Threshold-independent method for single-shot readout of spin qubits in semiconductor quantum dots

Rui-Zi Hu(胡睿梓), Sheng-Kai Zhu(祝圣凯), Xin Zhang(张鑫), Yuan Zhou(周圆),Ming Ni(倪铭), Rong-Long Ma(马荣龙), Gang Luo(罗刚), Zhen-Zhen Kong(孔真真),Gui-Lei Wang(王桂磊),4, Gang Cao(曹刚), Hai-Ou Li(李海欧),†, and Guo-Ping Guo(郭国平),5

1CAS Key Laboratory of Quantum Information,University of Science and Technology of China,Hefei 230026,China

2CAS Center For Excellence in Quantum Information and Quantum Physics,University of Science and Technology of China,Hefei 230026,China

3Key Laboratory of Microelectronics Devices and Integrated Technology,Institute of Microelectronics,Chinese Academy of Sciences,Beijing 100029,China

4Beijing Superstring Academy of Memory Technology,Beijing 100176,China

5Origin Quantum Computing Company Limited,Hefei 230026,China

Keywords: quantum computation,quantum dot,quantum state readout

1.Introduction

Spin qubits in gate-defined silicon quantum dots (QDs)are promising for realizing quantum computation due to their long coherence time,[1,2]small footprint,[3]potential scalability,[4]and industrial manufacturability.[5,6]In isotopically purified Si devices, the single-qubit gate fidelity has attained 99.9%,[7,8]and a two-qubit gate fidelity above 99%has been reported.[9–11]However, the corresponding readout fidelities are lower than 99%, which significantly reduces the overall fidelity of the gate operation.The Elzerman singleshot readout method is usually used,[12,13]which utilizes the spin-state-dependent tunneling rate or quantum capacitance to measure the spin qubits with extra charge sensors or dispersive sensing techniques.[14–16]The spin states are distinguished by comparing the readout tracesxwith the threshold voltagesxtwithin the readout timetr.However, this process is sensitive toxtandtr,and will lower the overall fidelity of the gate operations.

Several methods have been developed to optimize the readout fidelityFRand visibilityVR, as well as to find out the corresponding optimal threshold voltagextand readout timetr,e.g.,wavelet edge detection,[17]the analytical expression of the distribution,[18,19]statistical techniques,[21]neural network,[20]digital processing,[22]and the Monte–Carlo method.[23]Among these methods, the Monte–Carlo method is now widely used to numerically simulate the distributions of the experimental data in Si-MOS QDs,[24]Si/SiGe QDs,[25]Ge QDs,[26]single donors,[27]and nitrogen-vacancy centers.[28]A high-fidelity readout in a silicon single spin qubit has recently been achieved.[29]Even so, the readout visibilityVRis limited by the environment and experimental setup, e.g., the external magnetic field relative to the electron temperature(Bext/Te),relaxation time(T1),tunneling rate(Γin,out), measurement bandwidth, sample rate (Γs), and filter frequency.[30]

Here,we describe a threshold-independent method for the single-shot readout of semiconductor spin qubits.By considering the rate equations[30–33]and the Monte–Carlo method,we simulate the single-shot readout process and extractVRas a function of readout timetrand threshold voltagext.We demonstrate that the measured probability of the excited spin state (PM↑) is linearly dependent onVRin Eq.(5).Since the slope is the prepared probability of the excited spin state(PI↑) and is robust totrandxt, it is convenient to usePI↑instead ofPM↑to realize a threshold-independent data processing method.We then analyzed the error of the fitting process,finding that the error from the bin edges caused a discrepancy between the result and the expected value.We ensured accurate extrapolated probability by choosing readout timetrand threshold voltagext.Moreover, we use an effective area(Aeff) to show the effectiveness of the threshold-independent method,which is approximately 60 times larger than the commonly used method,i.e.,the threshold-dependent method.Finally, we discussed the influence ofTeon the effective area of both the threshold-independent method and the thresholddependent method with a fixed external magnetic field, and also provide a preliminary demonstration for a single-shot readout at 0.7 K/1.5 T in the future.

2.Result and discussion

2.1.Single-shot readout

Figure 1 outlines the processes of single-shot readout.The double quantum dots (DQDs) in our experiment resemble the device in Ref.[34].(NL,NR) in the charge stability diagram in Fig.1(a) represent the electrons occupied in the left and right QD.To measure the spin state of the first electron in the left QD,we deploy consecutive three-stage pulses,which consist of “empty”, “load” & “wait” and “readout” at(0,0)–(1,0)transition line,illustrated by“E”,“W”and“R”in black circles in Fig.1(a).Figure 1(b)shows the corresponding energy states.Here, we assume that the excited spin state is|↑〉.The location of the readout stage is carefully calibrated to ensure that the Fermi level of the reservoir is between the electrochemical potentials of spin-up and spin-down states.

The readout traces were measured by amplifying the single-electron transistor(SET)current(IS)with a room temperature low noise current amplifier(DLCPA-200)and a JFET preamplifier(SIM910),and then low-pass filtering the amplified signal using an analog filter (SIM965) with a bandwidth of 10 kHz.The blue and red curves represent spin up and spin down traces measured at point R, respectively.We can distinguish the different spin states by comparing the maximum value of each readout trace with the threshold voltage within the readout time.

The state-to-charge (STC) conversion is realized by distinguishing two different traces in the readout phase,as shown in Fig.1(c).Single electron tunneling onto or off the QD causes a change in the readout tracex.We distinguish the different spin states in the QD by comparing readout tracexwith the threshold voltagext.Ifxremains below the threshold during the readout phase,we assume it is a|↓〉state,and vice versa.We then emptied QDs by raising the electrochemical potential and waited for enough time.After the empty stage,we loaded a new electron with a random spin state and waited for the next readout stage.

We measured the electron spin relaxation time by repeating this three-stage pulse and changing the waiting time in the loading stage.Figure 1(d)shows typical exponential decay of the measured spin-up probabilityPM↑=ρ·e−t/T1+α, whereρis the amplitude andαis the dark count.Additionally, we can manipulate the spin qubit by using a similar pulse and a microwave pulse,as reported in Ref.[35].

Fig.1.(a) Charge stability diagram of the DQD measured by differentiating the single-electron transistor(SET)current(IS)as a function of the VB and VP gate voltages.The pulse sequence for measuring the spin relaxation time (T1) via the (0,0)–(1,0) charge transition line is overlaid on the data.(b)Illustration of the energy states of the pulse sequence: the measurement starts by emptying the electron in the QD at point E,it then injects a random spin into it and waits for a time at point W, and finally moves to point R for STC conversion.(c)The readout traces(x)are achieved by amplifying IS with a room temperature low noise current amplifier(DLCPA-200)and a JFET preamplifier(SIM910),and then low-pass filtering the amplified signal using an analog filter(SIM965)with a bandwidth of 10 kHz.The blue and red curves represent spin up(|↑〉)and spin down(|↓〉)traces at point R,respectively.By comparing the maximum of each readout trace(xmax)with the threshold voltage(xt)within the readout time(tr),we can distinguish the different spin states.(d)A typical exponential decay of spin-up probability with 1000 repeated measurements for each point.The exponential fit for 1/T1 is 112±6 s−1.

2.2.The readout visibility

We demonstrated a maximum visibilityVR= 85.4%while measuring the spin relaxation time, as shown in Fig.2(a).is calculated from the simulated data via the Monte–Carlo method.FR↓andFR↑are the readout fidelities of|↓〉 and|↑〉.In Fig.2(b), we simulated the distribution of the single-shot signal with a high accuracy(Rsquared≈0.98) of the corresponding fitting process (the right-hand inset in Fig 2(b)).The insets in Fig 2(b)show the fitting results of the averaged readout traces(¯x)and the probability density function(PDF)of the maximum traces(xmax)of the readout phase for every single measurement.The details of the simulation process and the insets in Fig.2(b) are discussed in Section 1 of the supplementary materials.[23,27,30–32]By fitting ¯xwith the rate equations, we obtained the tunneling rates of the STC conversion as shown in the left-hand inset of Fig.2(b):Hz and=1.39±0.04 kHz.

Fig.2.(a) The fidelities of spin-up and spin-down state (FR↑,FR↓) and the related readout visibility (VR) versus xt.(b) The fitting model for the distribution of the experimental data has a high value of R squared(≈0.98), which indicates the good accuracy of the results.The insets show fitting results for the averaged traces(¯x)and the probability density function(PDF)of the maximum value of each readout trace(xmax).(c)The state-to-charge(STC)conversion visibility(VSTC)versus tr and the corresponding maximum value VR.(the bottom arrow)and(the top arrow)are not equal.(d)The electrical detection visibility(VE)and the corresponding optimal threshold voltage xt as a function of readout time tr.

Here,readout fidelitiesandare given by

the corresponding readout visibilityVRis obtained from the Monte–Carlo method and the STC visibilityVSTCis calculated as above.Thus,we have the electrical detection visibilityVE=VR/VSTCindirectly.Figure 2(d)shows the maximumVEand the corresponding optimal threshold voltagextversus readout timetr.Considering the increasing or constant nature of the maximum valuextastrincreases in each readout trace,it can be inferred that the optimal threshold voltagextfollows a monotonically non-decreasing pattern.Meanwhile,the longertrwe consider,the more noise is added to the trace.Therefore,it is obvious thatVEwill decrease astrincreases.

Fig.3.(a)and(b)The VR and the amplitude(ρ)as a function of tr and xt, showing the consistency in the scaled color image.(c) The qubits are prepared in |↑〉 with probability .Throughout the STC conversion , the electron tunnels out from QD with probability P(t).Similarly, the electrons are measured as |↑〉 with throughout the electrical detection process).

2.3.Relation betweenand VR

We now focus on the details of readout visibilityVR.First, we illustrate the relationship betweenVRand readout timetralong with threshold voltagextvia the Monte–Carlo method in Fig.3(a).As mentioned in Subsection 2.2,xtcorresponding to the maximum electrical detection visibilityVEincreases astrincreases.The STC visibilityVSTCis a uni-modal function oftr.Therefore,VRincreases first and then decreases along thetraxis, andxtof the maximumVRincreases along thextaxis astrincreases.

We then drew the spin-up probabilityρin the same range in Fig.3(b)for comparison.The expressionαis obtained by fitting the experimental data of the spin relaxation process with an exponential functionα.Here,is the dark count of the spin relaxation process.Figures 3(a)and 3(b)show that the readout visibilityVRand probabilityρare consistent in the scaled color images.

To analyze this consistency, we focus on the details of Eq.(1).As shown in Fig.3(c),we note the spin-up probability at the beginning of the readout phase as the prepared probabilityPI↑.Throughout the STC conversion, the probability of the tunneling events detected within readout timetr(P(t))depends on the condition probability that electrons inortunnel out:

Similarly,by comparing the maximum of each readout tracesxmaxwith threshold voltagext, the electron is measured withPM↑throughout the electrical detection(FE↑,FE↓):

We factorized Eq.(4) into sectors with and withoutPI↑and substituted Eq.(2) into Eq.(4) to obtain the expressionα=1−FR↓.The relation between the measured probabilityPM↑,prepared probabilityPI↑,and readout visibilityVRcan then be extracted by substituting Eq.(1)into Eq.(4)

The details of derivation are discussed in Section 6 of the supplementary materials.

Fig.4.(a)The bias of the expected probability(PE↑)relative to PI↑(PE↑/PI↑−1)as a function of readout time tr and threshold voltage xt.The regions between the blue, cyan, and green curves represent Aeff where |PE↑/PI↑−1| ＜1%,5% and 10%, respectively.For comparison, the red star represents the position of the maximum readout visibility VRmax, and the points in the pink shadow region satisfy VR ＞80%.(b) The map of the PDF of xmax (maximum of the readout trace).The monotonically non-decreasing feature of the maximum function causes the positions of the two peaks to increase as tr increases.The outline of the valley between two peaks is similar to that of (a).(c) The extrapolated probability PE↑ figured out from the thresholdindependent methods at VRmax (red circles)and minimum between two peaks in PDF of xmax (green triangles).The measured probabilities PM↑ (blue rectangles) are obtained from the exponential decay process at VRmax directly.The solid curves with the same color illustrate the corresponding exponential fitting results.twait is logarithmic,showing that the threshold-independent method suppresses dark count α and improves probability ρ.(d) The lefthand y-axis shows that Aeff is a function of the electron temperature Te in both threshold-independent method and threshold-dependent readout.The righthand y-axis shows the corresponding readout visibility VR.When Te exceeds 100 mK,VR and Aeff of the threshold-independent methods will decrease as Te increases.The Aeff of the threshold-independent method is higher than that of the threshold-dependent method until Te=0.7 K as VR fixed at 0.5.

Equation(5)reveals that the measured probabilityPM↑linearly depends on the readout visibilityVRand dark countα.Here,the slope is the prepared probabilityPI↑and the intercept is the dark count.By substituting Eq.(5)into the definition of the probabilitywe gotSincePI↑only depends on the “wait” process, the probabilityρis proportional to the readout visibilityVRin the readout process, and this proportional relation betweenρandVRexplains their consistency in the scaled color images shown in Figs.3(a)and 3(b).SincePI↑only depends on the“wait”process, we try to apply the threshold-independent data process methods in Subsection 2.4.

2.4.Threshold-independent data process

We calculate the extrapolated probability(PE↑)for eachtrandxtby applying Eq.(5)directly

In Section 5 of the supplementary materials,we will use the simulated traces to illustrate that the region1%almost covers the whole considered region under ideal circumstances.We calculated the cumulative error(CE)and absolute value of error between the distribution of experimental and simulated traces as a function of threshold voltagext.The shape of the CE curve demonstrates that the error in the fitting process comes from the bin edges.The features of the bin edge error are shown in the contour map ofin Figs.22(b)and 22(c)in the supplementary materials.

In the valley region between the two peaks, errors from bin edges are minimal,as shown in the map of the distribution ofxmaxin Fig.4(a).The outline of the valley region in Fig.4(b)resembles that of＜1% and 5% in Fig.4(a).As a compromise method,we can choosextaround the minimum between two peaks of the distribution ofxmaxinstead ofxtat.

By using the threshold-independent method to processing the experimental data, we calculated the extrapolated probabilityPE↑trand threshold voltagextat,along

Finally,we tried to use the area where%asAeffto quantify the accuracy and efficiency of different data processing methods.For comparison,we use the area of whereasAefffor the commonly used thresholddependent method.TheAeffof the threshold-independent method is 60 times larger than that of the threshold-dependent method,meaning that we can choose threshold voltagextand readout timetrin a 60 times larger range and maintain 99%accuracy.In addition,the threshold-independent method is more robust and can calibrate the measured result from the interference of the experimental hardware limitation.

2.5.Influence of the electron temperature

We try to characterize the influence ofTe.Reference[30]assumes that the tunneling rate follows a Fermi distribution

wheref(ε±Ez/2,Te) is the Fermi–Dirac function with−for|↓〉 and + for|↑〉,Γout(Γin) is the maximum tunnel out(in)rate,are tunneling rates of corresponding spin state, andεis the energy splitting between the Fermi energy of the electron reservoir and the average energy of|↑〉 and|↓〉electrons in QD.Therefore,ε+Ez/2(ε −Ez/2)represents the energy splitting between the Fermi energy of the electron reservoir and the energy state of|↑〉(|↓〉)electron in QD.

We extracted the maximum tunneling ratesΓoutandΓinby substitutingεinto Eq.(7).

We then tried to simulate the single-shot readout process at different electron temperaturesTe.Assuming that tunneling ratesΓoutandΓinare not associated withTe, we directly substitutedTeinto Eq.(7)to calculate the tunneling rates.We used the Monte–Carlo method to generate the simulated traces with a fixed external magnetic fieldBext=1.5 T at differentTe.The left-handy-axis in Fig.4(d) shows theAeffof both the threshold-independent method and the threshold-dependent method at differentTe.The right-handy-axis shows the corresponding readout visibilityVR.The simulation results show that theAeffof the threshold-independent methods is 60 times greater than that of the threshold-dependent method whenTe＜0.1 K.AsTeincreases,Aeffof the threshold-independent methods decreases.It is larger than that of the thresholddependent method untilTe=0.7 K.WhenTe＞0.7 K,the correspondingVR＜0.5.Here,we can give the boundary condition of the threshold-independent method asTe=0.7 K whenBext=1.5 T.

3.Summary

We described a threshold-independent method of the single-shot readout data process based on the linear dependence of measured probabilityPM↑with the corresponding readout visibilityVRand dark countα.Due to the error during the fitting process from bin edges,the extrapolated probability deviates from the prepared probability.For compromise, the region of readout timetrand threshold voltagextare reduced to the minimum of the distribution of the maximum of each readout tracexmaxto ensure that the accuracy loss of extrapolated probabilityPE↑is less than 1%.We then usedAeffto quantify the efficiency of the threshold-independent method and the threshold-dependent readout.The result shows that theAeffof the threshold-independent method is more than 60 times larger than that of the threshold-dependent method.Moreover, we simulated the single-shot readout process at different electron temperaturesTe.We broaden the boundary condition of the single-shot readout to 0.7 K withBext=1.5 T,whereVR=0.5.The significance of employing the thresholdindependent method will progressively increase as the experiment advances towards the fault-tolerance threshold of the logic qubit, particularly when operating under high electron temperature conditions.[36–38]

Acknowledgements

Project supported by the National Natural Science Foundation of China(Grant Nos.12074368,92165207,12034018,and 62004185),the Anhui Province Natural Science Foundation (Grant No.2108085J03), the USTC Tang Scholarship,and this work was partially carried out at the USTC Center for Micro and Nanoscale Research and Fabrication.