Dispersive qubit readout with machine learning

Enrico Rinaldi
University of Michigan, Ann Arbor MI, USA
and Theoretical Quantum Physics Laboratory, RIKEN, Japanand Center for Quantum Computing, RIKEN, Japanerinaldi.work@gmail.com
Roberto Di Candia
Department of Communications and Networking, Aalto University, Finlandrob.dicandia@gmail.com
Simone Felicetti
Istituto di Fotonica e Nanotecnologie, Consiglio Nazionale delle Ricerche (IFN-CNR), Italy
felicetti.simone@gmail.com
Fabrizio Minganti
Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), Switzerland
and Theoretical Quantum Physics Laboratory, RIKEN, Japanfabrizio.minganti@gmail.com

Abstract
Open quantum systems can undergo dissipative phase transitions, and their criticalbehavior can be exploited in sensing applications. For example, it can be used
to enhance the fidelity of superconducting qubit readout measurements, a central
problem toward the creation of reliable quantum hardware. A recently introduced
measurement protocol, named “critical parametric quantum sensing”, uses theparametric (two-photon driven) Kerr resonator’s driven-dissipative phase transition
to reach single-qubit detection fidelity of 99.9% [arXiv:2107.04503]. In this work,
we improve upon the previous protocol by using machine learning-based classification algorithms to efficiently and rapidly extract information from this critical
dynamics, which has so far been neglected to focus only on stationary properties.
These classification algorithms are applied to the time series data of weak quantum
measurements (homodyne detection) of a circuit-QED implementation of the Kerr
resonator coupled to a superconducting qubit. This demonstrates how machinelearning methods enable a faster and more reliable measurement protocol in criticalopen quantum systems.

1

Introduction

A sensing device is a system (weakly) coupled to a second system, the target, characterized by an
unknown parameter. By observing the response of the sensor to the coupling, one can estimate
the value of the parameter of the target [1]. The larger the response of the sensor, the better theestimation of the unknown parameter. For this reason, quantum systems around criticality havebeen proposed as sensors, because the diverging susceptibility characterizing second-order phasetransition empowers precise parameter estimation. Such a metrological advantage of quantum phasetransitions, however, is limited by a practical consideration: a diverging susceptibility is associatedFourth Workshop on Machine Learning and the Physical Sciences (NeurIPS 2021).

with a critical slowing down, i.e., the emergence of an infinitely-long timescale. Thus, the price to
pay for increased precision is a diverging measurement time [2]. Similarly to thermal and quantum
phase transitions, an open quantum system can develop second-order Dissipative Phase Transitions(DPTs) [3, 4] associated with a diverging susceptibility [5]. Very little has been done in investigating
the metrological properties of DPTs [6, 7].
Two-photon Kerr resonators are remarkably simple open quantum systems that are at the centerof intense experimental research in quantum optics and information [8, 9]. The two-photon Kerr
resonator undergoes a second-order DPT [10, 11] where the photon number hâ† âi has a non-analyticalchange. In Ref. [12], the authors introduced a quantum measurement protocol called “criticalparametric quantum sensing” using the two-photon Kerr resonator. Using this DPT, the state of aqubit (two-level quantum system) coupled to the sensor can be inferred with very high precision.
Indeed, if the qubit is in the state |↑i, after a quick transient dynamics, the resonator is almost empty,
while if the state is |↓i the resonator contains many photons. In a first approximation, a homodyne
measurement of the photon field allows assessing the state of a qubit with extremely high fidelity (upto 99.9%).
This protocol, however, relies on a measurement of the long-time (stationary) properties of the Kerr
resonator, thus ignoring all the information coming from the short-time dynamics of the system. Froma physical point of view, the importance of short-time measurement is clear when considering both
non-ideal dispersive measurement and including qubit dissipation and decoherence (both sources ofnoise in current NISQ-era quantum hardware). Thus, in realistic systems, a measurement protocol
must be performed within the qubit coherence time. More generally, growing efforts are dedicated tothe development of sensing protocols based on dynamical properties of critical quantum systems [13–17].
From a theoretical point of view, characterizing the average dynamics of an open quantum system
requires nontrivial computations of the spectral properties of the generator of the dissipative dynamics[18], making it a formidable challenge. This is even more true when discussing individual trajectories,
which describe the outcomes of single experimental realizations. Indeed, the complexity of thestochastic noise induces nontrivial correlation properties in the dynamics that cannot be captured by
the average dynamics of a Lindblad master equation [19–21]. For this reason, an efficient dynamic
critical parametric quantum sensor cannot rely on simple estimation strategies based on average
properties of the system. On the other hand, machine learning algorithms are well suited for such atask and are known to work well even in the presence of experimental noise [22].
In this paper, we present a novel machine learning approach to critical parametric quantum sensingthat improves the protocol metrological power by taking into account the dynamical aspect of the
DPT. The motivation for this improvement is the following. Even though the protocol using thesteady state reaches high fidelity, it is based on the assumption of a perfect qubit, and as such it
requires waiting for a sufficiently long time before the Kerr resonator reaches its steady state. In
actual experimental realizations, however, time is a resource, and the shorter a measurement takes, theless the qubit is prone to errors. The procedure we propose is a machine learning-based classificationof measurement time series, allowing to extract information on the qubit state from the quantum
fluctuations of the Kerr resonator dynamics, even at short times.

2

Two-photon Kerr resonator

The Hamiltonian of a two-photon driven Kerr-resonator coupled to a qubit isĤKerr = ωâ† â + ωq σ̂ + σ̂ − + (â†2 + â2 ) + χâ†2 â2 + g â† σ̂ − + âσ̂ + ,2(1)

where â(â† ) is the bosonic annihilation(creation) operator and σ̂ − (σ̂ + ) is the lowering(raising)spin operator. This model can be realized in various photonic platforms, such as circuit-QED
implementation, where a driven resonator is coupled with a superconducting quantum interference
device (SQUID) element [23, 24]. We define the pump-resonator detuning ω, the effective pumppower , and the SQUID-induced non-linearity χ.
Photons will continuously escape the Kerr resonator, and the number of photons emitted is directly
proportional to the number of photons in the Kerr resonator. This is described by the Lindblad master2

equation for the system density matrix [1]∂ρ̂(t) = −i[Ĥ, ρ̂(t)] + Γ[2âρ̂(t)â† − {â† â, ρ̂(t)}]∂twhere Γ ≥ 0 is the dissipation rate, and it is used as a characteristic timescale.

(2)

Dispersive-readout protocols assume the qubit-resonator coupling to be in the linear dispersive regime,
where the qubit-resonator detuning ∆ = |ωq − ωr |  |ωq + ωr |/2. In such a regime, the passage
of excitation between the qubit and the resonator is suppressed, and the overall effect of the qubiton the resonator is to introduce an effective shift on the frequency of the resonator by δω = g 2 /∆.
Indeed, if the qubit is |↑i (|↓i), a photon inside the resonator has energy ω + δω/2
√ (ω − 1δω/2). In
this dispersive regime, a transition occurs at a critical frequency, such that ω = 2 − Γ2 . As such,the small change δω can determine if the system is in the phase with a few photon or in that withmany photons, as shown in Fig. 1, where on the left we show the photon number inside the cavity
and on the right we plot the qubit state (see also the discussion in [12]). Consequently, by collecting
the number of photon emitted, we can determine the state of the qubit.
20

1.0

15

0.5

hσ̂z ihâ†âi

| ↑i

10

| ↓i

5

| ↑i

0.0

| ↓i

−0.5
−1.0

0
0

5

10

15

tΓ

0

5

10

15

tΓ

Figure 1: The number of photons and the qubit state averaged over 1000 quantum trajectories forthe two different initial qubit states at  = 1.67 and δω = 2.3. The steady-state solution is reachedaround tΓ ≈ 6.
There is, however, another unwanted effect stemming from the coupling to the Kerr resonator. Indeed,
the Kerr resonator induces an effective dissipation on the qubit, introducing additional unwanted
errors in the measurement. Since the probability that a quantum jump induced by the qubit-resonator
coupling increases with time, the shorter the required measurement time, the better the outcome. For
example, in Fig. 2 we show that the qubit flips due to the coupling to the cavity. Therefore, if we areto consider only the long-time dynamics, in both cases we would wrongly guess the initial state of
the qubit. This is the first reason to consider the Kerr resonator dynamics at short time.
Furthermore, from a metrological point of view, the main theoretical reason for this work is the fact
that, contrary to quantum phase transitions, the critical slowing down of DPTs only affects certain
observables. For instance, at the critical point of the two-photon Kerr resonator, the critical slowing
down only affects hâi, but leaves unaffected hâ† âi. This is a common feature of DPTs, and it wasdemonstrated also for the XYZ model [20]. Since the protocol proposed in Ref. [12] uses photon
number to estimate, e.g., the state of a qubit, it is possible to obtain metrological advantages fromthe diverging susceptibility of a DPT avoiding the critical slowing down. Consequently, the systemshould display critical fluctuations even at short times, allowing to extract information on the qubit
state from the emitted field of the resonator.

3

Methods

These critical fluctuations can be clearly seen in actual experimental realizations (or their numericalsimulations) as shown in Fig. 2 at short time. By comparing this single-shot dynamics in the left
panel, we see important fluctuations of the Kerr-resonator state, far larger than those in the averaged
dynamics (for instance, a small peak around tΓ = 1 when the qubit is initialized in |↓i = |0i). This1

To be precise, the DPT occurs in the “thermodynamic” limit χ → 0 [11]. Nevertheless, for small χ, a sharp
crossover occurs.

3

information, however, can be quite challenging to extract from an analytical point of view, because it
would require to obtain the Liouvillian superoperator eigenvalues and eigenmatrices (i.e., the spectral
decomposition of the generator of the dynamics) [11]. For this reason, we will employ a machinelearning algorithm to guess the qubit state exploiting these dynamical critical fluctuations.
Since we are interested in simulating actual experimental measurement outcomes, we resort tohomodyne quantum trajectories. Instead of the average dynamics of the system in Eq. (2), we
simulate a set of stochastic Schrödinger equations where noise is sampled at each time step [1]. Fora fixed set of parameters in ĤKerr we simulate 1000 quantum trajectories of the system startingfrom the two different states of the qubit |0i = |↓i and |1i = |↑i using QuTiP [25, 26]. We save thehomodyne quantum evolution for several observables related to the resonator (sensor), such as thenumber of photons hâ† âi, and to the qubit (target), such as hσ̂z i.
1.0

20

0.5

hâ†âi25

hσ̂z i

15
10
5
0
0

5

10

−0.5

15

−1.0

tΓ

| ↓i

0.0

| ↑i
| ↓i

| ↑i

0

5

10

15

tΓ

Figure 2: The number of photons and the qubit state for a single quantum trajectory where one of
the qubit states flips (fixed  = 1.67 and δω = 2.3). A measurement protocol based on the long-time(stationary) properties would fail to recognize that one qubit was in the |↑i state. Machine learningclassifiers can be accurate even with short-time data, when qubits have not had enough time to bealtered by the environment.
In circuit-QED experiments, the parameter Γ is fixed by the manufacturing process, the non-linearity
χ is set by the SQUID characteristics, while the detuning ∆ and the drive intensity  can be easily
tuned in real time. We choose to save data with a maximal time resolution of ∆t = 10−3 /Γ and up
to tΓ = 15, corresponding to state-of-the-art homodyne measurement frequencies. The other relevanttime scales, in units of Γ, are the total time of measurement tf and the instrument measurementsmoothing scale τ , corresponding to a time interval over which individual values can not be obtained
and are averaged over by the measurement instrument. These two time parameters can be optimizedby the protocol to achieve the highest fidelity.
We choose to analyze the quantum trajectories of the observable x̂ using two simple machine learningclassifiers for time series. Both classifiers employ a Support Vector Machine algorithm (SVC)with radial basis function kernels (rbf) to discover non-linearity in the feature space. The firstclassifier (named TAB+SVC(rbf)) is applied to a feature space identified with all the time pointsof the trajectory, therefore neglecting the time ordering and time correlations. On the other hand,the second classifier (named RIFE+SVC(rbf)) acts on a meta-feature space which is constructedfrom random interval features (RIFE) [27], such as the average value or the slope of a random timeinterval of the trajectory. To estimate the error on the classification accuracy we utilize repeatedcross-validation folds, with 100 repetitions and 5 folds.
Our choice of classifiers is limited and dictated by simplicity. Similar results should hold irrespectiveof the classifier algorithm as long as the time series dynamics can be captured. Ideally, classifiers
requiring less data to achieve high accuracy would be preferred because that would improve theexperimental setup. Higher accuracy can be achieved by including careful feature engineering andfeature selection steps, which we have left for future investigations.

4

Results

We want to tune the Kerr resonator to be the best possible measurement instrument. In Candia et al.
[12], such an optimal point was inferred from analytical considerations on the steady state. Here
4

10−1

10−1

TAB+SVC(rbf)

TAB+SVC(rbf)

RIFE+SVC(rbf)

RIFE+SVC(rbf)

Error

10−2

Error

10−2

10−3

10−4

10−3

5

10

10−4

15

tf Γ

5

10

15

tf Γ

Figure 3: The error (1-accuracy) made by different classifiers as a function of the measurement
time tf Γ (fixed  = 1.67 and δω = 2.3). The left panel is for τ Γ = 10−3 and the right panel forτ Γ = 10−1 .
instead we consider the dynamical properties. We scanned the parameter set {, ω} (which can beeasily controlled in standard experimental implementations) and verified the accuracy provided byour choice of classifiers for the state of a qubit initialized either in |↑i or |↓i, where the accuracy is
the ratio between the number of correct results over the total number of simulations (the variability of
the accuracy in our cross-validation procedure is always of the order of 0.002 or less). This procedure
is repeated for different total measurement time tf Γ and for different measurement smoothing scalesτ Γ: note that this procedure amounts to changing the quality and quantity of samples used in the
supervised learning classification task.
In Fig. 3, we plot the results obtained at various tf Γ. In the left panel, where τ = 10−3 /Γ, we seethat for very short times the algorithm is capable of determining the qubit state in a very short time
tf Γ = 2 and very high accuracy (errors ' 1 − 9 × 10−4 ). If we increase τ = 10−1 /Γ, we can obtain
similar accuracy (' 1 × 10−3 ), but we need to wait for longer times (tf Γ = 9).
These results demonstrate that, exploiting critical fluctuations, it is possible to obtain a performing
measurement instrument, capable of determining the state of a qubit with extreme high precisionand in very short time. In the future, we plan to further investigate the properties of critical open
quantum system on a more mathematical ground, and to test this protocol on actual experimental
data, including measurement noise and qubit dissipation effects.

Acknowledgments and Disclosure of Funding
We acknowledge the use of computing time on the RIKEN Hokusai BigWaterfall cluster.
E. R. is supported by Nippon Telegraph and Telephone Corporation (NTT) Research. R. D. acknowledges support from the Marie Skłodowska Curie fellowship number 891517 (MSC-IF GreenMIQUEC).

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