Dynamically protected cat-qubits: a new paradigm for universal quantum computation
Mazyar Mirrahimi, Zaki Leghtas, Victor V. Albert, Steven Touzard, Robert J. Schoelkopf, Liang Jiang, Michel H. Devoret
DDynamically protected cat-qubits:a new paradigm for universal quantum computation
Mazyar Mirrahimi , , Zaki Leghtas , Victor V. Albert , , StevenTouzard , Robert J. Schoelkopf , , Liang Jiang , , Michel H.Devoret , INRIA Paris-Rocquencourt, Domaine de Voluceau, B.P. 105, 78153 Le ChesnayCedex, France Department of Applied Physics, Yale University, New Haven, Connecticut 06520,USA Department of Physics, Yale University, New Haven, Connecticut 06520, USAE-mail: [email protected]
Abstract.
We present a new hardware-efficient paradigm for universal quantumcomputation which is based on encoding, protecting and manipulating quantuminformation in a quantum harmonic oscillator. This proposal exploits multi-photondriven dissipative processes to encode quantum information in logical bases composedof 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 oftwo-component Schr¨odinger cat states. While such a scheme ensures a protection ofthe logical qubit against the photon dephasing errors, the prominent error channel ofsingle-photon loss induces bit-flip type errors that cannot be corrected. Therefore, weconsider a second scheme based on a four-photon driven dissipative process whichleads 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 photonnumber parity measurements. Next, applying some specific Hamiltonians, we providea 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 withrespect to various decoherence channels of participating quantum systems. Finally,we also propose experimental schemes based on quantum superconducting circuits andinspired by methods used in Josephson parametric amplification, which should allowto achieve these driven dissipative processes along with the Hamiltonians ensuring theuniversal operations in an efficient manner. a r X i v : . [ qu a n t - ph ] D ec
1. Introduction
In a recent paper [1], we showed that a quantum harmonic oscillator could be usedas a powerful resource to encode and protect quantum information. In contrast tothe usual approach of multi-qubit quantum error correcting codes [2, 3], we benefitfrom the infinite dimensional Hilbert space of a quantum harmonic oscillator to encoderedundantly quantum information while no extra decay channels are added. Indeed, thefar dominant decay channel for a quantum harmonic oscillator, for instance, a microwavecavity field mode, is photon loss. Hence, we only need one type of error syndrome toidentify the photon loss error. In this paper, we aim to extend the proposal of [1] asa hardware-efficient protected quantum memory towards a hardware-efficient protectedlogical qubit with which we can perform universal quantum computations [4].Before getting to this extension, we recall the idea behind the proposal of [1]. Westart by mapping the qubit state c | (cid:105) + c | (cid:105) into a multi-component superposition ofcoherent states of the harmonic oscillator | ψ (0) α (cid:105) = c | (cid:105) L + c | (cid:105) L = c |C + α (cid:105) + c |C + iα (cid:105) ,where |C ± α (cid:105) = N ( | α (cid:105) ± |− α (cid:105) ) , |C ± iα (cid:105) = N ( | iα (cid:105) ± |− iα (cid:105) ) . Here, N ( ≈ / √
2) is a normalization factor, and | α (cid:105) denotes a coherent state of complexamplitude α . By taking α large enough, | α (cid:105) , |− α (cid:105) , | iα (cid:105) and |− iα (cid:105) are quasi-orthogonal(note that for α = 2 considered in most simulations of this paper, |(cid:104) α | iα (cid:105)(cid:105)| < − ).Such an encoding protects the quantum information against photon loss events. Inorder to see this, let us also define | ψ (1) α (cid:105) = c |C − α (cid:105) + ic |C − iα (cid:105) , | ψ (2) α (cid:105) = c |C + α (cid:105) − c |C + iα (cid:105) and | ψ (3) α (cid:105) = c |C − α (cid:105) − ic |C − iα (cid:105) . The state | ψ ( n ) α (cid:105) evolves after a photon loss event to a | ψ ( n ) α (cid:105) / (cid:107) a | ψ ( n ) α (cid:105)(cid:107) = | ψ [( n +1)mod4] α (cid:105) , where a is the harmonic oscillator’s annihilationoperator. Furthermore, in the absence of jumps during a time interval t , | ψ ( n ) α (cid:105) deterministically evolves to | ψ ( n ) αe − κt/ (cid:105) , where κ is the decay rate of the harmonicoscillator. Now, the parity operator Π = exp( iπ a † a ) can act as a photon jump indicator.Indeed, we have (cid:104) ψ ( n ) α | Π | ψ ( n ) α (cid:105) = ( − n and therefore the measurement of the photonnumber parity can indicate the occurrence of a photon loss event. While the paritymeasurements keep track of the photon loss events, the deterministic relaxation of theenergy, replacing α by αe − κt/ remains inevitable. To overcome this relaxation of energy,we need to intervene before the coherent states start to overlap in a significant mannerto re-pump energy into the codeword.In [1], applying some tools that were introduced in [5], we illustrated that simplycoupling a cavity mode to a single superconducting qubit in the strong dispersiveregime [6] provides the required controllability over the cavity mode (modeled as aquantum 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 acoupling regime, both qubit and cavity frequencies split into well-resolved spectral linesindexed by the number of excitations in the qubit and the cavity. Such a splittingin the frequencies gives the possibility of performing operations controlling the jointqubit-cavity state. For instance, the energy re-pumping into the Schr¨odinger cat stateis performed by decoding back the quantum information onto the physical qubit and re-encoding it on the cavity mode by re-adjusting the number of photons. However such aninvasive control of the state exposes the quantum information to decay channels (suchas the T and the T decay processes of the physical qubit) and limits the performanceof the protection scheme. Furthermore, if one wanted to use this quantum memory asa protected logical qubit, the application of quantum gates on the encoded informationwould require the decoding of this information onto the physical qubits, performing theoperation, and re-encoding it back to the cavity mode. Once again, by exposing thequantum information to un-protected qubit decay channels, we limit the fidelity of thesegates.In this paper, we aim to exploit an engineered coupling of the storage cavity mode toits environment in order to maintain the energy of the encoded Schr¨odinger cat state. Itis well-known that resonantly driving a damped quantum harmonic oscillator stabilizesa coherent state of the cavity mode field. In particular, the complex amplitude α ofthis coherent state depends linearly on the complex amplitude of the driving field. Incontrast, it has been proposed that coupling a quantum harmonic oscillator to a bathwhere any energy exchange with the bath happens in pairs of photons, one can drive thequantum harmonic oscillator to the two aforementioned two-component Schr¨odinger catstates |C + α (cid:105) and |C − α (cid:105) [7, 8, 9, 10, 11]. In Sec. 2 and Appendix A, we will exploit such a two-photon driven dissipative process and extend the results of [7, 8, 9, 10] by analyticallydetermining the asymptotic behavior of the system for any initial state. In particular,we will illustrate how such a two-photon process can lead to take the Schr¨odinger catstates |C + α (cid:105) and |C − α (cid:105) (or equivalently the coherent states |± α (cid:105) ) as logical | (cid:105) and | (cid:105) of aqubit 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 Schr¨odinger cat states |C (0mod4) α (cid:105) = N ( |C + α (cid:105) + |C + iα (cid:105) ) and |C (2mod4) α (cid:105) = N ( |C + α (cid:105) − |C + iα (cid:105) ) (or equivalently the states |C + α (cid:105) and |C + iα (cid:105) ) become a naturalchoice of logical | (cid:105) and | (cid:105) . Thus, we end up with a logical qubit which is protectedagainst photon dephasing errors and for which we can also track and correct errorsdue to the dominant single-photon loss channel by continuous photon number paritymeasurements [1].In Sec. 3, we present a toolbox to perform universal quantum computation withsuch protected Schr¨odinger cat states [12]. Applying specific Hamiltonians that shouldbe easily engineered by methods inspired by those used in Josephson parametricamplification, and in the presence of the two-photon or four-photon driven dissipativeprocesses, we can very efficiently perform operations such as arbitrary rotations aroundthe Bloch sphere’s X axis and a two-qubit entangling gate. These schemes can bewell understood through quantum Zeno dynamics [13, 14, 15] where the strong two-photon or four-photon processes project the evolution onto the degenerate subspace ofthe logical qubit (also known as a decoherence-free subspace [16]). In order to achievea full set of universal gates, we then only need to perform a π/ Y or Z axis. This is performed by the Kerr effect, induced when wecouple the cavity mode to a nonlinear medium such as a Josephson junction [17, 18].We will illustrate that these gates remain protected against the decay channels of allinvolved quantum systems and could therefore be employed in a fault-tolerant quantumcomputation protocol.Finally, in Sec. 4, we propose a readily realizable experimental scheme to achievethe two-photon driven dissipative process along with Hamiltonians needed for universallogical gates. Indeed, we will illustrate that a simple experimental design basedon circuit quantum electrodynamics gives us enough flexibility to engineer all theHamiltonians and the damping operator that are required for the protocols relatedto the two-photon process. Focusing on a fixed experimental setup, we will only need toapply different pumping drives of well-chosen but fixed amplitudes and frequencies toachieve these requirements. Moreover, comparing to the experimental scheme proposedin [11] (based on the proposal by [19]) our scheme does not require any symmetries inhardware design: in particular, the frequencies of the modes involved in the hardwarecould be very different, which helps to achieve an important separation of decaytimes for the two modes. As supporting indications, similar devices with parametersclose to those required in this paper have been recently realized and characterizedexperimentally [18, 20]. An extension of this experimental scheme to the case of the four-photon driven dissipative process is currently under investigation and we will describethe starting ideas.
2. Driven dissipative multi-photon processes and protected logical qubits
Let us consider the harmonic oscillator to be initialized in the vacuum state and letus 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, oneeasily observes that the photon number parity is conserved. More precisely, we considerthe master equation corresponding to a two-photon driven dissipative quantum harmonicoscillator (with ˙ ρ being the time derivative of ρ )˙ ρ = [ (cid:15) a † − (cid:15) ∗ a , ρ ] + κ D [ a ] ρ, (1)where D [ A ] ρ = AρA † − A † Aρ − ρA † A. When ρ (0) = | (cid:105)(cid:104) | , one can show that the density matrix ρ converges towards a pureeven Schr¨odinger cat state given by the wavefunction |C + α (cid:105) = N ( | α (cid:105) + |− α (cid:105) ), where α = (cid:112) (cid:15) /κ and N is a normalizing factor. Similarly, if the system is initiated ina state with an odd photon number parity such as the Fock state | (cid:105)(cid:104) | , it convergestowards the pure odd Schr¨odinger cat state |C − α (cid:105) = N ( | α (cid:105) − |− α (cid:105) ). Indeed, the setof steady states of Eq. (1) is given by the set of density operators defined on the two | + X i ≈ | α i|− X i ≈ |− α i | + Z i = |C + α i|− Z i = |C − α i Y (a) | + Z i = |C (0 mod α i|− Z i = |C (2 mod α i| + X i ≈ |C + α i|− X i ≈ |C + iα i Y (b) Figure 1. (a)
The two-photon driven dissipative process leads to the choice of evenand odd Schr¨odinger cat states |C + α (cid:105) and |C α − (cid:105) as the logical | (cid:105) and | (cid:105) of a qubitnot protected against the single-photon loss channel. In this encoding, the | + X (cid:105) and |− X (cid:105) Bloch vectors approximately correspond to the coherent states | α (cid:105) and |− α (cid:105) (the approximate correspondence is due to the non-orthogonality of the twocoherent states which is suppressed exponentially by 4 | α | ; While the coherent statesare quasi-orthogonal, the cat states are orthogonal for all values of α ; Since the overlapbetween coherent states decreases exponentially with | α | , the two sets of states canbe considered as approximately mutually unbiased bases for an effective qubit for | α | (cid:38) (b) The four-photon driven dissipative process leads to the choice of four-component Schr¨odinger cat states |C (0 mod α (cid:105) and |C (2 mod α (cid:105) as the logical | (cid:105) and | (cid:105) of a qubit which can be protected against single-photon loss channel by continuousphoton number parity measurements. Here |C (0 mod α (cid:105) = N ( |C + α (cid:105) + |C + iα (cid:105) ) correspondsto a 4-cat state which in the Fock basis is only composed of photon number statesthat are multiples of four. Similarly |C (2 mod α (cid:105) = N ( |C + α (cid:105) − |C + iα (cid:105) ) corresponds to a4-cat state which in the Fock basis is composed of states whose photon numbers arethe even integers not multiples of 4. In this encoding, | + X (cid:105) and |− X (cid:105) Bloch vectorsapproximately correspond to the two-component Schr¨odinger cat states |C + α (cid:105) and |C − α (cid:105) . dimensional Hilbert space spanned by {| α (cid:105) , |− α (cid:105)} [10]. For any initial state, the systemexponentially converges to this set in infinite time, making the span of {|− α (cid:105) , | α (cid:105)} theasymptotically stable manifold of the system. However, the asymptotic states in thismanifold are not always pure states. One of the results of this paper is to characterizethe asymptotic behavior of the above dynamics for any initial state (see Appendix A).In particular, initializing the system in a coherent state ρ (0) = | β (cid:105)(cid:104) β | , it converges tothe steady state ρ ∞ = c ++ |C + α (cid:105)(cid:104)C + α | + c −− |C − α (cid:105)(cid:104)C − α | + c + − |C + α (cid:105)(cid:104)C − α | + c ∗ + − |C − α (cid:105)(cid:104)C + α | , (2)with c ++ = 12 (cid:16) e − | β | (cid:17) , c −− = 12 (cid:16) − e − | β | (cid:17) ,c + − = iαβ ∗ e −| β | (cid:112) | α | ) (cid:90) πφ =0 dφe − iφ I ( | α − β e iφ | ) , where I ( . ) is the modified Bessel function of the first kind. For large enough | β | , thepopulations of the even and odd cat states |C ± α (cid:105) , c ++ and c −− respectively, equilibrate (a) 𝑛 = 2 (b) 𝑛 = 4 (c) 𝑛 = 9 (d) 𝑛 = 25 −1.0 ℜ(𝛽) ℑ ( 𝛽 ) Figure 2.
Asymptotic (infinite-time) behavior of the two-photon driven dissipativeprocess given by Eq. (1) where the density matrix is initialized in a coherent state. Herea point β in the phase space corresponds to the coherent state | β (cid:105) at which the process isinitialized. The upper row illustrates the value of the Bloch sphere X -coordinate in thelogical basis {|C + α (cid:105) , |C − α (cid:105)} ( ≈ (cid:104) α | ρ s | α (cid:105) − (cid:104)− α | ρ s | − α (cid:105) ) where α = √ ¯ n = (cid:113) (cid:15) ph /κ ph for ¯ n = 2 , , |± α (cid:105) . The lower row illustrates the purity of the steady state towhich we converge (tr { ρ ∞ } ) for various initial coherent states. Besides the asymptoticstate being the pure |± α (cid:105) away from the vertical axis, one can observe that theasymptotic 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 pureSchr¨odinger cat state |C + α (cid:105) . to one-half. At large enough α (see Fig. 2 top row), if one initializes with a coherentstate away from the vertical axis in phase space, then the system will converge towardsone of the two steady coherent states |± α (cid:105) (with the sign depending on whether oneinitialized to the right or the left of the vertical axis). This suggests that if we choosethe states |C + α (cid:105) and |C − α (cid:105) as the logical qubit states (see Fig. 1(a)), the two Bloch vectors | + X (cid:105) ≈ | α (cid:105) and |− X (cid:105) ≈ |− α (cid:105) are robustly conserved. Therefore, we will deal with aqubit where the phase-flip errors are very efficiently suppressed and the dominant errorchannel 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 channelfor the quantum harmonic oscillator. In the presence of dephasing with rate κ φ , butno single-photon decay (we will discuss this later), the master equation of the drivensystem is given as follows˙ ρ = [ (cid:15) a † − (cid:15) ∗ a , ρ ] + κ D [ a ] ρ + κ φ D [ a † a ] ρ. (3)Such a dephasing, similar to the photon drive and dissipation, does not affect the photonnumber parity. Therefore the populations of the cat states |C + α (cid:105) and |C − α (cid:105) , or equivalentlythe | + Z (cid:105) and |− Z (cid:105) states in the logical basis, remain constant in the presence of suchdephasing. This means that such an error channel does not induce any bit flip errors onthe logical qubit. It can however induce phase flip errors. But as shown in the AppendixA, the rate at which such logical phase flip errors happen is exponentially suppressedby the size of the cat. Indeed, for κ φ (cid:28) κ , the induced logical phase flip rate is givenby γ phase-flip ≈ κ φ | α | | α | ) → | α | → ∞ . The two-photon driven dissipative process therefore leads to a logical qubit basis whichis very efficiently protected against the harmonic oscillator’s dephasing channel. It is,however, well known that the major decay channel in usual practical quantum harmonicoscillators is single-photon loss [21]. While the two-photon process fixes the manifoldspanned by the states |C ± α (cid:105) as the steady state manifold, the single-photon jumps, thatcan be modeled by application at a random time of the annihilation operator a , lead toa bit-flip error channel on this logical qubit basis. Indeed, the application of a on |C ± α (cid:105) sends the state to |C ∓ α (cid:105) . Such jumps are not suppressed by the two-photon process anda single-photon decay rate of κ leads to a logical qubit bit-flip rate of | α | κ . It isprecisely for this reason that we need to get back to the protocol of [1] recalled in Sec. 1. In order to be able to track single-photon jump events, we need to replace the logicalqubit states |C ± α (cid:105) by the Schr¨odinger cat states |C (0mod4) α (cid:105) and |C (2mod4) α (cid:105) . 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 anyexchange of energy with the bath happens through quadruples of photons, we get thefollowing master equation:˙ ρ = [ (cid:15) a † − (cid:15) ∗ a , ρ ] + κ D [ a ] ρ. (4)The steady states of these dynamics are given by the set of density operators defined onthe 4-dimensional Hilbert space spanned by {|± α (cid:105) , |± iα (cid:105)} where α = (2 (cid:15) /κ ) / .In particular, noting that the above master equation conserves the number of photonsmodulo 4, starting at initial Fock states | (cid:105) , | (cid:105) , | (cid:105) and | (cid:105) , the system converges,respectively, to the pure states C (0mod4) α = N ( |C + α (cid:105) + |C + iα (cid:105) ) , C (1mod4) α = N ( |C − α (cid:105) − i |C − iα (cid:105) ) , C (2mod4) α = N ( |C + α (cid:105) − |C + iα (cid:105) ) , C (3mod4) α = N ( |C − α (cid:105) + i |C − iα (cid:105) ) . By keeping track of the photon number parity, we can restrict the dynamics to the evenparity states, so that the steady states are given by the set of density operators defined onthe Hilbert space spanned by {|C (0mod4) α (cid:105) , |C (2mod4) α (cid:105)} . Similar to the two-photon process,these two states will be considered as the logical, now also protected, | (cid:105) and | (cid:105) of aqubit (see Fig. 1(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 suppressedby the size of the Schr¨odinger cat state (see numerical simulations of Appendix A).Note that probing the photon number parity of a quantum harmonic oscillator ina quantum non-demolition manner can be performed by a Ramsey-type experimentwhere the cavity mode is dispersively coupled to a single qubit playing the role ofthe meter [22]. Such an efficient continuous monitoring of the photon number parityhas recently been achieved using a transmon qubit coupled to a 3D cavity mode inthe strong dispersive regime [23]. Furthermore, we have determined that this photonnumber parity measurement can be performed in a fault-tolerant manner; the encodedstate can remain intact in the presence of various decay channels of the meter. Thedetails of such a fault-tolerant parity measurement method will be addressed in a futurepublication [24].In summary, we have shown that one can achieve a logical qubit basis of cat states {|C + α (cid:105) , |C − α (cid:105)} through a two-photon driven dissipative process. A photon dephasing errorchannel is translated to a phase-flip error rate which is exponentially suppressed bythe size of the cat states. A single-photon decay channel, however, leads to a bit-fliperror channel whose rate is | α | times larger than the single-photon decay rate. Inorder to protect the qubit against such a prominent decay channel, we introduce thesimilar four-photon driven dissipative process whose logical qubit basis is given by theSchr¨odinger cat states {|C (0mod4) α (cid:105) , |C (2mod4) α (cid:105)} . Once again, the photon dephasing errorchannel is replaced by a phase-flip error channel whose rate is suppressed exponentiallyby the size of the Schr¨odinger cat state. A single-photon decay channel leads to thetransfer of quantum information to a new logical basis given by odd Schr¨odinger catstates {|C (3mod4) α (cid:105) , |C (1mod4) α (cid:105)} . However, we can keep track of single photon decay bycontinuously measuring the photon number parity. Therefore, the cat-state logical qubitcan be protected against single photon decay while also having photon dephasing errorsexponentially suppressed.
3. Universal gates and fault-tolerance
The proposal of the previous section together with the implementation scheme of thenext one should lead to a technically realizable protected quantum memory. Havingdiscussed how one can dynamically protect from both bit-flip and phase-flip errors, weshow in this section that such a protection scheme can be further explored towards anew paradigm for performing fault-tolerant quantum computation. Having this in mind,we will show how a set of universal quantum gates can be efficiently implemented onsuch dynamically protected qubits. This set consists of arbitrary rotations around the X axis of a single qubit, a single-qubit π/ Z axis, and a two-qubitentangling gate.The arbitrary rotations around X -axis of a single qubit and the two-qubit entanglinggate can be generated by applying some fixed-amplitude driving fields at well-chosenfrequencies, leading to additional terms in the effective Hamiltonian of the pumpedregime. In order to complete this set of gates, one then only needs a single-qubit π/ Y or Z axes. Here we perform such rotation around the Z axisby 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 tothe Josephson junction, providing the non-linearity needed for the multi-photon process.Finally, we will also discuss the fault-tolerance properties of these gates. Let us start with the case of the two-photon process where the quantum information isnot protected against single-photon loss. The parity eigenstates |C + α (cid:105) and |C − α (cid:105) areinvariant states when the exchange of photons with the environment only happensthrough pairs of photons. Here, we are interested in performing a rotation of an arbitraryangle θ around the X axis in this logical basis of {|C + α (cid:105) , |C − α (cid:105)} : X θ = cos θ ( |C + α (cid:105)(cid:104)C + α | + |C − α (cid:105)(cid:104)C − α | ) + i sin θ ( |C + α (cid:105)(cid:104)C − α | + |C − α (cid:105)(cid:104)C + α | ) . 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. Weshow that the simplest Hamiltonian that ensures such a transfer of population is adriving field at resonance with the quantum harmonic oscillator. The idea consists ofdriving the quantum harmonic oscillator at resonance where the phase of the drive ischosen to be out of quadrature with respect to the Wigner fringes of the Schr¨odingercat state. Furthermore, the amplitude of the drive is chosen to be much smaller thanthe two-photon dissipation rate. This can be much better understood when reasoningin a time-discretized manner. Let us assume α to be real and the quantum harmonicoscillator to be initialized in the even parity cat state |C + α (cid:105) . Applying a displacementoperator D ( i(cid:15) ) = exp( i(cid:15) ( a + a † )) with (cid:15) (cid:28) D ( i(cid:15) ) |C + α (cid:105) = N ( e − i(cid:15)α |− α + i(cid:15) (cid:105) + e i(cid:15)α | α + i(cid:15) (cid:105) )Following the analysis of the previous section, the two-photon process re-projects thisdisplaced state to the space spanned by {|C + α (cid:105) , |C − α (cid:105)} without significantly reducingthe coherence term: the states |− α + i(cid:15) (cid:105) and | α + i(cid:15) (cid:105) are close to the coherent states |− α (cid:105) and | α (cid:105) . Therefore, the displaced state is approximately projected on the statecos( (cid:15)α ) |C + α (cid:105) + i sin( (cid:15)α ) |C − α (cid:105) . This is equivalent to applying an arbitrary rotation gateof the form X (cid:15)α on the initial cat state |C + α (cid:105) . This protocol can also be understoodthrough quantum Zeno dynamics. The two-photon process can be thought of asa measurement which projects onto the steady-state space spanned by {|C + α (cid:105) , |C − α (cid:105)} .Continuous performance of such a measurement freezes the dynamics in this space whilethe weak single-photon driving field ensures arbitrary rotations around X -axis of thelogical qubit defined in this basis.In order to simulate such quantum Zeno dynamics, we consider the effective masterequation ˙ ρ = − i(cid:15) X [ a + a † , ρ ] + (cid:15) [ a † − a , ρ ] + κ D [ a ] ρ. (5)0Here, taking (cid:15) = ¯ nκ / (cid:15) X (cid:28) κ , we ensure the above Zeno dynamics inthe space spanned by {|C − α (cid:105) , |C + α (cid:105)} , with α = √ ¯ n (here, the choice of the phase of (cid:15) fixes α to be real). By initializing the system in the state |C + α (cid:105) and letting the systemevolve following the above dynamics, we numerically simulate the equivalent of a Rabioscillation’s experiment. We monitor the population of the states |C + α (cid:105) and |C − α (cid:105) (the | + Z (cid:105) and |− Z (cid:105) states) during the evolution. Fig. 3(a) illustrates the result of suchsimulation over a time of 2 π/ Ω X where Ω X the effective Rabi frequency is given byΩ X = 2 (cid:15) X √ ¯ n. This effective Rabi frequency can be found by projecting the added driving Hamiltonian (cid:15) X ( a + a † ) on the space spanned by {|C − α (cid:105) , |C + α (cid:105)} : (cid:15) X (cid:16) Π |C + α (cid:105) + Π |C − α (cid:105) (cid:17) ( a + a † ) (cid:16) Π |C + α (cid:105) + Π |C − α (cid:105) (cid:17) = ( α + α ∗ ) (cid:15) X (cid:0) |C + α (cid:105)(cid:104)C − α | + |C − α (cid:105)(cid:104)C + α | (cid:1) = Ω X σ Lx , where Π |C ± α (cid:105) = |C ± α (cid:105)(cid:104)C ± α | . One can note in Fig. 3(a) (where we have chosen (cid:15) X = κ / κ /(cid:15) X , which adds higher order terms to the above effective dynamics. Indeed,similar computations to the one in Appendix A can be performed to calculate theeffective dephasing time due to these higher order terms. In practice, this induceddecay can be reduced by choosing larger separation of time-scales (smaller (cid:15) X /κ ) atthe expense of longer gate times. However, even a moderate factor of 20 ensures gatefidelities in excess of 99 . t = 0, t = π/ X , t = π/ X and t = π/ X and, as illustrated in Fig. 3(c), we observe rotations of angle 0, π/ π/ π around the logical X axis for the qubit states defined as |C + α (cid:105) and |C − α (cid:105) .Let us now extend this idea to the case of the four-photon process where quantuminformation can be protected through continuous parity measurements. For the two-photon process, a population transfer from the even cat state |C + α (cid:105) to |C − α (cid:105) , is ensuredthrough a resonant drive ensuring single-photon exchanges with the system. For thefour-photon case, such a rotation of an arbitrary angle around the Bloch sphere’s X -axis necessitates a population transfer between the two states |C α (cid:105) and |C α (cid:105) . Thestate |C α (cid:105) correspond to a four-component Schr¨odinger cat state which in the Fockbasis is only composed of states with photon numbers that are multiples of 4. Similarly,the state |C α (cid:105) corresponds to a four-component Schr¨odinger cat state which in theFock basis is only composed of photon number states that are even but not multiples of4. Therefore, in order to ensure a population transfer from |C (0mod4) α (cid:105) to |C (2mod4) α (cid:105) , weneed to apply a Hamiltonian that adds/subtracts pairs of photons to/from the system.This can be done by adding a squeezing Hamiltonian of the form (cid:15) X ( e iφ a + e − iφ a † ) tothe Hamiltonian of the four-photon process (for a real α , we take φ = 0 in order to bein correct quadrature with respect to the Wigner fringes):˙ ρ = − i(cid:15) X [ a + a † , ρ ] + (cid:15) [ a † − a , ρ ] + κ D [ a ] ρ. (6)1 (a) Time ( π / Ω X ) Population of |C α + >Population of |C α − > (b) IQ (c) XXXX YYYY |C + α i|C + α i|C + α i|C + α i |C − α i|C − α i|C − α i|C − α i t = 0 t = π/ X t = π/ X t = π/ X Figure 3.
Quantum Zeno dynamics as a tool for performing rotations of an arbitraryangle around the X -axis of the logical qubit space spanned by {|C + α (cid:105) , |C − α (cid:105)} . Thequantum harmonic oscillator is driven at resonance in the Q -direction while the two-photon driven dissipative process is acting on the system. (a) Simulations of Eq. (5)illustrate the Rabi oscillations around the Bloch sphere’s X -axis in the logical qubitspace at an effective Rabi frequency of Ω X = 2 (cid:15) X √ ¯ n . Here (cid:15) X = κ ph /
20 and ¯ n = 4. (b) Wigner representation of the state at times t = 0, t = π/ X , t = π/ X and t = π/ X . We can observe the shifts in the Wigner fringes while the state remainsa coherent superposition with equal weights of |− α (cid:105) and | α (cid:105) . (c) The tomography atthese times t = 0, t = π/ X , t = π/ X and t = π/ X illustrate rotations of angles0, π/ π/ π around the logical X axis. In direct correspondence with the two-photon process, we initialize the system in thestate |C (0mod4) α (cid:105) and we simulate Eq. (6). Here (cid:15) = ¯ n κ / {| α (cid:105) , |− α (cid:105) , | iα (cid:105) , |− iα (cid:105)} , with α = √ ¯ n is asymptotically stable. Since all theHamiltonians and decay terms correspond to exchanges of photons in pairs or quadruplesand since we have initialized in |C α (cid:105) , we can restrict the dynamics to the subspacespanned by even Fock states. In this subspace, the asymptotic manifold is generatedby |C (0mod4) α (cid:105) and |C (2mod4) α (cid:105) . We also take (cid:15) X to be much smaller than κ . Simulationsshown in Fig. 4(a) (for ¯ n = 4 and (cid:15) X = κ /
20) illustrate the Rabi oscillations atfrequency Ω X = 2 (cid:15) X ¯ n around the Bloch sphere’s X axis in this logical basis. This Rabi frequency can also beretrieved by projecting the squeezing Hamiltonian onto the qubit subspace: (cid:15) X (cid:16) Π |C (0 mod α (cid:105) + Π |C (2 mod α (cid:105) (cid:17) ( a + a † ) (cid:16) Π |C (0 mod α (cid:105) + Π |C (2 mod α (cid:105) (cid:17) =( α + α ∗ ) (cid:15) X (cid:0) |C (0mod4) α (cid:105)(cid:104)C (2mod4) α | + |C (2mod4) α (cid:105)(cid:104)C (0mod4) α | (cid:1) = Ω X σ Lx . As shown in Figures 4(b),(c), we efficiently achieve an effective single-qubit gatecorresponding to rotations of an arbitrary angle around the Bloch sphere’s X -axis forthe logical qubit spanned by {|C (0mod4) α (cid:105) , |C (2mod4) α (cid:105)} .2 (a) (a) Time ( π / Ω X ) Population of | C α (0 mod 4) >Population of | C α (2 mod 4) > (b) (c) XXXX YYYY | C + α i| C + α i| C + α i| C + α i | C + iα i| C + iα i| C + iα i| C + iα i t = 0 t = π/ Z t = π/ Z t = π/ Z (b) IQ (c) XXXX YYYY |C α i|C α i|C α i|C α i |C α i|C α i|C α i|C α i t = 0 t = π/ X t = π/ X t = π/ X Figure 4.
Quantum Zeno dynamics as a tool for performing rotations ofarbitrary angles around the Bloch sphere’s X -axis of the logical qubit basis of {|C (0 mod α (cid:105) , |C (2 mod α (cid:105)} . A squeezing Hamiltonian is applied on the quantum harmonicoscillator while the four-photon driven dissipative process is acting. (a) Rabioscillations around the X axis with an effective Rabi frequency of Ω X = 2 (cid:15) X ¯ n . Here (cid:15) X = κ ph /
20 and ¯ n = 4. (b) Wigner representation of the state at times t = 0, t = π/ Z , t = π/ Z and t = π/ Z , with different fringe patterns associated torotations with different angles. (c) The tomography at these times t = 0, t = π/ Z , t = π/ Z and t = π/ Z illustrate rotations of angles 0, π/ π/ π around thelogical X axis. Here we show that the same kind of idea can be applied to the case of two logical qubitsto produce an effective entangling Hamiltonian of the form σ Lx ⊗ σ Lx . We start with thecase of two harmonic oscillators (with corresponding field mode operators a and a ),each one undergoing a two-photon process. Let us assume we can effectively couple thesetwo oscillators to achieve a beam-splitter Hamiltonian of the form (cid:15) XX ( a a † + a a † ),where (cid:15) XX (cid:28) κ , , κ , (we will present in the next section an architecture allowingto get such an effective beam-splitter Hamiltonian between two modes). In order toillustrate the performance of the method, we simulate the two-mode master equation:˙ ρ = − i(cid:15) XX [ a a † + a a † , ρ ] + (cid:15) , [ a † − a , ρ ] + (cid:15) , [ a † − a , ρ ] (7)+ κ , D [ a ] ρ + κ , D [ a ] ρ. (8)Simulations in Fig. 5(a) are performed by initializing the system at the logical state | + Z , + Z (cid:105) = |C + α (cid:105) ⊗ |C + α (cid:105) and letting it evolve under Eq. (7). These simulationsillustrate that two-mode entanglement does occur, reaching the Bell states |B +2 ,α (cid:105) =( |C + α (cid:105) ⊗ |C + α (cid:105) + i |C − α (cid:105) ⊗ |C − α (cid:105) ) / √ |B − ,α (cid:105) = ( |C + α (cid:105) ⊗ |C + α (cid:105) − i |C − α (cid:105) ⊗ |C − α (cid:105) ) / √
2. Indeed,by projecting the beam-splitter Hamiltonian (cid:15) XX ( a a † + a a † ) on the tensor productof the spaces spanned by {|C + α (cid:105) , |C − α (cid:105)} , we get as the effective Hamiltonian, that of atwo-qubit entangling gate: (cid:15) XX Π |C + α (cid:105) , |C − α (cid:105) ⊗ Π |C + α (cid:105) , |C − α (cid:105) ( a a † + a a † )Π |C + α (cid:105) , |C − α (cid:105) ⊗ Π |C + α (cid:105) , |C − α (cid:105) =2 | α | (cid:15) XX (cid:0) |C + α (cid:105)(cid:104)C − α | + |C − α (cid:105)(cid:104)C + α | (cid:1) ⊗ (cid:0) |C + α (cid:105)(cid:104)C − α | + |C − α (cid:105)(cid:104)C + α | (cid:1) = Ω XX σ ,Lx ⊗ σ ,Lx , XX = 2¯ n(cid:15) XX . Once again the decay of the fidelity to the Bell states is due to higher order terms inthe above approximation of the beam-splitter Hamiltonian by the projected one on thequbit’s subspace. This decay can be reduced by taking a larger separation of time-scalesbetween (cid:15) XX and κ , , κ , . However, as can be seen in the simulations, even witha moderate ratio 1 /
20 of (cid:15) XX /κ , and (cid:15) XX /κ , , we get a Bell state with fidelity inexcess of 99%. (a) π / Ω XX )Population of | B α + >Population of | B α − > (b) π / Ω XX )Population of | B α + >Population of | B α − > Figure 5.
Quantum Zeno dynamics as a tool for performing a two-qubitentangling gate for the two cases of the two-photon process, with the logicalqubit basis {|C + α (cid:105) , |C − α (cid:105)} , and the four-photon process, with the logical qubit basis {|C (0 mod α (cid:105) , |C (2 mod α (cid:105)} . (a) Considering the two-photon process and initializing theeffective two-qubit system in the state | + Z , + Z (cid:105) = |C + α (cid:105)⊗|C + α (cid:105) , we monitor continuouslythe fidelity with respect to the Bell states |B ± ,α (cid:105) = √ ( |C + α (cid:105) ⊗ |C + α (cid:105) ± i |C − α (cid:105) ⊗ |C − α (cid:105) ). Thesimulation parameters are the same as in previous figures and the effective entanglingHamiltonian is given by Ω XX σ ,Lx ⊗ σ ,Lx with Ω XX = 2¯ n(cid:15) XX ( (cid:15) XX = κ ph / (b) Similar simulation for the four-photon process, where the effective two-qubit system isinitialized in the state | + Z , + Z (cid:105) = |C (0 mod α (cid:105)⊗|C (0 mod α (cid:105) and we monitor continuouslythe fidelity with respect to the Bell states |B ± ,α (cid:105) = √ ( |C (0 mod α (cid:105) ⊗ |C (0 mod α (cid:105) ± i |C (2 mod α (cid:105) ⊗ |C (2 mod α (cid:105) ). For the case of the four-photon process, in order to achieve an effective Hamiltonianof the form σ Lx ⊗ σ Lx for the logical qubit basis of {|C (0mod4) α (cid:105) , |C (2mod4) α (cid:105)} , one needsto ensure exchanges of photons in pairs between the two oscillators encoding theinformation. This is satisfied by replacing the beam-splitter Hamiltonian with (cid:15) XX (cid:16) a a † + a a † (cid:17) . Once again, we initialize the system in the state | + Z , + Z (cid:105) = |C (0mod4) α (cid:105) ⊗ |C (0mod4) α (cid:105) and we let it evolve following the two-mode master equation:˙ ρ = − i(cid:15) XX [ a a † + a a † , ρ ] + (cid:15) , [ a † − a , ρ ] + (cid:15) , [ a † − a , ρ ] (9)+ κ , D [ a ] ρ + κ , D [ a ] ρ. (10)4Simulations of Fig. 5(b), illustrate the two-mode entanglement reaching the Bell states |B +4 ,α (cid:105) = ( |C (0mod4) α (cid:105) ⊗ |C (0mod4) α (cid:105) + i |C (2mod4) α (cid:105) ⊗ |C (2mod4) α (cid:105) ) / √ |B − ,α (cid:105) = ( |C (0mod4) α (cid:105) ⊗|C (0mod4) α (cid:105)− i |C (2mod4) α (cid:105)⊗|C (2mod4) α (cid:105) ) / √
2. By projecting the Hamiltonian (cid:15) XX ( a a † + a a † )on the tensor product of the spaces spanned by {|C (0mod4) α (cid:105) , |C (2mod4) α (cid:105)} , we get as theeffective Hamiltonian, that of a two-qubit entangling gate: (cid:15) XX Π |C (0 , mod α (cid:105) ⊗ Π |C (0 , mod α (cid:105) ( a a † + a a † )Π |C (0 , mod α (cid:105) ⊗ Π |C (0 , mod α (cid:105) =2 | α | (cid:15) XX (cid:0) |C (0mod4) α (cid:105)(cid:104)C (2mod4) α | + |C (2mod4) α (cid:105)(cid:104)C (0mod4) α | (cid:1) ⊗ = Ω XX σ ,Lx ⊗ σ ,Lx , where Ω XX = 2¯ n (cid:15) XX . π/ -rotation around Z -axis In order to achieve a complete set of universal gates, we only need another single-qubitgate consisting of a π/ Y or Z axis. Together with arbitraryrotations around X -axis, such a single-qubit gate enables us to perform any unitaryoperations on single qubits and along with the two-qubit entangling gate of the previoussubsection, 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 thecase of the two-photon process with the logical qubit basis {|C + α (cid:105) , |C − α (cid:105)} . The processrenders the two qubit states |± X (cid:105) ≈ |± α (cid:105) highly stable and tends to prevent any transferof population from the vicinity of one of these states to the other one. This is triviallyin contradiction with the aim of the π/ Y or Z axis. This simplefact suggests that performing such a gate is not possible in presence of the two-photonprocess. Here, we propose an alternative approach, consisting of turning off the two-photon process during the operation (possible through the scheme proposed in the nextsection) and applying a self-Kerr Hamiltonian of the form − χ Kerr ( a † a ) . In the nextsection, we will see how such a Kerr Hamiltonian is naturally produced through thesame setting as the one required for the two-photon processes.It was proposed in [17] and experimentally realized in [18] that a Kerr interactioncan be used to generate Schr¨odinger cat states. More precisely, initializing the oscillatorin the coherent state | β (cid:105) , at any time t q = π/qχ Kerr where q is a positive integer, thestate of the oscillator can be written as a superposition of q coherent states [21]: | ψ ( t q = πq χ Kerr ) (cid:105) = 12 q q − (cid:88) p =0 2 q − (cid:88) k =0 e ik ( k − p ) πq | βe ip πq (cid:105) . In particular, at t = π/ χ Kerr , the states |± α (cid:105) evolve to 1 / √ |± α (cid:105) − i |∓ α (cid:105) ).Therefore, in the case of the logical qubit basis {|C + α (cid:105) , |C − α (cid:105)} , this is equivalent to a( − π/ Z -axis.Analogously for the case of four-photon process, initializing the oscillator in the two-component Schr¨odinger cat state | + X (cid:105) ≈ |C + α (cid:105) , obtains the state 1 / √ (cid:0) |C + α (cid:105) − i |C + iα (cid:105) (cid:1) at5 Table 1.
List of Hamiltonians and decay operators providing protection and a set ofuniversal gates Two-photon protection Four-photon protectionDecay operator κ ph a κ ph a Driving Hamiltonian i(cid:15) ph ( a † − a ) i(cid:15) ph ( a † − a )Arbitrary rotations around X (cid:15) X ( a † + a ) (cid:15) X ( a † + a ) π/ Z − χ Kerr ( a † a ) − χ Kerr ( a † a ) Two-qubit entangling gate (cid:15) XX ( a a † + a a † ) (cid:15) XX ( a a † + a a † ) time t = π/ χ Kerr . Thus, we have a ( − π/ Z -axis for the logicalqubit basis of {|C (0mod4) α (cid:105) , |C (2mod4) α (cid:105)} , . The proposed set of Hamiltonians allows one to obtain a set of universal quantumgates for the two cases of two-photon and four-photon processes (see Table 1). In thissubsection, we consider a logical qubit encoded by the four-photon driven dissipativeprocess and protected against single-photon decay through continuous photon-numberparity measurements. We will discuss the fault-tolerance of the above single and twoqubit gates with respect to the decoherence channels of the single-photon decay andthe photon dephasing. Indeed, we will not discuss here the tolerance with respect toimprecisions of the gates themselves as we believe such errors should not be put on thesame 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 environmentis crucial to ensure a scaling towards many-qubit quantum computation, the degreeof perfection of gate parameters, such as the angle of a rotation for instance, can beregarded as a technical and engineering matter.More precisely, we show that the error rate due to the photon loss channel doesnot increase while performing the quantum operations of the previous subsections andthat the continuous parity measurements during the operations enable the protectionagainst such a decay channel. Furthermore, arbitrary rotations of the single-qubitaround X -axis as well as the two-qubit entangling gate are performed in presence of thefour-photon process and therefore the qubit will also remain protected against photondephasing channel. For the single-qubit π/ Z -axis, as long as the KerrHamiltonian strength χ Kerr is much more prominent than the dephasing rate (which isthe case in most current circuit QED schemes), turning on the four-photon process afterthe operation will correct for the phase error accumulated during the operation.
Single-qubit X θ gate and two-qubit entangling gate. These operations would beperformed in concurrence with the four-photon process, which continuously and stronglyprojects to the state space generated by {|± α (cid:105) , |± iα (cid:105)} . Consider the case of single-qubit X θ gate (with the same kind of analysis valid for the two-qubit entangling gate).6Starting with the state | + Z (cid:105) = |C (0mod4) α (cid:105) and in the absence of single-photon jumps,the system evolves at time t to | ψ ( t ) (cid:105) = cos(Ω X t ) |C (0mod4) α (cid:105) − i sin(Ω X t ) |C (2mod4) α (cid:105) . Withthe additional presence of one single-photon jump during this time, this state becomes a | ψ ( t ) (cid:105) = cos(Ω X t ) |C (3mod4) α (cid:105) − i sin(Ω X t ) |C (1mod4) α (cid:105) . More precisely, after a single-photonjump has occurred, the Zeno dynamics of Eq. (6) keeps ensuring the rotation aroundthe X -axis of the new logical qubit basis {|C (3mod4) α (cid:105) , |C (1mod4) α (cid:105)} corresponding to theodd photon number parity manifold. Note that after two and three photon jumps, werespectively get back to the even and odd parity manifolds but altering the basis elements(equivalent to a bit-flip). Finally, after four jumps, we end up in the initial logical basisas if no jump has occurred. This simple reasoning indicates that a continuous photonnumber parity measurement during the operation should ensure the protection of therotating quantum information against the single-photon decay channel. The simulationsof Fig. 6 confirm the fact that performing such a single qubit X θ gate, in the presence ofthe single-photon decay channel, does not increase the decay rate or lead to new decaychannels. Continuous photon number parity measurements should therefore correctfor such loss events and protect the qubit while the operation is performed. Thesesimulations correspond to the master equation:˙ ρ = − i(cid:15) X [ a + a † , ρ ] + (cid:15) [ a † − a , ρ ] + κ D [ a ] ρ + κ D [ a ] ρ. We take (cid:15) X = 0 and (cid:15) X = κ /
20 respectively in Figures 6(a) and (b) and κ = κ /
200 for both plots. As can be seen through these plots, the decay rate remainsthe same in absence or presence of the two-photon driving field ensuring the arbitraryrotation around the X -axis. Additionaly, the probability of having more than one jumpduring the operation time remains within the range of 1%, indicating that with suchparameters one would not even need to perform photon-number parity measurementsduring the operation and that a measurement after the operation would be enough toensure a significant improvement in the coherence time. Single-qubit π/ -rotation around Z -axis. In order to show that the Kerr effect canbe applied in a fault-tolerant manner to perform such a single-qubit operation, we applysome of the arguments of the supplemental material of [1]. We need to consider the effectof photon loss events on the logical qubit during such an operation.We note first that the unitary generated by the Kerr Hamiltonian does not modifythe 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 theKerr effect. Now, let us assume that a jump occurs at time t during the operation: thestate after the jump is given by a e itχ Kerr ( a † a ) | ψ (cid:105) = e itχ Kerr a † a e itχ Kerr ( a † a ) a | ψ (cid:105) , where we have applied the commutation relation a f ( a † a ) = f ( a † a + I ) a , f being anarbitrary analytic function. This means that up to a phase space rotation e i tχ Kerr a † a ,the effect of a photon jump event commutes with the unitary generated by the KerrHamiltonian. Assuming much faster parity measurements than the Kerr dynamics and7 (a) π / Ω X ) Fidelity to | ψ >Fidelity to a| ψ >/||a | ψ >||Fidelity to a | ψ >/||a | ψ >||Fidelity to a | ψ >/||a | ψ >|| (b) π / Ω X ) Fidelity to | ψ (t)>Fidelity to a| ψ (t)>/||a | ψ (t)>||Fidelity to a | ψ (t)>/||a | ψ (t)>||Fidelity to a | ψ (t)>/||a | ψ (t)>|| Figure 6.
Decay of the unprotected qubit (no photon number parity measurements)encoded in the 4-cat scheme due to single-photon loss channel. (a) thequbit is initialized in the state | ψ (cid:105) = |C (0 mod α (cid:105) and no gate is appliedon the qubit. The decoherence due to the single-photon loss channel leadsto a decay in the fidelity with respect to the initial state and creates amixture of this state with the three other states a | ψ (cid:105) / (cid:107) a | ψ (cid:105)(cid:107) = |C (3 mod α (cid:105) , a | ψ (cid:105) / (cid:107) a | ψ (cid:105)(cid:107) = |C (2 mod α (cid:105) and a | ψ (cid:105) / (cid:107) a | ψ (cid:105)(cid:107) = |C (1 mod α (cid:105) . (b) in presenceof the squeezing Hamiltonian performing the X θ operation, this decoherencerate remains similar and mixes the desired state | ψ ( t ) (cid:105) = cos(Ω X t ) |C (0 mod α (cid:105) − i sin(Ω X t ) |C (2 mod α (cid:105) with the states a | ψ ( t ) (cid:105) / (cid:107) a | ψ ( t ) (cid:105)(cid:107) = cos(Ω X t ) |C (3 mod α (cid:105) − i sin(Ω X t ) |C (1 mod α (cid:105) , a | ψ ( t ) (cid:105) / (cid:107) a | ψ ( t ) (cid:105)(cid:107) = cos(Ω X t ) |C (2 mod α (cid:105) − i sin(Ω X t ) |C (0 mod α (cid:105) and a | ψ ( t ) (cid:105) / (cid:107) a | ψ ( t ) (cid:105)(cid:107) = cos(Ω X t ) |C (1 mod α (cid:105) − i sin(Ω X t ) |C (3 mod α (cid:105) . The photonjumps inducing such mixing of the quantum states are however tractable throughcontinuous photon number parity measurements. keeping track of both the number of parity jumps p and the times of their occurrences { t k } pk =1 , the state after the operation is fully known. In particular, the four-componentSchr¨odinger cat state is rotated in phase space by an angle of 2( (cid:80) pk =1 t k ) χ Kerr . Wecan take this phase space rotation into account by merely changing the phase of thefour-photon drive (cid:15) in the four-photon process.
4. Towards an experimental realization within a circuit QED framework
In this subsection, we propose an architecture based on Josephson circuits whichimplements the two photon driven dissipative process. Using the coupling of cavitymodes to a Josephson junction (JJ), single photon dissipation, and coherent drives, weaim to produce effective dynamics in the form of Eq. (1). These are the same tools used inthe Josephson Bifurcation Amplifier (JBA) to produce a squeezing Hamiltonian [25] andhere we will show that, by selecting a particular pump frequency, we can achieve a two8
INPUTOUTPUT ω p ω b ReadoutCavity (b)Low Q
JJMemory Cavity (a)High Q ~ Figure 7.
Proposal for a practical realization of the two photon driven dissipativeprocess. Two cavities are linked by a small transmission line in which a Josephsontunnel junction is embedded. This element provides a non-linear coupling between themodes of these two cavities. A pump tone at frequency ω p is applied to the readoutcavity. If we set ω p = 2˜ ω a − ˜ ω b (˜ ω a and ˜ ω b are the shifted frequencies of the modes a and b in the presence of all couplings and the pump), we select an interaction term ofthe form a b † + c.c., where a and b are the annihilation operators for the fundamentalmodes of the cavities. Combining this interaction with a drive and strong single-photondissipation of mode b leads to the desired dynamics for mode a of the form Eq. (1).In this way, quantum information can be stored and protected in mode a . photon driven dissipative process. Furthermore, in the next subsection, we show thatby choosing adequate pump frequencies, we may engineer the interaction terms neededto perform the logical gates described in subsections 3.1, 3.2 and 3.3. An architecturesuitable for the four photon driven dissipative process is subject to ongoing work.The practical device we are considering is represented in Fig. 7. Two cavities arelinked by a small transmission line in which a Josephson Junction is embedded. Thisprovides a non-linear coupling between the modes of these two cavities [18, 20]. TheHamiltonian of this device is given by [26] H = (cid:88) k (cid:126) ω k a † k a k − E J cos (cid:18) Φ φ (cid:19) , Φ = (cid:88) k φ k ( a k + a † k ) , (11)where E J is the Josephson energy, φ = (cid:126) / e is the reduced superconducting fluxquantum, and φ p is the zero point flux fluctuation for mode p of frequency ω p . Here weare only concerned by the dynamics of the fundamental modes of the two cavities andwe assume that all other modes are never excited. We denote a and b the annihilationoperators of these two modes and ω a , ω b their respective frequencies. We assume that | Φ /φ |(cid:28) b with twofields: a weak resonant drive (cid:15) b ( t ) and a strong off-resonant pump (cid:15) p ( t ). The frequencies9of modes a and b are shifted by the non-linear coupling. The dressed frequencies arenoted ˜ ω a and ˜ ω b and we take (cid:15) b ( t ) = 2 (cid:15) b cos(˜ ω b t ) and (cid:15) p ( t ) = 2 (cid:15) p cos( ω p t ) with : ω p = 2˜ ω a − ˜ ω b . We place ourselves in a regime where rotating terms can be neglected and the remainingterms after the rotating wave approximation constitute the effective Hamiltonian1 (cid:126) H = g ( a b † + a † b ) − (cid:15) b ( b † + b ) + χ aa a † a ) + χ bb b † b ) + χ ab ( a † a )( b † b ) . (12)While the induced self-Kerr and cross-Kerr terms χ aa , χ bb and χ ab can be deduced fromthe Hamiltonian of Eq. (11) through the calculations of [26], one similarly finds g = (cid:15) p ω p − ˜ ω b χ ab / . More precisely, this model reduction can be done by going to a displaced rotating framein which the Hamiltonian of the pumping drive is removed. Next, one develops thecosine term in the Hamiltonian of Eq. (11) up to the fourth order and removes thehighly oscillating terms in a rotating wave approximation.Physically, the pump tone (cid:15) p allows two photons of mode a to convert to a singlephoton of mode b , which can decay through the lossy channel coupled to mode b . Thedrive tone (cid:15) b inputs energy into mode b , which can then be converted to pairs of photonin mode a . The last three terms in Eq. (12) are the Kerr and cross-Kerr couplingsinherited from our proposed architecture. Although these are parasitic terms, we showthrough numerical simulations that their presence does not deteriorate our scheme.Taking into account single-photon decay of the mode b , the effective masterequation is given by:˙ ρ = − i (cid:126) [ H , ρ ] + κ b D [ b ] ρ . (13)Neglecting the Kerr and cross Kerr terms and assuming that g , (cid:15) b (cid:28) κ b , weadiabatically eliminate mode b [27, 7] and find a reduced dynamics for mode a ofthe form of Eq. (1) where (cid:15) = 2 (cid:15) b g κ b , κ = 4 g κ b and α = (cid:113) (cid:15) b /g . One can check the validity of this model reduction by comparing the numericalsimulation of Eq. (1) to the master equation Eq. (13). Fixing κ b = 1 ∗ we take χ aa = 0 . , χ bb = 0 . , χ ab = 0 .
033 and (cid:15) p / (˜ ω b − ω p ) = 3, and hence g = 0 . (cid:15) b = 4 g (to fix the average number of photons in the target cat to 4). In Figure 8,we compare the fidelity to the target cat state of solutions of Eq. (13) (blue solid line)and solutions of Eq. (1), starting in vacuum. The two curves both converge to a fidelityclose to one, which indicates that the steady state of Eq. (13) is hardly affected by thepresence of Kerr and cross Kerr terms and by the finite ratio of g , (cid:15) b to κ b . ∗ We have intentionally avoided to provide the units to only focus on the separation of time-scales;however, all these parameters could be considered in the units of 2 π × MHz and they will be within thereach of current circuit QED setups. Time in units of κ b F i de li t y t o t a r ge t c a t s t a t e Two modes modelOne mode reduced model
Figure 8.
Numerical simulation of Eq. (13) (full blue line) and Eq. (1) (dashedred line). We represent the fidelity of the state w.r.t the state |C + α (cid:105) , where N is anormalization factor, and α = (cid:113) (cid:15) b /g ph . The dashed and full curves have comparableconvergence rates and converge to the same state. This indicates that the reducedmodel of Eq. (1) is a faithful representation of the complete model Eq. (13). The finitediscrepancy is due to the finite ratio between g ph , (cid:15) b and κ b , and the presence of nonzero Kerr and cross Kerr terms. IQ ω b1 ω b2 ω p1 ω p2 ω XX IQ ~ ~ Figure 9.
Architecture for coupling two qubits protected by the two-photon drivendissipative processes. Two modules composed of a pair of high and low cavitiesare connected through a central JJ. This JJ provides a nonlinear coupling betweenthe two storage modes a and a of each module. Adding a pump at frequency ω XX = (˜ ω a − ˜ ω a ) / a a † + c.c , thus allowingfor the entangling gate detailed in Sec. 3.2 Rotations of arbitrary angles around the X -axis for the logical qubit {|C + α (cid:105) , |C − α (cid:105)} : simply adding a drive of amplitude (cid:15) a resonant with mode a will add a term proportionalto (cid:15) ∗ a a + (cid:15) a a † in Eq. (1). In the limit where | (cid:15) a | (cid:28) κ ph , this will induce coherentoscillation between the two states around the Bloch sphere’s X -axis, as explained inSec. 3.1. Entangling gate between two logical bits: we propose the architecture of Fig. 9 to coupletwo qubits protected by a two photon driven dissipative process. Two modules, eachcomposed of a pair of high and low Q cavities, are coupled through a JJ embedded ina waveguide connecting the two high Q cavities. This JJ provides a nonlinear coupling,which, together with a pump at frequency ω ZZ = (˜ ω a − ˜ ω a ) /
2, induces an interactionof the form e iφ pump a a † + c.c . Such a term performs an entangling gate between twological qubits, as described in Sec. 3.2. π/ -rotation around Z -axis: as mentioned throughout the previous subsection, themere fact of coupling the cavity mode to a JJ induces a self-Kerr term on the cavitymode. As proposed in Sec. 3.3, this could be employed to perform a π/ Z -axis in a similar manner to [18]. One only needs to turn off all the pumping drivesand wait for π/χ aa . φ φ φ Φ ext L J L J L J L J LL L L Φ ext Φ ext Φ ext φ −−++ +− − XYZ φ φ φ ext L J L J L J L J LL L L ext Φ ext Φ ext φ + (a) (b) (c) Figure 10.
Josephson ring modulators (JRM) providing desired interactions betweenfield modes. (a)
JRM developed to ensure quantum limited amplification of a quantumsignal or to provide frequency conversion between two modes; The signal and idler arerespectively coupled to the X and Y modes, as represented in (c) and the pump driveis applied on the Z mode. (b) A modification of the JRM to ensure an interaction ofthe form Eq. (14). Such an interaction should allow us to achieve the driven dissipativefour-photon process without adding undesired Hamiltonian terms. (cid:126) H = g ( a b † + a † b ) + (cid:15) b ( b + b † ) . Taking into account the single-photon decay of the mode b of rate κ b such that g , (cid:15) b (cid:28) κ b , we can adiabatically eliminate the mode b and find a reduced dynamicsfor mode a of the form Eq. (4). The problem is therefore to engineer in an efficientmanner the Hamiltonian H .Indeed, the same architecture as in Fig. 7 together with a pump frequency of ω p = 4˜ ω a − ˜ ω b should induce an effective Hamiltonian term of the form g ( a b † + a † b ).One can easily observe this by developing the cosine term in Eq. (11) up to the sixthorder in Φ /φ and by applying a rotating wave approximation, leading to an effectivecoupling strength of g = E J (cid:126) (cid:15) p ω p − ˜ ω b φ a φ b φ . However, such an architecture also leads toother significant terms limiting the performance of the process. In particular, throughthe same sixth order expansion, one can observe an amplified induced Kerr effect on themode a : χ pumped aa ( a † a ) with χ pumped aa = E J (cid:126) (cid:15) p ( ω p − ˜ ω b ) φ a φ b φ = (cid:15) p ω p − ˜ ω b g .Inspired by the architecture of the Josephson ring modulator [28, 29], whichensures an efficient three-wave mixing, we propose here a design which should inducevery efficiently the above effective Hamiltonian while avoiding the addition of extraundesirable interactions. The Josephson ring modulator (Fig. 10(a)) provides a couplingbetween the three modes (as presented in Fig. 10(c)) of the form H JRM = E L Φ X φ + Φ Y φ + Φ Z φ ) − E J (cid:20) cos Φ X φ cos Φ Y φ cos Φ Z φ cos Φ ext φ + sin Φ X φ sin Φ Y φ sin Φ Z φ sin Φ ext φ (cid:21) , where E L = φ /L , Φ X,Y,Z = φ X,Y,Z ( a X,Y,Z + a † X,Y,Z ) and Φ ext /φ is the dimensionlessexternal flux threading each of the identical four loops of the device. Furthermore, thethree spatial mode amplitudes Φ X = φ − φ , Φ Y = φ − φ and Φ Z = φ + φ − φ − φ are gauge invariant orthogonal linear combinations of the superconducting phases of thefour nodes of the ring (Fig. 10(c)).In the same manner the design of Fig. 10(b), for a dimensionless external flux ofΦ ext /φ = π/ ext /φ = 3 π/ H (cid:48) JRM = E L Φ X φ + Φ Y φ + Φ Z φ ) − √ E J sin Φ X φ sin Φ Y φ (cid:20) sin Φ Z φ + cos Φ Z φ (cid:21) . (14)Similarly to [29], 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 modeto the low-Q b mode, and we drive the X mode by a pump of frequency 4˜ ω a − ˜ ω b (˜ ω a and˜ ω b are the effective frequencies of the modes a and b ). By expanding the Hamiltonian3of Eq. (14) up to sixth order terms in φ = ( Φ X φ , Φ Y φ , Φ Z φ ), the only non-rotating termwill be of the form H eff = − √ √ n pump E J φ Z φ Y φ X φ ( e iφ pump a b † + e − iφ pump a † b ) , where φ pump is the phase of the pump drive and n pump is the average photon number ofthe coherent state produced in the pump resonator [30]. Acknowledgments
This research was supported by the Intelligence Advanced Research Projects Activity(IARPA) W911NF-09-1-0369 and by the U.S. Army Research Office W911NF-09-1-0514.MM acknowledges support from the Agence National de Recherche under the projectEPOQ2 ANR-09-JCJC-0070. ZL acknowledges support from the NSF DMR 1004406.VVA acknowledges support from the NSF Graduate Research Fellowships Program. LJacknowledges support from the Alfred P. Sloan Foundation, the Packard Foundation,and the DARPA Quiness program.
Appendix A. Asymptotic behavior of the two- and four-photon processes
Appendix A.1. Asymptotic state for arbitrary initial state of the two-photon process
As stated in Sec. 2.1, all initial states evolving under the two-photon driven dissipativeprocess from Eq. (1) will exponentially converge to a specific (possibly mixed)asymptotic density matrix defined on the Hilbert space spanned by the two-componentSchr¨odinger cat states {|C + α (cid:105) , |C − α (cid:105)} with α = | α | e iθ α . In order to characterize the Blochvector of this asymptotic density matrix ρ ∞ [Eq. (2)], it is sufficient to determinethree degrees of freedom: the population of one of the cats ( c ++ = (cid:104)C + α | ρ ∞ |C + α (cid:105) )and the complex coherence between the two ( c + − = (cid:104)C − α | ρ ∞ |C + α (cid:105) ). There existconserved quantities J ++ , J + − corresponding to these degrees of freedom [31] such that c ++ = tr { J † ++ ρ (0) } and c + − = tr { J † + − ρ (0) } for any initial state ρ (0). These conservedquantities are given by J ++ = ∞ (cid:88) n =0 | n (cid:105)(cid:104) n | (A.1) J + − = (cid:115) | α | sinh (2 | α | ) ∞ (cid:88) q = −∞ ( − q q + 1 I q ( | α | ) J ( q )+ − e − iθ α (2 q +1) , (A.2)where I q ( . ) is the modified Bessel function of the first kind and J ( q )+ − = (cid:0) a † a − (cid:1) !!( a † a + 2 q )!! J ++ a q +1 q ≥ J ++ a † | q |− ( a † a )!!( a † a + 2 | q | − q < . n !! = n × ( n − J evolves under Eq. (1) in the Heisenbergpicture, i.e., ˙ J = 12 κ (cid:0)(cid:2) α (cid:63) a − α a † , J (cid:3) + 2 a † J a − a † a J − J a † a (cid:1) . (A.3)For the case of Eq. (A.1), it is easy to see that ˙ J ++ = 0 since the two-photon systempreserves photon number parity and J ++ is merely the positive parity projector. The off-diagonal quantity from Eq. (A.2) is an extension of J (0)+ − , the corresponding conservedquantity for the non-driven ( α = 0) dissipative two-photon process (first calculatedin [32]; see also [31]). Each J ( q )+ − term in the sum for J + − evolves under Eq. (A.3) as˙ J ( q )+ − = 12 κ (2 q + 1) (cid:104) α J ( q − − − α (cid:63) J ( q +1)+ − − qJ ( q )+ − (cid:105) . The above equations of motion for J ( q )+ − mimic the recurrence relation | α | (cid:2) I q − (cid:0) | α | (cid:1) − I q +1 (cid:0) | α | (cid:1)(cid:3) + 2 qI q (cid:0) | α | (cid:1) = 0satisfied by the Bessel functions in J + − and both can be used to verify that J + − isindeed conserved. The square root in front of the sum for J + − is chosen such thattr { J † + − |C + α (cid:105)(cid:104)C − α |} = 1, which can be verified using (cid:104)C − α | J ( q ) † + − |C + α (cid:105) = (cid:115) | α | sinh (2 | α | ) I q (cid:0) | α | (cid:1) e iθ α (2 q +1) (A.4)as well as the identity (see Eq. (5.8.6.2) from [33]) ∞ (cid:88) q = −∞ ( − q q + 1 I q ( | α | ) I q ( | α | ) = sinh (2 | α | )2 | α | . (A.5) Appendix A.2. Asymptotic state for an initial coherent state of the two-photon process
The conserved quantities { J ++ , J + − } are sufficient to calculate the population c ++ = (cid:104)C + α | ρ ∞ |C + α (cid:105) and coherence c + − = (cid:104)C − α | ρ ∞ |C + α (cid:105) of the asymptotic state for any initialstate ρ (0). Letting ρ (0) = | β (cid:105)(cid:104) β | with β = | β | e iθ β , the respective terms are c ++ = tr { J † ++ ρ (0) } = 12 (1 + e − | β | ) (A.6) c + − = tr { J † + − ρ (0) } = iαβ (cid:63) e −| β | (cid:112) | α | ) (cid:90) πφ =0 dφe − iφ I (cid:0)(cid:12)(cid:12) α − β e iφ (cid:12)(cid:12)(cid:1) . (A.7)Eq. (A.6) is the same simple result as the non-driven case (e.g. Eq. (3.22) in [31]). Toderive Eq. (A.7), we first apply Eq. (A.2) to obtain the sum c + − = √ αβ (cid:63) e −| β | (cid:112) sinh (2 | α | ) ∞ (cid:88) q = −∞ ( − q q + 1 I q (cid:0) | α | (cid:1) I q (cid:0) | β | (cid:1) e i q ( θ α − θ β ) . (A.8)5This sum is convergent because the sum without the 2 q + 1 term is an addition theoremfor I q (Eq. (5.8.7.2) from [33]). To put the above into integral form, we use the identity(derivable from the addition theorem) I q (cid:0) | α | (cid:1) I q (cid:0) | β | (cid:1) = 12 π (cid:90) πφ =0 dφe iq ( φ + π ) I (cid:0)(cid:12)(cid:12) | α | − | β | e iφ (cid:12)(cid:12)(cid:1) . Plugging in the above identity into Eq. (A.8), interchanging the sum and integral(possible because of convergence), evaluating the sum (which is a simple Fourier series),and performing a change of variables obtains Eq. (A.7).When α = 0, Eq. (A.7) reduces to Eq. (14) from [32]. Assuming real α andusing Eq. (5.8.1.15) from [33], one can calculate limits for large | β | along the real andimaginary axes in phase space:lim β →∞ c + − = 12 erf( √ | α | ) √ − e − | α | | α |→∞ −→
12 and lim β → i ∞ c + − = − i
12 erfi( √ | α | ) √ e | α | − | α |→∞ −→ , where erf( . ) and erfi( . ) are the error function and imaginary error function, respectively.Both limits analytically corroborate Fig. 2 and show that the two-photon system issimilar to a classical double-well system in the combined large α, β regime. Appendix A.3. Influence of dephasing on the two-photon process
Equation (2) implies that while the states |C ± α (cid:105) define the basis of our logical qubit,the expectation values of the conserved quantities determine the state of the qubit(or equivalently its Bloch vector). Now let’s consider adding the photon dephasingdynamics κ φ D (cid:2) a † a (cid:3) to Eq. (1) and estimate what would happen to the qubit basiselements and more importantly the conserved quantities (determining the effect on theencoded information).Since dephasing preserves parity, the positive parity projector J ++ remainsconserved and the corresponding population of the cat-state c ++ thus remainsunchanged. The quantity representing the coherence ( J + − ) to first order decaysexponentially at a rate proportional to κ φ . Noting that the population of the states |± Z (cid:105) = |C ± α (cid:105) are conserved, this means that photon dephasing induces only phase-flip errors on our logical qubit. However, this phase-flip rate is itself exponentiallysuppressed with increasing the number of photons in the cat state | α | . To see this,we evaluate the first-order perturbative correction due to dephasing on the asymptoticmanifold. Since |C + α (cid:105)(cid:104)C − α | and J + − are right and left eigenvectors of the super-operatorfrom Eq. (1) and since dephasing preserves parity, the first order decay rate γ phase-flip is γ phase-flip = κ φ tr { J † + − D (cid:2) a † a (cid:3) |C + α (cid:105)(cid:104)C − α |} = κ φ (cid:104)C − α |D (cid:2) a † a (cid:3) J † + − |C + α (cid:105) . In the above, we have re-arranged for the adjoint of D to act on J + − instead of |C + α (cid:105)(cid:104)C − α | and used D † (cid:2) a † a (cid:3) = D (cid:2) a † a (cid:3) because a † a is Hermitian. Since J ( q )+ − consist of matrixelements | n (cid:105)(cid:104) n + 1 + 2 q | for n = 0 , , ... , each term in the sum for J + − has the simpleequation of motion D (cid:2) a † a (cid:3) J † ( q )+ − = − κ φ (2 q + 1) J † ( q )+ − . |𝛼| (a) Two-photon process (b) Four-photon process 𝛾 p h a s e − f l i p / 𝜅 𝜙 𝛾 p h a s e − f l i p / 𝜅 𝜙 |𝛼| Figure A1. (a)
Plot versus | α | of the eigenvalue γ phase-flip (scaled by κ φ /
2) ofthe evolution operator of Eq. (3) associated with the decay of J + − ( J + − ( t ) = J + − (0) e − γ phase-flip t ). The plot includes the analytical estimate from Eq. (A.9) aswell as two numerical plots for various κ φ /κ ph . One can see that the eigenvalueexponentially converges to zero with increasing the photon number in the cat state | α | . (b) Similar plot for Eq. (4) with the addition of κ φ D (cid:2) a † a (cid:3) , i.e., the eigenvalueof the evolution operator associated to the decay of J encoding the coherence term |C (0 mod α (cid:105)(cid:104)C (2 mod α | of the four-photon process qubit. The phase-flip rate is nowscaled by 2 κ φ which represents the rate for the case of α = 0 . The subsequent evaluation of the trace and sum results in the rate γ phase-flip = − κ φ | α | sinh(2 | α | ) (A.9)given in Sec. 2.1. We have numerically confirmed [Fig. A1(a)] that this is indeed thefirst-order correction to the asymptotic manifold. In the Figure, we plot versus | α | themagnitude of the eigenvalue of the evolution operator from Eq. (3) associated with thedecay rate of J + − (which is precisely the phase-flip rate γ phase-flip ). For small values of κ φ /κ , the numerical result approaches our analytical estimate.It is worth noting that under the effect of dephasing, the cat-states that comprisethe logical qubit basis elements will acquire a small random phase ( |C ± α (cid:105) becomes |C ± αe iφ (cid:105) where φ is a small random phase). Indeed, as an ensemble-averaged result, one canobserve that each of the two-dimensional Gaussian peaks that represent the cat statein the phase space slightly smear. However, this smearing merely changes the structureof our qubit basis elements and does not affect the encoded quantum information(represented by J ++ and J + − ). Appendix A.4. Asymptotic behavior of the four-photon process
The asymptotic manifold of the four-photon process from Eq. (4) is given by densitymatrices defined on the four-dimensional Hilbert space spanned by {|C ( µ mod α (cid:105)} (with µ = 0 , , , {|C (0 mod α (cid:105) , |C (2 mod α (cid:105)} comprising our logical qubit’s basis. Thecorresponding conserved quantity for the populations of |C (0 mod α (cid:105) and |C (2 mod α (cid:105) is once again identical to the non-driven case [31], J = (cid:80) ∞ n =0 | n (cid:105)(cid:104) n | . While ananalytical expression for the other conserved quantity J remains to be found, here weprovide a numerical analysis of the influence of the photon dephasing on the four-photonprocess.Fig. A1(b) shows a plot similar to Fig. A1(a), but now for γ phase-flip of the logicalqubit of the four-photon process. With the exception of a slight delay in the exponentialsuppression of the induced phase-flip rate, one observes that this suppression is almostidentical to the case of the two-photon process. References [1] Z. Leghtas, G. Kirchmair, B. Vlastakis, R.J. Schoelkopf, M.H. Devoret, and M. Mirrahimi.Hardware-efficient autonomous quantum memory protection. Phys. Rev. Lett., 111, 2013.[2] P. Shor. Scheme for reducing decoherence in quantum memory. Phys. Rev. A, 52:2493–2496,1995.[3] A. Steane. Error correcting codes in quantum theory. Phys. Rev. Lett, 77(5), 1996.[4] M.A. Nielsen and I.L. Chuang. Quantum Computation and Quantum Information. CambridgeUniversity Press, 2000.[5] Z. Leghtas, G. Kirchmair, B. Vlastakis, M.H. Devoret, R.J. Schoelkopf, and M. Mirrahimi.Deterministic protocol for mapping a qubit to coherent state superpositions in a cavity. Phys.Rev. A, 87, 2013.[6] D.I. Schuster, A.A. Houck, J.A Schreier, A. Wallraff, J.M. Gambetta, A. Blais, L. Frunzio, J. Majer,B. Johnson, M.H. Devoret, S.M. Girvin, and R. J. Schoelkopf. Resolving photon number statesin a superconducting circuit. Nature, 445:515–518, 2007.[7] H.J. Carmichael and M. Wolinsky. Quantum noise in the parametric oscillator: From squeezedstates to coherent-state superpositions. Phys. Rev. Lett., 60:1836–1839, 1988.[8] L. Krippner, W.J. Munro, and M.D. Reid. Transient macroscopic quantum superposition statesin degenerate parametric oscillation: Calculations in the large-quantum-noise limit using thepositive P representation. Phys. Rev. A, 50:4330–4338, 1994.[9] E. Hach III and C.C. Gerry. Generation of mixtures of Schr¨odinger-cat states from a competitivetwo-photon process. Phys. Rev. A, 49:490–498, 1994.[10] L. Gilles, B.M. Garraway, and P.L. Knight. Generation of nonclassical light by dissipative two-photon processes. Phys. Rev. A, 49:2785–2799, 1994.[11] M.J. Everitt, T.P. Spiller, G.J. Milburn, R.D. Wilson, and A.M. Zagoskin. Cool for Cats. 2012.arXiv:1212.4795.[12] A. Gilchrist, K. Nemoto, W.J. Munro, T.C. Ralph, S. Glancy, S.L. Braunstein, and G.J. Milburn.Schr¨odinger cats and their power for quantum information processing. J. Opt. B: QuantumSemiclass. Opt., 6(8), 2004.[13] P. Facchi and S. Pascazio. Quantum Zeno subspaces. Phys. Rev. Lett., 89(8):080401, 2002.[14] J.-M. Raimond, C. Sayrin, S. Gleyzes, I. Dotsenko, M. Brune, S. Haroche, P. Facchi, andS. Pascazio. Phase space tweezers for tailoring cavity fields by quantum zeno dynamics. Phys.Rev. Lett., 105:213601, 2010.[15] J.M. Raimond, P. Facchi, B. Peaudecerf, S. Pascazio, C. Sayrin, I. Dotsenko, S. Gleyzes, M. Brune,and S. Haroche. Quantum Zeno dynamics of a field in a cavity. Phys. Rev. A, 86:032120, 2012.[16] D.A. Lidar, I.L. Chuang, and K.B. Whaley. Decoherence-free subspaces for quantum computation.Phys. Rev. Lett., 81(12):2594–2597, 1998.8