Qubit Parity Measurement by Parametric Driving in Circuit QED
Baptiste Royer Shruti Puri Alexandre Blais




Institut quantique and Départment de Physique, Université de Sherbrooke, 2500 boulevard de l'Université, Sherbrooke, Québec J1K 2R1, Canada

Department of Applied Physics, Yale University, PO BOX 208284, New Haven, CT 06511



Institut quantique and Départment de Physique, Université de Sherbrooke, 2500 boulevard de l'Université, Sherbrooke, Québec J1K 2R1, Canada

Canadian Institute for Advanced Research, Toronto, Canada






 
Multi-qubit parity measurements are essential to quantum error correction. Current realizations of these measurements often rely on  ancilla qubits, a method that is sensitive to faulty two-qubit gates and which requires significant experimental overhead. We propose a hardware-efficient multi-qubit parity measurement exploiting the bifurcation dynamics of a parametrically driven nonlinear oscillator. This approach takes advantage of the resonator's parametric oscillation threshold  which is a function of the joint parity of dispersively coupled qubits, leading to high-amplitude oscillations for one parity subspace and no oscillation for the other. We present analytical and numerical results for two- and four-qubit parity measurements with high-fidelity readout preserving the parity eigenpaces. Moreover, we discuss a possible realization which can be readily implemented with the current circuit QED experimental toolbox. These results could lead to significant simplifications in the experimental implementation of quantum error correction, and notably of the surface code.
 

Qubit Parity Measurement by Parametric Driving in Circuit QED
    Alexandre Blais
    December 30, 2023
=============================================================










§ INTRODUCTION

Quantum error correction (QEC) protects fragile quantum information from decoherence and will play a vital role in large-scale quantum computations. 
Typical QEC codewords are defined in a given eigenspace of multiple parity operators. When an error occurs, the state of the qubits leaves the codespace, something that is revealed by measuring the parity operators.
Since these measurements have to be performed repeatedly, it is crucial that they be of high fidelity. Moreover, to avoid introducing errors, these measurements should leave the parity subspaces intact, i.e. states within a given parity subspace should remain unperturbed by the measurement. 

In practice, parity measurement strategies can be broadly classified as direct or indirect. The latter approach, used in recent demonstrations of small-scale error correction <cit.>, relies on a series of two-qubit entangling gates between the data qubits and an additional ancilla qubit which is subsequently measured <cit.>.
Drawbacks of this strategy are the accumulation of errors due to faulty two-qubit gates and the experimental overhead which could become an impediment to the implementation of larger QEC codes.


Faulty gates and overhead issues can be addressed by using direct parity measurements. The central idea in this approach is to map the parity information onto the state of a common mode coupled to the data qubits and which is then measured. For example, a possible strategy to realize direct measurements of two-qubit parity in circuit quantum electrodynamics (QED) is by monitoring the response of a resonator dispersively coupled to the qubits. In this situation, the frequency of the oscillator, and therefore its response to a drive, becomes dependent on the joint-qubit parity <cit.>.
A challenge with this method is to design and implement a protocol which preserves the parity eigenspaces. In other words, in an ideal parity measurement, the common mode and its environment gain information only about which parity subspace (even or odd) the qubits state belongs to. 
Possible improvements to overcome this eigenspace dephasing were introduced in Refs. <cit.>, but require quantum-limited amplifiers with unit efficiency <cit.> or high-efficiency single microwave photon detectors <cit.>. 

Here, we introduce a scheme for direct, high-fidelity parity measurements that leaves the parity subspaces intact. Our approach is based on dispersively coupling multiple qubits to a nonlinear resonator driven by a two-photon parametric pump. This situation leads to a qubit parity-dependent parametric oscillation threshold. 
When the qubits are in the even subspace, the amplitude of the two-photon drive is below the parametric oscillation threshold and the resonator state remains close to vacuum. On the other hand, in the odd subspace, the parametric drive is above threshold and the resonator bifurcates to a high-amplitude state.
We show that by monitoring the amplitude of the resonator output field with standard homodyne detection, it is possible to infer the parity of the qubit ensemble with high fidelity while preserving both even and odd parity subspaces. 
Importantly, we show that the photon number in the high amplitude state can be increased by reducing the resonator nonlinearity, leading to an increased signal-to-noise ratio (SNR) at constant eigenspace dephasing.
These ideas are generalized to more than two qubits by using a multi-tone parametric drive targeting the multiple dispersive shifts corresponding to the same parity subspace.

These ideas can be applied to different types of qubits coupled to oscillators. For concreteness, here we present a circuit QED implementation <cit.> based on transmon qubits <cit.> that can be easily implemented with the current circuit QED toolbox <cit.>.



§ RESULTS AND DISCUSSION








 §.§ Parametrically Driven Non-linear Resonator

Before introducing our proposal for multi-qubit parity measurements, we present its main component: a resonator of frequency  and Kerr non-linearity K. In the presence of a resonant parametric two-photon drive 
 of frequency ω_p = 2
and in a frame rotating at , this system is described by the Hamiltonian (ħ = 1)

    _R = /2 ( + ) - K/2,

where  and  denote the resonator's annihilation and creation operators, respectively. When the drive is turned off, = 0, the steady state of the system is the vacuum state.
Below the parametric oscillation threshold, < κ/2 with κ the single-photon loss rate of the resonator, this system corresponds to the widely used Josephson Parametric Amplifier (JPA) <cit.> with a vacuum-squeezed steady state.
Above > κ/2, this system bifurcates into one of two states of equal amplitude but opposite phases characterized by ⟨⟩_ss = ±α_o with <cit.>

    |α_o|    = (^2 - κ^2/4/K^2)^1/4,
    θ_o≡Arg[α_o]    = 1/2tan^-1(κ/√(4^2-κ^2)).

 Since both the Hamiltonian _R and the dissipation are symmetric under the transformation → - (see Methods), in steady state the resonator occupies either of the two states with equal probability, leading to a null average displacement of the resonator field. However, a single shot homodyne measurement of the resonator steady state will always reveal a large amplitude |α_o|. Once the resonator has latched onto one of its two steady states, tunnelling to the other is highly suppressed for large values of |α_o| <cit.>. In the limit where the two-photon drive is well above the parametric oscillation threshold ≫κ/2, the two steady states are coherent states. 

If the parametric drive is detuned such that -ω_p/2=δ, the system Hamiltonian becomes

    _R,δ = δ+_R.

At large detunings δ^2>^2 - κ^2/4, the vacuum-squeezed state is a steady state of the system, with the squeezing axis governed by the sign of the detuning δ <cit.>. The degree of squeezing decreases as the ratio |δ|/ increases and, for |δ|≫, the steady state is very close to the vacuum state.











 §.§ Two-qubit Parity Measurement

We now turn to the core of our proposal, first considering two-qubit parity measurements. More precisely, we aim to distinguish the odd subspace spanned by the two-qubit states {|01⟩,|10⟩} from the even subspace spanned by {|00⟩,|11⟩}. To this end, we take two qubits dispersively coupled with equal strength χ to the parametrically driven nonlinear resonator. In a frame rotating at , this system is described by the Hamiltonian

    _2qb = χ ( + ) + _R,

where  is the Pauli Z operator for the i^th qubit. Under this dispersive coupling, the resonator frequency becomes qubit-state dependent. 
We note that single-qubit readout in a similar setup was proposed in Ref. <cit.> and experimentally demonstrated in Ref. <cit.>.

The above Hamiltonian, combined with the discussion of the previous section, immediately suggests an approach for multi-qubit parity measurement. Indeed, in <ref>, the qubits induce a dispersive shift of the resonator frequency that will change the parametric oscillation threshold of the two-photon pump in a parity-dependent manner.
More precisely, if the state of the qubit lies in the odd subspace, |ψ_o⟩ = c_01|01⟩+c_10|10⟩, the two dispersive shifts cancel as illustrated in <ref>a. With δ = 0, the system then behaves as a resonantly driven nonlinear resonator. Consequently, in the odd subspace, the resonator bifurcates to a large amplitude state as illustrated in <ref>b. The combined qubits-resonator system thus evolves from the initial state, |Ψ(0)⟩ = |ψ_o⟩⊗|0⟩, to one of the two steady state |Ψ(t)⟩ = |ψ_o⟩⊗|±α_o⟩. Importantly, the phase of the oscillations, Arg[⟨⟩_o] = θ_o, θ_o + π, is independent of the state of the qubits within the odd subspace. In this situation, monitoring the output field of the resonator using standard homodyne measurement of the X_θ_o = <-i θ_o + i θ_o> quadrature reveals a large photon population in the resonator, |⟨⟩_o|^2=|α_o|^2. Note that during the homodyne measurement, the field can in principle switch between the two steady states ±α_o, something that can reduce the measurement fidelity. However, these switching events are rare for large |α_o| <cit.>.

On the contrary, in the even subspace, |ψ_e⟩ = c_00|00⟩+c_11|11⟩, the dispersive shifts of the two qubits add up and the two-photon drive is off-resonant by δ = ± 2χ. For dispersive shifts |2χ| ≫√(^2 - κ^2/4) <cit.>, the vacuum state remains a stable steady state even after activation of the two-photon drive as schematically depicted in <ref>b. That is, the system remains in the initial state, |ψ_e⟩⊗|0⟩. In this case, tracking the output of the resonator with homodyne measurement results in a null amplitude |⟨⟩_e|=0.

In practice, because the dispersive shifts are finite, the resonator state will deviate from vacuum when the qubits are in the even subspace and will become slightly vacuum-squeezed under the action of the off-resonant two-photon drive. The axis of squeezing, schematically represented in the inset of <ref>b, depends on the sign of the parametric pump detuning, and is therefore different for the two even states |00⟩ and |11⟩. This results in slow dephasing within the even parity subspace at rate γ_e = κ(/2χ)^2 (see Methods). This dephasing can be made small by limiting the amplitude of the two-photon drive /2χ≪ 1. Crucially, this does not limit the SNR of the measurement since |α_o| can be made large by reducing the resonator nonlinearity, K, as shown by <ref>. In other words, the measurement SNR and the eigenspace dephasing rate γ_e can be optimized separately.
This is in stark contrast with schemes based on coherent drives where, for a fixed dispersive coupling χ, the eigenspace dephasing increases with the SNR <cit.>. 

To numerically evaluate the performance of this measurement scheme, we simulate the evolution of <ref> under a stochastic master equation (see Methods) <cit.>.
We first compute 2000 trajectories where the qubits are initialized in the odd (even) subspace. For each trajectory, we integrate the resulting homodyne current and categorize it as odd (even) if the absolute value of the signal is above (below) an optimized threshold value.
The resulting measurement fidelity ℱ_m(τ) = 1/2[P(e|e) + P(o|o)] is shown as a function of time in <ref>a. Starting at ℱ_m(0) = 0.5 corresponding to a random parity guess, the fidelity steadily increases towards 1. For the realistic parameters K/κ = 0.175, χ/κ = 25, /κ = 2.5 and τ = 5/κ, we find a large measurement fidelity ℱ_m = 99.9%. In these simulations, the steady state photon number (in the odd subspace) is set to |α_o|^2 ≈ 14, leading to a high SNR once the resonator reaches steady state. For these parameters the measurement time is thus limited by the bifurcation time to the steady state, which scales as ∼ 1/( - κ/2) (see Methods). This could potentially be shortened by shaping the two-photon pulse (t) or with further parameter optimization. Moreover, the measurement fidelity might be improved further by using more sophisticated signal analysis methods such as machine learning techniques <cit.>. 

Starting with an unentangled superposition of the odd and even states, this parity measurement collapses the qubits to an entangled Bell state within one of the two subspaces. To study the creation of entanglement and assess the importance of eigenspace dephasing, we initialize the system in an unentangled state with both qubits in the +1 eigenstate of  and the resonator in the vacuum state, |++⟩⊗|0⟩. We again compute 2000 realizations of the evolution and register the qubits state conditioned on the measurement record, ρ_c. <Ref>b shows the 
concurrence of ρ_c as a function of the measurement time τ. From the initial unentangled state, the qubits are rapidly projected on one of the two parity subspaces, leading to a high concurrence at moderate times. At longer times, the concurrence conditioned on an odd parity measurement approaches unity and, in the even subspace, it slowly decreases due to the slow dephasing γ_e (not apparent on the scale of <ref>b). In order to study the properties of the measurement process only, we considered ideal qubits (T_1,T_2 →∞) and perfectly matched dispersive shifts.
In practice, these imperfections will cause the concurrence to slowly decrease and, in the case of relaxation errors during the measurement (T_1), will decrease the measurement fidelity ℱ_m.



After the measurement, the resonator is reset by turning off the two-photon drive and waiting for a few resonator lifetimes 1/κ or, alternatively, by adiabatically ramping down the parametric drive <cit.>. Since the resonator ends up in a state close to a coherent state, this process can also be sped up using active reset techniques <cit.> 







 §.§ Four-qubit Parity Measurement

We now turn to a generalization of the above approach to four qubits. This is motivated by the many QEC codes that require frequent parity measurements of more than two qubits. This is the case, for example, of the surface code which relies on four-qubit parity measurements <cit.>. Because of the larger Hilbert space, it is now challenging to extract the measurement fidelity and study the entangle creation from numerical simulations. As a result, in this section, we focus on the underlying concepts and on analytical results.

Building on the results for two qubits scenario presented above, we now consider four qubits dispersively coupled to a single nonlinear resonator, where we aim to distinguish between two parity subspaces which are eightfold degenerate. In the even subspace, the dispersive shift can take three different values δ_e = 0, ± 4χ (blue lorentzians) while, in the odd subspace, it can take two different values δ_o = ± 2χ (red lorentzians) as schematically illustrated in <ref>a. Accordingly, a naive generalization of the two-qubit scheme presented above is to excite the resonator with a two-tone two-photon drive ^(2ω) at frequencies 2(± 2χ), as shown by the two sets of orange double arrows in <ref>a.
As in the two-qubit case, this two-tone drive leads to a situation where the parity information is encoded in the amplitude of the resonator field: a high amplitude corresponds to the odd subspace and a null amplitude to the even subspace. When the two tones of the two-photon drive are of equal amplitude, the amplitude of the output field does not depend on the two possible dispersive shifts within the odd subspace δ_o = ± 2χ. However, the frequency of the output field directly depends on δ_o, leading to fast dephasing inside the odd parity subspace at a rate γ_o = κ |α_o|^2.
A possible solution introduced for linear drive schemes <cit.> and also applicable here is to use such a two-tone drive ^(2ω) in combination with a detector that is sensitive exclusively to the amplitude of the output field, i.e. a broadband, high-efficiency photon detector. However, the realization of this type of detector in the microwave domain remains challenging. 
Alternative proposals also offer solutions to this frequency distinguishability problem, but at the cost of higher experimental complexity <cit.>.

Here, we introduce a simpler, hardware-efficient approach to four-qubit parity measurements where the nonlinear resonator is coupled to a low-Q, “filter” resonator of frequency ω_f through a tunable coupling element. As we show, this effectively implements a “frequency erasure” channel that converts resonator photons at ± 2χ to a single frequency ω_f. As a result, only the parity information remains in the output field, i.e. the output field contains no information about the different dispersive shifts δ_o within the odd subspace. Crucially, this allows to infer multi-qubit parity using standard homodyne detection without inducing dephasing within that subspace. 

In order to implement this frequency erasure channel, we consider a two-tone modulation g^(2ω) of the resonator-filter coupling at frequencies Δ_f ± 2χ, where Δ_f ≡ - ω_f.
This multi-tone coupling modulation is schematically illustrated in <ref>a (dark green arrows), where one modulation tone (full lines) brings the δ = ± 2χ resonator peaks (blue) in resonance with the filter mode (purple) while the other coupling modulation tone (dashed lines) is off-resonant by ∓ 4χ and has only a small effect. Irrespective of the dispersive shift δ = ± 2χ, resonator photons are then converted to a single frequency ω_f. 
In a frame rotating at ± 2χ for the resonator, ω_f for the filter resonator and neglecting for now off-resonant terms, the above situation is described by the Hamiltonian (see Methods)

    _4qb,o^(± 2χ) =    _R + g/2[  + ],

where  and  denote the annihilation and creation operators of the filter mode, respectively. 




<Ref> crucially shows that the resonators' dynamic does not depend on the state of the qubits within the odd qubit subspaces, δ_o = ± 2χ.
Consequently, similarly to the two-qubit case, four-qubit parity information can be inferred without eigenspace dephasing by monitoring the amplitude of the output field of the filter mode using homodyne detection.


Expanding further the simple analysis leading to <ref> reveals that, in the odd parity subspace, the filter also emits in a qubit-state dependent sideband ω_f ± 4χ as illustrated by the dashed dark green lines in <ref>a. Consequently, a small portion of the “which-frequency” information is present in the output field, causing a slow dephasing at a rate γ_o^eff = κ_eff |α_o|^2/(1+(8χ/κ_f)^2) inside the odd subspace where κ_eff = g^2/κ_f (see Methods). Taking a measurement time κ_effτ = 5, a steady state photon number |α_o|^2 = 10 and a ratio χ/κ_f=20, this leads to an approximate error probability γ_0^effτ = 0.2 % which is below the threshold for quantum error correction with the surface code <cit.>.
Internal photon loss of the nonlinear resonator at a rate κ_int will also induce dephasing inside the odd subspace at a rate γ_o^int = κ_int |α_o|^2, something which should ideally be minimized.

As a side note, we mention that instead of modulating the coupling, an alternative solution leading to a similar frequency-erasure channel consists in modulating the nonlinear resonator frequency ω_r(t) and adjusting the two-photon drive accordingly. Moreover, we considered above that all qubits had the same dispersive coupling χ to the nonlinear resonator. As long as the absolute value of the dispersive coupling stays homogeneous, its sign could vary amongst the qubits, χ→ -χ, with sole consequence to exchange even and odd in the discussion above. Finally, an added advantage of introducing the filter mode is that it acts naturally as a Purcell filter for the qubits <cit.>.







 §.§ Circuit QED Implementation

Realization of the above ideas is natural in different quantum systems and, as a concrete example, we now describe a possible circuit QED <cit.> implementation with transmon qubits <cit.>.



  §.§.§ Two Qubits

<Ref>c shows the circuit for a two-qubit parity measurement, where two transmon qubits (red) are capacitively coupled to a nonlinear quarter-wavelength resonator (green). 
Taking the transmons to be far detuned from the resonator, the qubit-resonator coupling takes the dispersive character shown in <ref>. The dispersive couplings χ are adjusted to be of equal magnitude and we assume the
transmon qubits to be detuned from each other in order to avoid qubit-qubit interaction mediated by the resonator. 
The resonator nonlinearity K is induced in part by a superconducting quantum interference device (SQUID) located at the end of the resonator and in part by a qubit-induced nonlinearity.
The two-photon drive is induced by applying a flux modulation at twice the resonator frequency (orange). In short, the circuit we propose consists in two transmon qubits dispersively coupled to a Josephson Parametric amplifier (JPA) parametrically driven above threshold and is well within reach of current experimental capabilities.



  §.§.§ Four Qubits

<Ref>b shows a possible implementation of the four-qubit parity measurement. It consists of a nonlinear, quarter-wavelength coplanar resonator (green) capacitively coupled to four transmon qubits capacitively coupled to it (red). To erase the which-frequency information, the nonlinear resonator is coupled to a linear filter resonator (purple) by a tunable coupling element (dark green).
Multiple circuits allow for the necessary coupling modulation <cit.> and here we follow Ref. <cit.>. With this approach, the two-tone coupling modulation g^(2ω) is activated by driving the linear resonator (purple) with a three-tone coherent drive on the filter mode (dark green).






§ CONCLUSION


To summarize, we have introduced a scheme for qubit parity readout exploiting the bifurcation dynamics of a nonlinear oscillator. For two qubits, this leads to a high-fidelity readout that preserves the parity eigenspaces. We also presented an extension of this scheme to the parity readout of four qubits using a multi-tone parametric drive in combination with a multi-tone modulation of the coupling between a nonlinear resonator and a filter mode. 
Both schemes have a simple circuit QED implementation which could be realized with current devices.
This work paves the way for a hardware-efficient implementation of quantum error correction codes such as the surface code in circuit QED.







§ METHODS



 §.§ Stability of Resonator Vacuum State

When parametrically driven on resonance, the classical equations of motion for the field quadratures of the nonlinear resonator x = ⟨ + ⟩/ 2 and y = -i ⟨ - ⟩/ 2 are given by

    ẋ   = K (x^2 + y^2)y +  y - κ/2 x,
    ẏ   = -K (x^2 + y^2)x +  x - κ/2 y.

Computing the eigenvalues of the evolution matrix linearized around vacuum (x,y) = (0,0), we obtain λ_± = ± - κ/2. Small fluctuations around vacuum will thus make the system leave this unstable point on a timescale given by λ_+^-1 = ( - κ/2)^-1.



 §.§ Dephasing in the Two-Qubit Parity Measurement

In the odd qubit subspace, the dispersive shifts shown in <ref> cancel out and the qubits decouple from the resonator. Consequently, there is no dephasing in that subspace. On the other hand, in the even subspace, the two-photon parametric drive leads to a qubit-state dependant resonator field. More precisely, and as schematically illustrated in the inset of <ref>b, when the dispersive shifts are much larger than the two-photon drive and the resonator decay rate, 4χ≫, κ, the resonator field is in the slightly squeezed state | r iθ⟩. The squeezing parameter is r ≈/4χ and the squeezing angle θ≈ 0  or π/2 is qubit-state dependent <cit.>. The overlap of these squeezed pointer states is ⟨ r  | riπ/2⟩ = 1/√(cosh 2r). The corresponding measurement-induced dephasing in this subspace is then roughly given by γ_e∼κ (1-⟨ r  | riπ/2⟩) ∼κ(/4χ)^2 for small r.

A more rigorous derivation of this rate can be found in the Supplemental Material.







 §.§ Simulations

In order to model the back-action of the homodyne measurement chain, we simulate multiple realizations of the evolution of the system under the stochastic master equation <cit.>

    dρ = -i [, ρ] dt + κ𝒟[] dt + √(κ)ℋ[-i θ_o]ρ  dW,

where 𝒟[]∙ = ∙ - 1/2 {, ∙} is the dissipation superoperator and ℋ[M̂]∙ = M̂∙ + ∙M̂^† - Tr[M̂∙ + ∙M̂^†]∙ is the homodyne measurement back-action superoperator. Moreover, dW is a Wiener increment, which has statistical properties E[dW] = 0, E[dW^2] = dt with E[∙] denoting the ensemble average.
The results of <ref> were obtained using <ref> with the Hamiltonian <ref>.
<Ref> shows that the Hamiltonian and dissipation (first two terms) are symmetric under the transformation → -. This symmetry is broken by the homodyne measurement backaction (last term), i.e. by conditioning the state on the measurement record. In other words, although the average displacement of the resonator is null, conditioning the state on the measurement record makes it collapse onto ±α_o.

The homodyne current resulting from the stochastic master equation is given by j_h(t) = √(κ)⟨-iθ_o + iθ_o⟩ + dW/dt. For a given measurement time τ, the dimensionless integrated signal is given by s(τ) = √(κ)∫_0^τ dt  j_h(t).



 §.§ Effective Four-qubit Hamiltonian

As mentioned in the main text, we consider four qubits dispersively coupled to a nonlinear resonator under a two-tone two-photon drive ^(2ω)(t) = cos[2( - 2χ)t] + cos[2( + 2χ)t]. Coupling the nonlinear resonator to a harmonic filter through a two-tone modulation g^(2ω) = g cos[(Δ_f + 2χ) t] + g cos[(Δ_f - 2χ) t], this system is described by the Hamiltonian

    _4qb =     + χ∑_i=1^4 σ̂_zi - K/2  + ω_f 
        + ^(2ω)(t)[ + ] + g^(2ω)(t) [  + ].

For the circuit of <ref>, this two-tone coupling modulation is obtained by driving the filter mode with a three-tone linear drive at frequencies ω_d1, ω_d2, ω_d3. Setting ω_d1 - ω_d2 = Δ_f - 2χ and ω_d1 - ω_d3 = Δ_f + 2χ results in the desired two-tone modulation as well as AC-Stark shifts of the resonator and filter mode frequencies (see Supplemental Material). 

In order to go from <ref> to <ref> of the main text, we restrict the qubits state to the one-excitation subspace spanned by {|0001⟩,|0010⟩,|0100⟩,|1000⟩}, leading to a dispersive shift δ_o = -2χ. We then go to a frame rotating at -2χ for the nonlinear resonator and at ω_f for the filter mode and, neglecting fast-rotating terms, _4qb takes the form

    _4qb,o^(-2χ) =    /2( -K/2 + )+ g/2[  + ]
        + /2[i 8χ t + -i 8χ t]
        + g/2[i 4χ t + -i 4χ t].

The first line corresponds to the effective Hamiltonian <ref>. The second line is the off-resonant two-photon drive tone and has a small effect on the resonator. The third line leads to a small photon emission in the filter sideband ω_f - 4χ and, consequently, to a 
dephasing rate κ_eff |α_o|^2/(1+(8χ/κ_f)^2) (see Supplemental Material). 
The effective Hamiltonian _4qb,o^(+2χ) in the three-excitation subspace with dispersive shift δ_o = 2χ is obtained in the same way.


This research was undertaken thanks in part to funding from NSERC, the Canada First Research Excellence Fund and the Vanier Canada Graduate Scholarships. SP acknowledges financial support by the National Science Foundation under Grant DMR-1609326 and Army Research Office under Grant W911NF1410011.

















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