Fast high fidelity quantum non-demolition qubit readout via a non-perturbative cross-Kerr coupling

R. Dassonneville, T. Ramos, 3 V. Milchakov, L. Planat, É. Dumur, F. Foroughi, J. Puertas, S. Leger, K. Bharadwaj, J. Delaforce, C. Naud, W. Hasch-Guichard, J. J. Garćıa-Ripoll, N. Roch, and O. Buisson Univ. Grenoble-Alpes, CNRS, Grenoble INP, Institut Néel, 38000 Grenoble, France Institute of Fundamental Physics, IFF-CSIC, Calle Serrano 113b, 28006 Madrid, Spain Centro de Óptica e Información Cuántica, Facultad de Ciencias, Universidad Mayor, Chile (Dated: November 21, 2019)


I. INTRODUCTION
In Noisy Intermediate Scale Quantum (NISQ) devices [1], measurements are usually the last step of the algorithm. Here, a high-fidelity readout is an interesting asset that reduces the overhead in error mitigation [2] and in the characterization of gate fidelities [3]. However, highfidelity quantum non-demolition (QND) single-shot measurements become a requirement once we consider scaling up quantum technologies [4] to large devices, using quantum error correction [5,6] and fault-tolerant quantum computation [7,8]. In this context, lowering the readout and QND-errors is as important as decreasing the single-and two-qubit gate errors below the scaling thresholds.
A fast and high-fidelity QND measurement demands a strong coupling to the measurement device combined with a good preservation of the qubit state. In trapped ion qubits, this dilemma is solved by encoding information in two long-lived states, only one of which couples to incoming radiation [9]. Fluorescence counting gives a projective measurement with errors below 1 %, limited by the collection time [10]. Cavity-QED [11][12][13] setups follow a different strategy. Inserting the qubit inside a cavity allows to generate a strong coupling between the qubit and the cavity electro-magnetic (EM) field but also to increase the collection efficiency. An optical or microwave signal probes the resonator, implementing an indirect projective QND readout of the qubit polarizationσ z [12,14]. In these cavity-QED experiments it is very important to engineer the qubit-resonator coupling so as to maximize measurement's (i) single-shot readout fidelity, (ii) speed and (iii) QND-ness-preservation of the qubit's excited and ground state probabilities.
To illustrate this point, we consider the ubiquitous transmons qubit [15],Ĥ q ω qq †q − α qq † 2q2 1 2 ω q σ z , a slightly anharmonic oscillator with frequency ω q and anharmonicity strength α q . Three types of couplings, summarized in Table I, will be discussed. Qubits and resonators are usually coupled together via the electrical interaction between the dipole moment fieldq of the qubit and the field ampltitudeĉ of the resonator. This field -field interaction is known as the transverse coupling resulting in the Jaynes-Cummings model with the coupling term g x (q †ĉ +qĉ † ) in the Hamiltonian [11,13].
In the dispersive limit [15], the qubit-cavity detuning ∆ = ω c − ω q exceeds the coupling strength |g x | |∆|, and the cavity experiences an effective energy-energy interaction ∼ gx 2 αq ∆(∆−αq)σ zĉ †ĉ , known as the dispersive or cross-Kerr interaction. It gives rise to a qubit-dependent frequency shift, mapping the state of the qubit to the signal phase probing the resonator and thus providing a good QND projective measurement [14,16]. This transverse coupling has been extensively used in most circuit-QED experiments. State-of-the-art measurement fidelities and speeds using this standard dispersive technique are summarized in the first row of Table I. However, the dispersive readout is fundamentally limited by unavoidable higher order corrections to perturbation theory, which distort the qubit dynamics [17][18][19], and induce additional decay channels [20].
Several works have investigated how to overcome these limitations, designing new quantum circuits [21][22][23][24][25][26][27][28]. Implementing a coupling scheme that involves natively the energy of the qubit -as opposed to an effective energy interaction -resolves these limitations. Along this line, the longitudinal coupling ∼ g zq †q (ĉ † +ĉ) [cf. second row of Table I] is remarkable. It induces a qubit-dependent displacement of the cavity fieldĉ [27]. When combined with a parametric modulation g z (t) at the cavity frequency ω c , this interaction results in a faster separation of the pointer states with a QND-ness as high as Q = 98.4 % [29,30].  TABLE I. State-of-the-art parameters for three different coupling types between an harmonic readout mode and a superconducting qubit. The second column shows the direct coupling terms between the qubit, described as an anharmonic oscillator with ladder operators (q,q † ), and an harmonic read-out mode described by (ĉ,ĉ † ). Column three shows the effective coupling obtained after the rotating wave approximation (RWA), and two-level system approximation for all couplings plus the dispersive approximation in the case of the transverse coupling. Notice that the present experimental work implements two non-perturbative cross-Kerr couplings of the type presented in this table since two polariton modesĉu andĉ l are used for the readout [See Fig. 1 and Sec. II B for more details].
scheme based on a non-perturbative cross-Kerr interaction [cf. third row of Table I]. It leads to an alternative readout mechanism for superconducting qubits. This new process is fast, has a large single-shot fidelity, maximizes the QND nature of the process, and does not require any parametric modulation. The proposed setup is based on an artificial molecule with one emergent qubitlike degree of freedom and a bosonic ancilla that couples to the readout cavity [cf. Fig. 1a]. The qubit develops a Kerr-type interaction with the ancilla-cavity polariton branches [cf. Fig. 1b]. This interaction enables a detection scheme analogous to the standard dispersive measurement. Nevertheless, unlike transverse dispersive, since the coupling is not perturbative, it does not imply FIG. 1. Schematics of the circuit QED setup with the transmon molecule used for a high fidelity and fast qubit QND readout. (a) A cavity modeĉ is strongly and transversally coupled to an ancilla systemâ, which in turn couples diagonally to the qubitσz as ∼ gzzσzâ †â . (b) The strong hybridization between cavity and ancilla is manifested by two orthogonal polariton modesĉu andĉ l , which couple to the qubit with a non-perturbative cross-Kerr couplings ∼σz(χuĉ † uĉu +χ lĉ † lĉ l ) (see text). This allows us to infer the state |g or |e of the qubit by measuring the resonance shifts of the polaritons at the cavity transmission output.
any cavity-mediated excitations or decay. Moreover, the strength of the readout shift can be made large, and is independent of the detuning, allowing to neglect any spurious qubit-resonator coupling via an increased detuning. This results in a very efficient single-shot QND readout of the qubit even in its first demonstration: it has a record QND-ness of 99 %, a fidelity of 97.4 %, while only requires a short measurement time of 50 ns. This readout mechanism can be combined with other paradigms of direct qubit-qubit interactions [31], as an upgrade to existing quantum computing and simulation architectures.

II. TRANSMON MOLECULE INSIDE A CAVITY
In this section we give details on the physical mechanisms for the qubit readout using a non-perturvative cross-Kerr coupling. The setup demonstrating this new readout mechanism uses a transmon molecule (two coupled transmons) circuit [cf. Fig. 2c] inserted inside a cavity. We start by introducing the specific experimental system in Sec. II A, and then, in Sec. II B, we write down the theoretical model describing the open quantum dynamics of the system. We consider the strong coupling regime between cavity and ancilla, getting two strongly hybridized polariton modes. The qubit then couples strongly to these two polaritons via non-perturbative cross-Kerr couplings χ j . This allows for an efficient readout of the qubit state via the transmission output of the cavity as shown below.

A. Physical implementation
The device consists of an aluminium Josephson circuit, which is deposited on an intrinsic silicon wafer and inserted in a 3D copper cavity [cf. Fig. 2a]. An optical image of the molecule circuit is shown Fig. 2d, which implements the lumped element circuit of Fig. 2c. The molecule is realized by coupling two identical transmon qubits through a large inductance. The two small Josephson junctions of the transmons are shunted by capaci- tance C s between the two external pads and the central one. The two transmon qubits are coupled by a large inductance L a , obtained by a chain of 10 small SQUIDs of surface S SQU ID . Therefore this coupling inductance is tunable by an external flux Φ s . The circuit contains a second loop of surface S linking the two transmon Josephson junction and the inductance L a . It defines a second external flux Φ with Φ = rΦ s and r = S/S SQU ID 26. An additional capacitance C t is coupled the two external pads. As already discussed in previous work [21], the quantum dynamics of the transmon molecule in the case Φ = nΦ 0 (with n an integer) can be described bŷ where the first line describes the qubit, the second line the anharmonic oscillator called hereafter ancilla mode and the last one the coupling between them. Here, the phase average and phase difference between the two transmon Josephson junctions are denoted byx q andx a , respectively, whereas their conjugate charge operators, normalized by a Cooper pair charge 2e, are denoted bŷ n q andn a . The quantity E Cq = e 2 /2C s corresponds to the charging energy of a single transmon, whereas, E Ca = e 2 /2(2C t + C s ) is also related to the capacitance C t between the two external pads. E J is the Josephson energy of the transmons, and L J = ( Φ0 2π ) 2 1 E J the Josephson inductance. Except for the first two lines, we derived Eq. 1 by expanding to fourth order inx q andx a .
To measure the transmon molecule, we insert the silicon chip inside a 3D copper cavity with a volume 24.5 mm × 5 mm × 35 mm (length × height × width) along the a c , b c and c c directions, respectively (Fig. 2b). The cavity mode considered hereafter is the fundamental TE 101 mode with the microwave electric field aligned along the b c direction. It is modeled as an harmonic oscillator with frequency ω c and annihilation operatorĉ.

B. Qubit-polaritons model
The first line in Eq. (1) corresponds to the Hamiltonian of a transmon qubit rewritten asĤ q = ω qq †q − α qq †2q2 , with frequency ω q and anharmonicity α q . The second line in Eq. (1) describes the ancilla modeâ, with frequency ω a and anharmonicity U a . Due to the presence of the coupling inductance L a and capacitance C t , the ancilla anharmonicity U a is much weaker than the qubit anharmonicity α q . In our experiments, the ancilla will be weakly populated ( â †â 2), allowing us to safely neglect the anharmonicity U a , and regard the ancilla as an harmonic oscillatorĤ a = ω aâ †â [cf. appendix B]. Interesting nonlinear and bistability effects arise when the ancilla is strongly populated ( â †â 1), but these effects will be discussed elsewhere. Because of the circuit symmetry, there is neither a field-field (transverse) nor a field-energy (longitudinal) coupling between qubit and ancilla. The lowest order coupling corresponds to the last term in Eq. (1) and it is a direct consequence of the non-linearity of the Josephson junctions [32]. It is this term which will produce the cross-Kerr coupling between the qubit and the polariton modes.
To obtain such effect, we hybridize strongly the ancilla and the cavity mode by aligning the sample direction b s to the cavity direction b c . In this way, we maximize the coupling g a between the ancilla and the cavity. When neglecting residual asymmetry between the two transmons and misalignment between the sample and the cavity, the qubit-cavity transverse coupling is zero. This is guaranteed by the symmetry of the transmon molecule and of the TE 101 mode of the cavity. Consequently the cavity Hamiltonian and its interaction with the molecule circuit takes the simple form:Ĥ cav = ω cĉ †ĉ + g a (â † +â)(ĉ † +ĉ).
The total Hamiltonian of the system which includes the transmon molecule and the properly oriented cavity is then given by (cf. appendix B for details): In our experiment cavity and ancilla modes are close to resonance and thus strongly hybridized. This leads to two new normal modes called upper and lower polariton modes,ĉ u andĉ l , which are a linear combination of ancilla and cavity fields,â † +â andĉ † +ĉ. In the rotatingwave approximation (RWA), they are given by a rotation c u = cos(θ)â + sin(θ)ĉ, andĉ l = cos(θ)ĉ − sin(θ)â where the cavity-ancilla hybridization angle reads tan(2θ) = 2g a /(ω a − ω c ). In terms of these polariton modes, the total Hamiltonian takes the form [cf. appendix B] where ω u sin 2 (θ)ω c + cos 2 (θ)ω a + sin(2θ)g a and ω l cos 2 (θ)ω c +sin 2 (θ)ω a −sin(2θ)g a are the frequencies of the upper and lower polariton modes, respectively. In addition,σ z = 2q †q − 1 is the Pauli matrix of the transmon in the two-level system approximation, which interacts with the upper and lower polariton via non-perturbative cross-Kerr couplings χ u = g zz cos 2 (θ) and χ l = g zz sin 2 (θ), respectively. Each polariton is in some proportion cavitylike and therefore can be used as readout mode. Similarly, each polariton is also ancilla-like and therefore develop a cross-Kerr coupling with the qubit. We retrieve here the coupling between a qubit and a readout mode presented in the third row of Table I. It is important to note here that these cross-Kerr coupling strengths χ j are non-perturbative, meaning they do not depend on the qubit-resonator detuning but only on the hybridization angle θ and the initial ancilla-qubit cross-Kerr coupling g zz .

C. Conditional polaritons spectroscopy
Inspecting Eq. (3) we see that, except for dissipation and dephasing effects treated in appendix B, the population of the qubit σ z t0 remains constant during the dynamics, with t 0 the intial time. The qubit's main effect is thus simply to shift the transition frequencies of each polariton modeĉ j as ω j → ω j − χ j σ z t0 . The shift of the polariton resonances can be measured by shining a weak continuous coherent drive on the cavity and recording the amplitude of the field at the transmission output a out ss [cf. Fig. 1]. Fig. 3 shows typical spectroscopic measurements as a function of the driving frequency ω d , with the blue and red curves corresponding to the case the qubit is prepared in states |g ( σ z t0 ≈ −1) and in |e ( σ z t0 ≈ +1), respectively.
A complete description of the system, including losses and dephasing of the qubit, ancilla, and cavity, can be obtained by a master equation formalism as shown in appendix B. Using this formalism we derive a compact expression describing the transmission amplitude of output field, which reads Here, Ω = κinPin ω d is the strength of the microwave drive, with κ in the coupling to the input port and P in the input power. In addition, κ out describes the coupling to the output of the cavity and κ = κ out + κ in is the total cavity decay. The resonances of the lower and upper modes appear at the two driving frequencies ω d that satisfy δ eff c = 0, where δ eff c is the effective detuning of the cavity given by Here, Γ a is the ancilla decoherence, including dissipation and pure dephasing as shown in appendix B. On the other hand, the widths and heights of the resonances are determined not only by the cavity decay κ, but also by the ancilla effective decoherence Γ eff a given explicitly by In Fig. 3 the transmitted signal is measured using a 500 ns square microwave pulse applied immediatly after preparing the qubit in |g or |e states. Two resonance peaks are observed corresponding to the two polariton modes and qubit state dependent frequency shifts are clearly visible. The lineshapes are fitted using Eq. (5). The peaks of the lower and upper polariton branches are indeed shifted by ∼ 2χ j , up to small erros in the calibration and initial state preparation of the qubit states |g and |e .

A. Individual measurement records and quantum trajectories
Readout is performed using a standard microwave setup including a high saturation-power Josephson parametric amplifier made from a SQUID chain [33]. Next we consider the readout performance at zero flux measuring the signal transmitted through the lower polariton j = l. To readout the qubit state a coherent microwave tone is applied at frequency ω d /2π = (ω l −2χ l )/2π = 7.028 GHz. The amplitude of the readout tone is n l = c † l c l 2 based on a calibration using AC-Stark shift [34,35]. Since the polariton resonance frequency is conditioned to the qubit state, the coherent tone is detuned by χ l /π ≈ 10 MHz when the qubit is in |g and in resonance in |e . Therefore the transmitted signal presents weak or large amplitude conditioned to the qubit state |g and |e , respectively. The amplifier is operated in phase-sensitive mode leading to squeezed signal at the amplifier output. We define I(t) and Q(t) the in-phase and the quadrature microwave signal. Its phase has been adjusted so the information about the qubit state is only contained in I(t).
One thousand individual trajectories have been measured when the qubit is prepared either in |g or |e state. Four typical individual records are plotted in Fig. 4. The duration pulse is 1000 ns acquired over a larger time window (around 1300 ns). These measurement records give an insight on the real time dynamics of the qubit from single-shot trajectories. Notice that after a time of few κ −1 l ∼ 15 ns, the qubit state can already be inferred from a single trajectory, and that in Fig. 4(b) a quantum jump [36] of the qubit appears clearly. In addition to the individual trajectories, the mean value averaged over the one thousand trials, as well as the related standard deviation, is plotted as function of time. Due to qubit relaxation, the averaged excited state response (red solid line) decays towards the ground state response, while its corresponding standard deviation (red shaded area) grows in time. This finite qubit lifetime, limits the distinguishability for long measurement and highlights the need for a fast readout. The qubit decay under drive T 1,drive is similar to the one measured without drive T 1 3.3 µs. Therefore, the measurement does not disturb the qubit relaxation, indicating a QND measurement. and absence (c) of a quantum jump. Blue and red points refer to the case the qubit is initially prepared in states |g and |e , respectively (t = 0). The readout pulse with amplitude n l = 2 starts at t = 0 ns and stops at t = 1000 ns. Each point is measured with a 30 ns integration, corresponding to the resonator rising time 2κ −1 l . An average over 1000 measurement records is plotted in solid blue and red lines, as well as their standard deviation represented by corresponding shaded areas.

B. Quantum non-demolition fidelity
To check the QND-ness of the measurement, we quantify the repeatability of successive measurements. We now consider only the measurement records between time 10κ −1 l ∼ 150 ns and 1000 ns to be in the steady state regime of the applied squared pulse. It corresponds to the ground state if I(t) < I th or to the excited state if I(t) > I th with I th = 15.5 mV. We define four conditional probabilities, P α,β , the probability to measure α in the first measurement and β in the second measurement, where α, β = g, e can correspond to ground or excited states. From these probabilities, the QND fidelity [29] is obtained to be Q = Pg,g+Pe,e 2 = 99 %. Here, the imperfect value of P e,e = (98.3 ± 0.7) % is ex-plained by the relaxation during measurement, and the value P g,g = (99.6 ± 0.02) % is justified by the thermal excitations during measurement. Moreover, each probability has an extra uncertainty due to finite counting of ±0.6 %. These results are comparable to the QND fidelity obtained in Touzard et al [29] using a parametric modulation scheme and corresponds, to the best of our knowledge, to the state-of-the-art values.

C. Single-shot readout fidelity
In the early days of circuit-QED, averaging was necessary to infer the qubit state with high fidelity. However, thanks to the advent of Josephson-based amplifier [37][38][39], high fidelity, single shot discrimination of the qubit state is now possible [40]. Since then, works have been performed on Purcell filters and amplifiers in an attempt to increase further the readout fidelity [16,[41][42][43], which is now culminating at 99.6 % in 88 ns [14]. Readout fidelity is currently limited by the balance between the time needed to discriminate the qubit state and the qubit T 1 .
To quantify the readout fidelity, we perform heralding [44] by applying first a 50 ns square readout pulse. In the analysis, we keep only the sequences where the qubit is found in the ground state for this first measurement. After this pulse, we wait 300 ns ∼ 20κ −1 l for the resonator to decay back into its vacuum state before preparing the qubit in the ground or in the excited state. Then another 50 ns square readout pulse is applied. The two measurement pulses correspond to a steady state amplitude of n l 2. In Fig. 5, histograms of 24 · 10 3 single shot readouts are plotted as the function of the in-phase amplitude when the qubit is prepared in |g and |e states. A weight function is used to maximize the distinguishability between the two qubit states [14]. The histograms are fitted by the sum of two Gaussians (colored solid lines). The intersection of these two fitted histograms defines a threshold I th (vertical dash line) distinguishing the two qubit state. The readout fidelity is defined as F = 1 − (P (e|g) + P (g|e))/2 1 − ( g + e )/2, where P (x|y) is the probability of reading out x while having prepared the state y. In addition, g and e are the fraction of measured events of detecting I ≥ I th when the qubit was prepared in g and I ≤ I th when the qubit was prepared in e, respectively. We obtained a readout fidelity of F = 97.4 % affected by the imperfections g = 1.0 %, and e = 4.3 %. Here, we distinguish different sources of error as = o + a = o + p + t where o is the overlap error (green shaded area), a is the assignment error (red and blue shaded areas) which can be decomposed into p , the preparation error and t , the transition during measurement error. We also have overlap errors o,g = o,e = 0.4 %. In total, we have an assignment error of a,e = 3.9 % in which we expect ∼ 1.5 % due relaxation during measurement and ∼ 1.9 % error due to imperfections in the π-pulse. The leftover errors are within the uncertainty due to finite counting of ± 0.7 %, but may be attributed to a not perfect heralding procedure or possibly to measurement-induced transitions [18]. We believe that the readout fidelity can be further increased by implementing pulse envelop optimization such as DRAG pulse [45] to have less excited state preparation error, or CLEAR pulse [46] to achieve better discrimination in a shorter integration time and therefore reduce error due to relaxation during measurement.

D. Coherence and readout quality factor
Both QND-ness and single-shot readout fidelity are limited by the finite T 1 of the qubit. To understand qubit lifetime limitations, we have measured its relaxation for several fluxes [cf. Fig. 6]. We found a T 1 ranging from 3.3 µs at zero flux to 0.9 µs at Φ = 9Φ 0 . We identified two sources of imperfections in our system that create parasitic residual transverse coupling leading to a Purcell-limited qubit T 1 . The first source is the asymmetry of critical current in the Josephson junctions and the second is the possible misalignment of the sample inside the cavity. The effect of these two imperfections is discussed in detail in Appendix C. We computed the Purcell-limitation due to these residual transverse couplings, and obtained the red shaded area in Fig. 6, which Despite this limited T 1 , we achieve a good steady state signal-to-noise ratio (SNR) per photon number and a good readout quality factor defined as Q r = 4χ 2 /(κ 2 /4 + χ 2 )κT 1 427. Indeed, the optimal steady state SNR is given by [47] SN R = ηnQ r with n the photon number and η the quantum efficiency. As a comparison, Ref. [16] reports Q r = 540 and Ref. [14] Q r = 1075. In the future, our currently Purcel-limited T 1 can be largely improved, without sacrifying on κ and χ j , by a better optimization of parameters. In this way, we believe that an increase of one order of magnitude in Q r is within reach. Moreover, we expect the photon number limitation to be less restrictive for the non-perturbative cross-Kerr coupling than for the perturbative one and thus the steady state SNR can be further improved.

IV. CONCLUSION AND OUTLOOKS
In conclusion, we have developed an original qubit readout scheme relying on a non-perturbative cross-Kerr coupling, in contrast to the usual cross-Kerr coupling that is perturbatively obtained from the transverse coupling in the dispersive regime. Therefore, our new experimental measurement design does not suffer from cavity-mediated excitations or decay, and the strength of the readout shifts can be made large and independent of the detuning. This allows for a fast readout of the qubit, with a large single-shot fidelity, and a maximization of the QND-nature of the measurement. The qubit and readout performances are currently limited because of residual qubit-cavity transverse couplings. However, no fundamental reason prevents further suppression of this transverse coupling. In fact, we can obtain the same readout shifts 2χ j of the dispersive readout, but having a much larger qubit-polaritons detuning, which may allow us to significantly reduce the unwanted consequences of the residual transverse coupling in the future.
According to our readout error budget and to our QND-ness analysis, the measurement-induced qubit state mixing is particularly low compared to the standard literature. This could be explained by the non-perturbative nature of our cross-Kerr coupling and will be the topic of future investigations. In this section we describe the measurement setup shown in Fig. 7. Qubit and readout pulses are sent through the same input line. The transmitted signal passes through three circulators and a directional coupler before being amplified via the Josephson Parametric Amplifier (JPA). Then it passes through additional amplification stages before it is downconverted to DC voltages via an IQ mixer and digitized at 1 GS/s using an ADC. Finally, the signal is digitally integrated.
The JPA [33] is used in the phase-sensitive regime and thus phase stability is a key feature in this setup. The pump and cancellation drives need to be tuned at the same amplitude with opposite phases [14,29]. Moreover, the phase of the JPA also needs to be tuned to amplify the wanted quadrature. The JPA gain (20 dB) and its pump cancellation are tuned with a VNA and spectrum analyzer regardless of the sample.
whereq andâ are the annihilation operators for qubit and ancilla modes, satisfyingx q = (q +q † )/ √ 2 and x a = (â +â † )/ √ 2. In case of no asymmetry, the Hamiltonian parameters are given in term of microscopic circuit parameters asω . We can write the above Hamiltonian in normal ordering and perform the RWA, provided the couplings are much smaller than the free frequencies, i.e. K q , K a , g zz ω q ,ω a , and obtain Here, we have done the identifications ω q =ω q − g zz − 3K q , ω a =ω a − 2g zz − 3K a , U a = 3K a /2, and α q = 3K q /2. Since in our experiments qubit and ancilla are largely detuned, |ω q − ω a | g zz , we can neglect the last term in Eq. (B2) applying an additional RWA. In addition, since the population and anharmonicity of the ancilla are small for the parameter regime considered throughout this work, i. e. a † a 1 and U a g zz , α q , respectively, we can also neglect the ancilla anharmonic term in the first line of Eq. (B2). Regarding the coupling between ancilla and cavity mode, H cav = ω cĉ †ĉ + g a (ĉ † +ĉ)(â † +â), we also apply the RWA provided g a ω c ∼ ω a . After all the above approximations, the total HamiltonianĤ tot =Ĥ mol +Ĥ cav can be written aŝ Finally, we notice that in our experiments the cavity and ancilla are close to resonance, so that these two modes become strongly hybridized into upper and lower polariton modes defined asĉ u = cos(θ)â + sin(θ)ĉ, and c l = cos(θ)ĉ − sin(θ)â, respectively, with tan(2θ) = 2g a /(ω a − ω c ). Re-expressing the total Hamiltonian (B3) in terms of terms of these porlation modes, and doing the two-level approximation in the qubit subspacê σ z = 2q †q − 1 due to its large anharmonicity α q , we obtain the final Hamiltonian in Eq. (3) of the main text. On the other hand, a realistic circuit setup will also present dissipation and dephasing of the different components, so that the full dynamics of the open system is governed by the master equation, Here, κ is the decay of the cavity mode, γ a and γ φ a the decay and dephasing of the ancilla, and finally γ q and γ φ q the decay and dephasing of the qubit. The total decoherence of ancilla and qubit are defined as Γ a = γ a /2 + γ φ a and Γ q = γ q /2 + γ φ q , respectively. Finally, notice that in addition to the total HamiltonianĤ tot , Eq. (B4) also includes a Hamiltonian for a coherent drive on the cavity, H drive = Ω(ĉe iω d t +ĉ † e −iω d t ), which is necessary to describe the pulses for the transmission and QND measurements performed throughout this work.
Using input-output relation,ĉ out (t) =ĉ in (t) − √ κ outĉ (t), we can find a simple expression for the amplitude of the cavity field at the transmission output port.
Taking expectation values on steady state, we obtain ĉ out ss = √ κ out ĉ ss , where ĉ ss can be easily obtained from the steady state solution of the master equation (B4). Indeed, in the case of weak driving Ω κ, one obtains a linear system of equations for ĉ ss , and â ss , and the solution of the former is shown in Eq. (5) of the main text.

Appendix C: Imperfections
We have identified two main sources of imperfections that can lead to a non-zero qubit-cavity transverse coupling, therefore limiting the readout performances.
The first one is the Josephson junction asymmetry d J = |E J1 − E J2 |/(E J1 + E J2 ) in the transmon molecule circuit. It is experimentally challenging to suppress fully this asymmetry. Considering this asymmetry, we need to add in the Hamiltonian (1) a new term −2E J d J sin(x q ) sin(x a ), where now E J is the mean Josephson energy of the two Josephson junctions. At first order, this new term corresponds to a tranverse coupling between the qubit and the ancilla. It is therefore important to be able to characterize it. We measured the room temperature resistances between each pad of the sample. These resistances have contributions from the Josephson junction resistances R J1 , R J2 , the resistance of the array of SQUID and resistances from the connecting wires. The wire resistances are estimated via measurement of wires-only test structures on a dedicated test-chip fabricated during the same process. In the end, we solve a set of 3 equations with 3 unknowns and found an asymmetry The second imperfection is a misalignment of the sample inside the 3D cavity. Considering the size of the cavity groove and of the sample, we estimate a misalignment angle up to θ m = ±5 deg. Assuming roughly that the ratio of transverse coupling g q /g a is given by tan(θ m ), we estimate a qubit-cavity transverse coupling of |g q |/2π 25 MHz g a .

Appendix D: Hamiltonian valid at all flux
When a non-integer quantum flux is applied, the symmetry is broken and other terms arise in the molecule circuit Hamiltonian, namelŷ To fit the energy spectrum versus flux, we numerically diagonalize, in a (8×8×8) basis, the total Hamiltonian H tot =Ĥ exp mol,tot +Ĥ cav whereĤ exp mol,tot is the Taylor expansion to fourth order of Eq. (D1) andĤ cav = ω cĉ †ĉ + g a (â † +â)(ĉ † +ĉ). This Taylor expansion is valid at integer quantum flux but becomes less valid around frustration points [48]. Nonetheless, the eigenenergies are fitted within 2 % errors.
Appendix E: System characterization 1. Qubit-polaritons spectroscopy Fig. 8(a) presents the single tone spectroscopy performed by measuring the cavity transmission versus magnetic flux Φ and frequency. The two resonant polariton modes are observed as two maximal transmission peak which strongly vary with Φ. It demonstrates a direct coupling to the traveling microwave signal. The bare cavity resonant frequency ω bare c /2π = 7.169 GHz of the fundamental mode has been measured at 4 K but it is no longer visible at this frequency. Indeed because of its strong hybridization with the ancilla mode, the cavity is now split in the two polariton modes. From the cavity, they inherit their direct coupling to traveling microwave signal. From the ancilla, they get a flux dependence. The two polariton frequencies vary rapidly in flux with a period given by flux quantization in the large circuit loop. In addition a slow variation is superimposed and this affects differently to the two modes. The two polariton modes present a non linear response inherited from the ancilla anharmonicity. When the input microwave power is large, the polariton dynamics shows a bi-stability behaviour. This regime is beyond the scope of this article and we consider here only the low input power in the linear regime.
No qubit resonance is directly detected via single tone spectroscopy. Therefore two-tone spectroscopy is needed to reveal it. One tone is swept between 5.5 GHz and 6.4 GHz in the vicinity of the qubit resonance. The second tone measures the transmission signal at the resonant frequency of one of the polariton modes. This two tone spectroscopy reveals the qubit flux dependence (cf. Fig.  8(b)]. We observed a flux dependence periodic in Φ b but without any superimposed slow variation.
The resonant frequencies of the qubit and the two polariton modes are extracted from the single and two tone spectroscopy and plotted in Fig. 8(c) as function of flux Φ. They are perfectly fitted by the numerical model presented in appendix D. The model precisely described the flux variation of the qubit as well as the two polariton modes. From the fit shown in Fig. 8(b), the circuit parameters are determined and are listed in appendix F. Their values are consistent with estimation based on HFSS simulation and room temperature resistance measurements of the Transmon Josephson junction and SQUID chains. At non-integer reduced flux (Φ/Φ 0 = n), the symmetry is broken and the molecule Hamiltonian in Eq. (1) must be considered, which takes into account additional coupling terms such asx 2 qxa [21]. These terms complexify the quantum dynamics of the system and understanding their effect is beyond the scope of this paper. Hereafter we will only consider the working points at integer reduced flux (Φ/Φ 0 = n). At these fluxes, the qubit frequency is set to ω q /2π = 6.280GHz ± 4MHz (with small variations due to frequency renormalization) and only the ancilla frequency, and thus the polariton frequencies, can vary.

Polaritons tunability
Interestingly, the different flux working points allow to tune the ancilla-cavity hybridization angle without affecting the qubit frequency [cf. Fig. 8]. Therefore, we can in-situ tune the parameters ω j and χ j in Eq. (3).
In Fig. 9(a), the two polaritons resonant frequencies are plotted at integer flux quantum n. They are quantitatively described by the lower and upper polariton modes An avoided crossing between ancilla and cavity can thus be seen. (b) Cross-Kerr strengths between qubit and lower (orange) and upper (purple) polaritons. Black lines are the expected cross-Kerr coupling using χ l = gzz sin 2 (θ) and χu = gzz cos 2 (θ) with gzz/π = 69 MHz. The grey diamonds are simulated points computed using Black Box Quantization [49] with EM simulation.
c l andĉ u previously discussed. Here we set the bare cavity frequency to the value measured at 4 K and the bare ancilla is extracted from the expression ω a = ω l +ω u −ω c . On resonance (ω a = ω c ), the two polaritons are maximally hybridized. We measure g a /2π = 295 MHz from the anti-level crossing. The hybridization weights sin 2 (θ) and cos 2 (θ) between cavity and ancilla are then fitted. At zero flux the upper polariton mode is mainly ancilla-like while the lower polariton is mainly cavity-like. When the cavity and ancilla are resonant, the hybridization weight is 50 %. Each polariton resonance is shifted by the cross-Kerr coupling strength 2χ j conditioned on the qubit state. The cross-Kerr coupling between the qubit and the two polariton modes are plotted in Fig. 9(b) as a function of integer flux quantum. A single tone spectroscopy is performed around the polariton resonance when a π-pulse is applied or not. Because of relaxation, these experiments are performed in the time domain with a 30 ns π-pulse immediately followed by a 500 ns readout pulse. The cross-Kerr coupling is quantitatively described by 2χ j as predicted by the effective polariton model. We measured large readout shifts χ j /π from 9 to 58 MHz thanks to the non-perturbative cross-Kerr coupling. These readout shifts are neither limited by the validity of the dispersive approximation nor by the multi-level aspects of the transmon. For instance, in Ref. [14] the effective coupling for readout has been optimized and is reported to be χ = (gx) 2 αq ∆(∆−αq) = 2π· 7.9 MHz. This is on the order or below of what we can achieve with the present setup without doing an intense optimization of our parameters. Interestingly, at zero flux, the upper polariton, which is further detuned from the qubit than the lower polariton, has a stronger readout shift than the lower polariton.
Appendix F: Circuit parameters ωq/2π ωa/2π ωc/2π ω l /2π ωu/2π (GHz) 6.284 7.780 7.169 7.038 7.911 gzz/π ga/2π χ l q /π χ u q /π (MHz) 69 295 9 57 We summarize in Table II the different frequencies of the modes and the different coupling strengths at zero flux. We also detail in Table III the decay and coherence rates of the modes at zero flux. The microscopic circuit parameters are displayed in Table IV.