# Modeling and Experimental Validation of the Intrinsic SNR in Spin Qubit
Gate-Based Readout and Its Impacts on Readout Electronics

Bagas Prabowo1,2<EMAIL_ADDRESS>Jurgen Dijkema2,3 Xiao Xue2,3 Fabio
Sebastiano1,2 Lieven M. K. Vandersypen2,3 Masoud Babaie1,2 1Department of
Quantum and Computer Engineering, Delft University of Technology, 2600 Delft,
GA, The Netherlands  2QuTech, Delft University of Technology, 2600 Delft, GA,
The Netherlands  3Kavli Institute of Nanoscience, Delft University of
Technology, 2600 Delft, GA, The Netherlands

(13-12-2023)

###### Abstract

In semiconductor spin quantum bits (qubits), the radio-frequency (RF) gate-
based readout is a promising solution for future large-scale integration, as
it allows for a fast, frequency-multiplexed readout architecture, enabling
multiple qubits to be read out simultaneously. This paper introduces a
theoretical framework to evaluate the effect of various parameters, such as
the readout probe power, readout chain’s noise performance, and integration
time on the intrinsic readout signal-to-noise ratio (SNR), and thus readout
fidelity of RF gate-based readout systems. By analyzing the underlying physics
of spin qubits during readout, this work proposes a qubit readout model that
takes into account the qubit’s quantum mechanical properties, providing a way
to evaluate the trade-offs among the aforementioned parameters. The validity
of the proposed model is evaluated by comparing the simulation and
experimental results. The proposed analytical approach, the developed model,
and the experimental results enable designers to optimize the entire readout
chain effectively, thus leading to a faster, lower-power readout system with
integrated cryogenic electronics.

RF gate-based readout, Quantum capacitance, Readout fidelity, Readout SNR,
Spin Qubits, Double Quantum Dot, Cryogenic, Electronics, Cryo-CMOS, Noise
temperature.

##  I Introduction

Performing practical, fault-tolerant quantum algorithms with quantum error
correction (QEC) will require thousands of quantum bits (qubits) in a quantum
computer to be manipulated and read out simultaneously [1]. In the case of a
semiconductor spin qubit quantum computer (QC), the QEC cycle must be able to
read, decode the error, and apply the correction to the data qubits far faster
than the qubit’s dephasing time (T2*) [2]. Owing to the short T2* of $\sim$20
$\mu$s [3, 4, 5], it is essential to reduce the readout, decode, and gate
operation time to be below the sub-microsecond range for the QEC cycle time to
be significantly faster than the dephasing time.

In the case of reducing the readout time of the quantum processor, the choice
in the readout technique plays a crucial role. With the requirement of reading
thousands of qubits in the future, the choice of the readout architecture must
be fast and scalable. One of the most promising readout techniques pursued
currently for semiconductor spin qubits is the gate-based readout technique,
as it allows the readout architecture to adopt frequency division multiplex
access (FDMA) feature to enable the simultaneous readout of multiple qubits by
a single receiver (RX) [6, 7, 8].

Figure 1: (a) A prototype of a readout architecture; (b) Corresponding probe
tones, transmission responses of the sample, and constellation diagrams when
the system is probed at $f_{1}$ and $f_{2}$; (c) BER versus SNR plot.

A simplified model of a gate-based readout architecture is illustrated in Fig.
1(a), where port-1 and port-2 indicate the input and output ports of the
readout system, respectively. In principle, the readout architecture has a
state-dependent transmission (or reflection) behavior dependent on the qubit’s
state. To read the qubit’s state, the system is probed with a radio-frequency
(RF) tone from port-1 with a certain time duration, $T_{int}$. Depending on
the frequency of the stimulus tone, the readout tone observed at port-2
exhibits a change in amplitude or phase, providing information about the
qubit’s state. As an example, Fig. 1(b) illustrates the readout system’s
response when probed at $f_{1}$ and $f_{2}$. When the system is probed at the
resonance frequency of the $\ket{0}$ state ($f_{1}$), the readout signal
observed at port-2 resembles a binary amplitude shift keying signal (BASK),
due to the attenuated $\ket{1}$ response. On the other hand, when the system
is probed at a frequency precisely between the $\ket{0}$ or $\ket{1}$
responses ($f_{2}$), the output signal at port-2 resembles a binary phase
shift keying signal (BPSK), where the $\ket{0}$ and $\ket{1}$ response differs
by 180 degrees. Regardless of how the output signal is modulated, the readout
signal can be denoted as $A_{sig}$ in the constellation diagram, which
describes the separation between the two states’ responses.

In general, the readout signal (i.e., $A_{sig}$) must be large enough to
accommodate the thermal noise contributed by the environment and the
additional noise of the subsequent readout electronics, which is responsible
for amplifying and down-converting the readout signal. To quantify the effect
of the noise on the readout signal, the readout signal-to-noise ratio (SNR)
must be calculated, which depends on both the power of the readout signal
($P_{Sig}$) and the total noise observed at the input of the readout
electronics ($P_{N}$). Mathematically, $P_{Sig}$ is related to $A_{sig}$ by

$\displaystyle
P_{Sig}=\frac{|A_{sig}|^{2}}{R_{L}}=P_{RF}\left|\left(S_{21,\ket{0}}(\omega_{r})-S_{21,\ket{1}}(\omega_{r})\right)\right|^{2},$
(1)

where $P_{RF}$ is the power delivered by the RF source to the input port of
the system (i.e., port-1), $\omega_{r}$ is the readout angular frequency and
$\left|\left(S_{21,\ket{0}}(\omega_{r})-S_{21,\ket{1}}(\omega_{r})\right)\right|^{2}$
is the state-separation factor, which depends on the $S_{21}$ behavior of the
system when it is in the $\ket{0}$ or $\ket{1}$ state. The noise power,
$P_{N}$, on the other hand, can be expressed as $N_{0}/T_{int}$, where $N_{0}$
is the noise power spectral density, which includes the noise of the
electronics referred to port-2, and $T_{int}$ is the integration time. By
defining the SNR as $P_{Sig}/P_{N}$, the readout SNR can be expressed as

$\displaystyle
SNR=\frac{P_{RF}}{N_{0}/T_{int}}\times\left|\left(S_{21,\ket{0}}(\omega_{r})-S_{21,\ket{1}}(\omega_{r})\right)\right|^{2}.$
(2)

However, the quality of the readout acquisition is commonly characterized
using metrics such as the bit error rate (BER) (equivalently, the readout
infidelity, $1-F$) rather than the SNR. The relationship between the SNR in
(2) with the BER can be described as

$\displaystyle BER=Q\left(\sqrt{SNR}\right),$ (3)

where $Q(\cdot)$ is the tail distribution function of the standard normal
distribution. Based on (3), the resulting BER for a given SNR performance is
shown in Fig. 1(c). In fault-tolerant QC with QEC, a target BER of 10-4 is
often desired, which corresponds to an SNR requirement of 11.5 dB.

The target SNR of 11.5 dB can be achieved in numerous ways, based on (2).
Given a readout system with a certain state-separation factor, one could
choose an arbitrary $P_{RF}$, $N_{0}$, and a $T_{int}$ that complies with the
desired SNR. However, in the context of QEC, $T_{int}$ is commonly fixed based
on the QEC requirements. Thus, only $P_{RF}$ and $N_{0}$ are left to be
optimized to achieve the required SNR. By maximizing $P_{RF}$, $N_{0}$ can be
made larger while still achieving the desired SNR. The larger tolerable
$N_{0}$ benefits the readout electronics by allowing them to be designed at a
lower power. With the limited power budget available at the cryogenic level of
the dilution refrigerator, this approach enables better integration of
cryogenic electronics suitable for future large-scale integration.

Experimental observations, however, have shown that increasing $P_{RF}$ does
not necessarily improve the readout SNR [9, 10, 11]. Theoretically, (2) shows
a boundless trade-off between $P_{RF}$ and $N_{0}$, where a large $P_{RF}$
allows $N_{0}$ to be increased proportionally for a given target SNR,
$T_{int}$, and state-separation factor, contradicting experimental
observation. Thus, the limit in which $N_{0}$ can be relaxed as $P_{RF}$
increases remains unclear. In this article, we investigate the validity of (2)
by first constructing a qubit model that considers its quantum mechanical
behavior during readout. The proposed model is used to simulate the behavior
of a readout system to investigate the validity of (2) and the limitations of
the trade-offs between $P_{RF}$ and $N_{0}$. To confirm the behavior shown by
our proposed model, the simulated results are compared with experimental
results.

The article is organized as follows. Section II first discusses the behavior
of the qubits during readout and how they can be modeled. Section III presents
the simulation results of the readout system when the proposed qubit model is
considered. Section IV compares the simulated results with experimental
results, verifying the validity and limitations of the model. Upon
experimental verification, Section V discusses the impact of the trade-offs
between $P_{RF}$ and $N_{0}$ on the readout architecture. Section VI concludes
the article.

##  II Spin Qubit Readout Theory

###  II.1 Double Quantum Dot

Figure 2: (a) Colored image of a DQD; (b) Sketch of the charge stability
diagram of the DQD. The red solid lines indicate the interdot regime.
Initialization, manipulation, and readout regimes are labeled correspondingly.

The double quantum dot (DQD) is the basic building block of a semiconductor
spin qubit. Similar to a transistor, the DQD is composed of several plunger
gates to control the potential landscape of the device to trap electrons in
each respective quantum dot (QD), as indicated by the green and purple circles
in Fig. 2(a). By varying the left ($V_{LP}$) and right ($V_{RP}$) plunger gate
voltages, individual electrons can be loaded in and out of each QD site from
the electron reservoir, as indicated by the crossed boxes in Fig. 2(a). The
tunnel coupling ($t_{c}$) gate shown in the figure controls the tunnel
coupling interaction between the two QDs and plays an essential role in tuning
the device for readout.

The DQD is typically characterized by the charge stability diagram, as shown
in Fig. 2(b). The number (NL,NR) indicates the number of electrons in the left
and right QD, respectively. The dotted black lines in the diagram indicate the
plunger gate voltages at which the electrons from the QD can be loaded or
unloaded to the reservoir. For quantum computation with spin qubits,
initialization can take place in the (0,2) regime, marked as ’I’ in Fig. 2(b).
Once initialized, qubit manipulation can be executed in the (1,1) charge
regime at point ’M’, where the electron spin in the left QD can be manipulated
independently while keeping the right QD unaffected such that it can be used
as a reference for readout [5]. To read the spin state of the left QD, the DQD
is biased at the interdot regime (highlighted in red in Fig. 2(b)). Depending
on the final spin state to which the DQD collapses, the electron may oscillate
or stay in each respective dot in response to an RF tone applied at the
readout gate (RG), leading to a state-dependent behavior, as discussed in
detail below.

Figure 3: (a) Energy levels of the singlet and triplet electron configuration;
(b) Corresponding electron occupation in the two lowest orbitals.

The spin states of the DQD and their energy levels in the (1,1)-(0,2) charge
regimes must be evaluated to understand how the state-dependent behavior is
manifested during readout. Generally, a two-electron system has four possible
spin states when subjected to a magnetic field within a QD site. The four
possible configurations are namely the singlet (S) and the three triplets (T-,
T0, and T+) states, as illustrated in Fig. 3(a). The difference in energy
between the T0 and the T- (T+) state is denoted as the Zeeman energy
($\Delta$Ez), while EST denotes the energy difference between the S and T0.

Fig. 3(b) depicts the corresponding orbital diagram for the two-electron
system for the singlet and triplet states. Only the two lowest orbitals in
which electrons can occupy are considered (i.e., O1 and O2). When the DQD is
biased in the (0,2) charge regime, these orbitals are spatially confined to
only the right QD site, while they spread out across the DQD when biased in
the (1,1) charge regime. Due to the Pauli exclusion principle, each orbital
can only be occupied by two electrons of opposite spins [12]. Consequently,
only the singlet state can contain two electrons occupying the same orbital.

The energy levels of the singlet and triplet configurations for the (1,1) and
(0,2) charge configurations are plotted together in Fig. 4(a). The sketch’s
left and right columns indicate the singlet and triplet energy levels for the
(1,1) and (0,2) charge configurations, respectively. The figure illustrates
when the DQD is at the interdot regime, equivalently described as the zero
detuning condition ($\epsilon$ = 0). Here, the detuning parameter ($\epsilon$)
is defined as the energy difference between the S(1,1) state and the S(0,2)
state. In case the DQD collapses to a singlet state, an electron from one QD
can freely tunnel to another at the zero detuning as the S(1,1) and the S(0,2)
states have similar energy levels. In contrast, an electron cannot oscillate
back and forth when the DQD is in a triplet state, as there are no overlapping
triplet states when $\epsilon$ = 0.

Figure 4: (a) Sketch of the singlet and triplet energy levels of the DQD for
the (1,1) and (0,2) charge configurations. Due to the smaller EST of the (1,1)
state compared to the (0,2) state, the lowest energy level of the DQD is
T-(1,1) when $\epsilon=0$ [12]; (b) Equivalent lump model of the DQD under RF
excitation.

As the electron moves between the QD sites due to an RF readout signal applied
at RG, quantum capacitance is generated [13]. Consequently, the DQD can be
modeled as two parallel capacitors during readout, as illustrated in Fig.
4(b), where the capacitance contribution comes from namely the geometric
capacitance ($C_{geo}$) and the state-dependent quantum capacitance ($C_{q}$).
Thus, modeling the readout behavior of the system relies on quantifying the
impact of quantum capacitance, which is the main focus of the subsequent
section.

###  II.2 Quantum Capacitance Theory

As defined in [14], the quantum capacitance exhibited by the DQD is expressed
as

$\displaystyle C_{q}$
$\displaystyle=-\beta^{2}\frac{\partial^{2}E_{\ket{\psi}}}{\partial\epsilon^{2}},$
(4)

where $E_{\ket{\psi}}$ is the energy level of the singlet or triplet state of
a DQD and $\beta$ is the lever arm of the DQD device, and is given by

$\displaystyle\beta$ $\displaystyle=|e|C_{g}/C_{\Sigma},$ (5)

where $|e|$ is the electron charge, $C_{g}$ is the capacitance of the readout
gate, and $C_{\Sigma}$ is the total geometric capacitance of the DQD.

To calculate the quantum capacitance contribution from the singlet or triplet
state expressed in (4), the energy level of the DQD for different detunings
must be calculated in each respective configuration. This can be achieved by
solving the DQD’s Hamiltonian using the Fermi-Hubbard model[15]. However,
solving the total Fermi-Hubbard Hamiltonian of the DQD can be too complex and
cumbersome. As we are only interested in the relevant features of the DQD
under readout conditions, we simplify the DQD model and approximate the
behavior by examining the Hamiltonian of a five-level system consisting of the
ground and excited states of the S(1,1)-S(0,2) state and the three (1,1)
triplet states. In general, the Hamiltonian for the S(1,1)-S(0,2) singlet
state can be expressed by

$\displaystyle H_{DQD,\ket{S}}$
$\displaystyle=\frac{\epsilon}{2}\tau_{z}+t_{c}\tau_{x},$ (6)

where $t_{c}$ is the tunnel coupling, and $\tau_{z,x}$ are the Pauli matrices
[16]. By solving the Hamiltonian with the time-independent Schrodinger
equation, one can obtain the energy levels of the DQD for the singlet state as
a function of detuning, expressed as

$\displaystyle E_{\ket{S_{g,e}}}$
$\displaystyle=\mp\frac{1}{2}\sqrt{\epsilon^{2}+4t_{c}^{2}}.$ (7)

Furthermore, the energy levels of the (1,1) triplet state as a function of
detuning can be approximated by [13]

$\displaystyle E_{\ket{T_{0}}}$ $\displaystyle=\frac{\epsilon}{2},$ (8)
$\displaystyle E_{\ket{T_{\pm}}}$ $\displaystyle=\frac{\epsilon}{2}\pm
g\mu_{B}B,$ (9)

where $g$ is the electron gyromagnetic ratio, $\mu_{B}$ is the Bohr magneton
constant, and $B$ is the applied magnetic field.

Figure 5: Energy level diagram of a DQD based on the two-level quantum system
approximation. Note that the triplet (0,2) energy levels are not plotted as
they are significantly higher than the (1,1) triplet energy levels.

The energy levels of the singlet and triplet states expressed in (7)-(9) as a
function of the detunings are plotted in Fig. 5. By plugging the energy
expression (7)-(9) to (4), the induced quantum capacitance of the singlet and
triplet states can be respectively derived as

$\displaystyle C_{q}=$
$\displaystyle+\beta^{2}\frac{2t_{c}^{2}}{(\epsilon^{2}+4t_{c}^{2})^{3/2}}$
for $\ket{S_{g}}$, (10) $\displaystyle C_{q}=$
$\displaystyle-\beta^{2}\frac{2t_{c}^{2}}{(\epsilon^{2}+4t_{c}^{2})^{3/2}}$
for $\ket{S_{e}}$, (11) $\displaystyle C_{q}=$ $\displaystyle 0$ for
$\ket{T_{0}}$ and $\ket{T_{\pm}}$. (12)

Note that only the singlet state contributes an additional quantum capacitance
during readout due to the curvature of the energy level. Moreover, the ground
singlet state produces a positive quantum capacitance, whereas the excited
singlet state produces a negative quantum capacitance. In contrast, no quantum
capacitance contribution exists when the DQD is in the triplet states, as
their energy levels are linearly dependent on $\epsilon$.

As previously mentioned, the right QD can be used as a reference spin for
readout, in which its spin is initialized to a spin down and is used to
compare the spin state of the left QD. Consequently, the two lowest energy
levels used as the computational basis are the $\ket{S_{g}}$ and $\ket{T_{-}}$
states [see Fig. 3]. For convenience, the two computational basis states used
for the analysis are referred to as the $\ket{S}$ and the $\ket{T}$ state for
the rest of the paper unless stated otherwise.

Based on the developed equations and theory for quantum capacitance, we can
discuss and model the DQD behavior during readout in the following subsection.

###  II.3 Quantum Capacitance During Readout

In practice, the readout operation involves averaging the signal response of
the system over many RF cycles to increase the SNR. Since the signal is
averaged during readout acquisition, the quantum capacitance that the DQD
contributes is also effectively averaged out. To illustrate this, based on
(10), the quantum capacitance profile versus $\epsilon$ is illustrated in Fig.
6(a). Under the assumption that the readout gate (i.e., RG) is driven by a
signal in the form of $V_{A}\sin(\omega_{RF}t)$, the detuning value changes
sinusoidally over each RF cycle:

$\displaystyle\epsilon=\beta V_{A}\sin(\omega_{RF}t),$ (13)

in which, the role of the lever arm ($\beta$) is to convert voltage quantities
to energy, with $V_{A}$ denoting the signal’s amplitude, and $\omega_{RF}$ as
the angular frequency of the applied signal. Consequently, as shown in Fig.
6(a), the value of the quantum capacitance varies along the shaded region due
to the sinusoidal excitation. Due to the readout acquisition’s averaging
nature, the quantum capacitance’s effective value seen during readout is the
weighted average of the quantum capacitance value across the shaded region. By
substituting (13) into (10), the effective quantum capacitance can be
expressed as

$\displaystyle C_{q,eff}$
$\displaystyle=\frac{1}{T_{int}}\int_{0}^{T_{int}}C_{q,\ket{S_{g}}}(\epsilon)\text{
d}t\bigg{\rvert}_{\epsilon\,=\,\beta V_{A}\sin(\omega_{RF}t)}.$ (14)

Provided that the integration time is an integer multiple of the RF readout
signal’s period (i.e., $T_{int}$ = $nT_{RF}$), a closed-form expression for
$C_{q,eff}$ can be obtained,

$\displaystyle C_{q,eff}=\frac{\beta^{2}t_{c}\times
E\left(2\pi,\frac{-\beta^{2}V_{A}^{2}}{4t_{c}^{2}}\right)}{2\pi(\beta^{2}V_{A}^{2}+4t_{c}^{2})},$
(15)

where, $E(\theta,k)$ is an elliptical integral expressed as

$\displaystyle E(\theta,k)=\int^{\theta}_{0}\sqrt{1-k^{2}sin\theta}d\theta.$
(16)

Figure 6: (a) Instantaneous quantum capacitance of the DQD with respect to the
detuning of the DQD; (b) Effective quantum capacitance versus the signal
amplitude at the readout gate (RG).

The effective capacitance for different readout amplitudes is plotted in Fig.
6(b), which is based on the result of (15). A smaller readout excitation is
observed to induce a larger effective quantum capacitance due to the smaller
averaging window, which only captures the peak of the quantum capacitance
profile. At small readout voltages, $C_{q,eff}$ converges to
$\beta^{2}/4t_{c}$, in line with (10) when $\epsilon$ = 0. Also, note that a
small readout excitation degrades the readout signal [see eq. (2)], which may
reduce the SNR for a given fixed $T_{int}$. Hence, a longer $T_{int}$ may be
required to regain the same SNR. In contrast, a larger readout excitation
implies a smaller effective quantum capacitance, which may, in effect,
decrease the frequency shift of the response between the $\ket{S}$ and
$\ket{T}$ states. This behavior is reflected in the state-separation factor
between the $\ket{S}$ and $\ket{T}$ and consequently may lower the SNR as
described in (2).

###  II.4 Optimum Tunnel Coupling

It should be noted that the tunnel coupling can significantly influence the
behavior of the qubit during readout. When the tunnel coupling is set too low,
the energy gap between the $\ket{S_{e}}$ band and the $\ket{S_{g}}$ band
diminishes [see Fig. 5]. As the $\ket{S_{e}}$ band gets closer to the
$\ket{S_{g}}$ state, the system can transition between the $\ket{S_{e}}$ and
$\ket{S_{g}}$ energy bands instead of staying in the $\ket{S_{g}}$ band, which
is necessary for the system to realize quantum capacitance. For the assumption
presented here to be valid, the system must be biased in the adiabatic limit,
bounded by $(2t_{c}/h)/f_{r}>1$ [13], where $f_{r}$ is the readout frequency
and $h$ is the Planck constant. The system is more adiabatic and less likely
to transition between the $\ket{S_{e}}$ and $\ket{S_{g}}$ state as the ratio
between the tunnel coupling and the readout frequency $f_{r}$ increases.

The $C_{q,eff}$ for different adiabaticity factors are plotted in Fig. 7.
While it is desired to operate below the adiabatic limit, it is observed that
larger quantum capacitance is observed by biasing the DQD closer to the limit,
which can increase the sensitivity of readout. Based on the observation, the
qubit must be biased close to the adiabatic limit (i.e., $(2t_{c}/h)/f_{r}=2$)
to maximize the quantum capacitance contribution.

Figure 7: $C_{q,eff}$ for different adiabaticity factors (i.e.,
$(2tc/h)/f_{r}$).

###  II.5 Lever Arm

The quantum capacitance can also be increased by means of the lever arm
($\beta$) to increase the readout sensitivity, as described by (10). The lever
arm can be increased by designing the QD devices with quantum wells closer to
the gate electrodes or by implementing the device in a silicon substrate,
similar to a CMOS process [17, 18]. While the approach aligns well with the
future goal of monolithic integration with CMOS electronics, these devices
tend to have faster decoherence times as the qubit is located closer to the
gate dielectric interface [18]. Thus, increasing the lever arm will need
careful consideration as it influences multiple factors other than the readout
performance.

##  III Readout Simulation

The quantum capacitance model introduced in the previous section can now be
applied to a qubit sample that has been measured previously [19]. The qubit
micrograph is shown in Fig. 8(a). The sample, realized in 28Si/SiGe
heterostructure, consists of 2 DQD sites separated by a 250$\,\mu$m half-
wavelength superconducting transmission line fabricated with NbTiN material.
The half-wavelength transmission line is AC coupled to the input and output
ports (port-1 and port-2) through the coupling capacitors $C_{c}$. The DQD2 is
biased at the zero detuning regime, thus exhibiting a geometric and quantum
capacitance during readout when DQD2 is at the $\ket{S}$ state. Moreover, the
DQD1 is tuned in the (0,0) charge regime and thus only contains a fixed
geometric capacitance. The sample is probed with an RF signal from port-1. The
RF tone’s corresponding phase or amplitude change can be detected by measuring
its in-phase (I) and quadrature (Q) signal components at port-2 using an I/Q
RX.

The complete electrical model of the sample is shown in Fig. 8(b). To model
the state-dependent capacitance of DQD2, a voltage-dependent capacitor
$C_{q}(V_{RG})$ is included in the circuit, and its value depends on the
voltage at the readout gate [see Fig. 2]. A resistor (i.e., $R_{TL}$) is also
included in series to the transmission line in the schematic to model the
system’s loss, which can originate from resistive and dielectric losses in the
sample [20]. When comparing the simulation with the measurement results,
$R_{TL}$ is used as a fitting parameter based on the measured resonator’s
bandwidth. In the simulation, the quantum capacitance contribution is realized
by a lookup table (LUT) VerilogA model, relating different readout gate
voltage levels to distinct $C_{q}$ values using (10) and DQD properties
summarized in Tab. 1.

Figure 8: (a) Micrograph of the qubit sample; (b) Equivalent circuit model of the sample.  Table 1: System parameters for simulation and verification. Quantity, Symbol | Value
---|---
Tunnel coupling, 2$t_{c}$/h | 14.1 GHz
Lever arm, $\beta$ | 102 meV/V
Characteristic resonator impedance, $Z_{TL}$ | 4.5 k$\Omega$
Characteristic impedance of system, $Z_{0}$ | 50 $\Omega$
Geometric DQD capacitance, $C_{geo}$ | 1.9 fF
Coupling capacitance, $C_{c}$ | 0.32 fF
Series resistance, $R_{TL}$ | 17 $\Omega$

###  III.1 Frequency Shift

The transmission (S21) behavior between port-1 and port-2 for different
$P_{RF}$ values is plotted in Fig. 9(a). The orange and blue lines in the plot
indicate the transmission response when the DQD is in the $\ket{T}$ and the
$\ket{S}$ state, respectively.

Figure 9: (a) Simulated magnitude and phase response of the transmission gain
(S21) of the readout system in Fig. 8 for $P_{RF}$ of -140 dBm to -60 dBm in
steps of 5 dBm; (b) The frequency shift between the $\ket{S}$\- and
$\ket{T}$-state resonant frequencies for different RF input power.

To gain more insight into the (S21) behavior, it is instructive to see the
behavior of the input impedance of the sample. For simplicity, a large
coupling capacitor $C_{c}$ and a lossless system are first considered such
that the input impedance of the sample can be approximated as

$\displaystyle Z_{in}\approx Z_{TL,eff}\frac{Z_{L}+jZ_{TL,eff}\tan{(\gamma
l)}}{Z_{TL,eff}+jZ_{L}\tan{(\gamma l)}},$ (17)

where $l$ is the physical length of the transmission line, and $Z_{L}$ is the
load impedance (equal to 50 $\Omega$). $Z_{TL,eff}$ is the effective
characteristic impedance of the transmission line defined as
$\sqrt{{L_{TL}^{\prime}}/{C_{TL,eff}^{\prime}}}$, where $L_{TL}^{\prime}$ is
the inductance per unit length and $C_{TL,eff}^{\prime}$ is the effective
capacitance per unit length of the transmission line, including the
capacitance of the DQD. Additionally, $\gamma$ is the phase constant defined
as

$\displaystyle\gamma=\omega\sqrt{L_{TL}^{\prime}C_{TL,eff}^{\prime}},$ (18)

where $\omega$ is the angular frequency where the expression is evaluated. On
resonance, (17) simplifies to 50 $\Omega$ as $\tan(\gamma l)$ approaches 0. In
effect, all the power provided by the source ($P_{RF}$) is delivered to port-1
and consequently also delivered to the load at port-2, resulting in a 0 dB
response in the $|S\textsubscript{21}|$ plot [see Fig. 9(a)]. Away from
resonance, $\tan(\gamma l)$ tends to be large, thus $Z_{in}$ approaches
$\sim$$Z_{TL,eff}^{2}/Z_{L}$. Due to the system’s large $Z_{TL,eff}$,
reflection at the input plane occurs, leading to an incomplete power transfer
to $Z_{L}$. This reflection contributes to the observed bandpass response in
the $|S\textsubscript{21}|$ plot.

The resonant condition requires that the $\gamma l$ term in (17) be equivalent
to $\pi$, such that $Z_{in}$ simplifies to 50 $\Omega$. Consequently, the
resonant frequency can be derived based on (18) and can be recast as

$\displaystyle\omega_{res}=\frac{\pi}{\sqrt{\left(L_{TL}^{\prime}l\right)\times\left(C_{TL,eff}^{\prime}l\right)}}\rightarrow\omega_{res}=\frac{\pi}{\sqrt{L_{TL}C_{TL,eff}}},$
(19)

where $L_{TL}$ and $C_{TL,eff}$ are the transmission line’s total equivalent
lumped inductance and capacitance, respectively. Note that $C_{TL,eff}$, has
two different loading conditions depending on the state of the DQD2, which can
be described as

$C_{TL,eff}=\begin{cases}C_{TL,\ket{T}}=C_{TL}+2C_{geo}&\text{$\ket{T}$
state,}\\\ C_{TL,\ket{S}}=C_{TL}+2C_{geo}+C_{q}&\text{$\ket{S}$
state,}\end{cases}$ (20)

where $C_{TL}$ is the equivalent lumped capacitance of only the bare
transmission line, excluding the DQD’s contribution.

When DQD2 is in the $\ket{T}$ state, $C_{TL,eff}$ is only affected by
$C_{geo}$. On the other hand, when DQD2 is at the $\ket{S}$ state,
$C_{TL,eff}$ is affected by $C_{geo}$ and $C_{q}$, which leads to a lower
$\omega_{res}$, as observed from Fig. 9(a). Additionally, when the system is
subjected to a strong RF readout signal, $C_{q}$ is averaged over a larger
span. This results in a lower $C_{q,eff}$, causing a smaller resonant
frequency separation between the $\ket{S}$ and the $\ket{T}$ response.

To further increase the separation of the resonant frequency between the
$\ket{T}$ and $\ket{S}$ state, it is necessary to increase the
$C_{TL,\ket{S}}/C_{TL,\ket{T}}$ capacitance ratio. Hence, $C_{q}$ must be
maximized while keeping $C_{TL}$ and $C_{geo}$ low. A higher $C_{q}$ can be
achieved by increasing the lever arm of the device with the drawbacks
mentioned in Section II. On the other hand, $C_{TL}$ can be kept low by
designing the transmission line with a large characteristic impedance.

The simulated resonant frequency shift difference (i.e., $\Delta F$) of the
$\ket{S}$-state relative to the $\ket{T}$-state for different input probe
power is summarized in Fig. 9(b). At low readout power, a $\sim$5 MHz
frequency shift is seen. However, the frequency shift becomes smaller at
higher probe power as $C_{TL,\ket{S}}$ approaches $C_{TL,\ket{T}}$, which
aligns with the presented theory.

###  III.2 Readout Signal and Probe Frequency

Figure 10: Simulated readout response: (a) the state-separation factor; (b)
the received power at port-2 of the transmission line (i.e., $P_{Sig}$).

To determine the readout SNR, the readout signal’s power at port-2 must be
calculated. However, the behavior of the readout signal depends on the
frequency at which the system is probed. This section explores the behavior of
the readout signal in two scenarios: (1) when the system is probed at the
resonant frequency of state $\ket{T}$ (i.e., 6.91 GHz) leading to a BASK
response, and (2) at the frequency halfway in between the resonant frequency
response of the $\ket{S}$ and $\ket{T}$ state, leading to a BPSK response.

As expressed in (1), the readout signal’s power ($P_{sig}$) is defined as the
product of the state-separation factor and the power applied to the readout
sample. The simulated state-separation factor (i.e.,
$|{S_{21,\ket{T}}}(\omega_{r})$ \- ${S_{21,\ket{S}}}(\omega_{r})|^{2}$) for
both cases are plotted in Fig. 10(a). In case-1, the state-separation factor
is shown to be approximately 1 for low readout probe power. In this condition,
the ${S_{21,\ket{S}}}$ term is significantly smaller than the
${S_{21,\ket{T}}}$ term when probed at the resonance of the $\ket{T}$ state.
Consequently, the $\ket{S}$ response is attenuated during readout compared to
the $\ket{T}$ response, resembling a BASK readout signal, as discussed
previously. In case-2, the state-separation factor is shown to be smaller than
case-1 at low readout power. In this condition, both the ${S_{21,\ket{T}}}$
and ${S_{21,\ket{S}}}$ terms are generally small when probed between the
$\ket{T}$ and $\ket{S}$ resonant frequencies. This results in a readout
response that is both attenuated and exhibits a 180-degree phase difference
between the $\ket{T}$ and $\ket{S}$ readout response, akin to a BPSK signal.
Interestingly, as the applied input power increases, the state-separation
factor converges, reflecting that the resonant frequencies of the $\ket{T}$
and $\ket{S}$ states converge.

The readout signal’s power is shown in Fig. 10(b). Remarkably, the plot shows
a saturation in the readout signal power due to the decrease in the state-
separation factor with the application of higher readout probe power. As the
readout signal power dictates the readout SNR [see eq. (2)], the simulation
implies that there is a maximum achievable SNR for a given qubit property.
Interestingly, in a different way and based on the time-dependent charge
population in the DQD under RF excitation, [21] also theoretically predicted
the saturation in the readout signal. However, the work in [21] did not
consider the power-dependent frequency shift and did not present any
experimental results to support the theory.

From this observation, performing a readout shown by case-1 is generally
recommended, as it yields a larger readout signal response even at lower probe
power, leading to a higher SNR even at lower probe power. Furthermore, from a
practical point of view, readout at the bare resonant peak of the resonator is
simpler as the readout frequency is fixed and does not need to be changed to
account for the power-dependent resonant shift. Based on this reasoning, the
experiments in this paper follow the readout behavior shown by case-1.

###  III.3 Noise Estimation

The power spectral density of the system’s noise ($N_{0}$) at the input of the
readout chain can be expressed as $kT_{N}$( W/Hz), where $k$ is the Boltzmann
constant, and $T_{N}$ is the effective noise temperature of the readout
system. Considering an ambient temperature of $T_{amb}$ for the DQD sample,
$T_{N}$ may be estimated by

$\displaystyle
T_{N}=T_{amb}+T_{TWPA}+\frac{T_{LNA}}{G_{1}}+\frac{T_{RX}}{G_{1}G_{2}},$ (21)

where $T_{TWPA}$, $T_{LNA}$, and $T_{RX}$ are the noise temperatures of the
traveling wave parametric amplifier (TWPA), low noise amplifier (LNA), and the
RX, respectively, while $G_{1}$ and $G_{2}$ are the gain of the TPWA and LNA,
respectively. Considering a TWPA gain of 28 dB ($G_{1}$), the readout system’s
noise performance is mainly dictated by the TWPA. Assuming that the TWPA is
quantum-limited, its noise temperature can be approximated by $T_{TWPA}$ =
$hf_{r}/k$ [22]. For an $f_{r}$ of 6.91 GHz, and a $T_{amb}$ of 20 mK, the
total noise temperature of the readout architecture ($T_{N}$) is estimated to
be 350 mK.

###  III.4 Normalized Signal-to-Noise Ratio

The behavior of the readout SNR can now be discussed. However, to properly
evaluate the trade-off between $P_{RF}$ and $N_{0}$, it is far more
instructive to define the normalized SNR: SNRN = $P_{Sig}/N_{0}$ =
SNR/$T_{int}$. In this way, we can compare the performance of the readout
scheme independent of the chosen $T_{int}$. The simulated SNRN, when assuming
a $T_{N}$ of 350 mK and considering all the system parameters in Tab. 1, is
plotted in Fig. 11. The SNRN firstly increases linearly with the probe power
but finally saturates at a maximum of 95 dB$\cdot$Hz. Considering a $T_{int}$
of 1 $\mu$s, and assuming a quantum-limited readout chain, a theoretical
maximum SNR of 35 dB is achievable.

With the theoretical framework for gate-based RF readout established, the
measurement verification of the proposed model can now be carried out.

Figure 11: Simulated normalized SNR, assuming a quantum-limited readout chain,
and $T_{N}$ = 350 mK.

##  IV Measurements

Figure 12: Measurement setup used to verify the quantum capacitance behavior.

###  IV.1 Measurement Setup

The complete measurement setup is shown in Fig. 12, similar to the ones used
in [23, 19, 20]. The sample is mounted at the mixing chamber plate and cooled
down to an ambient temperature of 20 mK. Alongside the qubit sample, a TWPA
with 28 dB gain is installed to minimize the noise of the readout chain, thus
allowing one to measure a low-power readout signal at various $T_{int}$. The
TWPA pump is generated at room temperature (RT) and fed to the TWPA by a
coupler at the mixing chamber plate. An isolator is used between the qubit
sample and the coupler to isolate the qubit from reflections due to the large
TWPA pump tone. At the 4 K plate, a high electron mobility transistor (HEMT)
LNA with a 26 dB of gain is used to amplify the readout signal further. An
additional CMOS RX chip is mounted on this plate for signal downconversion and
amplification [23]. At RT, two off-the-shelf 1 GS/s 8-bit ADCs are used to
quantize both the I/Q signals, allowing for further digital signal processing
at the baseband. A single LO source at RT is also used to drive the two mixers
that upconvert and downconvert the readout signal to ensure phase
synchronization. Moreover, a 75 dB attenuation in the input RF line is
intentionally employed to ensure a proper thermal noise level at the input of
the sample at the mixing chamber plate.

###  IV.2 Charge Stability Diagram

The stability diagram of the sample is measured and shown in Fig. 13(a). The
diagram is obtained when the qubit sample is read out at the bare resonance
frequency (i.e., 6.91 GHz) for different $V_{LP}$ and $V_{RP}$ values. A clear
distinction between different charge states can be observed, especially at
lower charge population regimes.

Figure 13: (a) Measured charge stability diagram of DQD2. $|V_{out}|$ obtained
from $I_{out}$ and $Q_{out}$ voltages at the RT plane [see Fig. 12]; (b)
Zoomed-in charge stability diagram at (2,1)-(1,2) charge regime. The next two
sub-figures are measured across the red dashed line: (c) $|V_{out}|$ versus
$V_{RP}$; (d) $|V_{out}|$ versus frequency at zero and non-zero detuning.

The behavior of $C_{q,eff}$ is verified in the (2,1)-(1,2) charge state.
Considering that the quantum capacitance is generated by forcing an electron
to oscillate across the DQD site under RF excitation, the behavior in the
(2,1)-(1,2) regime is similar to the (1,1)-(0,2) regime discussed previously
since the experiment isolates only the quantum capacitance contribution
behavior. The charge stability scan around the (2,1)-(1,2) charge state is
shown in Fig. 13(b). To understand the effect of the quantum capacitance
during readout, a line cut is taken across the interdot crossing, and its
corresponding result is depicted in Fig. 13(c). At point (A), the DQD is not
loaded by the quantum capacitance as it is far from the zero detuning
($V_{RP}$ = -620 mV). As shown previously, averaging the quantum capacitance
well away from the zero detuning results in a net zero effective quantum
capacitance, which emulates the behavior of the $\ket{T}$ state. In contrast,
point (B) is located where the system is at zero-detuning ($V_{RP}$ = -590
mV), where the maximum quantum capacitance contribution can be observed. Due
to the quantum capacitance contribution, the resonator’s resonance frequency
shifts to a lower frequency, which, in effect, lowers the observed output
voltage. As more positive $V_{RP}$ is applied to the DQD ($V_{RP}$ = -560 mV),
the system returns to the response observed at (A).

The shift in the resonance frequency of the resonator can also be observed in
Fig. 13(d). The data is obtained by monitoring the magnitude of the
downconverted I/Q voltages (i.e., $|V_{out}|$) when the frequency of the probe
signal is swept from 6.9 GHz to 6.915 GHz. For comparison, the simulation
results are also included in Fig. 13(d) using the dotted lines, which show a
good agreement with the measurements. The orange line in the figure depicts
the response when the system is away from zero detuning ($V_{RP}$ = -620 mV),
while the blue line depicts the system’s response at zero detuning ($V_{RP}$ =
-590 mV). The mapping of points (A), (B), and (C) from Fig. 13(c) are also
labeled in Fig. 13(d) for clarity. As predicted from the simulation, a 5 MHz
frequency shift is observed for the zero-detuning response at very low power.

###  IV.3 Frequency Shift

Figure 14: Observed frequency shift in simulation and measurements for
different $P_{RF}$.

Given that the quantum capacitance cannot be measured directly, we can only
observe the manifestation of the quantum capacitance through the frequency
shift response of the sample. To verify this, the frequency at which the
maximum transmission is observed is tracked at zero detuning and non-zero
detuning conditions while sweeping the probe tone power from -125 dBm to -89
dBm, referred to the port-1 plane in Fig. 12(c). The frequency difference
($\Delta F$) is defined as the difference between the resonance peaks of the
non-zero and zero detuning responses. As shown in Fig. 14, the frequency shift
of the DQD decreases for higher input power, following the trajectory dictated
by the developed theory and simulation results. Interestingly, the frequency
shift reaches 0 faster than in the simulation. This behavior is discussed in
the following section.

###  IV.4 Readout Signal

Figure 15: (a) I/Q trajectory of readout signal for different $P_{RF}$ at zero
and non-zero detuning regimes. The colored graded line indicates the expected
I/Q trajectory from the simulation, while the triangle and the square data
points indicate the measured value at non-zero and zero detuning,
respectively. (b) The magnitude and (c) phase response of $A_{sig}$.

The simulated and measured readout I/Q responses for zero and non-zero
detunings are illustrated in Fig. 15(a). The data is obtained by referring the
measured I/Q data sets to the port-2 plane when the system is probed at the
readout frequency of $\omega_{r}/2\pi$ = 6.91 GHz. The triangle and square
points indicate the non-zero and zero detuning responses, respectively. Note
that the solid line indicates the simulated I/Q voltages, and the measured
data points are color-graded based on the readout power applied to the sample.
Based on Fig. 15(a), the I/Q points resemble an amplitude-modulated signal at
lower readout power, where the zero-detuning response is close to the origin
of the I/Q plane due to the attenuation of the shifted resonator response. At
higher readout power, the zero-detuning I/Q response acquires an extra phase
component as the resonant frequency of the zero-detuning response gets closer
to the bare-resonance frequency of 6.91 GHz. The distance between the two
responses is also reduced for the higher power, implying a decreased readout
signal amplitude at a larger probe power.

As explained in Section I and shown in Fig. 15(a), the readout signal
($A_{sig}$) is the distance between the qubit’s responses in the constellation
diagram. The magnitude and phase of $A_{sig}$ are respectively shown in Fig.
15(b) and Fig. 15(c). The simulated and measured $|A_{sig}|$ and $\angle
A_{sig}$ are seen to be in good agreement up to $P_{RF}$ of $\sim$-100 dBm.
Beyond this point, a noticeable difference between simulation and measurement
is observed, especially the measured $\angle A_{sig}$. Compared to the
simulation, at large probe power, the measured $\angle A_{sig}$ approaches 0
degrees faster, and the measured $|A_{sig}|$ does not saturate but reduces
significantly. Both magnitude and phase behaviors imply that the resonant
frequency of the zero detuning response becomes the same as the bare resonance
frequency of the resonator, consistent with the qualitative trend observed for
$\Delta F$ in Fig. 14.

The discrepancy between the simulation and the measurement results at higher
probe power levels can be attributed to several factors. The primary factor
contributing to the reduced readout signal amplitude at large probe power
levels is suspected to be associated with Landau-Zener transitions [24]. At a
very large excitation, the rate at which the system’s detuning evolves with
time (i.e., $\partial\epsilon/\partial t$) is fast enough such that the
evolution is no longer adiabatic. Hence, the system can not be examined with a
time-independent Hamiltonian, as initially assumed. In this regime, the system
can undergo transitions between the $\ket{S_{g}}$ and $\ket{S_{e}}$ energy
bands, as illustrated in Fig. 5, leading to a reduced readout response.
Interestingly, this observation is consistent with the numerical simulation
shown in [21], which predicts that the quantum capacitance contribution can be
reduced to 0 when the system is assumed to be non-adiabatic. In addition to
the Landau-Zener transitions, the reduction in the readout amplitude may also
be caused by the resonator’s self-heating, which can cause the resonator’s
quality factor to degrade at a very large excitation.

###  IV.5 Readout Noise

The standard deviation of $|V_{out}|$ can be used to estimate $T_{N}$
experimentally. By measuring the standard deviation of $|V_{out}|$ at the non-
zero detuning regime between a $V_{RP}$ of -570 mV to -560 mV (i.e., Point C
in Fig. 13(c)), the integrated noise voltage can be calculated.
Mathematically, this is equivalent to $\sqrt{kT_{N}\cdot 50/T_{int}}$ (V).
Experimentally, $T_{N}$ is estimated to be 460 mK when referring to the input
of the readout chain.

The measured $T_{N}$ is close to what was previously observed for a readout
chain incorporating TWPA amplifiers [25, 26, 27]. However, it is 1.3$\times$
higher than the previous theoretical estimation (i.e., 350 mK). The
discrepancy between the theoretical and measured values may stem from
different contributions. First, $T_{N}$ is estimated by always assuming that
$T_{amb}$ is 20 mK, which may be higher in reality as reflected by the
electron temperature. Moreover, the estimation did not account for the losses
of the isolator and coupler in front of the TWPA.

###  IV.6 Normalized Signal-to-Noise Ratio

Figure 16: Measured normalized readout SNR for different $P_{RF}$ and
$T_{int}$. The black dotted line indicates the simulated SNRN, assuming a
$T_{N}$ of 460 mK.

Fig. 16 shows the measured and simulated SNRN. The black dotted line indicates
the simulated SNRN based on the previously simulated readout signal, with a
$T_{N}$ of 460 mK. As expected, the data shows that the measured SNRN overlaps
each other for different $T_{int}$ experiments. A maximum measured SNRN of
$\sim$89 dB$\cdot$Hz is observed at $P_{RF}$ = -95 dBm, and SNRN degrades
beyond this power level.

##  V Discussion

The previous section showed that the qubits’ resonant frequency shift is
power-dependent. This section explores the implications of the power-dependent
readout behavior, examines the associated trade-offs, and addresses the noise
requirements for the readout chain.

###  V.1 Frequency Multiplexing Consideration

The measurement and model indicate that the readout power can control the
qubit’s frequency shift. Thus, a higher readout power leads to the possibility
of reading out more qubits per given bandwidth, resulting in a higher spectrum
efficiency as the frequency spacing requirement between each qubit decreases.
As depicted in Fig. 14, the frequency shift is estimated to be on the order of
2 MHz at the peak of the SNRN with a $P_{RF}$ of -95 dBm. Assuming each qubit
has a frequency shift of 2 MHz with a 3 MHz resonator bandwidth, each qubit
has a channel bandwidth of $\sim$5 MHz. However, it should be noted the
frequency spacing between the qubits cannot be too small as the readout pulse
power, depending on the desired integration times, may leak into neighboring
qubits, which eventually may require pulse shaping for a dense FDMA readout
architecture.

It is important to note that the number of qubits that can be read out within
a specific frequency range can be increased by narrowing the resonator
bandwidth through a resonator design with a large characteristic impedance.
However, there are practical constraints on how much the resonator bandwidth
can be reduced. At very small resonator bandwidth, readout is mostly limited
by the ring-up time of the resonator when given an RF impulse. Consequently,
the readout response may not settle fully for a given $T_{int}$, thus reducing
the readout signal and, eventually, the SNR.

###  V.2 Optimum Tunnel Coupling and Readout Operating Frequency

Based on the discussion above, one could also ask about the preferable readout
frequency for gate-based readout. Given a readout frequency $f_{r}$, one must
bias the $t_{c}$ such that $(2t_{c}/h)/f_{r}$ = 2. Consequently, the tunnel
coupling should be proportionally increased when the qubit is read at a higher
$f_{r}$ to keep the system at the adiabatic limit. A lower frequency shift is
expected when the qubit is read out at higher $f_{r}$ due to the lower quantum
capacitance at higher $t_{c}$ bias. In addition, a higher readout frequency
increases the power consumption of the electronics significantly, which can
also limit the scalability of the readout system due to the limited power
budget in the dilution refrigerator. However, operating at a higher readout
frequency may decrease the footprint of the resonator, which can be beneficial
when the system is scaled up.

The opposite behavior is expected when the readout frequency is lowered. Lower
readout frequency implies a lower $t_{c}$. Hence, a larger quantum capacitance
is to be expected, which can increase the SNR. As the frequency of operation
is lowered, power consumption of the electronics is also expected to follow
suit. However, the larger quantum capacitance means that each qubit occupies a
larger channel and limits the number of qubits occupying a certain bandwidth.
Additionally, pushing the readout frequency lower leads to a larger area
required for the resonator and may limit scalability.

###  V.3 Noise Temperature Requirement for Readout Electronics

Figure 17: SNR contour plot for different $T_{int}$ and $T_{sys}$ values with
a $T_{amb}$ = 4 K and $P_{RF}$ = -95 dBm.

As discussed in Section I, an SNR larger than 11.5 dB is needed to achieve BER
$\leq$ 10-4. Assuming a $T_{int}$ of 1 $\mu$s, the maximum inherent SNR of the
measured gate-based readout is 29 dB, extrapolated from SNRN of 89 dB$\cdot$Hz
[see Fig. 16]. Hence, an excess SNR of 17.5 dB allows one to increase $N_{0}$
by 56$\times$ and still satisfy the BER specification, resulting in a maximum
$T_{N}$ of 25.8 K for the readout chain. Furthermore, assuming the readout
electronics are now operating at 4 K, the readout chain should exhibit an
input-referred noise temperature ($T_{sys}$) of $\leq$ 21.8 K to still comply
with a readout SNR of 11.5 dB — a significantly more relaxed noise requirement
in comparison to the performance offered by TWPA and the HEMT LNA.

Fig. 17 shows the SNR for different $T\textsubscript{sys}$ and $T_{int}$
values when the system is probed at its maximum achievable $A_{sig}$ ($P_{RF}$
= -95 dBm) while considering a $T_{amb}$ of 4 K. Assuming a modest $T_{sys}$
of 50 K based on the recently published cryo-CMOS RX design [23, 28, 29, 30,
31], the contour plot reveals that the prior-art cryo-CMOS receivers cannot
satisfy the 11.5 dB target SNR at a $T_{int}$ of 1 $\mu$s. As it is suspected
that the noise performance of the cryo-CMOS receivers is limited by the shot
noise [32] and the self-heating effect [33], passive amplification techniques
should be investigated in the future to avoid large biasing currents required
for active devices in LNAs [34, 35].

##  VI Conclusion

This work presents a theoretical framework that allows one to investigate the
impact of the receiver’s noise temperature, readout probe power, and
integration time on the SNR and fidelity of RF gate-based readout systems.
Considering the qubit’s quantum mechanical behavior, a semi-classical model
was developed that estimates the state- and power-dependent resonance
frequency shift and intrinsic readout SNR. Following the trajectory dictated
by the developed theory and simulation, the experimental results showed that
the frequency spacing between the resonance peaks corresponding to different
qubit states was reduced by increasing the power of the readout tone. This
affects the separation between the qubit states in the constellation diagram
and eventually limits the maximum achievable SNR of the gate-based readout
systems. Moreover, based on the outcome of this study, different trade-offs
among the qubit biasing, the readout SNR, the system’s footprint, the
receiver’s operating frequency, bandwidth, and noise are discussed.
Consequently, the presented guideline can help designers realize a scalable
and low-power readout chain while obtaining a readout fidelity sufficient for
fault-tolerant quantum computation.

## Acknowledgment

This research was funded by Intel Corporation. The authors would like to thank
G. Zheng and P. Harvey-Collard for fabricating the device, Z. Y. Chang, J.
Mensingh, O. Benningshof, N. Alberts, R. Schouten, R. Vermeulen from TU Delft
for measurement support, L. DiCarlo and his team for access to the 3He
cryogenic measurement setup, L. P. Kouwenhoven and his team for access to the
NbTiN film deposition, W. Oliver for providing the TPWA, and the members of
the Spin Qubit team, the Cryo-CMOS team, M. F. Gonzalez-Zalba, and S.
Pellerano from Intel Corporation for the useful discussions. Data repository
in [36].

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