# Dynamics of measurement-induced state transitions in superconducting qubits

Yuta Hirasaki  Department of Applied Physics, The University of Tokyo, Tokyo
113-8656, Japan.  Shunsuke Daimon<EMAIL_ADDRESS>Department of
Applied Physics, The University of Tokyo, Tokyo 113-8656, Japan.  Quantum
Materials and Applications Research Center, National Institutes for Quantum
Science and Technology, Tokyo 152-8550, Japan.  Naoki Kanazawa  Toshinari
Itoko  Masao Tokunari  IBM Quantum, IBM Research Tokyo, 19-21 Nihonbashi
Hakozaki-cho, Chuo-ku, Tokyo, 103-8510, Japan.  Eiji Saitoh  Department of
Applied Physics, The University of Tokyo, Tokyo 113-8656, Japan.  Institute
for AI and Beyond, The University of Tokyo, Tokyo 113-8656, Japan.  WPI
Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577,
Japan.  RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan.

###### Abstract

We have investigated temporal fluctuation of superconducting qubits via the
time-resolved measurement for an IBM Quantum system. We found that the qubit
error rate abruptly changes during specific time intervals. Each high error
state persists for several tens of seconds, and exhibits an on-off behavior.
The observed temporal instability can be attributed to qubit transitions
induced by a measurement stimulus. Resonant transition between fluctuating
dressed states of the qubits coupled with high-frequency resonators can be
responsible for the error-rate change.

Quantum computers are expected to perform computational abilities that far
exceed those of their classical counterparts [1]. Many researchers have been
making tremendous efforts to develop technologies that apply the principles of
quantum mechanics to information processing [2, 3, 4, 5]. Diverse materials or
artificial atoms can serve as qubits, such as trapped ions [6], quantum dots
[7], and many others. Among these candidates, superconducting qubits [8] have
gained significant attention for their relatively long coherence times and the
ability to implement fast gate operations [9, 2, 10]. A superconducting qubit-
resonator system is well described by the Jaynes-Cummings model [11]. Within
this framework, quantum nondemolition measurements are achieved by using the
dispersive interaction between the qubit and resonator photons.

Existing superconducting quantum computers cannot escape from noise [12, 13],
including qubit decoherence [14] and imperfect qubit operations [15] and
readout [16]. These factors ultimately limit the computational abilities and
degrade the fidelity of qubit outputs. Several surveys have revealed that
qubit errors show fluctuating behavior over time [17, 18, 19, 20, 21], which
presents a significant challenge for current error mitigation techniques [22,
23, 24]. It is imperative to investigate the error dynamics of superconducting
qubit systems.

In previous reports [25, 26], the mechanism of measurement-induced state
transitions (MIST) has been extensively studied. It was found that when the
measurement stimulus is injected into the readout cavity, qubit dressed states
come into resonance and the qubit is excited beyond the computational
subspace. In addition, it was reported that the resonance shows a noisy
behavior when measured repeatedly, which can be attributed to a fluctuating
offset charge. However, the early studies are limited to a system of
superconducting qubits coupled with low-frequency readout resonators. In
addition, the dynamics of MIST has not been fully investigated.

In this study, we conduct a time-resolved MIST experiment for qubits in an IBM
Quantum system coupled with high-frequency readout resonators. We perform the
experiment for a total duration of approximately $2.70\times 10^{5}$ seconds,
and we analyze the occurrence rate and the duration of MIST. We probe how
dressed state energy fluctuates over time and temporarily resonates with other
energy levels.

All the experiments were performed on ibm_kawasaki, an IBM Quantum system.
This processor had 27 transmon qubits (this device has been updated and now it
has 127 qubits.), and we used the qubit 7 in this processor. The qubit
frequency is ${\omega_{01}}/{2\pi}=5.457$ GHz, with the anharmonicity
$\eta/2\pi=-336.6$ MHz, and the energy relaxation time is $104.9\;\mathrm{\mu
s}$. The qubit is capacitively coupled to a readout resonator with its
fundamental frequency $\omega_{\mathrm{r}}/2\pi=7.117\;\mathrm{GHz}$, and the
decay rate $\kappa=1/(209\;\mathrm{ns})$. The coupling strength between the
qubit and the resonator is $g/2\pi=86.85\;\mathrm{MHz}$. The readout
assignment error for this qubit is $1.46\%$.

In this dispersive regime, we can measure the qubit quantum state by the
dispersive readout. The dressed frequency of the resonator
$\omega_{\mathrm{r},\ket{i}}$ depends on the qubit quantum state $\ket{i}$,
and we can infer the qubit state by probing the readout resonator with a
measurement stimulus. The qubit information is encoded in the amplitude
$A_{\mathrm{out}}$ and phase $\phi$ of the output microwave, as they are
dependent on the resonator frequency. This signal is digitized and represented
as a point in the in phase and quadrature (IQ) plane.

Figure 1:  (a) A qubit energy diagram. The qubit is excited with the pulses
whose frequencies are equal to the qubit transition frequencies. $\omega_{01}$
denotes the transition frequency between the $\ket{0}$ and the $\ket{1}$
states, $\eta$ denotes the anharmonicity, and $\mathrm{X}_{\pi}^{ij}$ denotes
the $\pi$ pulse between the $\ket{i}$ and $\ket{j}$ states. (b) The in phase
and quadrature (IQ) components of the output microwaves with the three
different qubit states. The qubit is prepared in either the ground ($\ket{0}$)
state, the first excited state ($\ket{1}$) or the second excited ($\ket{2}$)
state. The blue (red) dots show the obtained data corresponding to the qubit
$\ket{0}$ ($\ket{1}$) state. The green dots correspond to the $\ket{2}$ state.

We initialized the qubit into three different states, the ground ($\ket{0}$),
the first excited ($\ket{1}$), and the second excited ($\ket{2}$) state as
shown in Fig. 1(a). We then performed measurements to obtain three distinct
clusters corresponding to the three quantum states. As shown in Fig. 1(b), the
blue dots correspond to the qubit $\ket{0}$ state, the red dots correspond to
the qubit $\ket{1}$ state, and the green dots correspond to the qubit
$\ket{2}$ state. We note that the dispersive shift and the quality factor of
this resonator allow us to resolve up to the first three states. Signals
corresponding to the states higher than the $\ket{2}$ state are expected to
appear near those of the $\ket{2}$ state [27]. Hereafter we denote states
higher than the $\ket{1}$ state as $\ket{\mathrm{HS}}$.

In the following experiments, the IQ measurement was repeated at regular
intervals of several hundred microseconds over a period of several tens of
minutes. At each interval, the qubit is initialized into the ground state with
a reset operation. We experimentally confirmed that the reset operation leaves
the HS states unchanged and initializes only the first excited state. Since
the IQ signals distribute as shown in Fig. 1(b), we can discriminate the
$\ket{0}$ and the other states by focusing on the sign of the in phase
component. A negative (positive) in phase component corresponds to the signal
being converted to 0 (1). Henceforth, we denote the probability of obtaining
states other than the $\ket{0}$ state as $P_{1}$. We repeat the IQ
measurements $L$ times, convert the signals to the binary data based on the in
phase components, and then take the ensemble average of $N$ data to obtain a
time series of $P_{1}$.

Figure 2:  (a) Observed $P_{1}$ as a function of time. $P_{1}$ denotes the
probability of measuring states other than the ground state. The inset to the
top panel illustrates the used quantum circuit, where the qubit is initialized
into the $\ket{0}$ state and then measured. We iterated the initialization and
the IQ measurement every $500\;\mu\mathrm{s}$ for the entire $500$-second
duration. (b) The same experiment was conducted multiple times, and the
resulting time series are illustrated in the four panels. (c) The histogram of
the duration time of the observed anomaly. (d)(e) A comparison of the IQ plots
for the normal (blue) and abnormal (red) time periods indicated in (a). The
left (right) panel corresponds to the normal (abnormal) time period.

We conducted the IQ measurement for 500 s and calculated the time series of
$P_{1}$ with $N=16384$ as shown in Fig. 2(a). We interleaved several different
quantum circuits to simulate practical quantum computation (see Appendix A for
the details of the experimental setup). In Fig. 2(a), $P_{1}$ remains around
$0.01$ until 230 s (indicated by the blue line), consistent with the
measurement of the $\ket{0}$ state as shown in the inset. From 230 s to 470 s,
however, $P_{1}$ shows a sudden rise to around $0.08$ (indicated by the red
line). This sudden rise cannot be explained in terms of a sampling error since
the standard deviation is given by
$\sigma=\sqrt{\frac{P_{1}(1-P_{1})}{N}}\approx 7.8\times 10^{-4}$. This
behavior is frequently observed in multiple instances, as demonstrated in the
four panels of Fig. 2(b). We repeated this experiment for approximately
$2.7\times 10^{5}$ seconds to inspect the endurance of the observed anomaly.
The histogram of the duration time of the anomaly is shown in Fig. 2(c). The
duration varies from several seconds to around 200 seconds and the anomaly is
repeatedly observed. The anomaly appeared for a total of $1.66\times 10^{3}$
seconds throughout the entire experiments, which spanned $2.70\times 10^{5}$
seconds.

As a temporal increase in $P_{1}$ often results from a temporal signal
rotation in the IQ plane, we compare the IQ plots of the two time periods in
Fig. 2(a). The result is shown in Fig. 2(d) and (e). The left (right) panel
shows the IQ signals in the lower (higher) error time period, as indicated by
the blue (red) line of Fig. 2(a). In the left panel, most of the dots are
clustered around a center in the fourth quadrant, which indicates that the
qubit is in the $\ket{0}$ state. However, in the right panel, we observe
distinct dense signals near the origin in the first quadrant, suggesting that
the qubit is excited to $\ket{\mathrm{HS}}$ states. Excitations beyond the
qubit subspace contribute to the higher error time period depicted by the red
line in Fig. 2(a).

Figure 3:  Controlled experiment to specify the excitation origin. (a) The
qubit thermalizes to the ground state after the $1000\;\mathrm{\mu s}$ delay,
and then it is excited to the $\ket{1}$ state with the $\pi$ pulse. The reset
operation initialize the qubit $\ket{1}$ state into the $\ket{0}$ state,
followed by the qubit measurement. The resulting signals are plotted in the IQ
plane at the bottom. (b) The measurement stimulus is transmitted to the qubit
after the qubit is excited to the $\ket{1}$ state. The obtained signals are
shown at the bottom.

From the result in Fig. 2, it can be deduced that the temporal HS excitation
is triggered by either the reset operation or the measurement stimulus. To
specify the underlying mechanism of the observed excitation, we conducted the
experiment shown in Fig. 3. In this controlled experiment, the difference lies
in the measurement stimulus, allowing us to establish that any observed
difference between the two experiments is triggered by the measurement
stimulus. These two experiments are performed alternately with an interval of
several hundred microseconds. In Fig. 3(a), we first wait for
$1000\;\mathrm{\mu s}$ to initializes the qubit into the thermal ground state,
and apply a $\pi$ pulse to excite it to the $\ket{1}$ state. The $\ket{1}$
state is initialized into the $\ket{0}$ state with a reset operation, and the
qubit is finally measured. The corresponding signals are plotted in the IQ
plane at the bottom. In Fig. 3(b), we insert transmitting the measurement
stimulus after the qubit is excited to the $\ket{1}$ state with the $\pi$
pulse. The IQ signals obtained after the reset are plotted in the IQ plane at
the bottom of Fig. 3(b).

In Fig. 3(b), a sharp density near the origin is observed, while the majority
of the IQ signals are distributed in the fourth quadrant. This result shows
that the excitation in Fig. 2 is triggered by transmitting the measurement
stimulus to the qubit in the $\ket{1}$ state. Remarkably, transmitting the
measurement stimulus to the qubit in the $\ket{0}$ state does not result in
excitation to $\ket{\mathrm{HS}}$ states. These findings suggest that
measuring the $\ket{1}$ state excites the qubit beyond the computational
subspace. Note that this MIST is observed only during specific time periods,
as discussed in Fig. 2.

Figure 4:  (a) Experimental pulse sequence. The qubit is excited to the
$\ket{1}$ state with the $\pi$ pulse, and then a measurement stimulus is
transmitted to the readout resonator with a varied amplitude. The qubit
$\ket{1}$ state is initialized into the $\ket{0}$ state with the reset
operation, and finally, we measure the population of the $\ket{\mathrm{HS}}$
states (states higher than the first excited state). (b) Population of the
$\ket{\mathrm{HS}}$ states as a function of the measurement stimulus
amplitude. The left (right) panel shows the $\ket{\mathrm{HS}}$ population in
the normal (abnormal) time period. We denote the higher dressed states as
$\ket{k,m}$ in the insets.

The mechanism of the MIST is provided in the previous report [25]. We label
the energy eigenstates of a qubit-photon system as
$\ket{\mathrm{qubit},\mathrm{resonator}}$. The interaction between the qubit
and the resonator imparts a photon-number-dependent energy shift to the bare
states. As a result of this energy shift, the dressed state $\ket{1,n}$ can be
in resonance with a $\ket{k,m}$ state, where $k$ is an integer larger than 1
and $m$ denotes some photon number. The temporal fluctuations in Fig. 2 can be
explained by the temporal resonant transition between qubit dressed states.

We then varied the number of photons in the readout resonator and measured the
HS population. The pulse sequence of the experiment is illustrated in Fig.
4(a). The qubit is prepared in the $\ket{1}$ state with the $\pi$ pulse, and
we populate the readout resonator with photons using a measurement stimulus.
The amplitude of the stimulus is varied, corresponding to injecting different
numbers of photons into the resonator. The $\ket{1}$ state is initialized into
the $\ket{0}$ state with the reset operation. We finally measure the qubit and
estimate the population of the $\ket{\mathrm{HS}}$ state as shown in Fig. 4(b)
(see Appendix B for the details of the estimation). We interleaved a time
series $P_{1}$ measurement in Fig. 2 with this experiment, distinguishing
between an abnormal (higher error) time period and a normal (lower error) time
period.

The result shows that as the amplitude of the measurement stimulus is
increased, the $\ket{\mathrm{HS}}$ population also increases during the
abnormal time period. Moreover, the population distribution exhibits a
resonance-like peak, reaching its maximum value at an amplitude of
approximately $0.33$ (in arbitrary units). This peak amplitude is close to the
measurement amplitude of $0.28$ (in arbitrary units). Notably, this peak
structure is exclusively observed during the abnormal time period, as shown in
Fig. 4(b). This result suggests that the dressed higher state comes into close
proximity with the dressed $\ket{1}$ state only within a specific time
interval, as illustrated in the insets.

We can provide only speculations behind the observed temporal resonant
transition, but we attribute this to a fluctuating offset charge of the
transmon qubit [26, 28, 29, 30, 31]. The higher energy levels of the transmon
qubit are sensitive to a change in the offset charge, resulting in a temporal
resonant transition between qubit dressed state. The previously reported time
scale of the changes in the offset charge [28] is consistent with that of the
observed instability in this paper, supporting the discussion. We note that
though the resonant transition occurs between the first excited state and
higher states in this device, the transition between the qubit ground and
higher states can be triggered in a system with different qubit parameters or
a different measurement amplitude. In addition, in contrast to the prior study
[26], the switching behavior of the qubit dressed states is observed in a
superconducting qubit system where the readout frequency is larger than the
qubit transition frequency $\omega_{01}$.

In conclusion, our time-resolved MIST measurement for a superconducting qubit-
resonator system suggests anomalous excitation of the qubit beyond the
computational subspace during a specific time period. We showed that the MIST
shows temporal fluctuation in a variety of time windows ranging from sub-
second to a few hundred seconds. The population of the excited state exhibits
a peak behavior as a function of the measurement stimulus amplitude only
during a specific time interval. These experimental results suggest that
resonant transition between the qubit dressed states can be triggered due to a
fluctuating offset charge even in a superconducting qubit system coupled with
high-frequency resonators.

###### Acknowledgements.

The authors thank Ted Thorbeck and Matthias Steffen for providing valuable
comments on the manuscript. This work was supported by CREST (Nos. JPMJCR20C1,
JPMJCR20T2) from JST, Japan; Grant-in-Aid for Scientific Research (S) (No.
JP19H05600), Grant-in-Aid for Transformative Research Areas (No. JP22H05114)
from JSPS KAKENHI, Japan. This work is partly supported by IBM-UTokyo lab.

##  Appendix A TIME SERIES $P_{1}$ MEASUREMENT

In the experiment shown in Fig. 2(a), we executed several quantum circuits
alternately to capture changes in the error rates in practical quantum
computation. The circuits are illustrated in Fig. 5. We incorporated basic
operations in quantum computation such as the NOT gate, the CNOT gate, the
reset operation, and the measurement in the six circuits, and investigated a
temporal increase in the error rates. The six circuits depicted in the figure
were executed in the order
(a)$\to$(b)$\to$(c)$\to$(d)$\to$(e)$\to$(f)$\to$(a)$\to\cdots$. The time
series of $P_{1}$ from each of the six quantum circuits is shown in Fig. 6
with its corresponding letter. When using the CNOT gate, we analyzed the
output of the first qubit as depicted in Fig. 5.

Figure 5:  Quantum circuits used in the experiment in Fig. 2. The six circuits
were executed in the order
(a)$\to$(b)$\to$(c)$\to$(d)$\to$(e)$\to$(f)$\to$(a)$\to\cdots$.

In Fig. 6(a), a temporal increase in the error rate is observed from around
200 s to 500 s. In Fig. 6(b), this peak is not evident. This observation
aligns with the fact that the in phase component of HS states is positive, and
it is converted to 1. As mentioned in the main text, however, the reset
operation is unable to initialize the HS states into the ground state, and
this excitation leads to a temporal increase in the error rates in other
circuits. In Fig. 6(c), we checked the HS excitation of the first qubit
results in a higher error in the second qubit due to the CNOT gate. Comparing
Fig. 6(a), (c) and (d), we can see that the entanglement with another qubit
does not result in the change in the error rates. Notably, though a temporal
increase in the error rates is not evident in the first qubit in Fig. 6(e), we
observed a temporal increase in the error rates in the second qubit triggered
by the HS excitation of the first qubit. In Fig. 6(f), we measure the qubit
first excited state, and this results in the error in Fig. 6(a). Figure 2(a)
was reproduced from the output in Fig. 5(a).

Figure 6:  The time series of $P_{1}$ obtained in Fig. 5. (a) This figure was
reproduced from the output in Fig. 5(a), and the same for the other five
figures. (c)(d)(e)(f) These were reproduced from the output of the first
qubit.

##  Appendix B ESTIMATION OF HS POPULATION FROM IQ DATA

In Fig. 7, we illustrate the procedure for estimating the population of higher
energy states (HS). Linear Discriminant Analysis (LDA) was implemented on the
IQ data obtained in Fig. 1(b). In Fig. 7(a), the three clusters are classified
by LDA, as indicated by the black line. The trained LDA is then used to
estimate the population of HS states from the measured IQ data. In Fig. 7(b),
IQ dots obtained in the experiment in Fig. 4 are presented. Each panel
corresponds to the measured signals with a different measurement stimulus
amplitude. The amplitude of the measurement stimulus is shown in the inset,
and with increasing amplitude, dense signals in the HS region are observed. We
applied LDA to convert the IQ dots into the three quantum states and plotted
the HS population as a function of the measurement amplitude in Fig. 4(b).

Figure 7:  Estimation of the populations of each quantum state from IQ dots.
(a) Training IQ data and LDA. (b) IQ data obtained in the experiment in Fig.
4(b) and LDA.

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