⇦Chinese Physics Letters, 2021, Vol. 38, No. 11, Article code 110303 Improved Superconducting Qubit State Readout by Path Interference Zhiling Wang (王志凌)†, Zenghui Bao (鲍增晖)†, Yukai Wu (吴宇恺), Yan Li (李严), Cheng Ma (马程), Tianqi Cai (蔡天奇), Yipu Song (宋祎璞), Hongyi Zhang (张宏毅)*, and Luming Duan (段路明)* Affiliations Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China Received 24 September 2021; accepted 15 October 2021; published online 27 October 2021 Supported by the Beijing Academy of Quantum Information Science, the Frontier Science Center for Quantum Information of the Ministry of Education of China through the Tsinghua University Initiative Scientific Research Program, the National Natural Science Foundation of China (Grant No. 11874235), the National Key Research and Development Program of China (Grant Nos. 2016YFA0301902 and 2020YFA0309500), Y.K.W. acknowledges support from Shuimu Tsinghua Scholar Program and the International Postdoctoral Exchange Fellowship Program.
†These two authors contributed equally to this work.
*Corresponding authors. Email: hyzhang2016@tsinghua.edu.cn; lmduan@tsinghua.edu.cn Citation Text: Wang Z L, Bao Z H, Wu Y K, Li Y, and Ma C et al. 2021 Chin. Phys. Lett. 38 110303    Abstract High fidelity single shot qubit state readout is essential for many quantum information processing protocols. In superconducting quantum circuit, the qubit state is usually determined by detecting the dispersive frequency shift of a microwave cavity from either transmission or reflection. We demonstrate the use of constructive interference between the transmitted and reflected signal to optimize the qubit state readout, with which we find a better resolved state discrimination and an improved qubit readout fidelity. As a simple and convenient approach, our scheme can be combined with other qubit readout methods based on the discrimination of cavity photon states to further improve the qubit state readout. DOI:10.1088/0256-307X/38/11/110303 © 2021 Chinese Physics Society Article Text High fidelity measurement of qubit states is an essential requirement for numerous protocols in quantum computation[1] and quantum information processing[2–5] and it lies at the heart of many quantum technologies.[6–9] For example, in a noisy intermediate-scale quantum computer, a quantum circuit is expected to run for a lot of trails to mitigate possible errors.[10,11] A higher qubit readout fidelity would greatly improve the probability of successful trials, and thus leading to a higher computing efficiency.[12,13] Recently, determining the state of various kinds of physical qubits with high fidelity becomes feasible, even in a quantum non-demolition (QND) way.[14–17] Fast and QND qubit state readout in a single shot is essential for a fault tolerant quantum computer, where errors have to be corrected in situ once being detected.[5,18,19] In a superconducting circuit quantum electrodynamics (cQED) system, the most commonly used qubit readout scheme is based on the dispersive interaction between the qubit and the cavity field. The cavity's response to coherent state photons is dependent on the state of the qubit, which acts as an intra-cavity pointer state.[20,21] The qubit state can be readily determined if the two pointer states, corresponding to the qubit ground state $\left|g\right\rangle $ and the excited state $\left|e\right\rangle $, are well separated in the phase space. Ideally a probe tone with more photons, and thus larger separation of the two pointer states, is preferred to achieve a fast and high fidelity qubit state measurement.[22] However, in practice a strong probe would induce notable back-action to the qubit state, which is unfavorable for a QND measurement.[23] Therefore, it is highly desired to make full use of photons in a cavity to effectively read out the qubit state and ultimately achieve a quantum limit measurement.[24,25] For a symmetric cavity with identical out-coupling rates at both ends, one always losses half of the signal by measuring only the transmission or reflection of the cavity.[26] One solution is to use asymmetric cavity with the out-coupling rate of one end significantly larger than that of the other.[5,22] The cavity photons would mainly leak out from the end with larger out-coupling rate, and thus can be effectively collected to determine the qubit state. In this Letter, we provide a complementary solution and show that it is possible to extract all the photons from a symmetric cavity through interference, thus increasing the separation of the two pointer states to reach its maxima under the given incoming photons. To demonstrate the idea, experimentally we design a sample containing five qubits with varied interference conditions for microwave photons in the readout cavities. With a set of hybrid couplers we are able to carry out qubit state measurement simultaneously using the output signal from the transmission path ($T$), the reflection path ($R$) and the interference path ($T+R$). It turns out that when constructive interference condition is satisfied, the cavity response measured from $T+R$ is enhanced compared with that from $T$ or $R$. Further, qubit state discrimination and qubit readout fidelity also show clear improvement when measuring from the $T+R$ output. Finally, it is worth noting that since our method provides an effective extraction of cavity photons, all those readout schemes based on detecting qubit state dependent cavity response, including cavity driving, nonlinear qubit-cavity interaction,[26–30] etc., would benefit from the path interference method proposed in this work.
cpl-38-11-110303-fig1.png
Fig. 1. Qubit state readout with path interference. Reflection and transmission of the readout cavity are interfered with a beam splitter to fully extract photons in the cavity.
Theory. We consider a coplanar waveguide (CPW) cavity with symmetric out-coupling to the transmission line and dispersively coupled to a superconducting qubit, as illustrated in Fig. 1. Normally the qubit state is determined by detecting the cavity response to a probe tone from the output $T$ or $R$. Here we consider using a beamsplitter to combine the two outputs and generate interference of the cavity field. Intuitively, the cavity photons can be effectively extracted once the constructive interference condition is met, and thus leading to a better qubit readout performance. The Hamiltonian of the system can be written as $H=(\omega_{\rm r}+\chi\sigma_z)a^† a+\omega_q \sigma_z/2$, where $\omega_{\rm r}$ is the bare frequency of the cavity, $\omega_q$ is the qubit transition frequency and $\chi$ is the dispersive shift due to the qubit-cavity coupling. The change of the qubit state would shift the cavity resonance through the term $\chi\sigma_z$, resulting in a change of the cavity response when probing the system at a certain frequency. For output $T$ and $R$, the distance between the two pointer states corresponding to the qubit state $\left|g\right\rangle $ and $\left|e\right\rangle $ can be given as (see the Supplementary Information for details) $$\begin{split} &D^{\rm T(R)}=|\alpha_{\scriptscriptstyle {\rm T(R)}}(\omega_{\rm d},1)-\alpha_{\scriptscriptstyle {\rm T(R)}}(\omega_{\rm d},-1)|\\ ={}&\frac{4\kappa_{\rm c}\chi|\alpha_{\rm in}|}{\sqrt{[\kappa^2+4\chi^2-4(\omega_{\rm d}-\omega_{\rm r})^2]^2 +16\kappa^2(\omega_{\rm d}-\omega_{\rm r})^2}},\\ \end{split}~~ \tag {1} $$ where $\alpha_{\scriptscriptstyle {\rm T(R)}}[\omega_{\rm d},1(-1)]$ represents the intra-cavity pointer state measured from the output $T$ ($R$) with probing frequency $\omega_{\rm d}$ and qubit state $\sigma_z = 1$ or $-1$, $\kappa$ and $\kappa_{\rm c}$ are the cavity's total damping rate and the damping rate to the transmission line, respectively. If a beamsplitter is used to interfere output $T$ and $R$, photon state at the two output ports of the beamsplitter can be written as $$ \alpha_{\rm int}^\pm(\omega_{\rm d},\sigma_z)= \frac{\alpha_{\scriptscriptstyle {\rm T}}(\omega_{\rm d},\sigma_z) \pm e^{i\theta_{\scriptscriptstyle {\rm RT}}}\alpha_{\scriptscriptstyle {\rm R}}(\omega_{\rm d},\sigma_z)}{\sqrt{2}},~~ \tag {2} $$ where $\theta_{\scriptscriptstyle {\rm RT}}$ is the relative phase between $T$ mode and $R$ mode before interference. Therefore, the distance between the two pointer states measured from the interference output can be written as $$\begin{split} D^{\rm int}(\theta)={}&|\alpha_{\rm int}(\omega_{\rm d},1)\pm \alpha_{\rm int}(\omega_{\rm d},-1)|\\ ={}&\left|\frac{1\pm e^{i\theta_{\scriptscriptstyle {\rm RT}}}}{\sqrt{2}}\right| D^{\rm T(R)}. \end{split}~~ \tag {3} $$ The inset in Fig. 2(b) gives the calculated cavity response on the phase plane when sweeping the probe frequency across the cavity resonance. The colored dots represent pointer states when the qubit is at the ground state $\left|g\right\rangle $ and the excited state $\left|e\right\rangle $ for a fixed probing frequency with a dispersive shift $\chi$. From Eq. (1), distances between the two pointer states are equal for output $T$ and output $R$, which is unsurprising for a symmetric cavity. It also means that if only collecting from either $T$ or $R$, one would lose half of the cavity photons, which leads to a loss of information about the qubit state up to $1/2$.[24] From Eq. (3) it can be found that once the constructive interference condition is satisfied, the distance of the cavity responses from the interference output can be maximally enlarged by $\sqrt{2}$ times than that from $T$ or $R$. Consequently the pointer state discrimination can be improved, which is preferred for the qubit state readout, as illustrated in Fig. 2(b). Experimental Results. In the experiment, we use a sample containing five transmon qubits, each being coupled to a hanger type CPW cavity for qubit state readout. The hanger is a typical symmetric cavity with only one port coupled to the transmission line.[31] As shown in Eq. (3), the distance of the two pointer states from interference output depends on the relative phase $\theta_{\scriptscriptstyle {\rm RT}}$, which can be experimentally tuned via a cryogenic phase shifter.[32–35] Here to demonstrate the idea, we vary $\theta_{\scriptscriptstyle {\rm RT}}$ by placing the readout cavity at different positions of the transmission line. For photons from a certain cavity going through $T$ and $R$, the phase difference between the two paths depends on the position of the readout cavities on the transmission line, as illustrated in Fig. 2(a). Detailed information about the sample is listed in the Supplementary Information. We use three 180$^\circ$ microwave hybrid couplers as the beamsplitters to generate path interference.[36,37] Transmission or reflection of the cavities is firstly guided to a coupler and is split into two channels. One channel is directly sent to the amplification chain as the output $T$ or $R$. The other channel is sent to the third coupler for interference. One of the interference output is sent to the amplification chain as the output $T+R$. It is necessary to mention that since $T$, $R$, $T+R$ are outputted from three different amplification chains, careful calibrations must be carried out before any comparison among them. Details about the calibration work can be found in Part II of the Supplementary Information.
cpl-38-11-110303-fig2.png
Fig. 2. A demonstration of the proposed scheme. (a) An illustration of the sample and the interference circuit. The device contains five qubits, with the readout cavities equally spaced by about 2.5 mm along the transmission line. The cavity reflection or transmission is first split by hybrid couplers into two channels. One is sent to the corresponding amplification chain and the other is sent to a third hybrid coupler to generate interference signal. Only one interference output is sent to the amplification chain and the other one is terminated. With the help of this setup, we can realize qubit state readout through $T$, $R$ and $T+R$ simultaneously. (b) The steady state cavity responses measured from $T$ (red), $R$ (blue), $T+R$ (black) are plotted on the phase plane, when sweeping the probe frequency across the cavity resonance. The relative phase $\theta_{\scriptscriptstyle {\rm RT}}$ for this cavity is 0.11, with which the constructive interference condition can be approximately satisfied. The solid and dashed lines are measurement results when the qubit is set to $\left|g\right\rangle $ and $\left|e\right\rangle $. The colored dots on the circles correspond to cavity responses at probe frequencies used in the single shot measurements. The two-dimensional Gaussian profiles represent the corresponding coherent states. Inset of (b) shows the calculated cavity responses for $T$, $R$ and $T+R$, taking $\theta_{\scriptscriptstyle {\rm RT}}=0$.
We first characterize the relative phase $\theta_{\scriptscriptstyle {\rm RT}}$ for each of the readout cavities. For different output lines, the phase delays from hybrid couplers to the room temperature homodyne detector are not the same, therefore $\theta_{\scriptscriptstyle {\rm RT}}$ cannot be directly obtained from the measured cavity responses from room temperature electronics. Instead, for a certain output line, we use cavity responses corresponding to different qubit states as reference to calibrate $\theta_{\scriptscriptstyle {\rm RT}}$. To be more specific, we set the qubit state to $\left|g\right\rangle $ or $\left|e\right\rangle $, then record the corresponding cavity responses from $T$, $R$ and $T+R$. Here, $\theta_{\scriptscriptstyle {\rm RT}}$ can be deduced by comparing the cavity responses from the same output line when the qubit is in $\left|g\right\rangle $ or $\left|e\right\rangle $. Details of this method can be found in Part III of the Supplementary Information. For the sample used in this work we have achieved various $\theta_{\scriptscriptstyle {\rm RT}}$ from $-1.42$ to $2.60$ for the five readout cavities. In the following experiment, we use the readout cavity with $\theta_{\scriptscriptstyle {\rm RT}} = 0.11$ to investigate the effect of constructive interference.
cpl-38-11-110303-fig3.png
Fig. 3. The distance between the pointer states corresponding to the qubit states $\left|g\right\rangle $ or $\left|e\right\rangle $ on the phase plane when scanning the probe frequency across the cavity resonance. The probe frequency with the maximal distance can be used for qubit state readout, which shall yield the best signal-to-noise ratio. The relative phase $\theta_{\scriptscriptstyle {\rm RT}}$ for this cavity is 0.11, which is close to the ideal constructive interference condition. The maximum value of the interference signal is about $\sqrt{2}$ times as large as that from $T$ or $R$, as predicted from the theory. The inset is the interference gain $\beta$ as a function of $\theta_{\scriptscriptstyle {\rm RT}}$. The dashed line is the theoretical prediction and the scattered points are experimental results. The error bars of the experimental results are given by a standard deviation of the measured steady state output.
As mentioned above, qubit state readout relies on discriminating the cavity responses at a certain frequency, thus a larger distance $D(\omega_{\rm d})$ is preferred. In order to evaluate cavity interference effect, we sweep the probe frequency and record cavity responses from output $T$, $R$ and $T+R$ when the qubit state is in $\left|g\right\rangle $ or $\left|e\right\rangle $.[38] Figure 2(b) gives such a result for the cavity with $\theta_{\scriptscriptstyle {\rm RT}} = 0.11$, for which the constructive interference condition is almost met. Different from the theoretical result shown in the inset of Fig. 2(b), the measured cavity response when the qubit being in $\left|e\right\rangle $ is a smaller circle on the phase plane than that when the qubit is in $\left|g\right\rangle $. This is due to the qubit relaxation during the cavity response measurement. Detailed discussion can be found in the Supplementary Information. From the measured cavity response in Fig. 2(b) we can calculate the distance between cavity responses corresponding to the qubit state $\left|g\right\rangle $ and $\left|e\right\rangle $ with varied probe frequencies, as illustrated in Fig. 3. The probe frequency with the largest distance can be used for qubit state readout, which shall give the best signal-to-noise ratio. Here we introduce an enhancement factor $\beta = 2D_{\rm m}^{\rm T+R}/(D_{\rm m}^{\rm T}+D_{\rm m}^{\rm R})$ to represent the readout enhancement when measuring from $T+R$ compared with measuring $T$ or $R$, where $D_{\rm m}^{\rm T+R(T,R)}$ represents the largest distance calculated from the measured cavity response. The inset in Fig. 3 shows the experimental results of $\beta$ with scattered plots. Theoretically one would have $\beta=\sqrt{2}\cos(\theta_{\scriptscriptstyle {\rm RT}}/2)$ from Eq. (3), which is plotted as the dashed line. It can be seen that the experimental result fits well with the theoretical prediction. When the constructive interference condition is met, we obtain an enhancement factor $\beta \sim 1.4$.
cpl-38-11-110303-fig4.png
Fig. 4. Single shot measurement. Histogram plots of the measured cavity response from (a) $T$ path and (b) $T+R$ path, when setting the qubit state to either $\left|g\right\rangle $ or $\left|e\right\rangle $. The readout time is 900 ns. For a given qubit state, the histogram plot normally contains two Gaussian components due to the thermal population and qubit relaxation during the readout process. The measurement error is defined as the overlap of the two Gaussian distributions corresponding to the two pointer states, which is marked with yellow shadow. (c) The measurement errors as a function of readout time for $T$ and $T+R$. The scattered plots are experimental results and the solid lines are from the theoretical calculation. Inset of (c) gives the ratio between the measurement errors from $T+R$ and that from $T$. (d) The optimized total error as a function of the qubit relaxation time $T_1$ for $T$ and $T+R$. Inset of (d) gives the error increase when using $T$ for qubit readout, comparing with $T+R$ for varied $T_1$.
In order to evaluate the performance of qubit readout with cavity interference, we implement single shot measurement for the qubit with $\theta_{\scriptscriptstyle {\rm RT}} = 0.11$ from both $T$ and $T+R$. We set the qubit to either $\left|g\right\rangle $ or $\left|e\right\rangle $, then record the cavity response to a square pulse with a measurement time $t_{\rm m}$ in a single shot. Figures 4(a) and 4(b) give the histogram plots of the single shot measurement from $T$ and $T+R$, respectively. When setting the qubit state to either $\left|g\right\rangle $ or $\left|e\right\rangle $, the measured voltage distribution contains two Gaussian components, corresponding to a projected state of $\left|g\right\rangle $ or $\left|e\right\rangle $ by the measurement pulse. The qubit state is assigned to $\left|g\right\rangle $ or $\left|e\right\rangle $ by comparing the measured cavity response with a threshold voltage $V_{\rm th}$. One could find that there is considerable error probability of assigning the qubit to an incorrect state. The readout error is defined as $\varepsilon = P(e|g)+P(g|e)$, where $P(i|j)$ refers to the probability of assigning the qubit to $\left|i\right\rangle $ when the qubit is prepared to $\left|j\right\rangle $. Several reasons could contribute to the readout error, including the overlap of the measured cavity response, the qubit relaxation and the non-zero thermal population.[27,39] The thermal population and qubit relaxation induced readout errors can be suppressed by improving the thermal anchoring and the qubit quality,[40] which is beyond the interest of this work. Here we focus on the measurement error, which is induced by the nonzero overlap of the cavity responses and is critically related to the measurement circuit and parameters. In the experiment we define the measurement error as the overlap of the two Gaussian distributions corresponding to the qubit state which is prepared and measured both in $\left|g\right\rangle $ or both in $\left|e\right\rangle $, as illustrated in Figs. 4(a) and 4(b), from which we have the measurement error of $4.8\%$ for $T$, but only $1\%$ for $T+R$. It clearly shows the improvement on the fidelity of single shot qubit readout with path interference. To explicitly demonstrates the suppression of measurement error with path interference, we further compute the measurement error with varied measurement time for both $T$ and $T+R$, which is shown in Fig. 4(c) as scattered plots. Benefiting from the enlarged distance and thus smaller overlap between the two pointer states, the measurement error from $T+R$ is always smaller than that from $T$. The ratio between them is shown in the inset of Fig. 4(c). As the measurement time increases, the measurement error from $T+R$ reduces much faster than that from $T$. We further investigate the effect of path interference on the total readout error. Intuitively, if the distance of the cavity responses increases by a factor of $\beta$, the measure time can be reduced by a factor of $1/\beta^2$ to yield the same measurement error. Therefore the total readout error could be reduced considering the reduced qubit relaxation induced error. In fact, for a given measurement condition, there exists an optimized readout time, which shall be long enough to keep a small measurement error, while not leading to a significant qubit relaxation error, and thus a minimum total readout error. Figure 4(d) gives such an optimized readout error with varied $\left|e\right\rangle $ state relaxation time $T_1$, with a photon number in the readout cavity of about $20$. It can be seen that measuring the qubit state with path interference $T+R$ yields a considerably smaller readout error than that with transmission $T$. For a realistically achievable qubit relaxation time $T_1\sim 30$ µs,[41,42] one can reach a readout fidelity better than $99\%$ when measuring with $T+R$, whereas measuring with transmission $T$ bears up to $80\%$ larger readout error. In summary, we have introduced a new scheme to improve the superconducting qubit readout fidelity with path interference. Our scheme is based on the constructive interference of transmission and reflection of the readout cavity. The cavity photons can be effectively extracted, which leads to a larger separation between the pointer states corresponding to the qubit state $\left|g\right\rangle $ and $\left|e\right\rangle $. We have developed a method to precisely measure the relative phase between cavity transmission and reflection, which is a critical parameter for the interference process. For the single shot qubit state measurement, we have observed a significant suppression on both measurement error and total readout error when using cavity interference as the readout signal instead of the output from $T$. In order to realize simultaneous improvement for multiple cavities coupled to a common transmission line, one could take the distance between adjacent cavities along the transmission line as $\lambda/2$, where $\lambda$ corresponds to the mean value of the resonant frequencies of the cavities, and could use a cryogenic phase shifter to realize constructive interference.[32–35] As a general scheme to effectively extract the cavity photons, our method can be combined with other readout optimization methods to further improve the performance of superconducting qubit state readout. 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