Universal Gate Set for Continuous-Variable Quantum Computation with Microwave Circuits

We provide an explicit construction of a universal gate set for continuous-variable quantum computation with microwave circuits. Such a universal set has been first proposed in quantum-optical setups, but its experimental implementation has remained elusive in that domain due to the difficulties in engineering strong nonlinearities. Here, we show that a realistic microwave architecture allows to overcome this difficulty. As an application, we show that this architecture allows to generate a cubic phase state with an experimentally feasible procedure. This work highlights a practical advantage of microwave circuits with respect to optical systems for the purpose of engineering non-Gaussian states, and opens the quest for continuous-variable algorithms based on a few repetitions of elementary gates from the continuous-variable universal set.

Introduction.-The ability to control and manipulate quantum systems has reached an unprecedented level in the past decades [1,2]. Quantum computation stems as one of the most promising potential applications of this enhanced controllability of quantum systems [3][4][5]. As an alternative to the use of two-level systems for quantum information encoding, continuous-variable (CV) architectures have emerged, where the underlying hardware consists in quantized radiation, either with optical devices [1], microwaves [6], or in cavity optomechanics [7].
The theoretical setting for quantum computation with CV-based architectures has been laid down in a seminal paper by Lloyd and Braunstein [8]. There, universal quantum computation in CVs is defined as the ability of implementing any evolution corresponding to Hamiltonians that are arbitrary polynomials in the mode quadratures. The basic ingredients to achieve CV universality are a set of Gaussian gates and a single non-Gaussian gate, which can be chosen arbitrarily among the polynomials of degree higher than 2 in the quadratures of the quantized modes. The ability to perform arbitrary sequences of these elementary quantum gates ensures universal quantum computation [8].
Since then, the community of quantum opticians has devoted considerable theoretical as well as experimental effort toward developing the building blocks for CV universality. In this framework, the experimental challenge consists in achieving a non-Gaussian operation. Experimental effort has focused on photon subtraction [9][10][11][12][13][14][15][16], photon detection [17][18][19] and on the use of ancillary low photon-number states combined with homodyne detection [20,21] as ways to achieve probabilistic non-Gaussian transformations, resulting however in low success probabilities [22] and limited versatility. In particular, much of the effort has been dedicated to gen-erate a so-called "cubic phase state" [15,16], as availability of this state allows for engineering a cubic phase gate [15,16,[22][23][24][25], and thereby promotes the set of Gaussian operations to a universal set [8,26]. Having at disposal such a gate would allow in particular to engineer Gottesman-Kitaev-Preskill (GKP) states [27,28], which have been shown to yield fault-tolerance [27,29,30]. Despite these efforts, the generation of a cubic phase state, as well as the implementation of a cubic phase gate, have remained elusive, due to the weakness of the nonlinearities that are available in the optical regime. Alternatively, deterministic nonlinear gates in strongly coupled quantum electrodynamics (QED) setups [31] as well as the dissipative stabilization of cubic phase states in optomechanical systems have been proposed [32,33].
In microwave quantum optics, commonly referred to as circuit QED (cQED), nonlinear photon-photon interactions are made possible via the strong nonlinearity provided by Josephson junctions (JJs) [34]. The anharmonicity that JJs induce on linear resonators is the basis for the design of different types of superconducting artificial atoms, yielding quantum bits [35]. Direct mediation of interactions without the need of an artificial atom is also possible. This mediation combined with external current or magnetic flux modulations leads to parametric processes that have been employed for the amplification of microwave signals, single-and two-mode squeezing operations [36][37][38], the realization of the dynamical Casimir effect [39], and more recently, the generation of Schrödinger cat states [40][41][42][43] as well as GKP states [44]. These parametric processes also allow for engineering further customizable higher-order photon-photon interactions [45,46]. However, the CV notion of universality has not been studied thoroughly in these systems yet, nor has it ever been achieved experimentally. In this paper, we bridge between the optical and microwave approaches, and show explicitly that parametrically modulated microwave circuits allow for implementing a universal gate set for CV quantum computation, in the sense of the CV universality notion recalled above [8]. As an application, we show that a state-of-the-art microwave platform allows for the generation of a cubic phase state -a long sought-after aspiration for the quantum optics community.
Universal gate set in Continuous-Variables.-A universal gate set for CV quantum computation is provided by the following operations [8]: are the quadrature operators for mode k satisfying the canonical commutation relation [q k ,p l ] = iδ kl (from here on = 1 and we drop the mode index if only a single mode is relevant).
The operations in Eq. (1) excluding e iq 3 γ represent respectively the displacement, squeezing, beam-splitter and Fourier-transform operators (where s i ∈ R for all i), and are universal for Gaussian operations, i.e. they allow implementing any arbitrary quadratic Hamiltonian. Addition of the cubic phase gate e iq 3 γ , where one value of the cubicity γ is sufficient [8,25,28,[47][48][49], allows promoting the Gaussian set of operations to universal quantum computation. Indeed, following [8], if we can apply the HamiltoniansÂ andB for a time δt, then we can approximate the evolution under their commutator for a time The commutation of a polynomial inq andp withq and p themselves reduces the order of the polynomial by at least 1, commutation with quadratic Hamiltonians never increases the order, and commutation with a polynomial of order 3 or higher increases the order by at least 1. Therefore, judicious commutation of the Gaussian operations with an applied Hamiltonian of order 3 or higher allows the construction of arbitrary Hermitian polynomials of any order inq andp. As a consequence, the set in Eq. (1) is universal. Direct application of Eq. (2) may result in a significant number of operations in order to approximate a desired Hamiltonian. More efficient schemes involving nested operations as well as numerical optimization may provide shorter gate sequences for achieving the same approximate Hamiltonian evolution [50][51][52]. However, the approach described above will suffice for our purpose of establishing a proof of principle for universality with microwave circuits. We are now going to introduce a cQED architecture that is instrumental to implement the universal gate set in Eq. (1).
Microwave circuit for CV universal quantum computation.-Interactions between microwave pho-tons in superconducting circuits can be realized by coupling the modes of interest to Josephson junctions acting as non-linear, low-loss inductive elements with potential energy U (ϕ) = E J (1 − cos(ϕ)), where ϕ is the superconducting phase across the junction and E J is the Josephson energy [34,35]. When Josephson junctions are arranged in a loop configuration, as in a dc superconducting quantum interference device (SQUID), the Josephson potential energy depends on the magnetic flux threading the loop, allowing for in-situ static tuning of the potential, as well as its parametric modulation [53,54]. Photon-photon interactions have been demonstrated in resonators terminated by dc-SQUIDs by introducing suitable drives to resonantly select specific n-photon processes from the Josephson potential. This potential has even parity for both a single junction and a symmetric SQUID, resulting in mixing processes with an even number of photons, such as four-wave mixing [55]. As it has been recently demonstrated, an asymmetry between the SQUID junctions introduces an odd contribution to the potential, enabling three-wave mixing (as well as higher-order odd photon interactions) [45,46]. However, the even contribution still results in undesired terms, most notably, self-and cross-Kerr interactions, that contain an equal number of creation and annihilation operators and are consequently resonant (non-rotating) in any reference frame. To overcome this challenge, a different arrangement of Josephson junctions in a loop, known as the Superconducting Nonlinear Asymmetric Inductive eLement (SNAIL), was recently introduced [56][57][58] in the context of Kerr-free three-wave mixing and parametric amplification. Here we propose a tunable resonator design based on a SNAIL, and show that by a two-tone flux modulation we can resonantly select all processes comprising the cubic interaction (â +â † ) 3 as we will detail later. The SNAIL loop consists of n large Josephson junctions in parallel with a single smaller junction with Josephson energies E J and αE J , respectively (Fig. 1a). By threading an external magnetic flux Φ ext through the loop, the inductive energy of the SNAIL circuit can be written as [56] where ϕ is the superconducting phase across the small junction, ϕ ext = 2πΦ ext /Φ 0 is the reduced applied magnetic flux and Φ 0 is the magnetic flux quantum. The advantage of the SNAIL circuit over traditional SQUIDs is that through its design, resulting in specific parameters n, α, in addition to the external flux Φ ext , its potential landscape around a minimum ϕ min can be tailored. Then we can Taylor expand (3) around this value as The three-wave mixing capability of this device corresponds to setting the coefficient of the fourth-order term identical to zero (c 4 = 0) [56], so that the leading nonlinear term is the cubic one. Our proposed architecture is a quarter wavelength transmission line resonator of length d terminated with a SNAIL loop in one end (Fig. 1b). We describe the state of the resonator in terms of the superconducting phase field ϕ(x, t). For our purposes, the phase at the position of the SNAIL ϕ S = ϕ(d, t) is assumed to be small. This means that the current flowing through the JJs is smaller than their corresponding critical currents. If this holds, the Lagrange equations of motion which determine the normal modes of the resonator -SNAIL system can be linearized, allowing one to obtain the system Hamiltonian. The nonlinear corrections are reintroduced perturbatively [49]. The resonator is also weakly coupled to an input transmission line (Fig. 1b), which allows driving the resonator field.
We chose the SNAIL parameters n, α and Φ ext in order to operate the device free of fourth-order interactions. A similar device has already been exploited for parametric amplification [57,58]. Here, however, we propose to endow this device of flux tunability in order to fully exploit the third-order interaction. For this, we apply a periodically modulated external flux of the form where ϕ dc ext is the static part of the flux and ϕ ac ext (t) is the time-modulation. The modulation ϕ ac ext (t) satisfies |ϕ ac ext (t)| 1. This is required in order to remain near the equilibrium point and to not excite higher nonlinear processes. We consider the SNAIL potential up to second  [56] we represent this subcircuit by a snail-like symbol. b Sketch of our proposed architecture. The quarter wavelength transmission line resonator is terminated into a SNAIL at the right end and capacitively coupled to an input transmission line at the left end through which microwave signals for control can be fed. The SNAIL potential can be tuned and modulated through an external flux line. order in ϕ ac ext (t). As customary, we follow the canonical quantization recipe for the resonator [49]. Upon quantization, the Hamiltonian describing our tunable resonator isĤ Due to the modulation of the potential around its minimum, we gain linear and quadratic contributions in addition to the cubic potential. Here ω r is the resonance frequency of the transmission line resonator modified by the presence of the SNAIL. The time-dependence of g 1 and g 3 is proportional to ϕ ac ext (t) and that of g 2 to ϕ ac ext (t) 2 . Notice that the perturbative treatment of the nonlinearity leads to the hierarchy ω r |g i |. Engineering of Gaussian gates.-We start our demonstration of CV universality by showing that this architecture is capable of implementing the Gaussian operations in (1). For this, a modulation of the flux is not required, i.e. ϕ ac ext (t) = 0 in Eq. (4). In this case, the Hamiltonian (5) reduces toĤ = ω râ †â + g dc 3 â +â † 3 , with the coupling g dc 3 depending only on the microscopic parameters of the circuit as well as the static external flux. In order to engineer a squeezing operation, we apply an off-resonant microwave tone of frequency ω p = 2ω r through the input transmission line. As discussed in [58], in a frame rotating at the resonator frequency ω r the system is described by the effective Hamil-tonianĤ sq = − i 2 ξâ †2 − ξâ 2 where the parameter ξ depends on g dc 3 as well as on the amplitude of the drive. In particular, choosing ξ real allows us to obtain the squeezing operation in Eq. (1). This is the basis of SNAIL-based parametric amplification [57,58]. The Fourier transform e i π 4 (q 2 +p 2 ) follows trivially from the free evolution of the system, i.e., the evolution under the resonator Hamiltonian ω râ †â in the absence of any external modulation. As customary, a displacement operation is implemented by means of a microwave tone near resonance with modeâ. Finally, a tunable beam-splitter interaction can in principle be achieved by coupling two resonator-SNAIL units via a parametrically modulated dc-SQUID or a tunable gap qubit, or mediating a static nonlinear coupling via time-dependent drivings of both resonators [59][60][61].
Engineering a non-linear gate.-The power of our proposal relies on the realization of the interaction termq 3 which has been experimentally elusive so far. In order to engineer such a gate, we exploit the flux tunability of the SNAIL. In particular, we consider a two-tone modulation of the form ϕ ac ext (t) = λ [cos(ω r t) + cos(3ω r t)] , where λ 1 is a small modulation amplitude. This is justified by studying the cubic potential in Eq. (5) in the interaction picture with respect to the free resonator Hamiltonian ω râ †â . Because of the odd parity of the potential, there are no non-rotating contributions. The terms that are pure inâ ( †) rotate with frequency ∓3ω r while the mixed terms rotate with ±ω r . Thus, in order to select the full cubic interaction resonantly, the necessity to drive with two frequencies ω r and 3ω r arises. It must be pointed out that the drive at ω r also selects resonantly the linear and the quadratic terms in Eq. (5). However, in the Supplementary Material we show that for a realistic choice of parameters the quadratic term is sufficiently suppressed and can thus be neglected. This is not the case for the linear drive. However, its effect can be corrected via a displacement of the resonator field and thus we neglect it in the remainder of this letter.
Finally, following the above arguments and in the rotating frame, we isolate the desired cubic interaction where the coupling g ac 3 depends on the microscopic parameters of the circuit as well as on the modulation amplitude λ. For realistic parameters our theory predicts that it is possible to achieve g ac 3 /2π ≈ 0.5 MHz and ω r /2π ≈ 5.2 GHz while the coupling to (â +â † ) 4 is tuned to zero [49].
We expect the value of g ac 3 to be further increased by classical optimization of the circuit parameters, possibly including more SNAILs in an array configuration. Notice that ω r should be restricted such that the drive at 3ω r is sufficiently detuned from the plasma frequency of the JJs, which are typically on the order of a few ten of GHz [62]. Therefore, we have demonstrated that all of the operations in the set (1) can be implemented with our proposed architecture.
We emphasize that while universality is achieved within the gate set described above, many more gates are accessible through our proposal. For the goal of achieving universality, this does not matter. However, in a time where fault-tolerance has not yet been achieved and the number of gates that can be applied within the coherence time of the system is limited, having at disposal customizable high-order gates can lead to substantial advantages.
Generation of a cubic phase state.-We now address the generation of a cubic phase state |γ, r = e iγq 3 e r 2 (â †2 −â 2 ) |0 , where r is the real squeezing parameter, γ the cubicity of the cubic phase gate applied and |0 is the photon vacuum state [25]. Due to the weak coupling to the transmission line the main dissipation channel correspond to internal losses. We treat them within a Gorini-Kossakowski-Sudarshan-Lindblad master equation formalism with jump operatorL = √ κâ, where κ is the single photon loss rate. Evolving an initial squeezed state for a time t g with the Hamiltonian (7) results in the cubic phase state with cubicity γ = g ac 3 √ 8t g , where the factor √ 8 results from the normalization ofq. Fig. 2 shows the cubic phase state obtained from a master equation simulation [63] with κ/2π = 50 kHz (1/κ ≈ 3 µs). This corresponds to a quality factor Q of the order of 10 5 . As the initial state we chose a squeezed state with 6 dB of squeezing. After a time t g = 12 ns the obtained cubicity is γ ≈ 0.1, while the fidelity to the ideal state is 99.88 %.
The generated cubic phase state can be probed, e.g., by quantum state tomography. An established technique to do so uses a dispersively coupled qubit (not shown in Fig. 1b) for the parity readout of the resonator [64].
Conclusions.-In summary, we have proposed a microwave architecture that allows for the implementation of a universal gate set for continuous-variable quantum computation. Our architecture is based on a quarter wavelength transmission line resonator terminated by an array of Josephson junctions in a SNAIL configuration. The tunability of our device allows for engineering customized gates, and in particular the interactionq 3 , corresponding to a cubic phase gate. As an application, we have provided an experimentally realistic protocol for the generation of the cubic phase state, which is a resource state for CV quantum computation, and whose generation has not been experimentally achieved yet despite extensive effort undertaken with quantum optical setups.
On the one hand, our work opens the experimental quest for a cubic phase state with microwave circuits. Our proposal is within reach of current cQED technology in terms of resonator quality factors, that can be as high as 3×10 5 in 3D architectures [43], and the ability to tune the resonator field much faster than its corresponding lifetime, with pulse synthesis resolution within nanoseconds [65]. On the other hand, with the demonstration of the availability of a universal gate set based on microwave circuitry, we wish to sparkle the interest of the community working on CV quantum information, in optics as well as other systems, on the following question: what relevant quantum algorithms can be run with a universal gate set of CV operations, when a limited amount of gates can be sequentially implemented, and fault-tolerance is not required? Our work thereby paves the way for the research-area of "Continuous-Variable -Noisy Intermediate Scale Quantum" (CV-NISQ) devices, in resonance with the investigations recently emerged with qubits [66]. Indeed, quantum advantage in CV beyond a specific encoding has been addressed so far only in the context of sampling problems [28,[67][68][69][70][71][72], for the implementation of algorithms for optimization of continuous functions [73], or for numerical integration [74], leaving plenty of room for new applications yet to unveil.
We thank V. Shumeiko for useful discussions. G.F. acknowledges support from the VR Grant QuACVA. F. Q., G. J., S. G. and G. F. acknowledge support from the Wallenberg Center for Quantum Technology (WACQT). T. H. acknowledges support by the Deutsche Forschungsgemeinschaft via RTG 1995.
where c ac 3 (t) = −c ac 1 (t)/n 2 and c ac 4 (t) = −c ac 2 (t)/n 2 . We have explicitly neglected the quadratic contributions in Φ ac ext (t) for c ac 1 (t) and c ac 3 (t) as they will be off-resonant. As discussed in the main text, we choose Φ ac ext (t)/φ 0 = λ[cos(ω r t) + cos(3ω r t)] (λ 1) in order to retain the full cubic potential in an interaction picture (rotating frame). As stated above, in the presence of the modulation, the c 1 (t) coefficient is not zero. Then, there is a linear contribution ∝ Φ to the effective potential. This is the linear drive discussed in the main text which is resonantly selected by the drive component oscillating at ω r .
By a similar argument c ac 2 (t) and c ac 4 (t) contain no term that is linear in Φ ac ext (t). Indeed, one can show that for a non-zero c 2 (t) coefficient, there will be a contribution proportional to Φ 2 selected by the same frequency component. Higher resonances selected by the 3ω r frequency component will involve higher powers of the flux and, as we discuss in the next section, they can safely be neglected. In principle, one could choose a sufficiently small value of λ in order to get rid of this quadratic effect. Nevertheless, we are dealing with an open system and we require that the effective cubic coupling proportional to c 3 , and thus proportional to λ, exceeds the dissipation rate of the system by roughly one order of magnitude. As discussed in the main text, the SNAIL parameters α, n and Φ dc ext can be chosen so as to suppress the fourth-order contribution to the effective potential. In the following we will therefore consider only terms with m ≤ 3.

Transmission Line Resonator and SNAIL Lagrangian
Here we study a transmission line resonator of length d terminated in an array of M SNAILs. This is a generalization of the setup studied in the main text [Cf. Fig. 1b] which corresponds to the case M = 1. The Lagrangian of this system is with U eff , as specified in Eq. (S6), which approximates the SNAIL potential around its static minimum. The parameters c and l denote the capacitance and inductance per unit length of the resonator (respectively) and the state of the resonator is described by the generalized flux field Φ(x, t). Finally, Φ S = Φ(d, t) denotes the value of the latter at the SNAIL position. We neglect the capacitance of the SNAIL as the charging energy E C of each junction is negligible compared to the Josephson energy E J and furthermore assume that the flux drive at 3ω r is sufficiently detuned from the plasma frequency of the Josephson junctions. An extensive discussion of a similar architecture consisting of a half wavelength resonator interrupted by an array of SNAILs at its middle position is given in [1][2][3].
We follow now the standard quantization procedure [1,2,4,5]. To this end we first neglect all nonlinear terms in the Lagrangian and derive the normal mode structure of the system. We will then reintroduce the nonlinearities to obtain the Hamilton operator for the system. The linear part of the total Lagrangian reads The Euler-Lagrange equation results in the wave equation for the field in the transmission line resonator, where ν = 1/ √ c l is the phase velocity. The structure of the resonator normal modes is determined by the boundary conditions. We impose that the current at the left end (x = 0) of the resonator is identical to zero, i.e., −∂ x Φ(x, t)| x=0 /l = 0. On the other hand, the boundary condition at x = d is modified by the presence of the SNAIL array and can be determined by evaluating the Euler-Lagrange equation for x = d. From this we obtain We now restrict ourselves to a single mode and make a separation of variables ansatz (S15) Inserting this ansatz into the wave equation (S13) the problem decouples and we obtain with the linear dispersion relation kν = ω r . The boundary condition at x = 0 is satisfied by choosing f (x) = cos(kx). For small modulation amplitudes λ, it is justifiable to neglect the time-dependence in Eq. (S14) altogether such that the normal modes become time-independent. Then, inserting f (x) into the boundary condition (S14) at x = d and dropping any time-dependent terms, we obtain where ω 0 = (π/2)(ν/d) describes the bare resonance frequency of the resonator in the absence of the SNAIL array and Z c = l/c, the characteristic impedance of the resonator. Having obtained a solution for the spatial normal modes we evaluate the integral in Eq. (S11) which simplifies to (S19) Considering this together with Eq. (S17), the linear Lagrangian simplifies to with Φ S = φ(t) cos(kd).

Nonlinear Terms and the Hamiltonian
The nonlinear part of the Lagrangian is given by From the total Lagrangian L = L lin + L nl , we can derive the Hamiltonian from the Legendre transform and by introducing the conjugate variable As the Legendre transform leaves the nonlinear part of the Lagrangian invariant the Hamiltonian function reads The Hamiltonian is quantized by promoting the fields φ and N to the operatorsφ andn which satisfy the commutation relation [φ,N ] = i . It is convenient to express them in terms of the bosonic annihilation and creation operatorsâ andâ † (respectively) via the relations, Finally, we obtain the Hamilton operator of the resonator + SNAIL system aŝ H(t)/ = ω râ †â + g 1 (t) â +â † + g 2 (t) â +â † 2 + g 3 (t) â +â † 3 , with and φ ZPF = φ ZPF cos(kd) = 2ηω r cos(kd), with φ ZPF the zero point fluctuations. Using kd = (π/2)(ω r /ω 0 ), where ω 0 = (π/2)(ν/d), we can express the modified zero point fluctuations as Using the expression L −1 J = E J /φ 2 0 for the Josephson inductance, we express the coupling g 3 (t) as and the quadratic coupling as (S30)
We observe that the terms in the square bracket are equivalent to 3q 2 /2 + (ââ † +â †â ). These residual quadratic and linear terms can be removed or modified by using the Gaussian gates within our universal gate set. This holds e.g. in the case one would like to achieve in particular the cubic gate corresponding to theT -gate in GKP encoding [6] T = e In principle, this is possible by flux driving at ω r − δ and 3(ω r − δ) instead of ω r and 3ω r , respectively. The detuning δ is then chosen such that it cancels the additional term.
Also note that one vale of the cubicity is sufficient in order to attain any desired cubicity, provided arbitrary quadratic operations. Indeed, cubicity can be e.g. increased by "consuming" initial squeezing [6][7][8].
In the above analysis we have restricted ourselves to a single resonator mode. For this approximation to be valid it is necessary to avoid populating more than a single resonator mode as all of them are coupled through the SNAIL which may result in additional interaction terms in the Hamiltonian. Recalling that, for a standard quarter wavelength resonator, the frequency of the nth mode is given by ω n = (πν/d)(n + 1/2), we see that the frequency relation between the fundamental (ω 0 ) and the first mode (ω 1 ) is given by ω 1 = 3 ω 0 . Thus, flux driving at 3ω 0 would in principle lead to populating the mode at ω 1 . However, the presence of the SNAIL modifies this relation as shown above. Furthermore, if this modification is not sufficient one can apply impedance engineering as in [9,10] to obtain non-equidistant mode frequencies. We therefore conclude that restricting our analysis to a single mode is legitimate.

Parameter derivation
We now estimate the value of g ac 3 using realistic experimental parameters [1,2,11]. In particular, we will restrict to a single SNAIL with n = 3 large Josephson junctions. For its design parameters we choose E J such that L J = 95 pH and α = 0.25. Choosing α in such a way, results in a vanishing static fourth order coupling g dc 4 at Φ dc ext = 0.39 φ 0 . To obtain a resonance frequency ω r /2π in the range of 4 to 6 GHz, we consider a coplanar waveguide resonator with resonance frequency ω 0 /2π = 6 GHz. Assuming Z c = 50 Ω and numerically evaluating Eq. (S18) results in ω r /2π ≈ 5.2 GHz. For λ = 1/15 (∼ 0.07), g ac 3 /2π ≈ 0.5 MHz. This results in |g ac 3 /g ac 2 | ≈ 4. That this ratio is already sufficient to neglect the effect of the quadratic potential in (S25) becomes evident through a master equation simulation. For the parameters presented in the main text, the obtained fidelity with respect to the ideal state is 99.8 %.