Dynamically protected cat-qubits: a new paradigm for universal quantum computation

We present a new hardware-efficient paradigm for universal quantum computation which is based on encoding, protecting and manipulating quantum information in a quantum harmonic oscillator. This proposal exploits multi-photon driven dissipative processes to encode quantum information in logical bases composed of Schr\"odinger cat states. More precisely, we consider two schemes. In a first scheme, a two-photon driven dissipative process is used to stabilize a logical qubit basis of two-component Schr\"odinger cat states. While such a scheme ensures a protection of the logical qubit against the photon dephasing errors, the prominent error channel of single-photon loss induces bit-flip type errors that cannot be corrected. Therefore, we consider a second scheme based on a four-photon driven dissipative process which leads to the choice of four-component Schr\"odinger cat states as the logical qubit. Such a logical qubit can be protected against single-photon loss by continuous photon number parity measurements. Next, applying some specific Hamiltonians, we provide a set of universal quantum gates on the encoded qubits of each of the two schemes. In particular, we illustrate how these operations can be rendered fault-tolerant with respect to various decoherence channels of participating quantum systems. Finally, we also propose experimental schemes based on quantum superconducting circuits and inspired by methods used in Josephson parametric amplification, which should allow to achieve these driven dissipative processes along with the Hamiltonians ensuring the universal operations in an efficient manner.


Introduction
In a recent paper [1], we showed that a quantum harmonic oscillator could be used as a powerful resource to encode and protect quantum information. In contrast to the usual approach of multi-qubit quantum error correcting codes [2,3], our approach takes advantage of the infinite dimensional Hilbert space of a quantum harmonic oscillator by redundantly encoding quantum information without the introduction of additional decay channels. Indeed, the far dominant decay channel for a quantum harmonic oscillator, for instance, a microwave cavity field mode, is photon loss. Hence, we only need one type of error syndrome to identify the photon loss error. In this paper, we aim to extend the proposal of [1] as a hardware-efficient protected quantum memory towards a hardware-efficient protected logical qubit with which we can perform universal quantum computations [4].
Before getting to this extension, we recall the idea behind the proposal of [1]. We start by mapping the qubit state is a normalization factor, and α denotes a coherent state of complex amplitude α.
By taking α large enough, α , α − , α i and α −i are quasi-orthogonal (note that for α = 2 considered in most simulations of this paper, α α < − i 10 2 3 ). Such an encoding protects the quantum information against photon loss events. In order to see this, let us also define indicate the occurrence of a photon loss event. While the parity measurements keep track of the photon loss events, the deterministic relaxation of the energy, replacing α by α κ − e / t 2 , remains inevitable. To overcome this relaxation of energy, we need to intervene before the coherent states start to overlap in a significant manner to re-pump energy into the codeword.
In [1], applying some tools that were introduced in [5], we illustrated that simply coupling a cavity mode to a single superconducting qubit in the strong dispersive regime [6] provides the required controllability over the cavity mode (modeled as a quantum harmonic oscillator) to perform all the tasks of quantum information encoding, protection and energy re-pumping. The proposed tools exploit the fact that in such a coupling regime, both qubit and cavity frequencies split into well-resolved spectral lines indexed by the number of excitations in the qubit and the cavity. Such a splitting in the frequencies gives the possibility of performing operations controlling the joint qubit-cavity state. For instance, the energy re-pumping into the Schrödinger cat state is performed by decoding back the quantum information onto the physical qubit and reencoding it on the cavity mode by re-adjusting the number of photons. However, such an invasive control of the state exposes the quantum information to decay channels (such as the T 1 and the T 2 decay processes of the physical qubit) and limits the performance of the protection scheme. Furthermore, if one wanted to use this quantum memory as a protected logical qubit, the application of quantum gates on the encoded information would require the decoding of this information onto the physical qubits, performing the operation, and re-encoding it back to the cavity mode. Once again, by exposing the quantum information to un-protected qubit decay channels, we limit the fidelity of these gates.
In this paper, we aim to exploit an engineered coupling of the storage cavity mode to its environment in order to maintain the energy of the encoded Schrödinger cat state. It is wellknown that resonantly driving a damped quantum harmonic oscillator stabilizes a coherent state of the cavity mode field. In particular, the complex amplitude α of this coherent state depends linearly on the complex amplitude of the driving field. In contrast, by coupling a quantum harmonic oscillator to a bath where any energy exchange with the bath happens in pairs of photons, one can drive the quantum harmonic oscillator to the two aforementioned twocomponent Schrödinger cat states α +  and α −  [7][8][9][10][11]. In section 2 and appendix A, we will exploit such a two-photon driven dissipative process and extend the results of [7][8][9][10] by analytically determining the asymptotic behavior of the system for any initial state. In particular, we will illustrate how such a two-photon process allows us to treat the Schrödinger cat states α +  and α −  (or equivalently the coherent states α ± ) as logical 0 and 1 of a qubit which is protected against a photon dephasing error channel. Such a logical qubit, however, is not protected against the dominant single-photon loss channel. Therefore, in the same section, we propose an extension of this two-photon process to a four-photon process for which the end up with a logical qubit which is protected against photon dephasing errors and for which we can also track and correct errors due to the dominant single-photon loss channel by continuous photon number parity measurements [1]. While this work utilizes a combination of a drivendissipative process and continuous measurements to engineer and protect a qubit, another recent superconducting circuit proposal [12] utilizes a Hamiltonian-based qubit [13] and protected set of gates. It is interesting to note that both proposals protect their respective qubits from dephasing by engineering the qubit basis states.
In section 3, we present a toolbox to perform universal quantum computation with such protected Schrödinger cat states [14]. Applying specific Hamiltonians that should be easily engineered using methods similar to those in Josephson parametric amplification, and in the presence of the two-photon or four-photon driven dissipative processes, we can very efficiently perform operations such as arbitrary rotations around the Bloch sphereʼs x-axis and a two-qubit entangling gate. These schemes can be well understood through quantum Zeno dynamics [15][16][17] where the strong two-photon or four-photon processes continuously project the evolution onto the degenerate subspace of the logical qubit (also known as a decoherence-free subspace [18]). More precisely, the two-photon (resp. fourphoton) process could be considered as a strong measurement of the quantum harmonic oscillator, projecting the system on the space spanned by ). The idea of the quantum Zeno dynamics is then to add a Hamiltonian evolution with a time-scale much slower than the projection rate through the measurement. This will lead to a quasi-unitary evolution within the subspace fixed by the measurement. Furthermore, by increasing the separation of the time-scales between the Hamiltonian evolution and the measurement projection rate, this evolution approaches a real unitary. In order to achieve a full set of universal gates, we then only need to perform a π /2-rotation around the Bloch sphereʼs y-or z-axis. This is performed by the Kerr effect, induced when we couple the cavity mode to a nonlinear medium such as a Josephson junction (JJ) [19,20]. We will illustrate that these gates remain protected against the decay channels of all involved quantum systems and could therefore be employed in a fault-tolerant quantum computation protocol. Finally, in section 4, we propose a readily realizable experimental scheme to achieve the two-photon driven dissipative process along with Hamiltonians needed for universal logical gates. Indeed, we will illustrate that a simple experimental design based on circuit quantum electrodynamics gives us enough flexibility to engineer all the Hamiltonians and the damping operator that are required for the protocols related to the two-photon process. Focusing on a fixed experimental setup, we will only need to apply different pumping drives of well-chosen but fixed amplitudes and frequencies to achieve these requirements. Moreover, comparing to the experimental scheme proposed in [11] (based on the proposal by [21]) our scheme does not require any symmetries in hardware design: in particular, the frequencies of the modes involved in the hardware could be very different, which helps to achieve an important separation of decay times for the two modes. As supporting indications, similar devices with parameters close to those required in this paper have been recently realized and characterized experimentally [20,22]. An extension of this experimental scheme to the case of the four-photon driven dissipative process is currently under investigation and we will describe the starting ideas.
2. Driven dissipative multi-photon processes and protected logical qubits 2.1. Two-photon driven dissipative process Let us consider the harmonic oscillator to be initialized in the vacuum state and let us drive it by an external field in such a way that it can only absorb photons in pairs. Assuming furthermore that the energy decay also only happens in pairs of photons, one easily observes that the photon number parity is conserved. More precisely, we consider the master equation corresponding to a two-photon driven dissipative quantum harmonic oscillator (with ρ being the time derivative of ρ) , the asymptotically stable manifold of the system. However, the asymptotic states in this manifold are not always pure states. One of the results of this paper is to characterize the asymptotic behavior of the above dynamics for any initial state (see appendix A). In particular, initializing the system in a coherent state ρ β β = (0) , it converges to the steady state α − ≈ − X are robustly conserved. Therefore, we will deal with a qubit where the phase-flip errors are very efficiently suppressed and the dominant error channel is the bit-flip errors (which could be induced by a single-photon decay process). This could be better understood if we consider the presence of a dephasing error channel for the quantum harmonic oscillator. In the presence ofdephasing with rate κ ϕ , but no single-photon decay (we will discuss this later), the master equation of the driven system is given as follows The two-photon driven dissipative process therefore leads to a logical qubit basis which is very efficiently protected against the harmonic oscillatorʼs dephasing channel. It is, however, well known that the major decay channel in usual practical quantum harmonic oscillators is single-  (1) where the density matrix is initialized in a coherent state.
Here a point β in the phase space corresponds to the coherent state β at which the process is initialized. The upper row illustrates the value of the Bloch sphere xcoordinate in the logical basis where α ϵ κ =¯= n 2 2ph 2ph for¯= n 2, 4, 9 and 25. We observe that for most coherent states except for a narrow vertical region in the center of the phase space, the system converges to one of the steady coherent states α ± . The lower row illustrates the purity of the steady state to which we converge ρ ∞ ( ) { } tr 2 for various initial coherent states.
Besides the asymptotic state being the pure α ± away from the vertical axis, one can observe that the asymptotic state is also pure for initial states near the center of phase space. Indeed, starting in the vacuum state, the two-photon process drives the system to the pure Schrödinger cat state α +  . photon loss [23]. While the two-photon process fixes the manifold spanned by the states α ±  as the steady state manifold, the single-photon jumps, that can be modeled by application at a random time of the annihilation operator a, lead to a bit-flip error channel on this logical qubit basis. Indeed, the application of a on α ±  sends that state to α ∓  . Such jumps are not suppressed by the two-photon process and a single-photon decay rate of κ 1ph leads to a logical qubit bit-flip rate of α κ 2 1ph . It is precisely for this reason that we need to get back to the protocol of [1] recalled in section 1.

Four-photon driven dissipative process
In order to be able to track single-photon jump events, we need to replace the logical qubit . To this aim, we present here an extension of the above two-photon process to a four-photon one. Indeed, coupling a quantum harmonic oscillator to a driven bath in such a way that any exchange of energy with the bath happens through quadruples of photons obtains the following master equation: protected against the single-photon loss channel. In this encoding, the + X and − X Bloch vectors approximately correspond to the coherent states α and α − (the approximate correspondence is due to the non-orthogonality of the two coherent states which is suppressed exponentially by α 4 2 . While the coherent states are quasiorthogonal, the cat states are orthogonal for all values of α. Since the overlap between coherent states decreases exponentially with α 2 , the two sets of states can be considered as approximately mutually unbiased bases for an effective qubit for α ≳ 2).
corresponds to a 4-cat state which in the Fock basis is composed of states whose photon numbers are the even integers not multiples of 4. In this encoding, + X and − X Bloch vectors approximately correspond to the two-component Schrödinger cat states α The steady states of these dynamics are given by the set of density operators defined on the four-dimensional Hilbert space spanned by . In particular, noting that the above master equation conserves the number of photons modulo 4, starting at initial Fock states 0 , 1 , 2 and 3 , the system converges, respectively, to the pure states , .
By keeping track of the photon number parity, we can restrict the dynamics to the even parity states, so that the steady states are given by the set of density operators defined on the Hilbert . Similar to the two-photon process, these two states will be considered as the logical, now also protected, 0 and 1 of a qubit (see figure 2(b)). Once again, a photon dephasing channel of rate κ ϕ leads to a phase-flip error channel for the logical qubit where the error rate is exponentially suppressed by the size of the Schrödinger cat state (see numerical simulations in appendix A). Note that probing the photon number parity of a quantum harmonic oscillator in a quantum non-demolition manner can be performed by a Ramsey-type experiment where the cavity mode is dispersively coupled to a single qubit playing the role of the meter [24]. Such an efficient continuous monitoring of the photon number parity has recently been achieved using a transmon qubit coupled to a 3D cavity mode in the strong dispersive regime [25]. Furthermore, we have determined that this photon number parity measurement can be performed in a faulttolerant manner; the encoded state can remain intact in the presence of various decay channels of the meter. The details of such a fault-tolerant parity measurement method will be addressed in a future publication [26].

Universal gates and fault-tolerance
The proposal of the previous section together with the implementation scheme of the next one should lead to a technically realizable protected quantum memory. Having discussed how one can dynamically protect from both bit-flip and phase-flip errors, we show in this section that such a protection scheme can be further explored towards a new paradigm for performing faulttolerant quantum computation. Having this in mind, we will show how a set of universal quantum gates can be efficiently implemented on such dynamically protected qubits. This set consists of arbitrary rotations around the x-axis of a single qubit, a single-qubit π /2 rotation around the z-axis, and a two-qubit entangling gate.
The arbitrary rotations around x-axis of a single qubit and the two-qubit entangling gate can be generated by applying some fixed-amplitude driving fields at well-chosen frequencies, leading to additional terms in the effective Hamiltonian of the pumped regime. In order to complete this set of gates, one then only needs a single-qubit π /2-rotation around either the yor z-axes. Here we perform such a rotation around the z-axis by turning off the multi-photon drives and applying a Kerr effect in the Hamiltonian. Such a Kerr effect is naturally induced in the resonator mode through its coupling to the JJ, providing the nonlinearity needed for the multi-photon process. Finally, we will also discuss the fault-tolerance properties of these gates.
In other to ensure such a population transfer between the even and odd parity manifolds, one can apply a Hamiltonian ensuring single-photon exchanges with the system. We show that the simplest Hamiltonian that ensures such a transfer of population is a driving field at resonance with the quantum harmonic oscillator. The idea consists of driving the quantum harmonic oscillator at resonance where the phase of the drive is chosen to be out of quadrature with respect to the Wigner fringes of the Schrödinger cat state. Furthermore, the amplitude of the drive is chosen to be much smaller than the two-photon dissipation rate. This can be much better understood when reasoning in a time-discretized manner. Let us assume α to be real and the quantum harmonic oscillator to be initialized in the even parity cat state α with ϵ ≪ 1 brings the state towards Following the analysis of the previous section, the two-photon process re-projects this displaced state to the space spanned by  without significantly reducing the coherence term; the states α ϵ − + i and α ϵ + i are close to the coherent states α − and α . Therefore, the displaced state is approximately projected on the state  . This is equivalent to applying an arbitrary rotation gate of the form ϵα X on the initial cat state α +  . This protocol can also be understood through quantum Zeno dynamics. The two-photon process can be thought of as a measurement which projects onto the steady-state space spanned by Continuous performance of such a measurement freezes the dynamics in this space while the weaksingle-photon driving field ensures arbitrary rotations around the x-axis of the logical qubit defined in this basis.
In order to simulate such quantum Zeno dynamics, we consider the effective master equation , .
), the slight decay of the Rabi oscillations as a function of time. This is due to the finite ratio κ ϵ / X 2ph , which adds higher order terms to the above effective dynamics. Indeed, similar computations to the one in appendix A can be performed to calculate the effective dephasing time due to these higher order terms. In practice, this induced decay can be reduced by choosing larger separation of time-scales (smaller ϵ κ X 2ph ) at the expense of longer gate times. However, even a moderate factor of 20 ensures gate fidelities in excess of 99.5%.
As illustrated in figure 3(b), we can calculate the Wigner function at particular times during the evolution. This is performed for the times t = 0, π Ω  Wigner representation of the state at times t = 0, π Ω We can observe the shifts in the Wigner fringes while the state remains a coherent superposition with equal weights of α − and α . (c) The tomography at these times t = 0, π Ω illustrates rotations of angles 0, π /4, π /2 and π around the logical x-axis.
π Ω = / t 2 X and, as illustrated in figure 3(c), we observe rotations of angle 0, π /4, π /2 and π around the logical x-axis for the qubit states α Let us now extend this idea to the case of the four-photon process where quantum information can be protected through continuous parity measurements. For the two-photon process, a population transfer from the even cat state α drive which provides single-photon exchanges with the system. For the four-photon case, such a rotation of an arbitrary angle around the Bloch sphereʼs x-axis necessitates a population transfer between the two states α 0mod4  to the Hamiltonian of the four-photon process (for a real α, we take ϕ = 0 in order to be in correct quadrature with respect to the Wigner fringes): , .  Here we show that the same kind of idea can be applied to the case of two logical qubits to produce an effective entangling Hamiltonian of the form σ σ ⊗ x L x L . We start with the case of two harmonic oscillators (with corresponding field mode operators a 1 and a 2 ), each one undergoing a two-photon process. Let us assume we can effectively couple these two oscillators to achieve a beam-splitter Hamiltonian of the form ϵ 2,2ph (we will present in the next section an architecture allowing to get such an effective beam-splitter Hamiltonian between two modes). In order to illustrate the performance of the method, we simulate the two-mode master equation: rotations of angles 0, π /4, π /2 and π around the logical x-axis.

XX XX
Once again, the decay of the fidelity to the Bell states is due to higher order terms in the above approximation of the beam-splitter Hamiltonian by its projection on the qubitʼs subspace. This decay can be reduced by taking a larger separation of time-scales between ϵ XX and κ κ , 1,2ph 2,2ph . However, as can be seen in the simulations, even with a moderate ratio 1 20 of ϵ κ XX 1,2ph and ϵ κ / XX 2,2ph , we get a Bell state with fidelity in excess of 99%. For the case of the four-photon process, in order to achieve an effective Hamiltonian of the form σ σ ⊗ ). (b) Similar simulation for the four-photon process, where the effective two-qubit system is initialized in the state + + = ⊗

Kerr effect for π=2-rotation around z-axis
In order to achieve a complete set of universal gates, we only need another single-qubit gate consisting of a π /2-rotation around the y or z-axis. Together with arbitrary rotations around the x-axis, such a single-qubit gate enables us to perform any unitary operations on single qubits and, along with the two-qubit entangling gate of the previous subsection, provides a complete set of universal gates. However, this fixed angle single-qubit gate presents an issue not manifested in the other gates. To see this, consider the case of the two-photon process with the logical qubit basis α α The process renders the two qubit states α ± ≈ ± X highly stable and tends to prevent any transfer of population from the vicinity of one of these states to the other one. This is trivially in contradiction with the aim of the π /2-rotation around the y or zaxis. This simple fact suggests that performing such a gate is not possible in presence of the two-photon process. Here, we propose an alternative approach, consisting of turning off the two-photon process during the operation (possible through the scheme proposed in the next section) and applying a self-Kerr Hamiltonian of the form χ − † ( ) a a Kerr 2 . In the next section, we will see how such a Kerr Hamiltonian is naturally produced through the same setting as the one required for the two-photon process.
It was proposed in [19] and experimentally realized in [20] that a Kerr interaction can be used to generate Schrödinger cat states. More precisely, initializing the oscillator in the coherent state β , at any time π χ Kerr where q is a positive integer, the state of the oscillator can be written as a superposition of q coherent states [23]: ( )

Fault-tolerance
The proposed set of Hamiltonians allows one to obtain a set of universal quantum gates for the respective two-photon and four-photon processes (see table 1). In this subsection, we consider a logical qubit encoded by the four-photon driven dissipative process and protected against single-photon decay through continuous photon-number parity measurements. We will discuss the fault-tolerance of the above single and two qubit gates with respect to the decoherence channels of single-photon decay and photon dephasing. Indeed, we will not discuss here the tolerance with respect to imprecisions of the gates themselves as we believe such errors should not be put on the same footing as the errors induced by the decoherence of the involved quantum systems. While the protection against errors due to the coupling to an uncontrolled environment is crucial to ensure a scaling towards many-qubit quantum computation, the degree of perfection of gate parameters, such as the angle of a rotation for instance, can be regarded as a technical and engineering matter. More precisely, we will prove the first order fault-tolerance of all these quantum operations. Here, by first order fault-tolerance, we mean that we prevent the errors due to singlephoton loss or photon dephasing to propagate through various quantum operations (see e.g. definition 4 of [27]). Therefore such an error does not get amplified to produce more errors than can be corrected by the quantum error correction scheme. As we do not address issues such as combining the operations through a concatenation procedure, we do not deal here with a Arbitrary rotations around X ϵ threshold theorem. In other words, we show that the error rate due to the photon loss channel does not increase while performing the quantum operations of the previous subsections and that the continuous parity measurements during the operations enable the protection against such a decay channel. Furthermore, arbitrary rotations of a qubit around the x-axis as well as the twoqubit entangling gate are performed in presence of the four-photon process, thereby protecting the qubit against photon dephasing. For the single-qubit π /2-rotation around the z-axis, as long as the Kerr Hamiltonian strength χ Kerr is much more prominent than the dephasing rate (which is the case in most current circuit QED schemes), turning on the four-photon process after the operation will suppress for the phase error accumulated during the operation. Indeed, during the operation, the cavity state is exposed to pure dephasing (rate κ ϕ ) and energy damping (rate κ 1ph ).
This gate time is π χ Kerr (q = 2 for the two-photon process and q = 8 for the four-photon process). The coherent states forming the encoded quantum state, will have their amplitude reduced by the factor κ − , and their phase will drift by a random phase with standard deviation δφ κ = ϕ t q . When the pumping is switched back on at time t q , these errors will be reduced by a factor which grows exponentially with the cat size, as show in appendix A. Hence these errors can be suppressed in the limit where χ κ κ ≫ ϕ , Kerr 1ph , and for a sufficiently large α. Single-qubit θ X gate and two-qubit entangling gate. These operations would be performed in concurrence with the four-photon process, which continuously and strongly projects to the state space generated by . Consider the case of the single-qubit θ X gate. Starting with the state + = α ( ) Z 0mod4  and in the absence of single-photon jumps, the system evolves at With the additional presence of one single-photon jump during this time, this state becomes Although the qubit has changed basis from the even-parity cat states to their odd-parity counterparts , the Zeno dynamics ensures that the information preserved in the qubit continues to be rotated by θ X . After two-and threephoton jumps, we respectively get back to the even and odd parity manifolds, but this time with the order of the basis elements reversed (equivalent to a bit-flip). Finally, after four jumps, we end up in the initial logical basis as if no jump has occurred. This simple reasoning indicates that a continuous photon number parity measurement during the operation should ensure the protection of the rotating quantum information against the single-photon decay channel. The simulations of figure 6 confirm the fact that performing such a single qubit θ X gate, in the presence of the singlephoton decay channel does not increase the decay rate or lead to new decay channels. Continuous photon number parity measurements should therefore correct for such loss events and protect the qubit while the operation is performed. These simulations correspond to the master equation: , . for both plots. As can be seen through these plots, the decay rate remains the same in absence or presence of the two-photon driving field ensuring the arbitrary rotation around the x-axis.
Additionaly, the probability of having more than one jump during the operation time remains within the range of 1%, indicating that with such parameters one would not even need to perform photon-number parity measurements during the operation and that a measurement after the operation would be enough to ensure a significant improvement in the coherence time.
The same kind of analysis is valid for the two-qubit entangling gate. A single-photon loss event for one (or both) of the qubits will lead to switching the associated logical qubit basis of the entangling Hamiltonian Ω σ σ ⊗ XX x L x L 1, 2, , from the even parity manifold to the odd parity one. We can keep track of the encoding subspace for each qubit, by measuring the photon number parity of the two cavity modes after the operation.
Single-qubit π /2-rotation around z-axis. In order to show that the Kerr effect can be applied in a fault-tolerant manner to perform such a single-qubit operation, we apply some of the arguments of the supplemental material of [1]. We need to consider the effect of photon loss events on the logical qubit during such an operation.
We note first that the unitary generated by the Kerr Hamiltonian does not modify the photon number parity as this Hamiltonian is diagonal in the Fock states basis. Therefore, photon number parity remains a quantum jump indicator in presence of the Kerr effect. Now, let us assume that a jump occurs at time t during the operation: the state after the jump is given by . We can take this phase space rotation into account by merely changing the phase of the four-photon drive ϵ 4ph in the four-photon process.

Two-photon driven dissipative process
In this subsection, we propose an architecture based on Josephson circuits which implements the two-photon driven dissipative process. Using the coupling of cavity modes to a JJ, singlephoton dissipation, and coherent drives, we aim to produce effective dynamics in the form of equation (1). These are the same tools used in the Josephson bifurcation amplifier to produce a squeezing Hamiltonian [28] and here we will show that, by selecting a particular pump frequency, we can achieve a two-photon driven dissipative process. Furthermore, in the next subsection, we show that by choosing adequate pump frequencies, we may engineer the interaction terms needed to perform the logical gates described in sections 3.1, 3.2 and 3.3. An architecture suitable for the four-photon driven dissipative process is subject to ongoing work.
The practical device we are considering is represented in figure 7. Two cavities are linked by a small transmission line in which a JJ is embedded. This provides a nonlinear coupling between the modes of these two cavities [20,22]. The Hamiltonian of this device is given by [29] ∑ ∑ and We place ourselves in a regime where rotating terms can be neglected and the remaining terms after the rotating wave approximation constitute the effective Hamiltonian While the induced self-Kerr and cross-Kerr terms χ aa , χ bb and χ ab can be deduced from the Hamiltonian of equation (9) through the calculations of [29], one similarly finds ϵ ω ω χ = −g 2. More precisely, this model reduction can be done by going to a displaced rotating frame in which the Hamiltonian of the pumping drive is removed. Next, one develops the cosine term in the Hamiltonian of equation (9) up to the fourth order and removes the highly oscillating terms in a rotating wave approximation.
Physically, the pump tone ϵ p allows two-photons of mode a to convert to a single-photon of mode b, which can decay through the lossy channel coupled to mode b. The  inputs energy into mode b, which can then be converted to pairs of photon in mode a. The last three terms in equation (10) are the Kerr and cross-Kerr couplings inherited from our proposed architecture. Although these are parasitic terms, we show through numerical simulations that their presence does not deteriorate our scheme.
Taking into account single-photon decay of the mode b, the effective master equation is given by: Neglecting the Kerr and cross-Kerr terms and assuming that , we adiabatically eliminate mode b [7,30] and find a reduced dynamics for mode a of the form of equation (1)  ). The dashed and full curves have comparable convergence rates and converge to the same state. This indicates that the reduced model of equation (1) is a faithful representation of the complete model equation (11). The finite discrepancy is due to the finite ratio between ϵ g , b 2ph and κ b , and the presence of non-zero Kerr and cross-Kerr terms.
in vacuum. The two curves both converge to a fidelity close to one, which indicates that the steady state of equation (11) is hardly affected by the presence of Kerr and cross-Kerr terms and by the finite ratio of ϵ g , b 2ph to κ b .

Logical operations
Rotations of arbitrary angles around the x-axis for the logical qubit 2 , this will induce coherent oscillation between the two states around the Bloch sphereʼs x-axis, as explained in section 3.1.
Entangling gate between two logical bits: we propose the architecture of figure 9 to couple two qubits protected by a two-photon driven dissipative process. Two modules, each composed of a pair of high and low Q cavities, are coupled through a JJ embedded in a waveguide connecting the two high Q cavities. This JJ provides a nonlinear coupling, which, together with a pump at frequency ω ω ω =˜−( , induces an interaction of the form + ϕ † e a a c.c.
Such a term performs an entangling gate between two logical qubits, as described in section 3.2.
π /2-rotation around z-axis: as mentioned throughout the previous subsection, the mere fact of coupling the cavity mode to a JJ induces a self-Kerr term on the cavity mode. As proposed in section 3.3, this could be employed to perform a π /2-rotation around the z-axis in a similar manner to [20]. One only needs to turn off all the pumping drives and wait for π χ induces an interaction term of the form + † a a c.c. 1 2 , thus allowing for the entangling gate detailed in section 3.2 Taking into account the single-photon decay of the mode b of rate κ b such that , we can adiabatically eliminate the mode b and find a reduced dynamics for mode a of the form of equation (4). The problem is therefore to engineer in an efficient manner the Hamiltonian H 4ph .
Indeed, the same architecture as in figure 7, together with a pump frequency of ω ω ω =˜−4 . One can easily observe this by expanding the cosine term in equation (9)  . Inspired by the architecture of the (JRM) [31,32], which ensures an efficient three-wave mixing, we propose here a design which should induce very efficiently the above effective Hamiltonian while avoiding the addition of extra undesirable interactions. The JRM ( figure 10(a)) provides a coupling between the three modes (as presented in figure 10(c)) of the form  (12). Such an interaction should allow us to achieve the driven dissipative four-photon process without adding undesired Hamiltonian terms.    Similarly to [32], by decreasing the inductances L and therefore increasing the associated E L , one can keep the three modes of the device stable for such a choice of external fluxes. This however comes at the expense of diluting the nonlinearity. Now, we couple the z mode of the device to the high-Q storage mode a, its y mode to the low-Q b mode, and we drive the X mode by a pump of frequency ω ω where ϕ pump is the phase of the pump drive and n pump is the average photon number of the coherent state produced in the pump resonator [33].

Summary and conclusions
We have shown that one can achieve a logical qubit basis of cat states  through a two-photon driven dissipative process. A photon dephasing error channel is translated to a phase-flip error rate which is exponentially suppressed by the size α 2 of the cat states. A singlephoton decay channel, however, leads to a bit-flip error channel whose rate is α 2 times larger than the single-photon decay rate. In order to protect the qubit against such a prominent decay channel, we introduce the similar four-photon driven dissipative process whose logical qubit basis is given by the Schrödinger cat states continuously monitoring the photon number parity. Therefore, the cat-state logical qubit can be protected against single-photon decay while also having photon dephasing errors exponentially suppressed. Next, we have introduced a complete set of universal quantum gates that could be performed on the encoded and protected logical qubits. This set consists of arbitrary rotations around the x-axis of a single qubit, a two-qubit entangling gate, and a single-qubit π /2 rotation around the z-axis. The first two gates can be performed in presence of the driven dissipative process and through quantum Zeno dynamics. For the last single-qubit gate, we explore the induced Kerr effect of the quantum harmonic oscillator while the driven dissipative process is turned off. We illustrate that these gates can be performed without propagating errors due to photon dephasing or single-photon loss. They have therefore the potential of being integrated in a fault-tolerant quantum computation architecture.
Finally, we have also discussed the implementation of these tools within the framework of circuit quantum electrodynamics. Inspired by methods used in Josephson parametric amplification, we propose simple experimental schemes to achieve, effectively, the multiphoton driven dissipative processes and also various quantum gates introduced through the paper. In particular, we have implemented in our laboratory the system ensuring the two-photon driven dissipative process and the preliminary experimental results are in good agreement with the theory.
freedom [33] such that For the case of equation (A.1), it is easy to see that˙= ++ J 0 since the two-photon system preserves photon number parity and ++ J is merely the positive parity projector. The off-diagonal quantity from equation (A.2) is an extension of +− J ( ) 0 , the corresponding conserved quantity for the non-driven (α = 0) dissipative two-photon process (first calculated in [35]; see also [34]). This sum is convergent because the sum without the + q 2 1 term is an addition theorem for I q (equation (5.8.7.2) from [34]). To put the above into integral form, we use the identity (derivable from the addition theorem)  (14) from [35]. Assuming real α and using equation (5.8.1.15) from [36], one can calculate limits for large β 2 along the real and imaginary axes in phase space: where erf ( ) . and erfi ( ) . are the error function and imaginary error function, respectively. Both limits analytically corroborate figure 1 and show that the two-photon system is similar to a classical double-well system in the combined large α β , regime.

A.4. Asymptotic behavior of the four-photon process
The asymptotic manifold of the four-photon process from equation (4)  . While an analytical expression for the other conserved quantity J 02 remains to be found, here we provide a numerical analysis of the influence of the photon dephasing on the four-photon process. Figure A1(b) shows a plot similar to figure A1(a), but now for γ − phase flip of the logical qubit of the four-photon process. With the exception of a slight delay in the exponential suppression of the induced phase-flip rate, one observes that this suppression is almost identical to the case of the two-photon process.   of the four-photon process qubit. The phase-flip rate is now scaled by κ ϕ 2 which represents the rate for the case of α = 0.