Dispersive readout of Majorana qubits

We analyze a readout scheme for Majorana qubits based on dispersive coupling to a resonator. We consider two variants of Majorana qubits: the Majorana transmon and the Majorana box qubit. In both cases, the qubit-resonator interaction can produce sizeable dispersive shifts in the MHz range for reasonable system parameters, allowing for sub-microsecond readout with high fidelity. For Majorana transmons, the light-matter interaction used for readout manifestly conserves Majorana parity, which leads to a notion of QND readout that is stronger than for conventional charge qubits. In contrast, Majorana box qubits only recover an approximately QND readout mechanism in the dispersive limit where the resonator detuning is large. We also compare dispersive readout to longitudinal readout for the Majorana box qubit. We show that the latter gives faster and higher fidelity readout for reasonable parameters, while having the additional advantage of being fundamentally QND, and so may prove to be a better readout mechanism for these systems.

Majorana-based quantum computing requires a scheme for the measurement of MZMs [27][28][29][30][31][32][33]. Such measurements take on an especially important role in measurement-only approaches to topological quantum computation, where they replace braiding for implementing quantum logic gates [25,26,34,35]. Ideally, measurements will be fast, high-fidelity, and quantum non-demolition (QND). The QND property ensures that the measured observable is a conserved quantity, and constrains the post-measurement state to be an eigenstate of the observable, such that repeated measurements give the same outcome. This is a critical requirement for measurement-only topological quantum computation with MZMs, where the measurements determine the dynamics of the system.
Looking broadly at measurement schemes for solid state quantum computing, a standard approach has been to couple a qubit-state-dependent charge dipole to the electric field of a resonator, which is used as a measurement probe [36]. A QND readout scheme for Majorana qubits based on parametric modulation of a qubitresonator coupling was recently introduced in Ref. [30]. However, the workhorse of measurement schemes for solid state qubits is dispersive readout, which has been very successful for superconducting [37,38], semiconduct-ing [39][40][41], and hybrid semiconductor-superconductor qubits [42,43]. In these schemes the resonator is tuned far off-resonance from the qubit frequency, and acquires a qubit-state-dependent frequency shift. It is natural to ask whether such a dispersive readout scheme can offer similar advantages for Majorana qubits, and to what extent it can satisfy the stringent QND requirements that are demanded by measurement-only topological quantum computation.
In this paper, we investigate a dispersive readout scheme for two prototypical Majorana qubits: the Majorana transmon [24,[44][45][46] and the Majorana box qubit [25,26]. These two designs are distinguished by whether the two topological wire segments that host the four MZMs form two distinct superconducting charge islands or a single island with a uniform superconducting phase, respectively. We calculate the qubit-statedependent dispersive shift that arises when these Majorana qubits are capacitively coupled to the electric field of a readout resonator. The size of these dispersive shifts directly determine the rate at which one can perform qubit readout by driving the resonator and observing the phase shift of the reflected field [47]. It also therefore determines the clock frequency in measurement-only approaches to quantum computation with MZMs.
Majorana qubits differ from conventional superconducting charge qubits, such as the Cooper pair box and the transmon [48], as a dispersive shift for a Majorana qubits can arise despite the fact that the interaction with the resonator does not induce (virtual) transitions between the two logical qubit states. Instead, the shifts result from (virtual) transitions to excited states outside the qubit subspace. This is especially beneficial for the Majorana transmon, where dispersive shifts arise through a qubit-resonator interaction that conserves the Majorana charge parity. For the Majorana box qubit, the Majorana parity is only approximately conserved in a limit of large frequency detuning from the resonator. Topological superconductors (gray rectangles) host MZMsγi, and qubit readout corresponds to a measurement of iγ2γ3. Majorana wavefunctions can be made to overlap either by (a) direct tunneling or (b) tunneling to a proximal quantum dot (gray circle). The resonator (illustrated by an LC oscillator) is capacitively coupled to the island charge. Alternative resonator-island coupling geometries are possible, including coupling to the quantum dot in (b). (a) Majorana transmon qubit: the topological superconductors form two distinct superconducting islands, shunted by a Josephson-junction with energy EJ (one of the islands may be grounded). The level diagram illustrates low-energy eigenstates labeled |g, ± and |e, ± . The blue and red arrows indicate allowed transitions induced by the qubit-resonator interaction. These allowed transitions conserve Majorana parity. (b) Majorana box qubit: the topological superconductors are shunted by a trivial superconductor, and the device forms a single superconducting island. The level diagram labels dressed eigenstates of the coupled superconducting island-dot system. In this case, the resonator induces transitions between dressed states of different dot occupancy and Majorana parity.
qubits, we find that dispersive shifts can be comparable to those of conventional transmon qubits [38] and nanowire quantum dots [41]. Specifically, for reasonable system parameters, we predict dispersive shifts in the MHz range. Measuring the qubit necessarily requires lifting the MZM degeneracy, and the corresponding qubit frequency is in the range 1-2 GHz. Our results suggest that sub-microsecond high-fidelity QND readout is feasible for Majorana qubits.
The remainder of the paper is organized as follows. We give a high-level introduction to dispersive coupling between a Majorana qubit and a resonator in Section II. In Section III and Section IV we describe in detail the dispersive coupling for a Majorana transmon and a Majorana box qubit, respectively. For both qubit variants, we calculate the dispersive frequency shift of a readout resonator from second order Schrieffer-Wolff perturbation theory for a range of system parameters using numerical diagonalization. We also provide simple, approximate analytical expressions. We estimate the resulting measurement timescales and fidelities (in the absence of any unwanted qubit decoherence or noise in the system parameters, see e.g. [33,49,50]) in Section V. We compare the results for dispersive readout to the longitudinal readout scheme introduced in Ref. [30]. Finally, in Section VI we discuss the implications of our results for dispersive readout of Majorana qubits.

II. LIGHT-MATTER INTERACTION FOR MAJORANA QUBITS
We begin by presenting a high-level overview of the interaction between a Majorana qubit and an electromagnetic resonator, focusing on the dispersive coupling regime. Such a coupling provides the underlying physical mechanism that can be used for dispersive qubit readout [36].
The resonator-based readout schemes considered in this paper involve capacitive coupling of the charge degree of freedom of the measured system (the qubit) to the electric field of a nearby resonator. That is, we have an interaction Hamiltonian of the form whereQ = eN is a charge operator for the qubit system, is the voltage bias on the qubit due to the resonator, and λ quantifies the interaction strength.
For simplicity, we model the resonator by a single harmonic oscillator mode with annihilation (creation) operatorâ (â † ). The physics resulting from the coupling described by Eq. (1) depends on the internal level structure of the qubit system. Given that the charge number operatorN can have off-diagonal matrix elements in the qubit eigenbasis, the absorption or emission of a resonator photon can induce a transition between eigenstates in the qubit system. The internal level structure of the qubit can lead to selection rules where only certain transitions are allowed. We will show below that different Majorana qubit designs give rise to different selection rules, and discuss the consequences of this for QND readout.
We consider two distinct types of Majorana qubits, illustrated in Fig. 1. Each topological superconductor hosts a pair of Majorana edge modes. Due to charge conservation, a minimum of two topological superconductors are required to encode a Majorana qubit, for a total of four MZMs. We identify two broad classes of Majorana qubits, depending on whether these topological superconductors form one or two distinct superconducting charge islands. The Majorana transmon qubit is representative of a configuration where the two topological superconductors form two distinct islands, and the relevant charge degree of freedom for readout is the difference in charge between these two islands. Variations of this configuration can include grounding one of the two superconducting islands, and/or introducing a Josephson tunnel coupling between the islands and ground [24]. However, it should be noted that connecting one of the islands to ground in this manner could increase the rate of quasiparticle poisoning events [50]. For the Majorana box qubit (also referred to as the Majorana loop qubit), the two topological superconductors are shunted by a trivial superconductor to form a single superconducting island. In this case, a charge dipole can be formed by tunnel coupling to a proximal quantum dot, providing a mechanism for readout.
The physics of these devices is described in more detail in the following sections, and we here only give a high-level discussion of their internal level structure and selection rules. For the Majorana transmon in Fig. 1 (a), energy levels can be labeled |g, ± , |e, ± , . . . , where g, e, . . . denote a transmon-like ladder of eigenstates, and ± denotes the eigenvalue of iγ 2γ3 = ±1, the Majorana parity we wish to measure. As indicated in the level diagram in Fig. 1 (a), only transitions that conserve the Majorana parity are allowed. This means that the Majorana parity is conserved during readout and that the interaction is manifestly QND with respect to this quantity. This stronger-than-usual form of QND measurement stems from the fractional and non-local nature of the MZMs [25], and was dubbed topological QND (TQND) measurement in Ref. [30].
For the Majorana box qubit, MZMs are tunnel coupled to a proximal quantum dot, as illustrated in Fig. 1 (b). The dot might be formed naturally between two topological superconductors due to the boundary conditions set by the superconducting/semiconducting interface [51]. In our analysis we assume that this quantum dot has wellseparated energy levels, and for simplicity only a single level that is energetically accessible.
Charge tunneling between the topological superconducting island and the dot provides a mechanism for readout. Because the superconducting island charge is no longer conserved, the eigenstates of the qubit-dot sys-tem are dressed states where the Majorana edge modes are partially localized on the dot. These dressed states are illustrated in the level diagram in Fig. 1 (b). In a readout protocol, the tunnel coupling should be turned on gradually, such that the system evolves adiabatically from the bare to the dressed eigenstates, and we label the dressed eigenstates by the states they are adiabatically connected to in the absence of tunneling. Qubit readout corresponds to distinguishing the dressed states adiabatically connected to the degenerate qubit groundspace.
As indicated in the level diagram in Fig. 1 (b), the relevant transitions for coupling to the resonator involve a single charge transfer from the island to the dot, or vice versa. This charge transfer also flips the (dressed) Majorana parity. In this case, an approximately QND interaction can still be achieved in the dispersive regime, where the resonator is far detuned from any internal transition, such that the resonator-induced transitions indicated in Fig. 1 (b) are purely virtual. However, it is a notable difference between the Majorana transmon and the Majorana box qubit readout scheme that the former has the advantage of a manifestly QND interaction independent of whether the system is in the dispersive regime or not.
As an aside, this last point can be contrasted to the readout scheme proposed in Ref. [30], where modulation of a system parameter is used to activate a parity conserving qubit-resonator coupling. This coupling arises independently of the frequency detuning of the resonator from any internal qubit transition. With the scheme in Ref. [30] it is therefore possible to achieve strong qubitresonator coupling in a regime where all parity nonconserving processes are heavily suppressed and can be neglected. We return to a brief comparison with Ref. [30] in Section V and Appendix E.
For the purpose of qubit readout, real transitions between qubit-system eigenstates are undesirable. An effective interaction suitable for readout is recovered from Eq. (1) in the dispersive regime. This refers to a coupling regime where the resonator frequency is far offresonance from any relevant transitions between qubit states that are allowed by the selection rules. The transitions to higher energy levels indicated in Fig. 1 (a,b) are then only virtual transitions. In this situation, Eq. (1) can be treated perturbatively, leading to an effective interaction of the form (for both types of Majorana qubit) Here ω r is the resonator frequency, ω q is the energy splitting between the two eigenstates used to encode a qubit, andσ z is the corresponding logical Pauli-Z operator. In general ω r,q include Lamb shifts due to the qubit-resonator coupling. Finally, χ q is the qubitstate-dependent dispersive frequency shift of the resonator. Under this Hamiltonian the qubit states can be distinguished by detecting a phase shift of the resonator under a coherent drive at the resonator fre-quency [36,37,47,48]. The speed of such a measurement is set by the magnitude of the dispersive shift χ q . We give a derivation of Eq. (2) starting from Eq. (1), for a generic multi-level system, in Appendix A. Throughout this paper we compute χ q for three different qubit types labeled q ∈ {t, mt, mb}, for a conventional transmon, a Majorana transmon, and a Majorana box qubit, respectively.
It is important to emphasize that although Eq. (2) is QND with respect to the logicalσ z operator, this Hamiltonian is an approximation to the underlying light-matter interaction, Eq. (1). The TQND property of the Majorana transmon refers to the fact that parity protection is manifest at the more fundamental level of Eq. (1). As discussed briefly above, and in more detail in the following, dispersive readout for the Majorana box qubit is not TQND in the same strong sense as for the Majorana transmon. Both the Majorana transmon and the Majorana box qubit, however, share the feature that no transitions are allowed between the two lowest energy eigenstates used to form a qubit. Instead (virtual) transitions out of the qubit subspace are used to realize a readout mechanism. This is in contrast to conventional superconducting charge qubits, such as the transmon and the Cooper pair box, where the light-matter interaction causes transitions between the energy eigenstates that define the qubit [48]. In this case, the readout mechanism introduces a source of error in the form of Purcell decay, wherein the qubit may relax via emission of a photon via the resonator [52]. (We note that we have restricted our notion of measurement back-action to the readout mechanism itself. Additional unwanted effects that may be introduced such as quasiparticle poisoning [50] or heating are not treated here.) In the following sections, we describe the dispersive readout schemes for the Majorana transmon and the Majorana box qubit in detail.

III. MAJORANA TRANSMON QUBIT
A. Model for the qubit A Majorana transmon qubit, shown in Fig. 1 (a), consists of two distinct charge islands that are shunted by a Josephson junction. Each island, labeled α ∈ {L, R}, is in a topological superconducting phase and has electron number operatorN L,R and dimensionless superconducting phase operatorφ L,R , satisfying [N α , e iφ β /2 ] = δ αβ e iφ β /2 , with α, β ∈ {L, R}. The charging energy and conventional Cooper pair tunneling between the two islands is captured by a Hamiltonian whereN ≡ (N L −N R )/2 andφ ≡φ L −φ R , E C is the charging energy due to capacitive coupling of the two islands, n g represents an offset charge, and E J is the Josephson coupling due to Cooper pair tunneling across the Josephson junction. The transmon regime is characterized by E J E C [48]. Note that we here use a convention whereN counts the number of electrons rather than number of Cooper pairs, such that we have the following action on charge eigenstates: We have neglected the capacitances of each superconducting island to ground, and assume the long-island limit where MZMs located on the same island are wellseparated.
Variations of the Majorana transmon include grounding one of the two islands (such that we can set e.g. ϕ R = 0 andφ =φ L ), and/or introducing a Josephson coupling to a bulk superconductor in addition to the Josephson coupling between the two islands [24]. These variations are qualitatively similar, and our results extend to these cases without any significant modification.
To read out this qubit, the MZMs corresponding tô γ 2 andγ 3 are brought together and the combined parity iγ 2γ3 is measured. When these two MZMs are brought together (see Fig. 1), their interaction is governed by a tunneling Hamiltonian [4,24,44,46] where E M is proportional to the wavefunction overlap of the MZMs. This expression accounts for the fact that the qubit loop might enclose an external flux Φ x , where in we have defined ϕ x = 2πΦ x /Φ 0 with Φ 0 = h/2e the magnetic flux quantum. In Appendix B we compare the direct tunneling model Eq. (5) with a model where the two islands are coupled to a common quantum dot, acting as a mediator. The main outcome of this comparison is that, when the energy penalty to occupy the quantum dot becomes large, the two models are equivalent. The full Majorana transmon qubit Hamiltonian iŝ Since iγ 2γ3 commutes withĤ MT , the eigenstates can conveniently be labeled by two quantum numbers |j, a where j = g, e, f, . . . denote a transmon-like ladder of eigenstates and a = ± denotes the eigenvalue of iγ 2γ3 = ±1. The level structure is shown in Fig. 2 A simplified Hamiltonian can be found by following the standard approach of treating the transmon degree of freedom as a Kerr nonlinear oscillator [36,48]. To keep the discussion simple, we set the offset charge and external flux to zero, n g = 0, ϕ x = 0, for the remainder of this section. We can introduce ladder operators viâ Taylor expanding the cosφ term to fourth order inφ, substituting the expressions above, and dropping fast rotating terms, we obtain the standard re-sultĤ where ω t = √ 8E J E C − E C is the transmon energy and E C is the anharmonicity.
Repeating the steps for the Majorana termĤ M yieldŝ with coefficients Under the above approximations, we see that the energy splitting of the two lowest energy levels with unequal Majorana parity, |g, ± , which we will label ω mt , is On the other hand, the "transmon transitions" |g, + ↔ |e, + and |g, − ↔ |e, − (indicated in Fig. 2) have energy splittings respectively. As we will show, despite that there is no charge matrix element between the two logical qubit states |g, ± , the fact that the two transition frequencies ω ± are non-degenerate for E M > 0 nevertheless leads to a iγ 2γ3 -dependent dispersive shift of the resonator.

B. Dispersive interaction with a resonator
The Majorana transmon qubit can be read out via a resonator that is capactively coupled to the island charge, as schematically illustrated in Fig. 1. This interaction has the formĤ while the resonator Hamiltonian is given byĤ r = ω râ †â . Here λ 2(C c /C r )E C R K /4πZ r quantifies the capacitive coupling strength, with C c the coupling capacitance, C r the resonator capacitance, Z r = L r /C r the resonator characteristic impedance, and R K = h/e 2 the resistance quantum.
We numerically diagonalize the full Hamiltonian H MT , Eq. (6), to calculate the dispersive shift χ mt defined in Eq. (A5). To assist with interpreting our results, we first calculate dispersive shifts for a conventional transmon qubit, which corresponds to the limit E M = 0.
Conventional transmon.-In this case, the qubit is encoded in the two eigenstates |0 ≡ |g, a and |1 ≡ |e, a where the choice of a = ± is arbitrary. The spectrum is shown in Fig. 2 (a). There are two primary transitions that contribute to the conventional transmon dispersive shift χ t , defined in Eq. (A5). Namely, the qubit transition |g, a ↔ |e, a , with frequency ω t , and the transition |e, a ↔ |f, a with frequency approximately given by ω t − E C / . We numerically calculate χ t as a function of the detuning parameter ∆ t ≡ ω t − ω r in Fig. 3 (b). The singularities at ∆ t = 0 and ∆ t = E C / correspond to values of ω r where the resonator is resonant with the |g, a ↔ |e, a and |e, a ↔ |f, a transitions, respectively. The regime between these two singularities, where the dispersive shift changes sign, is known as the straddling regime [48].  to χ t , come from transitions |g, + ↔ |e, + with frequency ω + , and |g, − ↔ |e, − with frequency ω − , as indicated in Fig. 2. The frequencies ω ± are close to ω t , approximately given by Eq. (12). As 2E M ω mt increases and becomes comparable to ω t , χ mt approaches a comparable magnitude to χ t , the dispersive shift for a conventional transmon, as shown in Fig. 3 (a,c). We note that the strength of the dispersive shift χ mt also depends on the offset flux ϕ x , as shown in Fig. 3 (d). Care must be taken to ensure that ϕ x = π, where χ mt vanishes and changes sign.
We can also find an approximate analytical expression for the dispersive shift. To this end, we substitute Eq. (7) intoĤ int and make a rotating wave approximation to find where This form clearly shows how energy exchange with the resonator leads to transitions between transmon levels within the same parity sector (|g, ± ↔ |e, ± , |e, ± ↔ |f, ± , etc.). From Eq. (14) we find a simple approximate expression for χ mt (see Appendix C) This is compared to the result based on exact diagonalization of the qubit Hamiltonian in Fig. 3 (c). We emphasize that Eq. (16) is a somewhat crude approximation similar in accuracy to the standard approximation used for conventional transmon qubits (Eq. (3.12) in Ref. [48]).

IV. MAJORANA BOX QUBIT
A. Model for the qubit The Majorana box qubit, shown in Fig. 1 (b), is an alternative design for a qubit based on MZMs that has been studied in the context of measurement-only topological quantum computing [25,26]. In this qubit design, the topological superconductors are shunted by a (trivial) superconducting bridge instead of a Josephson junction. This model can be thought of as a limiting case to the Majorana transmon, where E J /E C → ∞. In this limit, we haveφ R →φ L , such that the previous charge and phase operators,N = (N L −N R )/2 andφ =φ L −φ R , are zero. Instead, the relevant degree of freedom is the total chargeN tot =N L +N R , and the corresponding conjugate phaseφ tot ≡ (φ L +φ R )/2. The device acts as a single island with charging energŷ where E tot quantifies the charging energy due to capacitive coupling of the island to ground (and to the resonator), which we neglected for the Majorana transmon. The charge and phase operatorsN tot ,φ tot act on charge eigenstates analogously to Eq. (4), whereN tot now counts the total charge on the superconducting island consisting of the two topological superconductors. The Majorana box qubit also comes in several qualitatively similar variations. When the two topological superconductors are aligned horizontally in series as in Fig. 1 (b) (formed from a single nanowire), the qubit is also refereed to as a Majorana loop qubit. Alternatively, the two topological superconductors can be arranged in parallel with the superconducting shunt perpendicular to the nanowires [25]. One can also consider additional MZMs per island, used as ancilla modes for measurement-only topological quantum computing. In this case, a Majorana box qubit with four MZMs is called a tetron, with six MZMs a hexon, and so on. Our results can be generalized to these variations.
The coupling of the two topological superconductors due to a non-zero overlap of the Majorana modes corresponding toγ 2 andγ 3 can be modeled using Eq. (5) withφ → 0. However, to properly account for the movement of charge that leads to a coupling to the resonator, we take one step back and explicitly include coupling to bound states in the semiconducting region between the two topological superconductors. In the limit of where the energy penalty to occupy these bound states is large compared to the tunnel coupling, the HamiltonianĤ M can be recovered as an effective description. Including such bound states as intermediate degrees of freedom is however necessary to correctly capture the coupling to the resonator that results from charge tunneling to the semiconductor. In the proposals in Refs. [25,26], this description is moreover very natural because a gate-defined quantum dot is explicitly introduced to mediate a tunable interaction between the nanowires.
We model the quantum dot between the two topological superconductors by a single fermionic operatord, satisfying {d,d † } = 1. This degree of freedom is illustrated in Fig. 1 (b). The dot is described by a Hamiltonian H d = εd †d , with ε the dot occupation energy. Tunneling between the island and the dot is modeled by a Hamiltonian [53] where t L,R ≥ 0 are the tunneling amplitudes between the two respective topological superconductors, and we have included the possibility of an external flux, ϕ x , threading the qubit loop. The full Majorana box qubit Hamiltonian is thusĤ We show the spectrum in the uncoupled case t L,R = 0 in Fig. 4 (a). As with the Majorana transmon qubit, each state shown in Fig. 4 (a) is two-fold degenerate, a degeneracy that splits when we include non-zero tunneling |t L,R | > 0, as shown in Fig. 4 (b).
The dot-island HamiltonianĤ MB conserves the total chargeN tot +d †d , such that the Hamiltonian can be diagonalized block by block, following Ref. [30]. After a unitary transformationĤ MB =Û †Ĥ MBÛ we havê The functions ε c (n), ε m (n) and E(n) are given in Eq. (D9). As is clear from Eq. (20), the eigenstates ofĤ MB can be labeled by three quantum numbers |N, n d , a : the island charge N ∈ Z, the dot occupancy n d = 0, 1 and the Majorana parity a = ±. The dressed eigenstates ofĤ MB , Eq. (19), are thus related to the bare charge states of the uncoupled system (i.e., when t L,R = 0) through  where the unitary transformation is defined in Appendix D 1. The labels on the left-hand side here designate hybridized degrees of freedom, in particular, when t L,R > 0, the Majorana fermion hybridizes with the dot, and a = ± refers to the corresponding "dressed" Majorana parity.
To keep the discussion simple, we from now on focus on the sector with zero total dot-island charge,N +d †d = 0, and set t L,R = t, n g = 0 for the remainder of this section. Within the zero total charge sector, Eq. (20) takes the formĤ where we have dropped a constant term, andĉ satisfies {ĉ,ĉ † } = 1 and describes the movement of an electron from the dot to the island within the zero total charge sector. The coefficients are given by with δ = E tot + ε the energy penalty for moving charge from the island to the dot. For small t/δ the second term in Eq. (22) moreover reduces to Eq. (5), since In the opposite limit, if the chemical potential of the quantum dot is tuned such that ε = −E tot and the energy penalty to occupy the dot δ = 0, then ε m ∼ t.

B. Dispersive interaction with a resonator
As with the Majorana transmon, we can read out the logical state by coupling the qubit to a resonator. There are essentially two options for engineering a dipolecoupling by capacitively coupling to the resonator. Either the resonator voltage can be (predominately) coupled to the dot, or (predominantly) to the superconducting island. The key requirement for readout is that the resonator must be sensitive to the movement of charge between the superconducting island and the dot, and the two coupling schemes are in that sense equivalent (as shown, e.g., in Ref. [30].) The two choices might, however, have different practical advantages and disadvantages; in particular, stronger coupling may be possible by coupling to the island. We here focus mainly on capacitive coupling to the superconducting island charge, for concreteness, but we emphasize that our results apply equally well to both schemes. The qubit-resonator interaction is thus, in analogy with Eq. (13), given bŷ We perform the same unitary transformation that led to Eq. (22), and again set t L,R = t, n g = 0 and project onto the subspace with zero overall charge, to find an interaction in this subspace of the form (see Appendix D 1) where We note that, in the case where the quantum dot is resonant with the island and δ = 0, then g c = g m = 0 and |g ± | = λ/2. For the alternative choice of coupling the resonator to the dot, simply replaceN tot →d †d in Eq. (25) and the above results still apply with a sign change λ → −λ in Eq. (27) [30]. From the second and third line of Eq. (26) we see that, in this frame, the resonator induces a transition that involves moving an electron from the dot to the island and flipping the Majorana parity iγ 2γ3 . The energy difference corresponding to this transition is ε c ± ε m = f ± , depending on the state of the Majorana degree of freedom.
Having diagonalized the Majorana box qubit Hamiltonian, it is straight forward to use the second order Schrieffer-Wolff formula Eq. (A4) to obtain an analytical expression for the qubit-state dependent dispersive shift. Under a rotating wave approximation for the resonatorqubit interaction we find where we assume ε c > ε m . We compare Eq. (28) to a numerical diagonalization of the qubit Hamiltonian, following the same procedure as for the Majorana transmon qubit, to extract the qubitdependent dispersive shift χ mb . In Fig. 5 (a) we show χ mb as a function of ∆/|g + | for different tunneling strengths t L,R = t, where ∆ ≡ f + / − ω r . As the Majorana modes hybridize with the dot, the qubit splitting and the dispersive shifts grow larger, also shown in Fig. 4 (b). As with the Majorana transmon, the dispersive shift χ mb depends on the offset flux ϕ x , as shown in Fig. 5 (c). Again, care must be taken to ensure ϕ x = π. However, in contrast to the Majorana transmon, the dependence is more favourable, leading to a wider range of ϕ x where χ mb is close to its maximum possible magnitude.
Our results show that the energy scale of the dispersive shifts for the Majorana box qubit are comparable to the Majorana transmon qubit for comparable ratios between resonator coupling strengths and detuning ∆/g. A more detailed comparison is made in Section V.

V. READOUT TIMES AND FIDELITIES
In this section we calculate estimates of the timescales and fidelties of the Majorana qubit readout schemes presented in the previous sections. The key results are presented in Fig. 6. It is important to note that these estimates have been obtained for an idealised situation where the dispersive approximation is assumed to be valid, and no noise or decoherence is included beyond the dephasing caused by the measurement itself. These results are therefore not meant to be quantitative predictions for the measurement fidelity in an experiment, but serves to compare the speed and fidelity for different qubit types: the conventional transmon, Majorana transmon and the Majorana box qubit. For the Majorana box qubit, we also compare dispersive readout to longitudinal readout, see Appendix E.
Our predictions of the dispersive shifts of the Majorana transmon and Majorana box qubit as functions of qubit frequency ω q are compared in Fig. 6 (a). To make this comparison, we fix the value of ∆/g = −10 where g ∈ {g t , g + }, see Eqs. (15) and (27c), and ∆ ∈ {ω + − ω r , f + / − ω r }, see Eqs. (12) and (23c), for the Majorana transmon and the Majorana box qubit, respectively. We also include the dispersive shift of a conventional transmon, for which we use g = g t and ∆ = ω t −ω r . In other words, g and ∆ quantify the relevant coupling strength and resonator detuning for each qubit type, respectively.
We observe that for smaller qubit frequencies (corresponding to weaker MZM interaction energies), the Ma-jorana box qubit produces larger dispersive shifts than the Majorana transmon for the same value of ∆/g. Nevertheless, both variants may achieve dispersive shifts in the MHz regime for reasonable parameters, comparable to conventional transmon qubits [38] and the recent demonstration of a nanowire quantum dot readout in Ref. [41].
The qubit-state-dependent phase shift that arises during dispersive coupling allows for readout of the qubit by probing the resonator at its resonant frequency. The size of the dispersive shift χ directly determines the rate at which this phase shift can be resolved to a given fidelity. We quantify this effect with the signal-to-noise ratio (SNR) for a heterodyne measurement of the resonator output field. To simplify the treatment, we consider an idealized situation with unit efficiency measurement and no additional noise or decoherence, such that the qubitdependent response of the resonator is Gaussian. In this case an analytical form for the SNR can be found [36,54]: where | | is the amplitude of the resonator drive and τ is the measurement time. The SNR will in general depend on the resonator damping rate κ, and we have set κ = 2χ to give the optimal SNR at long integration times [36,54]. The measurement fidelity can be related to the SNR through We emphasize that these results hold for the ideal, dispersive Hamiltonian Eq. (2). In other words, the dispersive approximation is assumed to be valid. It should be noted, however, that this approximation will break down for large photon numbers [36].
From these expressions, we calculate the expected measurement infidelities 1 − F for each qubit as a function of integration time τ at ω q /2π = 1 GHz in Fig. 6 (b). We have chosen the resonator drive strength such that n/n crit = 1/5, wheren = 2( /κ) 2 is the resonator photon number and n crit ≡ (∆/2g) 2 . The latter can be thought of as a rough measure of when the dispersive approximation is expected to break down [36].
For comparison we also show the infidelity of a longitudinal readout scheme for the Majorana box qubit in Fig. 6 (b). We have chosen parameters such that κ and n are equal between the dispersive and longitudinal cases, which for these parameters correspond to a modulation of the longitudinal coupling strength byg z /2π 10 MHz. As shown in Fig. 8 in Appendix E such a modulation can be achieved by a very modest modulation in either the tunnel coupling or external flux. It is noteworthy that longitudinal readout gives a much faster (and thus higher fidelity) readout for this modest value of parametric modulation. For example, doubling the modulation amplitude translates to a readout that is roughly twice as fast.
Finally, we calculate the measurement integration time required to achieve a measurement fidelity of 99.99% for the dispersive readout protocols as a function of qubit splitting ω q , shown in Fig. 6 (d). For the chosen system parameters and assumptions, both Majorana qubits may achieve high-fidelity dispersive measurements in a fraction of a microsecond. Furthermore, the Majorana box qubit, which produces a larger dispersive shift, benefits from a faster readout time at the same value of ∆/g and qubit frequency.

VI. CONCLUSIONS
Our results are very promising for dispersive readout as a means to measure Majorana qubits quickly and with high fidelity. We have calculated the qubit-dependent dispersive shifts of a readout resonator for Majorana transmons and Majorana box qubits, under a simple capacitive coupling of the resonator to the qubit. This dispersive shift can be used to readout the state of the qubit by measuring the phase shift of a resonant probe tone on the resonator. We find that the dispersive shift for Majorana qubits of both types can be in the MHz range for reasonable parameters. These results are encouraging, as they indicate that well-established and extremely successful readout techniques can be adopted from the circuit QED context [36,38].
There are some key differences in the QND nature of dispersive readout for a Majorana transmon compared to a Majorana box qubit. For the Majorana transmon, the qubit-resonator interaction manifestly preserves the Majorana parity, independent of the detuning of the readout resonator from the relevant transitions between qubit energy levels. This protection originates from the fact that bothĤ MT in Eq. (6) andĤ int in Eq. (13) commute with iγ 2γ3 . The Majorana parity is therefore preserved independently of whether the perturbative dispersive approximation H disp , Eq. (2), is valid. As a result, the dispersive readout of a Majorana transmon is quantum non-demolition in a stronger sense than for conventional charge qubits.
For the Majorana box qubit, the situation is different. Here, coupling to the resonator is induced by tunneling of charge from the qubit island to a nearby quantum dot. In a readout scheme, the tunnel coupling should be turned on adiabatically, such that the system evolves into dressed joint eigenstates of the qubit-dot system [see Eq. (20)]. However, the interaction with the resonator induces transitions between dressed eigenstates of different Majorana parity; see Eq. (26). A quantum nondemolition readout is therefore only approximately recovered in a limit where the relevant transition frequency for moving an electron between the island and the dot is far detuned from the resonator frequency, leading tô H disp in Eq. (2). The readout is therefore no longer QND when the dispersive approximation breaks down, which can happen, e.g., for large photon numbers. It is worth noting that the joint Majorana-dot parity is conserved by Eq. (26). If one can also perform high-fidelity, QND measurements of the dot, it may be possible to confirm the Majorana parity by using a subsequent dot measurement after decoupling the two systems [33].
These fundamental differences makes a quantitative comparison of readout fidelity and speed more challenging. In Section V we compared the two qubits for equal qubit energy splitting, and at a fixed value of coupling strength relative to resonator detuning, g/∆, and fixed value of resonator photons relative to n crit ≡ (∆/2g) 2 . With this choice, our results suggest that the Majorana box qubit produces larger dispersive shifts, and may therefore enjoy a faster readout. However, because the breakdown of the dispersive interaction will manifest itself differently for the two qubits, this comparison might not be fair. In particular, the performance of dispersive readout for large photon numbers requires further study.
We also draw attention to the functional dependence of the dispersive shift on flux that threads the relevant loops for each respective qubit, shown in Fig. 3 (d) and Fig. 5 (c). With the likelihood that offset fluxes are present in the system, and given some distribution of these between different qubits, a challenge for the scalability of the (measurement-based) approaches is the requirement to locally tune the flux for each qubit in order to maximize readout fidelity. From this perspective we find that the Majorana box qubits are favourable, since flux tuning is likely only necessary for a limited number of qubits that have offsets very close to ϕ x = π.
Finally, we note that the longitudinal readout protocol introduced in Ref. [30] is entirely independent of the resonator detuning, and can therefore be used in a regime where the parity breaking terms for the Majorana box qubit are negligible (corresponding to a regime where the dispersive shift is negligible). Our results moreover show that longitudinal readout may lead to even faster and higher fidelity readout in practice, given that a reasonable parametric modulation is possible.

ACKNOWLEDGMENTS
We thank Andrew Doherty and Torsten Karzig for useful discussions. This research was supported by the Australian Research Council, through the Centre of Excellence for Engineered Quantum Systems (EQUS) project number CE170100009 and Discovery Early Career Research Award project number DE190100380.  Fig. 3(a). Here ε/h = 20 GHz and tL,R = t is tuned such that the energy splitting between the two lowest levels, ωmt, is equal to the corresponding case from Fig. 3(a).
This "indirect" model, wherein the MZMs interact via virtual occupation of the quantum dot can be compared to a "direct" interaction, Eq. (5) [46]: whereφ =φ L −φ R . The two models agree when δ = ε + E C is large relative to the tunnel couplings t L,R , where E C is the charging energy due to capacitive coupling between the two topological superconductors, from Eq. (3). To demonstrate this, we have numerically plotted the spectrum and dispersive shifts of a Majorana transmon qubit for both interaction terms in Fig. 7.