Superconducting circuit protected by two-Cooper-pair tunneling

W. C. Smith,1, ∗ A. Kou,1 X. Xiao,1 U. Vool,1 and M. H. Devoret1, †

1Departments of Applied Physics and Physics, Yale University, New Haven, CT 06520, USA(Dated: January 22, 2020)

We present a protected superconducting qubit based on an effective circuit element that onlyallows pairs of Cooper pairs to tunnel. These dynamics give rise to a nearly degenerate ground statemanifold indexed by the parity of tunneled Cooper pairs. We show that, when the circuit elementis shunted by a large capacitance, this manifold can be used as a logical qubit that we expect to beinsensitive to multiple relaxation and dephasing mechanisms.

I. INTRODUCTION

A. Motivation

Superconducting circuits are widely recognized as apowerful potential platform for quantum computationand now stand at the frontier of quantum error correction[1]. Future progress will likely stem from two complemen-tary strategies: (i) active error correction characterizedby measurement-based [2–4] and autonomous [5–8] stabi-lization, and (ii) passive error correction characterized byprotected qubits [9, and references therein]. We addressstrategy (ii) in this article by designing an experimentallyaccessible protected qubit.

The transmon qubit [10] has proven to be a remark-ably successful prototype of a protected qubit. Its cir-cuit (depicted in Fig. 1a) contains a Josephson junctionwhose potential energy is U = −EJ cosϕ, with EJ beingthe tunneling energy and ϕ being the superconductingphase across the junction. The large shunt capacitanceimposes a large effective mass for the analogous “particlein a box,” confining the low-energy wavefunctions nearϕ = 0, the only minimum for ϕ ∈ (−π, π). This confine-ment suppresses the susceptibility of the qubit to offsetcharge noise and renders the energy spectrum approxi-mately harmonic with level spacing much smaller thanEJ (see Fig. 1b). On the other hand, circuit elementswith degenerate phase states that only allow tunneling ofpairs of Cooper pairs, meaning their potential energy isU = −EJ cos 2ϕ, have been developed in recent years as abuilding block for topologically protected qubits [11, 12].In this article, we propose a transmon-like qubit withadditional protection from environmental noise by com-bining the large shunt capacitance of the transmon withsuch a cos 2ϕ circuit element (the cross-hatched box inFig. 1c). In this case, the wavefunctions are localizednear ϕ = 0, π (see Fig. 1d), resulting in a nearly degener-ate harmonic level arrangement. While the detrimentaleffects of offset charge noise are similarly suppressed inthis circuit, sensitivity of the qubit to other decoherence

∗ Current Address: QUANTIC Team, INRIA Paris, 2 rue SimoneIff, 75012 Paris, France; william.smith@inria.fr† michel.devoret@yale.edu

(a) (b)

(c) (d)

ϕ

ϕ

EJ

EJ

Cshunt

Cshunt

FIG. 1. (a) Electrical circuit for the transmon qubit. (b) Po-tential energy of the transmon with energy levels and wave-functions for the first few eigenstates. (c) Electrical circuitfor the idealized protected qubit. The cross-hatched circuitelement comprises a capacitance in parallel with an induc-tive element that exclusively permits the tunneling of pairsof Cooper pairs. The superconducting island is indicated bycolor. (d) Potential energy of the ideal charge-protected qubitwith the lowest energy levels and wavefunctions.

mechanisms is also reduced, owing to the conservation ofCooper pair number parity.

In this article, we introduce a few-body transmon-type qubit where the charge carriers are exclusively pairsof Cooper pairs. Our central result is that there ex-ists an experimentally attainable parameter regime forwhich conservative predictions of relaxation and dephas-ing times exceed 1 ms, i.e. an order of magnitude higherthan those of typical transmons, given the same envi-ronmental noise [13, 14]. In the remainder of Sec. I,we describe a toy model for the protected qubit. Weproceed by analytically and numerically examining theHamiltonian for the full superconducting circuit in Sec.II. Our attention in Sec. III then turns to properties ofthe ground state manifold, which we envision using as aprotected qubit. A brief discussion about the concept ofprotection and examples of protected qubits, as well as

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our perspectives on readout and control, follows in Sec.IV. Finally, we summarize our results in Sec. V.

B. cos 2ϕ element

We first examine the advantages of the ideal circuit inFig. 1c as a protected qubit. This circuit can be viewedas a Josephson-junction-like element (the cross-hatchedbox) shunted by a capacitance. Pairs of Cooper pairs arethe only charge excitations permitted to tunnel throughthis element [11]. In the Cooper pair number basis, thepotential energy assumes the form

−1

2EJ

∞∑N=−∞

(|N〉〈N + 2|+ |N + 2〉〈N |) = −EJ cos 2ϕ,

where EJ is the effective tunneling energy of the pro-cess. This expression follows from the conjugacy rela-tion [ϕ,N ] = i, where N is the number of Cooper pairsthat have tunneled. The invariance of the potential un-der translations in ϕ by multiples of π implies that half-fluxons are able to traverse the element.

The shunt capacitance and other charging effects pro-duce a quadratic kinetic energy, yielding the Hamiltonian

H = 4EC(N −Ng)2 − EJ cos 2ϕ, (1)

where EC is the charging energy and Ng is the offsetcharge. This offset charge has been introduced due tothe periodicity of the Hamiltonian in ϕ, which reflectsthe presence of a superconducting island in the circuit(as colored in Fig. 1c).

Since the circuit element only allows pairs of Cooperpairs to tunnel, the parity of the number of Cooperpairs that have tunneled is preserved under the actionof the Hamiltonian. This property leads to two nearlydegenerate ground states |+〉 and |−〉, which only con-sist of even and odd Cooper pair number states, re-spectively [12]. Since these states have no overlap incharge space (equivalently, they have opposite period-icity in phase space—see Fig. 1d), we have 〈−|O|+〉 ≈ 0for any sufficiently local operator O. Furthermore, thestates [15] | 〉 = 1√

2(|+〉± |−〉) are respectively localized

near ϕ = 0, π (see Fig. 1d). Because these states havesuppressed overlap in phase space for large EJ/EC (i.e.they are roughly inversely periodic in charge space), wehave 〈 |O| 〉 ≈ 0 for similarly local O. These are pre-cisely the conditions for simultaneously suppressing spu-rious transitions and phase changes between the states[16] [resembling a Gottesman-Kitaev-Preskill (GKP) en-coding on a circle [17]].

More concretely, the ground state splitting obeys

∆E ≈ 16EC

√2

π

(2EJEC

)3/4

e−√

2EJ/EC cos(πNg),

for large EJ/EC (see App. A). The two ground stateenergies oscillate out of phase with one another in Ng.

Moreover, this shows that the splitting, as well as thecharge dispersion, is exponentially suppressed in EJ/EC .Thus, the role of the shunt capacitance is to decreasethe charging energy EC and hence mitigate offset chargenoise, much like in the transmon qubit [10].

II. SUPERCONDUCTING CIRCUIT

A. Hamiltonian

We now detail the superconducting circuit necessary topractically implement the sought-after cos 2ϕ Josephsonelement. This circuit, depicted in Fig. 2a, is composed oftwo identical arms, each containing a Josephson junctionin series with superinductances [18, 19], arranged in par-allel [20, 21] and shunted by a large capacitance. Thesesuperinductances are split in half and placed on eitherside of the respective Josephson junctions to avoid largecapacitances to ground. Kirchhoff’s current law allows usto treat the phases across the superinductances in seriesas equal. When the external magnetic flux threading theinductive loop reaches half of a flux quantum, i.e. when

(a)

(b)

ϕextϕ1 ϕ2

φ2/2

φ2/2φ1/2

φ1/2

Cshunt

FIG. 2. (a) Reduced electrical circuit for the physical pro-tected qubit. When ϕext = π, the two Josephson junctionsand superinductances collectively behave as the cross-hatchedelement. The superconducting island is indicated by color.(b) Contour plot of the potential energy U in Eq. 2 in theϕ1ϕ2-plane at ϕext = π for θ = 0. The numerically com-puted instanton trajectory between adjacent potential min-ima is overlaid in black. Importantly, this trajectory closelyresembles a sequence of straight lines.

ϕext = π, the loop approximates the cross-hatched box inFig. 1c. At this particular bias point, a Cooper pair canonly tunnel through one of the Josephson junctions if it isaccompanied by another Cooper pair tunneling throughthe other junction (in either direction). Conversely, afluxon traversing a single Josephson junction correspondsto a half-fluxon traversing the whole element.

We consider the symmetric and antisymmetric combi-nations of superconducting phase coordinates

φ = ϕ1 + ϕ2 ϕ =1

2(ϕ1 − ϕ2) θ =

1

2(φ1 − φ2),

and their conjugate charges {n,N, η} in numbers ofCooper pairs. Note that the prefactor in the definition ofϕ is chosen to bring the coordinate into agreement withthe phase drop across the inductive loop in the limit thatθ vanishes. The Hamiltonian reads

H = 4εC

[2n2 +

1

2(N −Ng − η)2 + xη2

]+ εL

[1

4(φ− ϕext)

2 + θ2

]− 2εJ cosϕ cos

φ

2, (2)

where εC and εJ are the single junction charging andtunneling energies, 2εL is the inductive energy of eachsuperinductance, x ≡ CJ/Cshunt is the ratio of the junc-tion capacitance to the shunt capacitance, and Ng is theoffset charge on the superconducting island (see Fig. 2a).

From this expression, it is clear that this circuit hasthree strongly coupled modes. The φ mode is flux de-pendent and is strongly and nonlinearly coupled, via theJosephson junctions, to the ϕ mode. The ϕ mode isoffset charge dependent and strongly but linearly capaci-tively coupled to the θ mode. Our analysis and the effectsobserved in the remainder of this article require the pa-rameter regime εL � εJ , εC . εJ , and x � 1 [22]. Inparticular, the parameters chosen for numerical simula-tions are listed in Tab. I and are similar to those of recentfluxonium devices [18, 23].

B. Semiclassical theory

In order to gain insight into the structure of the Hamil-tonian in Eq. 2, we briefly revisit its potential energy Uin the ϕ1ϕ2-plane, which is plotted in Fig. 2b for θ = 0.The cosine terms in the potential form a two-dimensional“egg carton” of wells. The minimum of the quadraticterm in the potential occurs at ϕ1 + ϕ2 = ϕext, whichgenerally falls between adjacent diagonal ridges of co-sine wells. At the special value of ϕext = π, these tworidges are degenerate. Near this value of the externalflux, we consider the path of the system between neigh-boring potential minima by numerically solving for thethree-dimensional instanton [24] trajectory (see App. B).

We then constrain the system to this maximally proba-ble tunneling path. For the parameters mentioned above,

εJ/h εC/h εL/h x

15 GHz 2 GHz 1 GHz 0.02

TABLE I. Circuit parameters used for numerical simulations.

this path is well-described by

φ =1

1 + z

(2∣∣∣ϕ− 2π round

ϕ

2π

∣∣∣+ zϕext

), (3)

where z ≡ εL/εJ is a small parameter [25]. Plugging thisexpression into the Hamiltonian in Eq. 2, approximatingthe resulting one-dimensional potential by the first fewterms in its Fourier series, and Taylor expanding aboutz = 0 yields H ≈ Heff with

Heff = 4εC

[1

4(1− z)(N −Ng − η)2 + xη2

]+ εLθ

2

− 16

3πεL(π − φext) cosϕ− εJ

(1− 5

4z

)cos 2ϕ (4)

to leading order. Here, φext =∣∣ϕext − 4π round ϕext

4π

∣∣ andwe have discarded terms higher than the second har-monic or O(εL) (see App. B for additional terms). Thistreatment exposes the “cos 2ϕ nature” of the potential atϕext = π, where the cosϕ term vanishes. By comparisonto Eq. 1, we see that the added complication is that theϕ mode is strongly coupled to the θ mode. The result-ing hybridization is a central ingredient to understandingproperties of the system beyond the ground state mani-fold (see Sec. II D).

We comment that this approximation neglects quan-tum fluctuations that are perpendicular to the path inEq. 3. This is consequently a semiclassical approxima-tion: we have minimized the energy of the system withrespect to the dynamical coordinate orthogonal to thetrajectory [24]. Moreover, from Fig. 2b, it is clear thatthe approximation we have made is that fluxons traversea single Josephson junction at a time.

C. Energy spectrum

From numerical diagonalization of Eq. 2 (see App. C),we obtain the dependence of the energy levels on externalflux as shown in Fig. 3a [26]. At ϕext = π (the dashedline in Fig. 3a), the spectrum resembles a doubled har-monic oscillator with energy

√16xεLεC . Once ϕext devi-

ates from π, half of the energy levels increase in energylinearly with slope ∼ 32

3 εL [27]. The other half of theenergy levels form a flux-independent harmonic ladder.

We can understand this level structure as that of twoemergent modes (see also App. D). The first mode is flux-dependent and its excitations correspond to the numberof magnetic flux vortices, or fluxons, enclosed by the in-ductive loop in Fig. 2a. In turn, the number of enclosedfluxons identically maps onto the magnitude and chirality

(a) (b)

(c)

FIG. 3. (a) Normalized transition energies E from the ground state of the Hamiltonian in Eq. 2 as a function of externalflux at Ng = 0. The essential feature is the presence of a flux-independent plasmon mode (maroon) with energy

√16xεLεC

and a flux-dependent fluxon mode (green). The coloring reflects the assigned quantum numbers. (b) Charge wavefunctions〈N |ψ〉 = 〈N |m±〉 of the first four eigenstates of Eq. 2, illustrating that the plasmon excitation index m = 0, 1, . . . indicatesthe functional form while the fluxon excitation index ± indicates charge parity. (c) Phase wavefunctions 〈ϕ, φ|ψ〉 of thefirst four eigenstates of Eq. 2, showing that the intrawell excitation number corresponds to the plasmon index and that thebonding/antibonding configuration corresponds to the fluxon index. All wavefunctions are computed at ϕext = π.

of the circulating persistent current in the inductive loop.The second mode is flux-independent and its excitationscorrespond to quantized charge density oscillations, orplasmons, across the inductive loop/shunt capacitance inFig. 2a. Each plasmon involves the two superinductances(energy 2εL in parallel) and the shunt capacitance (en-ergy xεC), and hence has energy

√16xεLεC . Hereafter,

we refer to these modes as the “fluxon mode” and the“plasmon mode,” respectively [28]. Additionally, we as-sign the labels |m • 〉 to the lowest-energy states, wherem denotes the number of plasmons and • (or ◦ ) denotesthe presence (or absence) of a fluxon excitation relativeto the ground state [29]. Note that the fluxon placeholderindices are defined by

• / ◦ =

/ , for ϕext mod 2π < π

−/+, for ϕext mod 2π = π

/ , for ϕext mod 2π > π

,

where |m±〉 = |m〉 ⊗ 1√2(| 〉 ± | 〉) (as in Sec. I B) and

the index represents the persistent current direction.This labeling serves the purpose of assigning quantumnumbers consistently for all external flux values (see thecolors in Fig. 3a [30]), except the particular case ϕext

mod 2π = 0 that we do not focus on.

D. Wavefunctions

Calculated charge wavefunctions 〈N |ψ〉 and phasewavefunctions 〈ϕ, φ|ψ〉, obtained from numerical diag-onalization of Eq. 2, are shown in Figs. 3b, 3c for thefour lowest-energy eigenstates at ϕext = π. Roughly, the

phase wavefunctions are computed by projection of theθ coordinate and a Fourier transform to the ϕφ-plane,while the charge wavefunctions are computed by projec-tion of the θ coordinate and constraint to the trajectoryin Eq. 3 (see App. C for details).

The charge wavefunctions are grid states with Fock-state envelopes [17, 31]. For fluxon excitation index +/−,these grid states are superpositions of even/odd Cooperpair number states [32]. Additionally, m corresponds tothe order of the Fock state envelope. Note that a logicalqubit encoded in |0+〉 and |0−〉 is protected from spuri-ous transitions except those mediated by operators thatflip Cooper pair parity.

On the other hand, the phase wavefunctions are ap-proximately Fock states localized within the potentialenergy wells (see Sec. II B) [23]. The fluxon index +/−denotes whether the state |m±〉 is a symmetric (bonding)or antisymmetric (antibonding) superposition of stateslocalized within opposite ridges of potential wells. Theseridges correspond to persistent currents of opposite chi-rality, and hence also to the absence/presence of a fluxonin the inductive loop of the circuit [33]. In this picture,m refers to the Fock order of the localized states. Finally,operators that flip Cooper pair parity correspond to oddfunctions of φ or functions of ϕ with period an odd di-vision of 2π, which can be seen from Fig. 3c to mediatethe transition |m+〉 ↔ |m−〉

III. QUBIT

In Sec. II, we analyzed the multi-mode Hamiltoniandescribing the superconducting circuit in Fig. 2a. Nu-

merical diagonalization of this Hamiltonian showed theemergence of a linear plasmon mode and a nonlinearfluxon mode [34]. In this section, we consider the proper-ties of the logical qubit formed by {|0+〉, |0−〉}, the twolowest-energy eigenstates at ϕext = π, which generalizesto {|0 ◦ 〉, |0 • 〉} away from ϕext = π.

A. Matrix elements

To better elucidate which types of operators can andcannot induce transitions between the two states of thequbit, we examine the relevant matrix elements corre-sponding to capacitive and inductive coupling. This dis-cussion is particularly relevant to understanding the ex-pected dominant loss mechanisms (Sec. III C) and de-signing a measurement and control apparatus that doesnot directly couple to the qubit (Sec. IV C).

For capacitive coupling, a generic voltage V couples tothe superconducting island of the circuit in Fig. 2a via agate capacitance Cg and will append the term

Hint =Cg

Cshunt + Cg(2eη)V

to the Hamiltonian in Eq. 2, in addition to dressing theshunt capacitance. This voltage may be a degree of free-dom of another mode in the embedding circuit, a noisesource, or an ac drive. We therefore see that the suscep-tibility of undergoing a transition from the ground state,due to capacitive coupling to the qubit island, is directlyrelated to the matrix element 〈ψ|η|0 ◦ 〉.

For inductive coupling, a generic current I couples tothe circuit via a small inductance Ls shared with theinductive loop, which adds the term

Hint =Ls

2L(φ0φ) I

to the Hamiltonian in Eq. 2. Here, φ0 = ~/2e is thereduced magnetic flux quantum and L is the superinduc-tance in each arm of the qubit (i.e. εL = φ2

0/L). Likethe voltage source, this current may represent an inter-nal or environmental degree of freedom. We see that thesusceptibility of undergoing a transition from the groundstate, due to inductive coupling to the inductive loop, isrelated to the matrix element 〈ψ|φ|0 ◦ 〉.

Limiting the Hilbert space to the six lowest-energyeigenstates {|m±〉 : m = 0, 1, 2}, we numerically com-pute the normalized matrix elements

|Oψ|2 ≡|〈ψ|O|0 ◦ 〉|2

〈0 ◦ |O†O|0 ◦ 〉

from the ground state |0 ◦ 〉 for the operators O = η, φ.Results are plotted in Fig. 4. Note that

∑ψ |Oψ|2 = 1

and |Oψ|2 > 0, so we may reasonably consider these astransition probabilities via O.

We see from Fig. 4a that transitions mediated by ca-pacitive coupling to the qubit island are only allowed

(a) (b)

FIG. 4. (a) Normalized charge matrix elements |ηψ|2 betweenthe ground state |0 ◦ 〉 and the excited state |ψ〉, showing im-munity of the qubit manifold to capacitive coupling. (b) Nor-malized phase matrix elements |φψ|2 between states |0 ◦ 〉 and|ψ〉. This matrix element is near-unity between the two qubitstates at ϕext = π, meaning the qubit manifold is primarilysusceptible to inductive loss.

from |0 ◦ 〉 to |1 ◦ 〉. These selection rules result from thedecoupling of the even and odd Cooper pair number par-ity manifolds (see App. D). Most importantly, transitionsbetween qubit states are forbidden, meaning capacitivecoupling offers a promising ingredient for indirect qubitmeasurement and control (see Sec. IV C). Conversely, in-ductive coupling to the inductive loop of the qubit per-mits transitions between |0 ◦ 〉 and |0 • 〉 in the vicinity ofϕext = π, as shown in Fig. 4b. This effect arises becausethe operator φ induces transitions between the Cooperpair parity manifolds, as can be seen from the Fourierseries for Eq. 3. As a consequence, we expect that relax-ation of the qubit will be primarily due to inductive lossin the superinductances (see Sec. III C).

B. Disorder

A highly symmetric superconducting circuit is usuallyfragile in view of unavoidable fabrication imperfections[33]. The symmetry of our circuit involving the two in-ductive arms in Fig. 2a may be broken in three parame-ters: the Josephson energies of the junctions, the capaci-tances of the junctions, or the superinductances. To an-alyze these effects, we numerically diagonalize Eq. 2 andexamine the energy splitting ∆E [35] as well as the chargedispersion ε = maxNg

∆E−minNg∆E of the {|0+〉, |0−〉}

manifold at ϕext = π. A dimensionless quantity δ ∈ [0, 1)is introduced to parameterize the extent of asymmetry inall three cases, and the δ-dependence of the energies ∆Eand ε is studied.

(a) (b)

FIG. 5. (a) Plot of the charge dispersion ε of the qubit transi-tion as a function of disorder parameters δ. Here, δJ , δC , andδL correspond to disorder in the characteristic energy scalesεJ , εC , and εL, respectively. Disorder in the area of the twoJosephson junctions is represented by δA. (b) Plot of theenergy splitting ∆E as a function of disorder parameters δ.The dashed lines and circles indicate the values of δL used forinductive disorder in Sec. III C, III D.

1. Disorder in εJ

We model disorder in the Josephson energies of thejunctions by allowing the values of εJ to deviate. Wetherefore set the left and right junction tunneling ener-gies to (1±δJ)εJ , respectively, where δJ is the aforemen-tioned asymmetry parameter. The Hamiltonian in Eq. 2is perturbed by the term

H ′ = 2εJδJ sinϕ sinφ

2.

See Fig. 5 for a plot of ∆E and ε as a function of δJ . Theimportant feature in these plots is that the charge disper-sion decreases exponentially while the splitting increasesexponentially with δJ [36]. These features arise from theeffective Hamiltonian in Eq. 4 being accompanied by 2π-periodic terms in the presence of disorder. In this caseof disorder in εJ , the approximations in Sec. II B lead toH ′ ≈ H ′eff with

H ′eff = − 16

3πεJδJ

(sinϕ− 1

5sin 3ϕ

),

which evidently permits the tunneling of single Cooperpairs across the element. The resulting qubit retainscharacteristics of the symmetric circuit as well as theasymmetric/transmon-like circuit.

2. Disorder in εC

Analogous to Sec. III B 1, we set the left and right junc-tion charging energies to εC/(1±δC), respectively. Aside

from dressing the charging energy, the Hamiltonian inEq. 2 inherits the term

H ′ = −8εCδC

1− δ2C

n(N −Ng − η),

whose effect on the qubit manifold is plotted in Fig. 5.This form for capacitive disorder is assumed due to

the fact that, if δJ = δC ≡ δA, then the product εJεCis kept constant for both junctions under the effects ofdisorder. This corresponds to the physical case where thejunction plasma frequencies are fixed by oxidation, buttheir areas differ due to fabrication imperfections. Here,the junction areas are (1 ± δA)A because A ∝

√εJ/εC .

The consequences of area disorder are plotted in Fig. 5.

3. Disorder in εL

Following the same procedure, we set the left and rightsuperinductance inductive energies to εL/(1±δL), respec-tively. This form is taken in order to fix the total linearinductance in the loop. Aside from dressing the inductiveenergy, the Hamiltonian in Eq. 2 is perturbed by

H ′ = εLδL

1− δ2L

(φ− ϕext)θ. (5)

Note in Fig. 5 that the charge dispersion and energy split-ting follow the same general trend for inductive disor-der as for the other three. The key difference is thatthe charge dispersion decreases more quickly than forany other form of disorder. Oppositely, the splittingis initially the same as for area disorder, but the slopedecreases in δL. We conclude that disorder allows usto engineer a circuit with a sufficiently non-degenerateground state manifold whose charge dispersion is largelysuppressed. For reasons that will become clear in Sec.III D, these features are extremely valuable for designinga qubit that is protected from dephasing.

C. Relaxation

We model loss due to an arbitrary channel usingFermi’s Golden Rule. This gives us the relaxation rate,including both emission and absorption, of

1

T1=

1

~2|〈0+|O|0−〉|2 [SEE(∆ω) + SEE(−∆ω)] . (6)

In this expression, O is the operator of the circuit cou-pling to a noisy bath variable E(t), the noise spectral den-sity of which is given by SEE(ω). Note that ∆E = ~∆ω.In this subsection, we calculate the relaxation rates forthe expected dominant loss mechanisms of the qubitbased on numerical diagonalization of the full Hamilto-nian in Eq. 2. We perform this calculation for variousdegrees of inductive disorder δL (see Fig. 5).

Loss channel O Quality factor T1 (ms)

δL = 0.0 δL = 0.3 δL = 0.6 δL = 0.9

Capacitive 2eNi Qcap ∼ 1× 106 [37] 780 000 17 000 1 000 18

Inductive φ0φi Qind ∼ 500× 106 [38] 0.61 0.79 1.1 1.4

Purcell 2eη Qcap ∼ 1× 106 [37] ∞ 2 500 380 470

Quasiparticle 2φ0 sin ϕi2

1/x∗qp ∼ 0.3× 106 [38] ∞ ∞ ∞ ∞

Dephasing channel λ Spectral density amplitude Tφ (ms)

δL = 0.0 δL = 0.3 δL = 0.6 δL = 0.9

Charge Ng

√ANg

∗ ∼ 1× 10−4 [39] 0.0037 0.15 74 3.3×106

Flux ϕext

√Aϕext/2π ∼ 3× 10−6 [40] 0.022 0.13 0.67 1.8

Shot np nth/Qcap ∼ 1× 10−7 [37] 4.6 4.8 5.3 8.7

Critical current εJ√AεJ /εJ ∼ 5× 10−7 [41] 210 40 8.2 2.9

TABLE II. Expected relaxation times T1 and pure dephasing times Tφ at ϕext = π through various channels. The operatorsO coupling to the bath and the noisy parameters λ are listed in addition to the associated quality factors and spectral densityamplitudes, respectively. Relaxation and dephasing times are shown for varying cases of inductive disorder: δL = 0.0, 0.3, 0.6, 0.9(see also Fig. 5). The operators Ni depict the number of Cooper pairs having tunneled across the two Josephson junctions,i.e. Ni = n± 1

2(N − η) Entries that read “∞” represent numerical infinity. *These values are reported for clarity but have no

bearing on the estimates shown (see Secs. III C, III D).

We consider four possible loss channels for the qubit:capacitive loss, inductive loss, Purcell loss, and quasipar-ticle tunneling [38]. Capacitive loss involves dielectricdissipation in the Josephson junction capacitances. Inthis case, we have O = 2eNi = 2e

[n ± 1

2 (N − η)]

forthe charge across the i-th junction and E = V for a bathvoltage with

SV V (ω) + SV V (−ω) =2~

CJQcap(ω)coth

~|ω|2kBT

. (7)

Here, CJ = e2/2εC is the junction capacitance, T is thetemperature, andQcap(ω) is a frequency-dependent qual-ity factor [42] with nominal value Qcap ∼ 1×106 [37] (seeApp. E for more details).

Depending on the specific implementation, inductiveloss may occur within the superinductances via quasi-particle tunneling [38]. This situation can be modeledby taking O = φ0φi for the flux across the i-th superin-ductance and E = I for a bath current with

SII(ω) + SII(−ω) =2~

LiQind(ω)coth

~|ω|2kBT

.

In this expression, Li = (1 ± δL)L is the i-th superin-ductance and Qind(ω) is a frequency-dependent qualityfactor with nominal value Qind ∼ 500 × 106 [38]. Asshown in App. E, this frequency dependence is includedto extrapolate to small qubit transition frequencies.

Loss due to the Purcell effect should mainly arise fromcoupling of the qubit to the plasmon mode, which wemodel as dielectric loss in the shunt capacitance. As such,we have O = 2eη and E = V for a bath voltage withthe same noise spectral density as Eq. 7, but with CJreplaced by Cshunt.

Finally, quasiparticle tunneling is expected to occuracross either Josephson junction and contribute to qubitrelaxation [38]. In this case, we have O = 2φ0 sin ϕi

2 forthe i-th junction and the noise spectral density

Sqp(ω) + Sqp(−ω) = 2~ωReYqp(ω) coth~ω

2kBT,

with ReYqp(ω) being the dissipative part of the Joseph-son junction admittance [43] (see App. E for the explicitform). Note that this admittance scales linearly in bothεJ and the quasiparticle density xqp defined relative tothat of Cooper pairs.

The calculated relaxation rates and the correspondingcomponents of Eq. 6 are shown in Tab. II for all fourloss channels at ϕext = π. For both capacitive and in-ductive loss, we note that measurements have not yetbeen performed using qubits with extremely small en-ergy splittings, and the calculation presented here relieson an extrapolation to such frequencies. A key feature isthe complete absence of quasiparticle loss, as in the flux-onium qubit [38]. Additionally, we see that the asym-metric qubit is marginally less susceptible to inductiveloss than the symmetric qubit. This improvement comesat the cost of the susceptibility to capacitive and Purcellloss. However, we emphasize that the lifetimes shownin Tab. II are conservative estimates that demonstrateT1 & 1 ms, which is at least competitive with state-of-the-art qubit implementations [13, 31, 38, 44–46].

D. Pure dephasing

In this subsection, we examine the dependence of thequbit transition energy ∆E on various system parame-

ters λ corresponding to different dephasing mechanisms.We estimate the dephasing times due to offset chargenoise, external flux noise, photon shot noise in the plas-mon mode, and critical current noise. The noise spectraldensities are all assumed to be 1/f and hence given bySλλ(ω) = 2πAλ/|ω|, where

√Aλ is the amplitude, with

the sole exception of shot noise whose spectral density isassumed to be Lorentzian.

1. Charge noise

In the case of perfect symmetry, the charge dispersionε is identically mapped to ∆E. As a consequence, re-silience to dephasing from offset charge noise demandsa high degree of degeneracy, making experimental im-plementation difficult. Fortunately, this issue can beavoided by introducing inductive disorder into the sys-tem (see Sec. III B 3). In the slow-varying limit for chargenoise, the pure dephasing rate is bounded by

1

Tφ=

π

(2e)2ε/~,

where e is Euler’s number [10]. In Tab. II, we see thatthis yields a strict bound on the decoherence time in thecase of perfect symmetry, which is greatly alleviated inthe presence of inductive disorder. Note that this esti-mate does not explicitly involve

√ANg because the slow-

varying limit has been taken where the offset charge as-sumes a random value for each measurement, but doesnot fluctuate within a given measurement.

2. Flux noise

At ϕext = π, the qubit manifold is only susceptible toexternal flux noise to second order. However, one caneasily obtain [47] the relation ∂2∆E/∂ϕ2

ext ∝ 1/∆E (atϕext = π), which shows that insensitivity to flux-noise-induced dephasing requires sufficiently weak degeneracy,at odds with the requirement for resilience to chargenoise. Once more, introducing inductive disorder avoidsthis issue. Neglecting components that depend on de-tails of the implementation, the pure dephasing rate [48]is bounded by

1

Tφ= Aϕext

∣∣∣∣∂2∆E

∂ϕ2ext

∣∣∣∣,where

√Aϕext

/2π ∼ 3×10−6 is the amplitude of the noisespectral density [40, 49] for ϕext. As in the case of chargenoise, this places a stringent limit on the decoherencetime of the qubit in the absence of inductive disorder, aswe see in Tab. II.

3. Shot noise

We expect the dominant contribution to dephasingdue to thermal photons to arise from coupling to theplasmon mode. This mode has an angular frequencyωp ≈

√16xεCεL/~ and an occupation np with mean given

by nth = 1/(e~ωp/kBT −1). To this end, we place a boundon the pure dephasing time Tφ using the expression [13]

1

Tφ= nthκ

χ2

χ2 + κ2,

where χ is the dispersive shift of the qubit on the plas-mon mode and κ = ωp/Qcap(ωp) is the linewidth ofthe plasmon mode. Note that, as in Sec. III C, we usethe frequency-dependent dielectric quality factorQcap(ω)with nominal value Qcap ∼ 1× 106 [37]. In all cases, thisform of noise is not expected to limit the decoherencetime of the qubit (see Tab. II).

4. Critical current noise

The final dephasing mechanism we investigate is criti-cal current noise, or fluctuations in the Josephson energyεJ . As in the case of flux noise, we discard factors that aresensitive to experimental details and take the dephasingrate to be bounded by [48]

1

Tφ=√AεJ

∣∣∣∣∂∆E

∂εJ

∣∣∣∣ ,where

√AεJ ∼ 5× 10−7εJ is the amplitude of the noise

spectral density [41] for εJ . As in the case of photon shotnoise, we do not expect critical current noise to place astrict bound on the decoherence time (see Tab. II).

At this point, it is clear that inductive asymmetry con-stitutes a necessary ingredient in the protection of thisqubit. In light of Eq. 5, we attribute this to the result-ing hybridization of the φ and θ modes, which are notdirectly coupled in the symmetric case (see Eq. 2). Thefluxon transition between the qubit states inherits somecharacter of the plasmon transition, thereby breaking thecorrespondence between ε and ∆E and reducing the fluxmatrix element 〈0−|φ|0+〉.

IV. DISCUSSION

A. Protection

We have proposed a superconducting circuit whoseground state manifold can be made protected, at theHamiltonian level, from multiple common sources ofnoise. Here we describe some of the other strategiesfor achieving protection. The simplest manifestations in-volve improving quality factors associated with couplingto different thermal baths [44, 50, 51] and reducing noise

spectral densities for different Hamiltonian parameters[13, 52, 53]. At a higher level, a popular approach tosuppressing qubit relaxation has been to localize wave-functions in disparate regions of phase space to lessentransition matrix elements [23, 31, 38, 54]. On the otherhand, delocalization of the same wavefunctions has beenshown to mitigate dephasing effects by reducing qubitsensitivity to Hamiltonian parameters [10, 18, 45, 55–57]. Superconducting circuits with multiple degrees offreedom whose qubit wavefunctions are both localizedand delocalized combine both of these approaches. Inthese circuits, quantum information is diffused amongconstituent local degrees of freedom, providing protectionfrom local perturbations. The many-body limit promisestopological protection [11, 16, 20, 58], in which globaloperators are necessary to manipulate logical qubits, butthe few-body case described here offers an experimen-tally realistic approximation [12, 21, 31, 33, 46, 49, 59].In the following subsection, we outline the key differencesbetween our circuit and similar proposals.

B. Comparison to other protected qubits

1. Heavy fluxonium

The heavy fluxonium [23, 54], in which a conventionalfluxonium qubit is shunted by a large capacitance, sharesmany similar features with the circuit in Fig. 2a. In fact,our circuit roughly reduces to that of the conventionalfluxonium in the extremely asymmetric limit of δL → 1,and the eigenstates of both circuits are similarly repre-sented in terms of persistent currents. The two maindifferences are that our circuit contains an additionalsmall Josephson junction and that the shunt capacitanceis placed across the junction along with a subset of thearray junctions. An analogy can be drawn between δL inour circuit and the shunt capacitance in the heavy flux-onium, because they both suppress the qubit transitionfrequency. Physically, the qubit eigenstates differ whenthese analogous parameters fall to small values. Whilethe eigenstates collapse to Cooper pair number paritystates in our circuit, this is not true for the heavy fluxo-nium because the variable ϕ is not compact.

2. Rhombus qubit

The circuit in Fig. 2a bears resemblance to the rhom-bus [60], with two central differences. First, in the rhom-bus, the superinductances are replaced by single Joseph-son junctions, or arrays of a few larger junctions [9, 31].This changes the parameter regime of the circuit fromεL � εJ to εL ∼ εJ , decreasing the amplitude of thecos 2ϕ term in the Hamiltonian. Second, the shunt ca-pacitance is replaced by a gate capacitance to a voltagesource [31]. This is akin to substituting an electrostaticgate for a shunt capacitance to obtain the Cooper pair

box [56] from the transmon [10]; overall suppression ofthe charge dispersion is traded for the ability to bias thecircuit at its charge sweet spot.

Finally, the rhombus by itself is not designed tobe a protected qubit. Rather, when multiple rhombiare arranged into a one-dimensional chain (or a two-dimensional fabric), the ground states are eigenstates ofa nonlocal operator, which provides topological protec-tion [9, 11, 16, 58]. On one hand, our qubit does notrequire such scaling to achieve protection. On the otherhand, the protection we predict is inherently susceptibleto local perturbations (see Secs. III C, III D).

3. 0-π qubit

The 0-π qubit [21] also has a similar superconductingcircuit to that in Fig. 2a, but with three essential distinc-tions. First, the pairs of superinductances on each arm ofthe circuit are combined, altering the embedding capac-itance matrix, in particular the capacitances to ground.Second, there is the addition of a second large capaci-tance shunting the inductive loop between its two hori-zontally oriented nodes. When this second capacitance isprecisely Cshunt, this permits the exact decoupling of theθ mode from the ϕ mode in Eq. 2. Third, the 0-π qubitis operated in a parameter regime where εL ∼ 0.01εJ , asopposed to our circuit, where εL ∼ 0.1εJ [33]. This ad-ditional order of magnitude in the superinductance mayprove marginally less accessible experimentally [18, 19].

Notably, the inductive loop in the 0-π qubit is threadedwith ϕext ≈ 0 at its working point instead of ϕext = π.This leads to a substantial change in the physics of theground state manifold; |0+〉 and |0−〉 are approximatelylocalized in distinct potential wells [33]. In our case, thesestates are approximately the symmetric and antisymmet-ric superpositions of the localized wavefunctions, whichare themselves localized in distinct Cooper pair numberparities.

C. Readout and control

Protected qubits face the serious obstacle of realiz-ing state manipulation and measurement while remainingsufficiently isolated from their environments to preservetheir coherence. We envision performing readout andcontrol using an ancillary mode structure, which enablescascaded dispersive readout and Raman indirect transi-tions, as outlined below.

For readout, we aim to exploit the sizable native dis-persive shift of the qubit on the plasmon mode, χ/2π ∼−20 MHz (see Sec. III D 3). Unfortunately, the small an-harmonicity, which is at most of order 10 MHz, of theplasmon mode makes readout of the plasmon mode us-ing a linear mode (e.g. a microwave cavity) difficult. As aremedy, we propose introducing an ancillary anharmonicmode by which to measure the plasmon state. In this

scheme, a dispersive interaction will be mediated by theancillary mode between the readout and plasmon modes,thereby enabling dispersive measurement with two read-out tones [61].

The above-mentioned ancillary anharmonic mode willalso be useful for control of the protected qubit. Asshown in Sec. III A, direct transitions mediated by ca-pacitive coupling to the qubit superconducting island arestrictly forbidden in the symmetric case, and weakly for-bidden in the asymmetric case. Note that this couplingmethod is chosen to minimize the contributions to de-coherence resulting from the introduction of the read-out/control circuit. For manipulation, we therefore pro-pose transitioning through the |1+〉 or |1−〉 state, as ina conventional Λ system. An additional complicationis that, even in the presence of inductive asymmetry,the joint fluxon-plasmon transitions |0+〉 ↔ |1−〉 and|0−〉 ↔ |1+〉 are forbidden.

Using an ancillary mode based on the SuperconductingNonlinear Asymmetric Inductive eLement (SNAIL) [62],it is possible to engineer selection rules where the jointfluxon-plasmon transitions are weakly permitted whilethe qubit transition is left unaffected, as has recentlybeen demonstrated in a fluxonium qubit [63]. This stemsfrom the feature of the SNAIL that allows, at specific fluxbiases, third-order nonlinearities without fourth-ordernonlinearities. If we parameterize the broken selectionrule using y ∈ [0, 1), the matrix element of |0+〉 ↔ |1−〉relative to |0+〉 ↔ |1+〉, then the gate speed via sequen-tial direct transitions and a stimulated Raman transitionis estimated to be (1 + y−1)tgate and 2∆y−1t2gate, respec-tively [64]. Here, ∆ ∼ 100 MHz is the detuning of the Ra-man drive from the intermediate state and tgate ∼ 10 nsis the π-pulse time [65]. This also introduces an addi-tional Purcell loss channel |0−〉 → |1+〉 → |0+〉 (andvice versa), which corresponds to the relaxation timeT1 ∼ y−2Qcap(ωp)/ωp. We therefore expect gate speedsto be slowed by a factor of y−1 < 10 while lifetimes arekept in the millisecond range.

V. CONCLUSION

In summary, we have designed a few-body supercon-ducting circuit in which the charge carriers are well-approximated by pairs of Cooper pairs at a particularbias point. The Josephson tunneling element that sup-ports these charge carriers is characterized by a cos 2ϕterm in the Hamiltonian, whose emergence we haveshown analytically. Our numerical simulations sup-plement these arguments and demonstrate protectionagainst a variety of common relaxation and dephasingsources. We find that this protection is substantially en-hanced in the presence of disorder. Finally, we comparedour circuit to similar proposals and offered our perspec-tives on readout and control.

As a final remark, we comment that engineering a cir-cuit whose potential energy is dominated by a cos 2ϕ

term opens the door to more exotic designs with po-tential energies of the form cosµϕ, with µ ∈ N, whichcould be obtained by introducing additional loops inthe circuit. These could be tremendously valuable forquantum simulation, realizing nearly degenerate groundstate manifolds with greater multiplicities, or performingdegeneracy-preserving measurements of photon numberparity [66].

ACKNOWLEDGMENTS

We thank J. Cohen, S. M. Girvin, L. I. Glazman, M.Mirrahimi, R. J. Schoelkopf, K. Serniak, and S. Shankarfor their insights. This work was supported by theArmy Research Office Grant No. W911NF-14-1-0011 andW911NF-18-1-0212. W.C.S. was supported by DoD-MURI award No. FP057123-C. Facilities use was sup-ported by the Yale Institute of Nanoscience and QuantumEngineering under National Science Foundation GrantNo. MRS1119826.

Appendix A: Mathieu equation

The time-independent Schrodinger equation for Eq. 1assumes the form

4EC

(i∂

∂ϕ+Ng

)2

ψ − EJ cos 2ϕψ = Eψ.

Using the trial function g ≡ e−iNgϕψ, this equation be-comes

g′′ +

(EJ

4ECcos 2ϕ+

E

4EC

)g = 0,

which is the Mathieu equation [10, 67]. The k-th eigenen-ergy is given by

Ek(Ng) = 4ECaNg+`(k,Ng)

(− EJ

8EC

),

where a denotes the Mathieu characteristic value and `is an integer that sorts the eigenvalues. The crucial dif-ference between this solution and that for the Cooperpair box is that the characteristic exponent 2(Ng + `) isreplaced by Ng + ` [10]. This gives rise to the nearlydegenerate ground states in the Hamiltonian in Eq. 1.In particular, for EJ � EC , we can use the asymptoticform of the Mathieu characteristic values to find

Ek(Ng) ≈ Ek(1/2)− 1

2εk cos(πNg)

with εk being the charge dispersion of the k-th level. Im-portantly, for even values of k, we have Ek = Ek+1 ≈(√

32EJEC − 2EC)k (neglecting constants and nonlinear

terms) and εk = −εk+1. Additionally, the expression forthe ground state splitting in Sec. I B is justified by

εk ≈ (−1)k4k+2

k!EC

√2

π

(2EJEC

) 2k+34

e−√

2EJ/EC ,

and the fact that ∆E = E1 − E0 = ε0 cos(πNg).

Appendix B: Instanton treatment

The average tunneling trajectory between potentialminima for the Hamiltonian in Eq. 2 can be found byexamining an equivalent classical problem. This is doneby inverting the potential and solving the classical equa-tions of motion for a particle that starts at one max-ima and ends at the other, each with zero kinetic energy[24]. Consequently, this classical trajectory requires aninfinite amount of time. The corresponding differentialequations,

~2

8εC

[2ϕ+

1

x(ϕ+ θ)

]= 2εJ sinϕ cos

φ

2

~2

16εCφ =

1

2εL(φ− ϕext) + εJ cosϕ sin

φ

2

~2

8xεC(ϕ+ θ) = 2εLθ,

are solved numerically. Carrying out the procedure inSec. II B, but retaining terms through the fourth har-monic and O(z2), results in the extended version of theeffective Hamiltonian

Heff = 4εC

{1

2

[1 +

1

(1 + z)2

]−1

(N −Ng − η)2 + xη2

}

+ εLθ2 − εL

(16

3π− 56

9πz

)(π − φext) cosϕ

− εJ{

1− 5

4z +

1

48[81− 2π2 − 6(π − φext)

2]z2

}cos 2ϕ

+ εL

(16

45π− 88

75πz

)(π − φext) cos 3ϕ

− εL(

1

12− 17

72z

)cos 4ϕ,

which clearly reduces to Eq. 4. At ϕext = π, we seethat the dominant correction term is of the form cos 4ϕ.Moreover, we observe that the odd Fourier terms exactlyvanish for the symmetric circuit at this flux bias, whilethe coefficients for the even Fourier terms decay roughlyin powers of z.

Appendix C: Numerical diagonalization

For numerical diagonalization of either Eq. 2 or Eq.4, we employ a charge basis for the ϕ mode in order

to efficiently capture the dependence of the Hamiltonianon Ng [68]. This is due to the exact periodicity of theHamiltonian in ϕ, which inhibits the use of a harmonicoscillator basis to capture the multi-well physics apparentin Fig. 2b. For the θ and φ modes, we use harmonicoscillator bases. Explicitly, the full Hamiltonian in Eq. 2can be written as

H =√

8εLεC a†a+

√16xεLεC b

†b

+ 2εC[N −Ng − iηzpf(b

† − b)]2

− 2εJ cosϕ cos[

12φzpf(a

† + a) + 12ϕext

](C1)

after discarding constants, where a†/a and b†/b arebosonic creation/annihilation operators for the φ and θmodes, respectively. Additionally, the zero point fluctu-

ation amplitudes φzpf =(

8εCεL

)1/4and ηzpf = 1

2

(εLxεC

)1/4have been introduced for convenience. The diagonaliza-tion basis is then

{|Npq〉 : |N | ≤ N0, p ≤ p0, q ≤ q0}

with |N〉 being the Cooper pair number eigenstates of theoperator N and |p〉/|q〉 being the photon number eigen-states of a†a/b†b. For numerical convergence, the dimen-sions N0 & 7, p0 & 7, and q0 & 30 are used, dependingon the desired truncation accuracy. The remaining oper-ators are written as

N −Ng =

N0∑N=−N0

(N −Ng)|N〉〈N |

cosϕ =1

2

N0∑N=−N0

(|N〉〈N + 1|+ |N + 1〉〈N |

)in this basis, while the final cosine term in Eq. C1 hasan exact matrix expression [26], which we do not repeathere. Although the above discussion applies to the caseof perfect symmetry, the discussion in Sec. III B allows astraightforward extension to the disordered case. Finally,we comment that, despite the high dimensionality of thematrix in Eq. C1, diagonalization can be made efficientby taking advantage of its sparsity.

Now we detail the method used to compute the wave-functions 〈N |ψ〉 and 〈ϕ, φ|ψ〉 (up to an overall phase) for|ψ〉 = |m±〉 plotted in Figs. 3b, 3c. First, the coordi-nate space wavefunction 〈ϕ, φ, θ|ψ〉 is constructed using〈ϕ|N〉 = e−iNϕ [68] and the expressions for 〈φ|p〉 and〈θ|q〉 as harmonic oscillator wavefunctions:

〈ϕ, φ, θ|ψ〉 =

N0∑N=−N0

p0∑p=0

q0∑q=0

cNpq〈ϕ|N〉〈φ|p〉〈θ|q〉,

where the complex coefficients cNpq are obtained by di-agonalization (and consistent choice of overall phase).Then, the θ-dependence is removed by projection ontoθ = 0 and normalization. This amounts to computing

〈ϕ, φ|ψ〉 =〈ϕ, φ, 0|ψ〉√∫ 2π

0dϕ∫∞−∞ dφ |〈ϕ, φ, 0|ψ〉|2

,

which is plotted in Fig. 3c. Next, the φ-dependence isremoved by invoking Eq. 3. In this step, we project φonto φ(ϕ) and normalize to obtain

〈ϕ|ψ〉 =〈ϕ, φ(ϕ)|ψ〉√∫ 2π

0dϕ |〈ϕ, φ(ϕ)|ψ〉|2

.

Finally, the Fourier transform of this function yields thecharge wavefunctions

〈N |ψ〉 =1

2π

∫ 2π

0

dϕ eiNϕ〈ϕ|ψ〉,

which are plotted in Fig. 3b.

Appendix D: Parity sectors

To explain the energy level structure at ϕext = π, wefactor Eq. 4 into the form [69]

Heff = H+ ⊕H−.

The parity sectors are governed by the Hamiltonians

H± = 4εC

[1

4(1− z)(2N + k± −Ng − η)2 + xη2

]+ εLθ

2 − εJ(

1− 5

4z

)cos ϕ.

In the above, k+ = 0 and k− = 1 while ϕ and N shouldbe viewed as conjugate operators corresponding to pairsof Cooper pairs. Note that this expression is an exactreformulation of Eq. 4. In the limit that εJ � εC , wemay Taylor expand the potential about ϕ = 0 and retainthe first few terms. This step discards the effects of theoffset charge, rendering H+ and H− identical. We thendecompose the linear part of the Hamiltonian into normalmodes, which yields

H± ≈ 4εC

(1

1− zN2n +

x

1 + zη2n

)+ εLθ

2n

+1

2εJ

(1− 3

4z

)ϕ2n −

1

24εJ (ϕn + zθn)

4(D1)

to leading order in z. The corresponding transformationis given by (

ϕ

θ

)=

(1 z

− 12 1

)(ϕn

θn

).

Eq. D1 reveals two weakly anharmonic modes: the plas-mon mode at the low frequency of

√16xεLεC/h and

a junction self-resonant mode at the high frequency of√8εJεC/h. These modes are coupled by a quartic non-

linearity, which has the primary effect of inducing a Kerrshift on the junction self-resonant mode. At frequencieslower than

√8εJεC/h, the energy level structure is that of

a two-fold degenerate harmonic oscillator, in agreement

with the simulated energy spectrum at ϕext = π (see thedashed line in Fig. 3a).

Appendix E: Coherence estimates

To expand on Secs. III C and III D, where relaxationpure dephasing times were calculated for four values ofδL, we plot in Fig. 6 the results for the entire span ofδL. In addition, in Fig. 6c we also plot the expectedrelaxation, pure dephasing, and decoherence times com-bining the effects of all considered channels. Here, theoverall decoherence time is defined using the relation1/T2 = 1/2T1 + 1/Tφ. We also note that, for the es-timate of the dephasing rate due to charge noise, we donot use the extrapolation shown in Fig. 5a for the casesof disorder in εJ , εC , and εA. Numerical instabilities arenot encounted for disorder in εL until δL ≈ 0.95 (which isoutside the plotted range for both Fig. 5a and Fig. 6b).

In the remainder of this appendix, we describe the as-sumed frequency dependence of the three quality factorsused for the relaxation time estimates in Sec. III C. First,for the dielectric quality factor, we use the form [38, 42]

Qcap(ω) = Qcap

(2π × 6 GHz

|ω|

)0.7

so that the nominal value Qcap ∼ 1 × 106 correspondsto measurements performed at the resonant frequency of6 GHz [37].

Second, for quasiparticle loss, the dissipative part ofthe admittance for a Josephson junction with tunnelingenergy εJ is given [43] to be

ReYqp(ω) =

√2

π

8εJRK∆

(2∆

~ω

)3/2

xqp

√~ω

2kBT

×K0

(~|ω|

2kBT

)sinh

~ω2kBT

, (E1)

where RK = h/e2 is the resistance quantum, ∆ is the su-perconducting gap, and K0 is the modified Bessel func-tion of the second kind. This expression holds underthe assumption that the quasiparticle bath is in thermalequilibrium at temperatures T � ∆/kB, which may notbe correct, depending on the implementation [70]. As wehave seen, the matrix elements for quasiparticle loss van-ish for this circuit, so this admittance has little bearingon our relaxation time estimates.

On the other hand, inductive loss has been suggestedto occur due to quasiparticle tunneling across Josephsonjunctions that compose superinductances [38]. If we ex-pect the frequency-dependence of the dissipative part ofthe admittance to agree between quasiparticle and induc-tive loss, then we arrive at

Qind(ω) = Qind

K0

(h×0.5 GHz

2kBT

)sinh

(h×0.5 GHz

2kBT

)K0

( ~|ω|2kBT

)sinh

( ~|ω|2kBT

)so that the nominal value Qind ∼ 500× 106 correspondsto measurements performed at the resonant frequency of0.5 GHz.

(a) (b) (c)

FIG. 6. (a) Calculated relaxation times T1 for the protected qubit as a function of inductive asymmetry δL (see Tab. II) forfour different loss mechanisms. The expected contribution due to quasiparticle loss across the small junctions is too small tobe seen on this plot. (b) Calculated pure dephasing times Tφ for the qubit for four noise channels. (c) Combined estimates ofthe coherence time T2 inferred using 1/T2 = 1/2T1 + 1/Tφ.

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