Mixed-mode RF reflectometry of quantum dots for reduction of crosstalk effects

Practical quantum computation would require millions of qubits. ^1,2) RF reflectometry is a promising technique to readout qubits in such a large-scale system, ^3–14) thanks to its capability for signal multiplexing ^3,15) and gate-based dispersive readout. ^4,7) Reflectometry signals from the resonance circuits connected to qubits can be multiplexed in the frequency domain ⁴⁾ as well as by phase encoding, using common readout circuitry outside the qubit chip. ¹⁶⁾ However, RF lines for such readout may suffer from on-chip crosstalk when the density of the RF lines becomes higher. ^17,18) The crosstalk will mix up signals, ^19–21) resulting in unwanted applications of RF signals to unintended qubits and indistinguishable signals reflected from multiple qubits.

In this study, we propose a mixed-mode RF reflectometry to reduce crosstalk effects. ^22–25) In the mixed mode, reflected signals are transmitted by differential and common modes of multiple transmission lines. By taking the coupling among transmission lines into account and superimposing the differential and common modes, we can obtain the reflection coefficient from a single line, hence reading out the individual qubit state, without crosstalk. We confirmed the effectiveness of the mixed mode in an electromagnetic field simulation software based on finite element method (Ansys HFSS). This paper is an extended version of Ref. 26 and includes a detailed explanation of mixed-mode transmission and simulation results. These results show crosstalk increases the impact of noise on reflectometry phase measurements, which can be suppressed by analyzing mixed-mode transmission. These considerations confirm the effectiveness of mixed-mode transmission in reducing the influence of crosstalk.

Mixed mode transmits differential and common modes simultaneously. Advantages of mixed-mode transmission include improved immunity against noise and sensitivity to small signals in the differential mode. In the differential transmission, waves with the opposite polarities are passed through microstrip lines (MSLs), and the difference between the two signals is utilized. In the field of information and telecommunication, the common transmission is used only to measure the symmetry of the two transmission lines. In this paper, we use both of the two transmissions to obtain (ideal) single-ended S parameters without crosstalk influence ${S}^{{\rm{s}}}$ by converting the mixed-mode S parameters ${{S}}^{{\rm{m}}}$ to single-ended S parameters. We note that the ideal single-ended S parameters ${S}^{{\rm{s}}}$ cannot be calculated only from single-ended S parameters including crosstalk effect ${S}^{{\rm{x}}},$ which are measured by standard RF reflectometry. In order to reduce the effect of crosstalk, the information on crosstalk coupling is indispensable.

For the S parameter conversion, we shall introduce the conversion matrix here. With a crosstalk influence, the conversion between mixed-mode S parameter (${S}^{{\rm{m}}}$) and single-ended S parameter (${S}^{{\rm{s}}}$) is expressed with the modulation of the MSL characteristic impedance by crosstalk taken into account, by the following equations: ²²⁾

$$\begin{eqnarray}&&\left[{{S}}^{{\rm{s}}}\right]={\left(\left[{{M}}_{1}\right]-\left[{{S}}^{{\rm{m}}}\right]\left[{{M}}_{2}\right]\right)}^{-1}\left(\left[{{S}}^{{\rm{m}}}\right]\left[{{M}}_{1}\right]-\left[{{M}}_{2}\right]\right),\end{eqnarray} \tag{ 1a }$$

$$\begin{eqnarray}\left[{{M}}_{1}\right]=\left[\begin{array}{cc}\frac{1+{{k}}_{{\rm{oo}}}}{2\sqrt{2{{k}}_{{\rm{oo}}}}} & -\frac{1+{{k}}_{{\rm{oo}}}}{2\sqrt{2{{k}}_{{\rm{oo}}}}}\\ \frac{1+{{k}}_{{\rm{oe}}}}{2\sqrt{2{{k}}_{{\rm{oe}}}}} & \frac{1+{{k}}_{{\rm{oe}}}}{2\sqrt{2{{k}}_{{\rm{oe}}}}}\end{array}\right],\end{eqnarray} \tag{ 1b }$$

$$\begin{eqnarray}\left[{{M}}_{2}\right]=\left[\begin{array}{cc}\frac{1-{{k}}_{{\rm{oo}}}}{2\sqrt{2{{k}}_{{\rm{oo}}}}} & -\frac{1-{{k}}_{{\rm{oo}}}}{2\sqrt{2{{k}}_{{\rm{oo}}}}}\\ \frac{1-{{k}}_{{\rm{oe}}}}{2\sqrt{2{{k}}_{{\rm{oe}}}}} & \frac{1-{{k}}_{{\rm{oe}}}}{2\sqrt{2{{k}}_{{\rm{oe}}}}}\end{array}\right],\end{eqnarray} \tag{ 1c }$$

$$\begin{eqnarray}\left[{{S}}^{{\rm{m}}}\right]=\left[\begin{array}{cc}{{S}}_{{\rm{dd}}} & {{S}}_{{\rm{dc}}}\\ {{S}}_{{\rm{cd}}} & {{S}}_{{\rm{cc}}}\end{array}\right],\end{eqnarray} \tag{ 1d }$$

where ${k}_{{\rm{oo}}}={Z}_{{\rm{d}}}/2{Z}_{0}$ and ${k}_{{\rm{oe}}}={2Z}_{{\rm{c}}}/{Z}_{0}$ represent the crosstalk effects (${Z}_{0},$ ${Z}_{{\rm{d}}},$ and ${Z}_{{\rm{c}}}$ are MSL characteristic impedances without crosstalk influence and differential- and common-mode characteristic impedances, respectively). In the absence of crosstalk, ${k}_{\mathrm{oo}}={k}_{\mathrm{oe}}=1.$ These Eqs. (1a )–(1d ) are used later in this article to obtain single-ended S parameters to check the suppression of crosstalk influence by mixed-mode transmission.

Based on the theoretical background described above, one can experimentally derive the single-ended S parameters with suppressed crosstalk effects from the following three steps. Firstly, we obtain characteristic impedances for differential and common modes (${Z}_{{\rm{d}}}$ and ${Z}_{{\rm{c}}}$) by simulations or measurements. ²²⁾ These characteristic impedances are important parameters to obtain ${k}_{{\rm{oo}}}$ and ${k}_{{\rm{oe}}}$ in Eqs. (1a )–(1d ). Secondly, one measures the mixed-mode S parameters ${S}^{{\rm{m}}}$ by RF reflectometry through a magic-T circuit. ²⁷⁾ Finally, the single-ended S parameters ${S}^{{\rm{s}}}$ are calculated by substituting these parameters into Eqs. (1a)–(1d ).

In what follows, we aim to simulate the influence of crosstalk on gate-based spin-qubit readout signals. We first quantify the effect of crosstalk for single-ended RF reflectometry, i.e., reflectometry without mixed-mode. We then demonstrate that signal degradation due to crosstalk will be suppressed in reflectometry with mixed-mode as theoretically discussed above. The simulation model used in this study is shown in Fig. 1(a). The model includes two adjacent MSLs. The parameter values used in the simulation are shown in Table I. Each MSL can be regarded as a transmission line for RF reflectometry to readout a qubit placed at the end of the line. Lumped resonance circuits consisting of inductance $L,$ capacitance $C,$ and resistance $R$ are located at the end of line as a synthetic impedance for a qubit device and its readout LC resonator. In this analysis, $L=10\,$nH, $C=100$fF, and $R=1\,$MΩ. ^13,28) $C$ and $R$ are determined based on typical values for real qubit systems; on the other hand, a small inductance is chosen for $L,$ so that the resonance frequency is in the order of GHz. Such a relatively high resonance frequency for a spin qubit readout will be desirable in future integrated qubit systems where dense frequency multiplexing and wide bandwidth are needed at the same time. In this condition, the carrier signal is almost completely reflected with a phase shift [see Fig. 1(b)].

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To simulate the S parameters ${S}^{{\rm{s}}}$ and ${S}^{{\rm{x}}}$ of the two-MSL model, we utilized two types of RF ports: wave port and lumped port. The wave port is compatible with simulations where electromagnetic fields are spread under two MSLs, meaning that it can be used to analyze mixed modes containing differential and common modes. By mixed-mode analysis ²²⁾ the single line reflection coefficient without crosstalk can be obtained. On the other hand, a lumped port is used to calculate crosstalk effects in the MSLs. By comparing results for the two types of ports, we reveal the reduction of crosstalk effects in the RF reflectometry by mixed mode.

To confirm the effectiveness of the mixed mode, we first simulated the amplitudes of ${S}_{11},$ $| {S}_{11}| ,$ with and without the mixed mode (i.e., $|{S}_{11}^{{\rm{s}}}|$ and $| {S}_{11}^{{\rm{x}}}|$). Here, ${S}_{11}$ represents the reflection at port 1, and the reduction in its amplitude indicates crosstalk effects. We performed this analysis as a function of the inter-MSL distance $d$ with a carrier frequency of 4.5 GHz. In the simulation, w = 0.3 mm and d is changed between 1 μm and 1 mm. While d much smaller than 0.3 mm would be only meaningful when w is also reduced, we fix w to 0.3 mm in this work to keep the MSL characteristic impedance to 50 Ω.

As seen in Fig. 2(a), without mixed mode, $| {S}_{11}|$ decreases as $d$ decreases, indicating the crosstalk effects. In contrast, as shown in Fig. 2(a), the mixed-mode analysis yields $| {S}_{11}|$ that is essentially independent of $d$ and almost equal to unity — an expected behavior without crosstalk. These results indicate an effectiveness of the mixed-mode detection.

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In some spin readout protocols such as quantum capacitance readout, RF reflectometry is used to read out a small capacitance change (∼1 fF) caused by the qubit state. ²⁹⁾ In the next simulation we simulate ${S}_{11}$ for the two capacitance conditions, 100 fF and 101 fF, to emulate the qubit readout signal in such scenarios. Hereafter, ${S}_{11}$ (${S^{\prime} }_{11}$) represents the reflection coefficient at $C=100$ fF ($101$ fF). Figures 2(b) and 2(c) show the results of the shift in ${S}_{11}^{{\rm{x}}}$ by the capacitance change as a function of $d.$ As seen in the figures, the amplitude is affected by crosstalk; on the other hand, the phase is insensitive to $d.$ These results indicate that phase readout is more robust against crosstalk than amplitude readout in our case, suggesting that it is more suitable for large-scale qubit systems with dense RF readout lines.

We simulated a spin readout measurement with crosstalk and revealed how mixed-mode detection can suppress the crosstalk influence. Here, we evaluate how this helps to readout the qubit state in the presence of noise. ³⁰⁾ To evaluate this effect, we assume a finite noise in the complex plane for the reflected signals and calculate a signal-to-noise ratio (SNR) dependence on the MSL distance $d.$ Here, we focus on the SNR for the RF phase, where we define the signal θ as the phase difference between the reflected signals for $C=100$ fF and $101$ fF, and the noise Φ as the phase broadening by noise $\sigma$ [Fig. 3(a)]. $\sigma$ is a standard deviation defined by $\sigma =\sqrt{\unicode{x03008}{\left|{S}_{11}^{* }-\unicode{x03008}{S}_{11}^{* }\unicode{x03009}\right|}^{2}\unicode{x03009}}$ and is assumed to be ∼0.07, ³¹⁾ where ${S}_{11}^{* }$ denotes the measurement outcome of ${S}_{11}$ for single-shot qubit readout in the presence of noise and $\unicode{x03008}\ldots \unicode{x03009}$ denotes the statistical average (note that $\unicode{x03008}{S}_{11}^{* }\unicode{x03009}={S}_{11}$). Φ is then given by ${\rm{\Phi }}={\rm{\arcsin }}{\rm{}}(\sigma /\left|{S}_{11}\right|),$ where $\left|{S}_{11}\right|$ is the reflection amplitude for $C=100$ fF (we neglect its change when $C$ is increased to $101$ fF). As shown in Fig. 2(c), θ does not change significantly by the crosstalk. On the other hand, since the absolute value of ${S}_{11}^{{\rm{x}}}$ is reduced by crosstalk [Fig. 2(a)], Φ increases by crosstalk for the same amount of noise σ. Taking into these two contributions, crosstalk has a negative impact on SNR for RF phase readouts.

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Figure 3(b) shows the calculated θ and Φ (with and without mixed mode analysis), where the model shown in Fig. 1 is used. Φ without mixed-mode detection is significantly affected as expected: it increases monotonically as the distance between MSLs becomes small. On the other hand, Φ with mixed-mode detection is almost independent of the distance. Comparing the Φ with and without the mixed-mode detection, the 14 degree increase of Φ for $d={10}^{-3}$ mm is suppressed down to ∼1 degree. This simulation result shows the effectiveness of mixed-mode detection on the crosstalk effect between densely integrated MSLs.

In this study, we proposed a mixed-mode transmission to reduce the influence of crosstalk between close RF lines which are most likely employed in large-scale integrated qubit systems. First, we modeled a system for RF crosstalk by using an electromagnetic field simulator, which has two adjacent RF lines with readout resonators. Thereby, we performed simulations of the reflection of the system as a function of the distance between the two RF lines, with two different circuit parameters which correspond to different qubit states. The simulations show that although crosstalk increases as the distance between the two RF lines decreases, the reflection amplitudes obtained by using mixed-modes analysis are independent of the distance between MSLs. To confirm how this crosstalk affects spin readouts measurement, we simulated the impact of noise for RF phase measurement. We confirmed in the simulation that the mixed-mode detection can suppress the influence of crosstalk on phase noise, with Φ reduced from 14 degrees to 1 degree in some cases. We anticipate that this technique will be helpful for the realization of large-scale qubit systems.

This work was supported by MEXT Quantum Leap Flagship Program (MEXT QLEAP) Grant No. JPMXS0118069228, JST Moonshot R&D Grant No. JPMJMS2065, JST CREST (JPMJCR1675), JST PRESTO (JPMJPR21BA), and JSPS KAKENHI (20H00237, 20K15114, 21K14485).