The quartic Blochnium: an anharmonic quasicharge superconducting qubit

The quasicharge superconducting qubit realizes the dual of the transmon and shows strong robustness to flux and charge fluctuations thanks to a very large inductance closed on a Josephson junction. At the same time, a weak anharmonicity of the spectrum is inherited from the parent transmon, that introduces leakage errors and is prone to frequency crowding in multi-qubit setups. We propose a novel design that employs a quartic superinductor and confers a good degree of anharmonicity to the spectrum. The quartic regime is achieved through a properly designed chain of Josephson junction loops that shows minimal quantum fluctuations without introducing a severe dependence on the external fluxes.


Introduction
Superconducting qubits are nowadays one of the leading hardware platforms for quantum information and quantum computation purposes [1,2,3,4].In a future universal quantum computer, errors that originate from the interactions with the environment will be ideally completely eliminated via implementation of quantum error correction schemes [5].In the present context of the so-called noise intermediate-scale quantum (NISQ) era, it is very important to mitigate errors at the hardware level and quantum design represents a major resource in this sense.Since the construction of the prototypical charge box [6,7], the most successful superconducting qubit has been with no doubt the transmon [8], thanks to its intrinsic simplicity and its weak sensitivity to charge fluctuations, owing to a large shunt capacitance.Remarkable performances have also been demonstrated by the fluxonium [9,10,11,12,13,14], that constitutes a highly coherent evolution of the flux qubit [15] showing reduced sensitivity to charge noise thanks to the employment of a superinductance [16], and represents a true competitor of the transmon.Multiple different designs of superconducting qubits have also been proposed, that aim at encoding an intrinsic protection in the wave function, and promising platforms definitely stand out, such as continuous variable qubits [17,18,19,20,21], 0 − π and parity-protected qubits [22,23,24,25,26,27,28,29,30,31,32], and the bifluxon [33].
In the family of the post-transmon setups employing superinductors [34,35,36,37,38,39], a particular instance is represented by the quasicharge superconducting qubit [40,41].Known also as Blochnium, it realizes the dual version of the transmon and achieves strong insensitivity to flux fluctuations thanks to a very large inductance, that plays the analogous role of a large shunt capacitance in the transmon.In addition, the presence of the superinductance renders the quasicharge degree of freedom dynamical, making the Blochnium also insensitive to charge fluctuations.This fundamental and intrinsic robustness comes at the expenses of a weak anharmonicity [42], that is directly inherited from the duality with the transmon.This aspect may become problematic due to frequency crowding in chips hosting several qubits, an issue that will become more and more important as the number of qubit on chip will increase, suggesting that novel approaches are necessary to improve the qubit performances.
In this work we suggest to employ a quartic superinductor to strongly increase the anharmonicity of the Blochnium spectrum, thus complementing a circuit that is already insensitive to charge noise and very well protected from flux noise.A quartic inductor has been first proposed in Ref. [43] and recently realized in the quarton [44] and the unimon [45] qubits.In all cases, the working principle is similar: a linear inductor in loop with a Josephson junction with a flux Φ 0 /2 threading the loop, with Φ 0 = h/2e, h the Planck constant and e the electron charge, cancels the quadratic term by destructive interference when the inductive energy matches the Josephson energy.In the unimon qubit this cancelation is achieved through an inductor long enough to match the inverse Josephson energy in the desired energy range, whereas in the quarton the inductor is formed by a chain of M Josephson junctions.In this case, the M junctions of the chain must have an energy M times larger than the one they are in loop with.
In our proposal, the quartic superinducting regime is achieved by designing a chain of quartons, that generalizes tunable non-linear superinductor designs previously proposed in the literature [34,35,36,39].
The latter achieve the quartic superinducting regime with long chains of flux qubits [15], that represent the M = 2 case.By employing a chain of N quarton loops in series in the classical regime we achieve the potential (1) that provides the system with an intrinsic nonlinearity, that awards a high degree of anharmonicity to the Blochnium spectrum.The quartic oscillator also induces a much larger spread of the wavefunction, with ⟨φ 2 ⟩ ≫ 1, that can be relevant to proposals for quantum computation based on grid states [18,46,47,30].Interestingly, since the Blochnium realizes the dual of the transmon, we effectively realize a quantum system described by a quartic kinetic term, something that is not very often encountered in physics.
As anticipated in Ref. [35], in a general quartic superinductor quantum fluctuations induce a finite, quantum-limited linear inductance in the system, that represents the long wavelength mode of the system.We indeed find an inductance scaling L ∝ M 1/3 N/ξ 4/3 J , showing how quantum fluctuations can represent a severe limitation to a purely quartic regime and need to be properly accounted for.
The work is structured as follows: in Sec. 2 we introduce the model circuit and describe how to get a quartic superinductor out of it.Sec. 3 focuses on the properties of a Blochnium circuit built out of a quartic superinductor, while the impact of fluctuations in the electromagnetic environment and imperfections on the quartic blochnium are estimated in Sec. 5. Sec.6 contains our main conclusions and perspectives.Some technical details can be found in appendix.

Quartic superinductor
A superinductor is an inductive circuital element showing perfect DC conduction, extremely low dissipation, and a very high impedance that is able to exceed the resistance quantum at the relevant (microwave) frequency.Superinductors have been recently experimentally realized through long chains of nominally equal Josephson junctions [16,48,34,49,50], geometrical inductances [36], and granular materials [51,52].
In order to obtain a quartic superinductor we consider the system of Josephson junction depicted in Fig. 1  in units of Φ 0 /2π.Introducing the total phase difference across the chain as φ = M n=1 φ n , the full Josephson potential of the unit cell reads Choosing φ x,c = π and by minimizing the potential for each phase φ n we find the superinductor solution φ n = φ/M , and expanding the potential for small φ the unit cell effective potential reads Clearly, for α = 1/M the quadratic term is switched off and we are left with a quartic potential.This is what is used in the quarton [44] as a qubit, together with a capacitive term that is associated to the effective junction.Assuming the latter to be negligible, we now consider a generic chain of N loops.Denoting the overall phase drop across the chain as φ, the latter is divided to a good approximation in equal parts φ/N across each loop and the effective superinductor potential takes the form (4) For α = 1/M we obtain the potential Eq. (1).The great advantage of this procedure is that we can achieve a large quartic inductance L (4) ≃ M N 3 without having N, M to be very large.In fact, this means that with N, M ≃ 10 we already enter the Blochnium regime.This is particularly important because α is the ratio between the junction areas and the values of M cannot be increased too much due to fabrication constraints.

Quantum quartic oscillator
For the choice α = 1/M the quadratic term is suppressed and we end up with a purely quartic potential for the phase φ.Upon shunting the chain on a capacitor C 0 and quantizing the circuit by imposing canonical commutation rules between phase φ and charge q, so that q = −2ei∂ φ , we obtain a model of a quantum quartic oscillator with E C = e 2 /(2C 0 ) and E (4) The quartic oscillator has one very interesting property, in that already in absence of the periodic potential we have squeezed oscillator states.The exact ground state of the Hamiltonian Eq. ( 5) is not known analytically (see Appendix A for an expression of the spectrum), but we can use harmonic oscillator states as ansatz variational wavefunctions to estimate the spread of the wave function, ξ ≡ ⟨φ 2 ⟩, and the spectrum.We introduce bosonic operators such that The value of ξ is obtained by minimizing the expectation value of the Hamiltonian on the ground state L = E Jc /2N , the ground state is exact and the spread of the wave function is , so that for a superinductor it grows as N 1/4 .For the quartic oscillator, through the variational wave function approach we obtain ξ = (4E C /3E (4) L ) 1/6 , such that the spread of the wave function scales as This way, we are able to change the behavior of the phase fluctuations from ⟨φ 2 ⟩ ≃ (N ) 1/2 to ⟨φ 2 ⟩ ≃ N .

Quartic Blochnium
We now consider the circuit formed by closing the chain on a Josephson junction characterized by capacitance C 0 and Josephson energy E J , as shown in Fig. 1b).Setting α = 1/M we switch the quadratic term off and we are left with the quartic potential L φ 4 .Again, the Hamiltonian is then quantized by imposing canonical commutation rules between phase φ and charge q, so that q = −2ei∂ φ , and the quartic Blochnium Hamiltonian reads where The Blochnium regime is achieved when in which case by neglecting the inductive term we obtain solutions of the Schrödinger equation in the form of Bloch waves, that represent quasicharge states.We then consider the action of the inductive term in generating a long wavelength modulation of the weight of these quasicharge states, in an envelope function approximation.The latter is achieved by writing the wave function as where ψ q,0 (φ) = e iqφ u 0q (φ) is a Bloch function satisfying For the lowest band we can approximate ϵ 0 (q) = −λ 0 cos(2πq)/2, with J /E C the bandwidth of the transmon charge dispersion.Neglecting the coupling between the s = 0 band and the s > 0 [40] bands, the weight v l,0 (q) satisfies the equation We clearly see that the effective Hamiltonian for the weights v l,0 (q) is dual to the transmon Hamiltonian and features a quartic derivative rather than a second one.
An analytic approximation of the spectrum can be obtained in the spirit of the quantum quartic oscillator previously outlined.We expand the cosine potential at small momentum and assume harmonic oscillator wave functions with a spread in quasicharge σ to be determined by energy minimization.If the spread ξ in phase is enhanced by the quartic potential, the spread Exact spectrum and low energy eigenstates σ in quasicharge is expected to shrink, and indeed we find . (10) We can then use the value of σ to estimate the energy difference between the ground state and the first excited state, obtaining ) Assuming E C ≃ E J , that is the relevant working regime for the Blochnium, we have λ 0 ≃ E C , E J (reasonable values are E C , E J ≃ 1 ÷ 10 GHz).The relevant working regime to have a transition frequency in the GHz spectrum is provided by the choice E Jc ≃ 100λ 0 , that is compatible with a choice of N, M ≃ 10, and α = 1/M .
In order to check the analytical predictions we numerically calculate the spectrum and the wave functions of the quartic Blochnium in the dual representation.We expand in Fourier series u 0 (q) = n e inq u 0,n and obtain the dispersion ϵ 0 (q) through numerical diagonalization of the transmon Hamiltonian with E C = 7.0 GHz and E J = 4.7 GHz.The resulting first quasicharge band is shown in Fig. 2a), thin gray curve, and is well approximated by a two-harmonic expression ϵ 0 (q) = −λ 0 cos(2πq)/2+λ 1 cos(4πq)/2.We then expand in Fourier series v l,0 (q) = m e −2πimq v l,m and numerically diagonalize the Hamiltonian Eq. ( 9) with the quartic potential depicted in Fig. 2(b) (thin gray, right axis) has a quartic derivative (care has to be paid to a possibly q-dependent phase of ψ q,0 (φ)).The resulting spectrum is shown in Fig. 2a): we clearly see an evident spectrum anharmonicity, as expected from the quartic character of the potential.Furthermore, given the parameter choice, the lowest states show a very flat dependence on the flux φ x threading the Blochnium circuit, in agreement with the expectation of a flux insensitive regime (without considering variations of φ x,c ).The higher excited states show reduced flatness and a weak sensitivity to the external flux.The ground state and first excited state wave functions are shown in Fig. 2(b).
In order to check the accuracy of the dual model Eq. ( 9), we numerically diagonalize the exact Hamiltonian Eq. ( 7) and report the spectrum for comparison in Fig. 2(c).The similarity is very good, confirming the validity of the dual representation for the parameter regime.The exact ground state and first excited state wave functions are shown in Fig. 2(d), that also confirm the validity of the model.
An estimate of the expected spread of the wave function in φ space gives ξ/(2π) ≃ 1.6, that is compatible with an eye inspection.An important point of the Blochnium physics, and that somewhat defines its working regime, is that the wave function spreads among several wells of the cosine potential.It is this property that confers robustness to flux noise.Interestingly, we notice that in the original Blochnium circuit employing a linear inductance E (2) L ∝ 1/N , assuming E C ∝ 1/N due to stray capacitances yields a spread ξ 0 that is N -independent and fixed by the sample parameters.In contrast, Eq. (6) shows that in our setup the effective spread ξ eff ∝ N 1/3 , showing that it is possible to enter the Blochnium regime by increasing N , regardless of the possible presence of stray capacitances.

Anharmonicity
It is instructive to study the spectrum of the Hamiltonian Eq. ( 7) as a function of E  L /λ 0 the Blochnium regime is gradually lost and all states become flux dependent.
It is customary to quantify the anharmonicity of a qubit by comparing the transition between the first and second excited states, ω 12 , with the transition between the ground and first excited state, ω 01 .In the quartic potential approximation Eq. ( 7), the anharmonicity of the spectrum is intrinsic.An estimate can be performed at the level of the dual Hamiltonian Eq. ( 9) by expanding the cosine potential for E (4) L /λ 0 ≪ 1 and by employing the spectrum of the quartic oscillator provided in Appendix A, that gives L /2λ 0 ) 1/3 (3/2 + 10n/3 + n 2 ).The result is an intrinsic anharmonicity giving about 50% qubit anharmonicity.As shown in Fig. 4, this rough estimate agrees very well with numerical calculation of the anharmonicity through full diagonalization of Eq. ( 7) up to a value E (4) L /λ 0 < 10 −4 , beyond which the Blochnium regime is lost.
In order to further emphasize the anharmonicity of the quartic Blochnium spectrum, in Fig. 5 we compare it with the spectrum of a conventional transmon characterized by E J /E C = 50, the spectrum of the quantum harmonic oscillator, and the spectrum of a quantum quartic oscillator, each normalized by its associated transition ω 01 .On the scale of six excited states we see that the transition ω 12 /ω 01 (see inset) is clearly resolved in the case of the quartic Blochnium and the case of a quartic oscillator, whereas it is hardly resolved in the case of a transmon with respect to the harmonic oscillator case.This shows that a quartic anharmonicity is sufficient to obtain a good addressability of the ω 01 transition and promotes quartic po-

Collective modes in the superinductor
We now assess the collective modes in the Josephson junction chain and their role in the dynamics of the Blochnium circuit.The collective modes will be studied by first discussing the behavior of the single unit cell, that realizes a single quarton circuit, and then by considering the entire structure composed by a chain of quartons.Finally an analysis of the impact of quantum fluctuations on the low energy degrees of freedom will be presented.

Quarton unit cell
The single unit cell of the chain constitutes a quarton qubit, that is formed by M Josephson junctions in series closed on a junction of size reduced by a factor α. The independent degrees of freedom are constituted by the phase differences φ n at the M junctions and the dynamics of the circuits is best spelt in terms of collective modes.The circuit is characterized by mainly three capacitances, the capacitance of the M equal junctions, C J , the one of the smaller junction, αC J , and the one to ground of each superconducting island, C g .Expanding the Josephson potential up to second order in the phases φ n we can study the spectrum of the plasma oscillations.
In order to single out the low energy mode, following Ref.[53] we introduce a set of specific collective modes, the average mode φ = M n=1 φ n and the remaining M − 1 difference modes γ l , so that we can express the local phase differences as with W µm = 2/M cos [πµ(m − 1/2)/M ] for m = 1, . . ., M and µ = 1, . . ., M − 1.The potential is then expanded at fourth order in φ and second order in γ µ , in a way that the Hamiltonian for the low energy mode {φ, q} can be written as (2) For the choice α = 1/M the quadratic term vanishes exactly and results in the quarton Hamiltonian [44].
The other M −1 modes are described by a quadratic Hamiltonian accounting for the higher energy plasma modes.In Fig. 6, we report the results for α = 1/M for different values of C g /C J and for the choice M = 8.The spectrum approximately follows the continuum expression [48,54], with the first mode pinned at zero energy for all values of C g /C J , as a result of the cancelation at flux Φ 0 /2.In addition, we see that for large values of C g /C J all modes experience a reduction of their energy, whereas small values of the ground capacitance keep all but the first mode close to the plasma frequency E p = 8E C J E Jc , with E C J = e 2 /(2C J ).In order to separate the dynamics of low and high energy modes, we ask the high energy modes to be as close as possible to the plasma frequency and obtain a constraint for the maximum in agreement with [9].Neglecting the capacitive coupling between the difference modes charges p µ and the average charge q in the C g /C J ≪ 1 limit, higher and low energy modes are coupled by a residual term that originates from the quartic interaction and is well approximated by  Assuming all difference modes to be not excited and by taking their expectation value on the Gaussian ground state (or thermal state), we immediately obtain a correction to the quadratic term of the low energy average mode φ, so that where ξ 2 J = 8E C J /E Jc .The true quarton Hamiltonian [44] is then obtained with a slightly smaller junction, characterized by α = (1 − ξ 2 J /4)/M , properly corrected by quantum fluctuations.This way, the unit cell Hamiltonian is well approximated by with and the other difference modes can be neglected.

Quarton chain
The very same analysis can be repeated for the chain of quartons that constitute the entire non-linear superinductor and is described by the Hamiltonian By neglecting a capacitive coupling to ground of each island separating two quartons, the system is composed by a collection of uncoupled quartic oscillators and the general wave function will be given by a product state with the χ l eigenfunctions of the Hamiltonian Eq. ( 18).In the ground state, we can assume the quartic oscillators to be all in a Gaussian state, described by a spread ξ q that is determined by energy minimization, When closing the chain on the Blochnium small Josephson junction, the collective modes become the relevant degrees of freedom.We then introduce the superinductance mode φ = N j=1 φ j and difference modes γ ν = j V νj φ j in analogy with Eq. (13).We can express the wave function in the new variables and since the difference modes are orthogonal to the superinductance mode, no mixing arises in the Gaussian state, that can be written as We clearly see that the spread of the wave function in the superinductance mode is rescaled as ξ 2 q → N ξ 2 q , in agreement with the typical behavior of the wave function in case of a linear superinductance.Although formally exact, this step represents a strong approximation for the wave function of the chain of quartic oscillators.
When closing the chain on the small Blochnium Josephson junction, the wave function keeps the factorizes form, with the Gaussian state for the difference mode γ ν unchanged, and the Gaussian state of the superinductance mode φ replaced by the Blochnium wave function with the Blochnium wave function localized on a length scale ξ given in Eq. ( 6) and the Gaussian wave function localized on a much shorter length scale ξ q ≪ ξ.This point is crucial and represents the core of the superinductance classical approximation, that is valid in the case E C J /E Jc → 0 and neglects the contribution of quantum fluctuation.

Quantum fluctuations
For non-negligible E C J , we expect that the non-linear coupling between the γ µ and φ modes introduces a correction to the effective quadratic term, as in the case discussed for the quarton in the previous section.
The Hamiltonian for the chain of quartons expressed in terms of the φ and γ ν modes reads and we have neglected odd parity coupling between the different modes.First of all, by neglecting the coupling between different modes we see that although the spectrum of the φ and γ ν modes is characterized by the same energy scale the spread of the wave function in the superinductance and difference modes show a different scaling with N , suggesting that a separation of scales does indeed occur.
In order to assess the impact of the coupling between the modes, we can once again assume the ground state to be described by a Gaussian wave function for the modes γ ν and calculate its spread by energy minimization.This way, we find It follows that due to the repulsive interaction between the modes, the spread of the wave function ξ γ shrinks.By taking the expectation value of the Hamiltonian on the Gaussian wave function of the γ ν modes we finally find showing how a quadratic term is generated for the superinductance mode, whose origin is purely due to quantum fluctuations.We then conclude that the quadratic term renormalization can be suppressed by reducing ξ J , that can be achieved by reducing the charging energy E C J through proper shunting capacitances.
At this point, we can make contact with previous and similar proposals of a non-linear superinductor realized in a similar way as a chain of flux qubits [34,35], that is a particular case of M = 2, 3 depending on the flux qubit realization.Analogously, we find a fluctuation induced inductive term that is due to higher energy modes that are coupled to the low energy one due to the non-linearities.The resulting fluctuationinduced inductance scales with the number of loops and junctions as N M 1/3 , offering us an insight to assess its overall impact.

Decoherence estimates
The spectrum of the quartic Blochnium shows strong insensitivity of the lowest energy states to the flux φ x threading the main loop that, together with the intrinsic insensitivity to charge fluctuations, renders the qubit particularly appealing.Nevertheless, it is important to assess the impact of fluctuations in the electromagnetic environment and imperfections of the chain.The latter statically couple to φ 2 and their overall magnitude statistically averages to zero.Much more important are fluctuations in the electromagnetic environment in the form of time-dependent variations of the fluxes that thread the loops of the quartic superinductor and the offset charges localized on the superconducting islands separated by the Josephson junctions.
In general, given the two lowest energy states of the qubit, |0⟩ and |1⟩, and given a classical randomly fluctuating variable λ i linearly coupled to the observable O i through the Hamiltonian H = i λ i O i , the relaxation rate at temperature lower than the qubit frequency can be estimated through Fermi golden rule [55] Γ where S i (ω) is the spectral function of the fluctuating variable The dephasing rate depends on the characteristic time scales of the spectral function, in that it is affected by the zero-frequency part of it.For well behaved S(ω = 0) the decay of the off-diagonal matrix element of the qubit density matrix is exponential, with a rate (30) In turn, for spectra singular at ω = 0 such as 1/f noise described by S λi (ω) = 2πA 2 /|ω|, the decay is Gaussian over a time scale dictated by the inverse of [8] that collects all rates independently.

Flux fluctuations
Time-dependently fluctuating fluxes in the loops of the superinductor chain enter the Hamiltonian via the perturbation δU, The operator φ affects the relaxation rate by inducing transitions between the two qubit states, but its odd character has zero expectation value on the qubit states, so it has no effect on the dephasing rate at lowest order.Fluctuations in the flux δφ x,j can arise either due to short wavelength fluctuations of the magnetic fields or due to long wavelength fluctuations in presence of random area variations among the loops, δφ x,j = 2π(AδB j +δA j B)/Φ 0 .Short wavelength fluctuations of the field are expected to be correlated on a short-range scale and add up incoherently, thus producing an overall factor N .The full relaxation rate is estimated as where S δΦ (ω) = dte −iωt ⟨δΦ x (t)δΦ x (0)⟩ is the spectral function of the fluctuating flux δΦ x = (Φ 0 /2π) j δφ x,j and the relevant matrix element is given by (see Appendix B) with σ given in Eq. (10).The transition matrix element thus scales with the inverse of the quasicharge spread of the wave function and is proportional to √ N .This can be understood as arising from the enhanced spread of the wave function, that yields a larger dipole matrix element.As a result its square cancels one power of N in the rate Eq.(33), that becomes For the case of the fluxonium, that is also composed by a chain of Josephson junctions, it is customary to study the decoherence rate as a function of the number of junctions by keeping constant the energy scales defining the problem [9].In our case, by fixing L , and setting α = 1/M , the relaxation rate due to fluctuations in the loops becomes where c = 1536(3 √ 2) 2/3 π 8/3 ≃ 0.85 × 10 5 , that predicts an increase of the relaxation rate with the fourth power of the number of loops.This result owes its origin to non-linear dependence of E L ≃ 10 −6 GHz and λ 0 ≃ E C , E J ≃ GHz we find that Γ 1 ≃ 10 −5 (N 4 /M 2 ) × GHz 2 × S δΦj /Φ 2 0 .As an estimate of the worst scenario case, we assume a 1/f noise power spectrum affecting global field fluctuations for which S Φ = ij ⟨δA i δA j /A 2 ⟩A 2 Φ /hω with A Φ = 10 −6 Φ 0 [56,11,14].Statistically averaging over an area error of 10% gives a factor ij ⟨δA i δA j /A 2 ⟩ = N (0.1) 2 , and for a qubit frequency in the range of GHz we obtain Γ 1 ≃ N 5 /M 2 × 10 −10 Hz, showing how any reasonable choice of N cannot limit the coherence of the qubit.

Quantum phase slips due to charge fluctuations
So far we have neglected the impact of the offset charge fluctuations localized on the superconducting islands.It is well known that in the Josephson regime the low energy dynamics of Josephson chains is dominated by quantum phase slips.The latter are sensitive to the random offset charges on the islands via the Aharonov-Casher effect [57,58,46,59,60].The dependence of the spectrum on the charge offsets arises due to the compactness of the phase variables across each Josephson junction.The total Hamiltonian of the system has the general form H = 1 2 (q + q g ) T C −1 (q + q g ) + U (φ), (37) where C is the total capacitance matrix of the circuit and the potential U is the total potential describing the N J + 1 junctions, with N J = N M , including also the Blochnium junction of energy E J .The potential U is 2π-periodic in all the phases.
As discussed in Ref. [61], in a superconducting circuit involving only Josephson junctions and capacitances, the compactness of the phases renders the wave function necessarily indistinguishable upon a 2π phase shift in anyone of the φ n phases, so that the full wave function Ψ α,q must be periodic in φ.This way, the wave function has the general form where Ψ Bloch l,q (φ) is Bloch function.The latter can be easily written in the tight-binding approximation, that is valid when the Josephson energy is much larger than the charging energy, and the resulting Hamiltonian reads where δh l,l ′ describes the matrix elements between qubit states mediated by a phase slip in the n-th phase φ n .By expanding the Hamiltonian in terms of collective modes and employing the resulting eigenstate as a tight-binding basis, we find that a 2π shift in n = (j − 1)M + k phase φ n yields fractional shifts in the collective mode described by w n .As a result, the phase-slip-mediated matrix elements can be written as where the overlap O (n) is given by ] kk , W µ,n the amplitudes of the collective modes, and By inspection of the capacitance matrix we see that η k ≥ 1, with η k → 1 for C g /C J → 0, so that larger values are obtained for larger ground capacitance C g /C J > 1.As a result we have e −π 2 η k /ξ 2 J ≤ e −π 2 /ξ 2 J , where the right hand side is the quantum phase slip rate of a junction characterized by ξ J .
The overlap between the Blochnium wave function can be estimated as e −π 2 σ 2 ≃ e −π 2 /N ξ 2 γ , from which it follows that the quantum phase slip rate of the k-th junction in the j-th loop is given by The rate is independent on the number of loops N but it depends on the number of junctions in each loop M .By comparing it to the quantum phase slip rate e −π 2 /ξ 2 J , we see that although the effect of the quartic potential results in a looser phase, that can slip out of the minimum with higher probability, the presence of a ground capacitance results in an overall reduced quantum phase slip rate Γ qps k .The qubit dephasing rate is estimated through Eq. (31) and it is zero at first order at the sweet charge spot q g = 0.In the worst case scenario in which all charges fluctuate around the value 1/4 (in units of 2e), by summing the contributions of the independent phase slips the dephasing rate becomes that increases with the number of junctions, in agreement with the result that the superfluid phase in a chain of Josephson junction is destroyed by proliferation of phase slips.In addition, we have that rate for an individual phase slip is slightly increased due to a looser phase in the quartic potential.When expressed by keeping E (4) L fixed, the dephasing rate becomes It follows that for fixed E (4) L the dephasing time exponentially increases with N in analogy with the fluxonium [9,60].

Comparison with the quadratic Blochnium
It is instructive to compare the dephasing rate of the quartic Blochnium to the one of the quadratic Blochnium.The dephasing rates can be estimated through a simplified yet powerful argument.The dependence of the spectrum on the offset charges arises due to the Aharonov-Casher effect, in which a fluxon can escape the qubit ring via tunneling through the junctions.Interference between different paths through difference junctions is sensitive to the total charge enclosed between the two junctions.In the quadratic Blochnium, a fluxon can tunnel away from the loop through the chain via one of the N J large Josephson junctions characterized each by spread ξ J .By independently summing the rates it follows that the quadratic Blochnium has a quantum phase-slip rate given by On the other hand, in the quartic Blochnium a fluxon has first to tunnel via one of the N junctions with Josephson energy E Jc /M , that have a quartic potential around each minima and associated spread ξ q = √ 2M 1/3 ξ 2/3 J , and then through one of the N M junctions with spread ξ J .This is schematically depicted in Fig. 7. Adding the rates we have In order to make a comparison meaningful, we assume all energy scales to be the same and only choose N M = N J in a way to have same qubit frequency E 01 .Choosing the specific values reported in Tab. 1, it follows that the quadratic Blochnium has frequency E 01 = E Jc λ 0 /(2N J ) ≃ 0.79 GHz, that is of the same order than the quartic Blochnium frequency.It follows that the ratio of dephasing rates of the qubits is given by where Γ ϕ and Γ ϕ are the dephasing rates for the quadratic and quartic Blochnium, respectively.It then follows that the quartic one has slightly reduced dephasing rate, because the chain of loops is more robust to charge fluctuations in that the flux has to entirely cross the quarton loops.

Conclusions
In this work we have theoretically discussed a quartic Blochnium, an anharmonic quasi charge superconducting qubit.The latter is realized through a quartic superinductor closed on a Josephson junction characterized by E J ≃ E C in the regime in which the quartic inductance energy scale is E The quartic superinductor is formed by a chain of N loops, each constituted by a short chain of M equal Josephson junctions in parallel with a junction whose Josephson energy is 1/M times the energy of the other M junctions and by properly threading the loop with half flux quantum Φ 0 /2.This single loop realizes the analogous circuit of the quarton qubit [44], that exploits a quartic Josephson potential to obtain an anharmonic qubit.By considering a short chain of N loops we effectively realize a quartic superinductor described by the potential Eq. (1).
The quartic Blochnium we propose shows an anharmonic spectrum, with a ladder of levels similar to the quantum quartic oscillator spectrum, thus solving the weak anharmonicity problem of the Blochnium, that is inherited by the transmon due to its dual character.The quartic anharmonicity is milder than the one achieved in the flux controlled regimes typical of the flux qubit and the fluxonium.Nevertheless, it is less sensitive to flux fluctuations, even accounting for the ones associated to the external flux introduced to achieve the quartic regime.An optimal choice of parameters of the system is provided in Table 1.The present design thus promotes the system as a viable and promising quantum computing platform.

A Quantum quartic oscillator spectrum
The spectrum of a quartic oscillator has no analytic form and we have to resort to numerical diagonalization.A good strategy is provided by taking the matrix elements of the Hamiltonian Eq. (5) on the harmonic oscillator states, ψ n (φ/ξ), nonlinearity of the spectrum is evident from the diagonal terms and the repulsion provided by the offdiagonal terms.

B Qubit matrix elements
We now need to calculate the matrix elements of the perturbation.The matrix element of the phase φ between Bloch states is given by M q,s;q ′ ,s ′ ≡ π −π dφ 2π ψ * q,s (φ)φψ q ′ ,s ′ (φ) = i d dq δ(q − q ′ )δ ss ′ + δ(q − q ′ )Ω s,s ′ (q) Ω s,s ′ (q) = i π −π dφ 2π u * qs (φ) with Ω s,s = 0.It follows that for the two lowest energy qubit eigenstates, that belong to the band s = 0, by neglecting the contributions coming from Ω we have that The integrals can be calculated by constructing approximate Bloch functions v l (q) in the tight-binding approximation through localized eigenfunctions ϕ l .The latter can be chosen as Gaussian eigenstates of the harmonic oscillator, v l,φx (q) = 1 √ N n e −iφxn ϕ l (q − n) (53) with the spread σ to self-consistently determined by minimization of the energy of the Hamiltonian Eq. (9).The result for the qubit dipole transition matrix element is 1 − e −1/4σ 2 σ 2 cos(φ x ) , (54) and analogously the other matrix elements.

C Chain capacitance matrix
The capacitance matrix C of the quarton is an (M + 1) × (M + 1) real, symmetric, positive-definite matrix,

5 '
(a).It is constituted by a chain of N units, each one constituted by a loop consisting of M + 1 Josephson junction, M of which with equal Josephson energy E Jc , and the last junction with energy αE Jc .Let us first consider the single unit cell.Denoting φ ′ the phase difference across the small junction and φ n the phase difference across each junction, flux quantization imposes that φ ′ + M n=1 φ n + φ x,c = 2πn, where φ x,c = 2πΦ x /Φ 0 is the flux threading the loop °5 0 t e x i t s h a 1 _ b a s e 6 4 = " 7 j F / a x i K Q 5 0 q c k 4 5 3 c B + a S 7 C D O 0

LFigure 1 :
Figure 1: (a) Quartic superinductor constituted by a chain of N unit cells, each one constituted by a loop with M Josephson junctions of energy EJc closed on a single Josephson junction with energy αEJc, with α = 1/M .Each loop is threaded by a flux φx,c = 2πΦx,c/Φ0 = π.(b) Quartic Blochnium circuits constituted by a quartic superinductor, realized as in (a) and described by a quartic inductive energy E (4) L , closed on a Josephson junction characterized by Josephson energy EJ and charging energy EC .(c) Effective Josephson potential of the quartic Blochnium experienced by the phase-particle in the circuit, for the choice EJ = 4.6 GHz, EJc = 100 GHz, M = 8, and N = 40 (in thin gray it is shown only the quartic dependence of the potential).

4 )
2C B ) is the effective charging energy in terms of the effective capacitance C B associated to the additional junction.The full Josephson potential is shown in Fig. 1(c) for an effective inductive energy E (Jc = 8 × 10 −6 GHz, obtained for a choice of parameter E Jc = 100 GHz, M = 8, and N = 40, and E J = 4.7 GHz.

Figure 2 :
Figure 2: (a) Blochnium low energy spectrum versus external flux obtained by diagonalization of the dual model Eq.(9).In light gray the first quasicharge band dispersion ϵ0(q) of the transmon-like problem.The parameters of the model are EJ = 4.6 GHz, EJc = 100 GHz, EC = 7.0 GHz, M = 8, and N = 40, for which E (4) L ≃ 8 × 10 −6 GHz.(b) Ground state and first excited state wave functions (left axis) and Josephson potential (thin gray, right axis).(c) and (d) same as (a) and (b), respectively, for the exact Hamiltonian Eq. (7).

( 4 )L
L , to check the validity of the Blochnium regime.In Fig.3we present the spectrum as a function of the external flux φ x for three values of E (4) L .We clearly see that the Blochnium /λ 0 = 2. × 10 −5

Figure 3 :
Figure 3: Spectrum of the quartic Blochnium as a function of the external flux φx, for three different values of E (4) L and by keeping constant EJ = 4.0 GHz and EC = 7.0 GHz, from which it follows that λ0 = 3.94 GHz.Superimposed are the first two Bloch bands of the parent transmon problem.

Figure 5 :
Figure5: Spectrum of the quartic Blochnium, shown together with the spectrum of a transmon characterized by EJ /EC = 50, the spectrum of a quantum harmonic oscillator, and the spectrum of a quantum quartic oscillator, each normalized to its ω01 transition.Inset: zoom on the ω12 transition, showing the anharmonicity induced by the different terms.

Figure 6 :
Figure 6: Quarton collective modes in units of Ep = 8EC J EJc for M = 8, α = 1/8, and three values of Cg/CJ .In the quadratic approximation the first mode has exactly zero energy.

Φ 0 QFigure 7 :
Figure 7: Schematics of the interfereing fluxon tunneling processes that are sensitive to the fluctuating charge on the islands due to the Aharonov-Casher effect.Their amplitude involves tunneling through two junctions, yielding the result Eq.(47).

Table 1 :
where p n = (m + 1)(m + 2) and diagonalizing the Hamiltonian upon truncation of the spectrum.The E J (GHz) E C (GHz) λ 0 (GHz) E Optimal device parameters.