse shape distortion, for design efficiencies ${\eta }_{\mathrm{d}}=0.99$ (blue curves) and $0.999$ (red curves). The solid curves show the case when only one warping parameter is nonzero (the results coincide); the dashed curves are for the case ${\alpha }_{\mathrm{e}}={\alpha }_{\mathrm{r}}$.

Figure 7

Inefficiency $1-\eta$ as a function of the width of a Gaussian filter $\sigma$ (in ns) for design efficiencies ${\eta }_{\mathrm{d}}=0.9$ (green dashed curve), $0.99$ (blue dot-dashed curve), and $0.999$ (red solid curve). We use $\tau =33.3\mathrm{ns}$, as in Fig. 2. The upper horizontal axis shows the normalized value $\sigma /\tau$.

Figure 8

Solid lines: inefficiency $1-\eta$ averaged over 100 random noise realizations, as a function of the dimensionless noise amplitude $a$, for the multiplicative noise (red lines, bottom), Eq. (40), and the additive noise (blue lines, top), Eq. (41); both with $\overline{{\xi }^2}=0.78$. The error bars show the standard deviation for some values of $a$. The results are shown for ${\eta }_{\mathrm{d}}=0.99$ and $0.999$. In the simulation we used the time step $dt=1\mathrm{ns}$ and parameters of the procedure in Fig. 2 ($\tau =33.3\mathrm{ns}$). Black dotted lines (practically coinciding with the solid lines) are calculated by replacing the noise with the effective increase of the leakage rates ${\kappa }_{\mathrm{e}\left(\mathrm{r}\right)}$ (see text).

Figure 9

Illustration of the back-reflected field $\left|F\right(t-t_{\mathrm{d}}\left)\right|$ reaching the emitting resonator at time $t$, for the round-trip delay time $t_{\mathrm{d}}=t_{\mathrm{f}}/2$ (blue dashed curve) and $t_{\mathrm{d}}=t_{\mathrm{f}}/5$ (red solid curve), assuming the round-trip phase shift $\phi =\pi /8$. The kinks represent multiple reflections of the field emitted at $t=0$. We assumed parameters of Fig. 2 (${\eta }_{\mathrm{d}}=0.999$, $\tau =33.3$ ns, $t_{\mathrm{f}}=460$ ns).

Figure 10

(a) Dependence of the inefficiency $1-\eta$ on the normalized delay $t_{\mathrm{d}}/\tau$ due to the round trip along the transmission line, for the design efficiency ${\eta }_{\mathrm{d}}=0.999$ and several values of the phase shift $\phi$ accumulated in this round trip. The kinks at $t_{\mathrm{d}}/\tau =13.8/n$ correspond to the integer number $n$ of the round trips within the procedure time $t_{\mathrm{f}}$. (b) The same as in (a) for a smaller range of $t_{\mathrm{d}}/\tau$ (the results for $t_{\mathrm{d}}/\tau <0.1$ were not calculated). Notice that the inefficiency accounting for multiple reflections does not exceed twice the design inefficiency, $1-\eta \le 2(1-{\eta }_{\mathrm{d}})$.

Figure 11

Inefficiency $1-\eta$ as a function of normalized detuning $\Delta \omega \tau$ (lower horizontal axis) for three design efficiencies, ${\eta }_{\mathrm{d}}=0.9,$ $0.99$, and $0.999$. The upper horizontal axis shows the unnormalized detuning $\Delta \omega /2\pi$ in MHz, using the values ${\omega }_{}/2\pi =6$ GHz and $|{\mathbf{t}}_{\mathrm{e},\max }|=|{\mathbf{t}}_{\mathrm{r},\max }|=0.05$, so that $\tau =33.3$ ns.

Figure 12

Red solid line: the resonator frequency detuning $-\Delta {\omega }_{\mathrm{e}\left(\mathrm{r}\right)}/2\pi$ caused by changing $|{\mathbf{t}}_{\mathrm{e}\left(\mathrm{r}\right)}|$ for a particular set of parameters of the coupler (see text). Blue dashed line: the corresponding value of the coupler mutual inductance $M_{\mathrm{e}\left(\mathrm{r}\right)}$. The arrows indicate the corresponding vertical axes.

Figure 13

Solid line: the phase of the transmission amplitude, $arg\left({\mathbf{t}}_{\mathrm{e}\left(\mathrm{r}\right)}\right)$, as a function of its absolute value $|{\mathbf{t}}_{\mathrm{e}\left(\mathrm{r}\right)}|$ for a particular set of coupler parameters (see text). Dashed line: the normalized detuning $-\Delta {\omega }_{\mathrm{e}\left(\mathrm{r}\right)}/{\kappa }_{\mathrm{e}\left(\mathrm{r}\right)}=-\pi \Delta {\omega }_{\mathrm{e}\left(\mathrm{r}\right)}/{\omega }_{\mathrm{e}\left(\mathrm{r}\right)}{\left|{\mathbf{t}}_{\mathrm{e}\left(\mathrm{r}\right)}\right|}^2$.

Figure 14

Inefficiency $1-\eta$ as a function of $|{\mathbf{t}}_{\mathrm{e},\max }|=|{\mathbf{t}}_{\mathrm{r},\max }|$ for the couplers with parameters described in the text. The solid lines are for the design efficiency ${\eta }_{\mathrm{d}}=0.999$, while the dashed lines are for ${\eta }_{\mathrm{d}}=0.99$. The red lines show the results without compensation of the frequency detuning $\Delta {\omega }_{\mathrm{e}\left(\mathrm{r}\right)}\left(t\right)$ caused by changing $|{\mathbf{t}}_{\mathrm{e}\left(\mathrm{r}\right)}\left(t\right)|$ and correspondingly changing $arg\left({\mathbf{r}}_{\mathrm{e}\left(\mathrm{t}\right)}^{\mathrm{in}}\right)$. The blue lines assume 90% compensation of this detuning, 95% compensation for the green lines, 99% compensation for the magenta lines, and full 100% compensation for the black lines. For the black lines, the extra inefficiency is caused only by changing phases of ${\mathbf{t}}_{\mathrm{e}\left(\mathrm{r}\right)}$. The upper horizontal axis shows the product $\Delta {\omega }_{\max }\tau$, corresponding to $|{\mathbf{t}}_{\max }|$.

Figure 15

A beam splitter with input classical fields $A$ and $B$ transformed into the output fields $\tilde{A}$ and $\tilde{B}$, with the main transformation $A\to \tilde{A}$ characterized by the amplitude $\sqrt{\eta }$ and phase shift ${\phi }_{\mathrm{f}}$. In the quantum formulation, the input state $|{\Psi }_{\mathrm{in}}\rangle$ is transformed into the output state $|{\Psi }_{\mathrm{out}}\rangle$. In particular, in Sec. A 1, we consider the input state $|{\Psi }_{\mathrm{in}}\rangle =|{\psi }_{\mathrm{in}}\rangle |0\rangle$, calculate $|{\Psi }_{\mathrm{out}}\rangle$, and then reduce it to the density matrix ${\rho }_{\mathrm{fin}}$ of the main output arm, by tracing over the other output arm.

Figure 16

Schematic of the tunable coupler of Refs. [20, 22] between the $\lambda /4$ microwave resonator (at the left) and the transmission line (at the right). A voltage taken at the distance $d$ from the resonator end is applied to a transformer with a negative mutual inductance $-M_g$ and a SQUID providing positive Josephson inductance $L_J$. External flux ${\Phi }_{\mathrm{ext}}$ controls $L_J$, thus controlling the effective mutual inductance $M=-M_g+L_J$. The wave impedances of the lines are $R_{\mathrm{r}}$ and $R_{\mathrm{tl}}$.

Figure 17

The simplified schematic of Fig. 16, with the $d$-long piece of the resonator replaced by inductance $L_e$, and the transformer in series with SQUID replaced by an effective transformer with mutual inductance $M$. An incident wave with voltage amplitude $\mathcal{B}$ creates voltages $V$ and $x$ across the inductors $L_1$ and $L_2$. The wave is reflected as ${\mathbf{r}}^{\mathrm{in}}\mathcal{B}$ and transmitted as ${\tilde{\mathbf{t}}}^{\mathrm{in}}\mathcal{B}$ (the superscript “in” indicates the wave coming from inside the resonator and the tilde sign indicates the actual transmission amplitude, as opposed to the effective amplitude $\mathbf{t}$). In our case, ${\mathbf{r}}^{\mathrm{in}}\approx -1$ and $|{\tilde{\mathbf{t}}}^{\mathrm{in}}|\ll 1$.