Top-transmon: hybrid superconducting qubit for parity-protected quantum computation

Qubits constructed from uncoupled Majorana fermions are protected from decoherence, but to perform a quantum computation this topological protection needs to be broken. Parity-protected quantum computation breaks the protection in a minimally invasive way, by coupling directly to the fermion parity of the system --- irrespective of any quasiparticle excitations. Here we propose to use a superconducting charge qubit in a transmission line resonator (a socalled transmon) to perform parity-protected rotations and read-out of a topological (top) qubit. The advantage over an earlier proposal using a flux qubit is that the coupling can be switched on and off with exponential accuracy, promising a reduced sensitivity to charge noise.

Qubits constructed from uncoupled Majorana fermions are protected from decoherence, but to perform a quantum computation this topological protection needs to be broken. Parity-protected quantum computation breaks the protection in a minimally invasive way, by coupling directly to the fermion parity of the system -irrespective of any quasiparticle excitations. Here we propose to use a superconducting charge qubit in a transmission line resonator (a socalled transmon) to perform parity-protected rotations and read-out of a topological (top) qubit. The advantage over an earlier proposal using a flux qubit is that the coupling can be switched on and off with exponential accuracy, promising a reduced sensitivity to charge noise.

I. INTRODUCTION
Condensed matter systems with quasiparticles that are Majorana fermions (equal to their own antiparticle) offer a promising platform for topological quantum computation [1,2]. Majorana fermions are non-Abelian anyons of the Ising type [3,4], for which topologically protected operations (braiding) are insufficient to implement a universal quantum computation. Bravyi and Kitaev [5] showed that two types of phase coherent operations without topological protection are needed in addition to braiding: single-qubit rotations and joint read out of up to two qubits.
There is a great variety of proposals how to implement the unprotected operations without losing phase coherence [6][7][8][9][10][11][12][13][14]. What most of these proposals have in common, is that they are sensitive both to decoherence of quasiparticle excitations and to charge noise. The latter is required to jointly read out two qubits [15], but the former can be avoided. Parity-protected quantum computation (PPQC) [8] relies on the coherent manipulation of the charge degree of freedom of small superconducting islands, without requiring quasiparticle coherence. Insensitivity to quasiparticle decoherence allows to perform quantum computations even when Majorana fermions coexist with thermally excited states [16], which typically have a very small excitation gap [17,18] (although there exist ways to increase that gap [19][20][21]).
The specific proposal for PPQC introduced in Ref. [8] is to use the Aharonov-Casher effect [22,23] to couple Majorana fermions in a nanowire [24,25] to a flux qubit [26]. The flux qubit is a nontopological superconducting qubit, which can reliably be rotated and read out by microwaves. The Aharonov-Casher effect couples to the charge modulo 2e of the nanowire. This coupling is insensitive to sub-gap excitations or Cooper pair tunneling events through Josephson junctions, which do not change the fermion parity of the nanowire. The remaining sensitivity to charge noise is minimized by decoupling the flux qubit from the topological qubit when the operation is completed.
In this paper we consider an alternative way to perform parity-protected operations on a topological qubit, with an expected reduced sensitivity to charge noise. Instead of a flux qubit, we propose to use the socalled transmon qubit, which is a superconducting charge qubit with large ratio of Josephson energy over charging energy [27][28][29].
(The transmon is placed in a transmission line resonator for read out, hence the name.) The transmon and flux qubit both couple to the fermion parity of the topological qubit, but while the coupling strength of the flux qubit can only be varied as a power law in the magnetic field, this variation is exponential in the field strength for the transmon qubit [30].
In the next section we introduce this hybrid topological-transmon qubit (abbreviated to toptransmon) by discussing the two building blocks separately. We show how the coupling can be switched on and off exponentially by tuning the magnetic field. Then, we outline how the coupling can be used to rotate the topological qubit, as well as to jointly read out sets of topological qubits. Together with braiding, these are the operations required for a universal quantum computer [5].
We also discuss how the top-transmon permits quantum state transfer between topological and nontopological qubits. This is an alternative to earlier proposals using a superconducting flux qubit [13] or normal-state charge qubit [31] as the non-topological qubit (which lacked parity protection or the possibility to switch the coupling off exponentially). At the end of the paper we give an estimate of the relevant time scales for these parity-protected operations.

II. TOP-TRANSMON
We consider a pair of superconducting islands coupled through a Josephson junction. A semiconductor nanowire with strong spin-orbit coupling (typically InAs) is placed on the islands. The superconducting proximity effect in a parallel magnetic field can produce midgap states at the end points of undepleted sections of the nanowire [24,25]. Each midgap state binds a Majorana fermion. The transmon qubit is formed by the superconducting islands, while the topological qubit is formed by sets of four Majorana bound states. A schematic diagram of this hybrid "top-transmon" qubit is shown in Fig. 1. We introduce the two building blocks in separate subsections.

A. Transmon qubit
The superconducting islands have Hamiltonian H = H C + H J , containing the capacitive energy due to a charge difference of the islands and the Josephson energy due to the tunneling of Cooper pairs between the islands [32,33].
The capacitive energy (for capacitance C) is given by The difference δN = (N 1 − N 2 )/2 of the number of Cooper pairs N i on each island changes by unity at each pair tunnel event. The induced (or offset) charge q ind remains constant during pair tunnel events, but can be varied externally by a gate voltage V (coupled to the islands via a capacitance C g ). Unpaired electrons (n i on each island) also contribute to q ind , The coefficient f C = 0 if the capacitance matrix has 1 ↔ 2 exchange symmetry, but is nonzero for an asymmetric structure. (See App. A for a calculation.) Regardless of the value of f C , the induced charge changes by ±e if an unpaired electron is transported from one island to the other.
The Josephson energy is with ϕ the (gauge-invariant) phase difference across the Josephson junction. We need a tunable Josephson energy E J , which can be achieved by replacing the single Josephson junction by a pair of identical Josephson junctions in parallel [32,33]. The Josephson energy then depends on the magnetic flux Φ enclosed by the two Josephson junctions (each with coupling energy E 0 ), The Josephson coupling is maximal when Φ is an integer multiple of the superconducting flux quantum Φ 0 = h/2e. In a quantum mechanical description the number δN and phase ϕ are conjugate operators, hence their commutator [δN, φ] = −i and in the phase basis δN = −i∂/∂ϕ. In terms of the ladder operators n ± = e ±iϕ the Josephson energy takes the form of a tunneling Hamiltonian, where n + tunnels a Cooper pair from island 1 to island 2 and n − = (n + ) † describes the reverse process. These processes govern the (differential) Cooper pair box [34].
Typically [33], in a Cooper pair box the charging energy E C = e 2 /2C is much larger than the Josephson energy E J . The spectrum is then given approximately by the eigenvalues of H C , The role of the Josephson energy is to remove degeneracies, for example, the crossing of E 1 and E 0 at q ind = e becomes an anticrossing with a gap E J . The spectrum as a function of q ind thus consists of bands with periodicity 2e. Close to the anticrossing of E 1 and E 0 the Hamiltonian can be approximated by a two-level system, with Hamiltonian This superconducting charge qubit [35,36] is deficient because of its sensitivity to fluctuations in q ind (charge noise). The transmon [27][28][29] removes the deficiency by operating in the regime E J ≫ E C . The band index n then no longer specifies the charge on the islands, since Cooper pairs can tunnel freely through the Josephson junction. The dependence of E n on q ind now has approximately a cosine form (see Fig. 2), with band width W n ∝ exp(− 8E J /E C ) and band spac- The sensitivity to charge noise can be made exponentially small by increasing E J (through a variation of the flux Φ), so that the bands become flat and fluctuations in q ind do not lead to an uncertainty in energy (which is the origin of dephasing).
In a typical device [28], the band spacing (E 1 − E 0 )/h varies in the range 3−5 GHz, for ratios E J /E C increasing from 10 to 30. The band width, in contrast, drops by two orders of magnitude from W/h ≃ 100 MHz to 1 MHz.
The state of the transmon qubit can be read out by sending a microwave probe beam through the transmission line resonator [30,37,38]. The resonance frequency ω res depends on whether the transmon is in the ground state (|E 0 , s z = −1) or in the first excited state (|E 1 , s z = +1), according to Here ω 0 /2π ≃ 10 GHz is the bare resonance frequency and g/2π ≃ 100 MHz the qubit-resonator coupling strength. The detuning ω 0 − (E 1 − E 0 )/ is large compared to g, so that the resonance is only slightly shifted. Such small shifts can be measured sensitively as a phase shift of the transmitted microwaves.

B. Topological qubit
A pair of Majorana bound states forms a two-level system and hence a qubit. The fermion parity is even in the lower state |0 and odd in the upper state |1 . A logical qubit is constructed from two of these fundamental qubits [2], hence from four Majorana bound states. Without loss of generality we may assume that the combined fermion parity is even. The state of the logical qubit is then given by |ψ = α|00 + β|11 , |α| 2 + |β| 2 = 1.
The read-out operation projects |ψ on the state |00 or |11 . This is a fermion parity measurement of one of the two fundamental qubits that encode the logical qubit [10]. For a universal quantum computation it is also needed to perform a joint parity measurement on one fundamental qubit from the logical qubit plus one additional ancilla qubit [5]. Hence the required parity measurements will involve either two or four Majorana bound states. We denote this fermion parity by n M ∈ {0, 1}.
The Majorana bound states are located at the end points of undepleted segments of the nanowire, initially all on one of the two superconducting islands. The voltage V is adjusted so that initially q ind = 0 (modulo 2e). Gate electrodes (not shown in Fig. 1) transport onto the other island the Majorana bound states that are to be measured. The induced charge changes as a result of this operation, q ind → en M (modulo 2e), and so directly couples to the required fermion parity.

III. PARITY-PROTECTED OPERATIONS
In this section we show how the transmon qubit can be used to perform parity-protected operations on the topological qubit. Topologically protected braiding operations can be performed by means of T-junctions of nanowires [39,40], and will not be considered here.
Common to all parity-protected operations is that the flux Φ through the Josephson junction is kept close to zero (modulo Φ 0 ) both before and after the operation. The ratio E J (Φ)/E C is then much larger than unity, hence the coupling between transmon and topological qubit is exponentially small. During the operation Φ is adjusted to a value close to Φ 0 /2 (modulo Φ 0 ), so that the transmon becomes sensitive to the fermion parity of the topological qubit.

A. Read out and phase gate
The operations of read out and phase gate (= singlequbit rotation) proceed in the same way as in Ref. [8], where the topological qubit was coupled to a flux qubit rather than to a transmon (with the key difference that there the coupling could not be switched off exponentially). We summarize the procedure.
To read out the topological qubit, two of the four Majorana fermions that encode the logical qubit are moved from one island to the other. (A joint parity measurement on two topological qubits can likewise be performed by moving four Majorana fermions to the other island.) Depending on the fermion parity n M , the level spacing ∆E = E 1 − E 0 of the transmon qubit is given by in view of Eq. (8) with q ind = en M (modulo 2e). A measurement of ∆E by microwave spectroscopy thus determines n M . The read-out operation projects the state |ψ in Eq. (10) onto either |00 or |11 . A single-qubit rotation is a unitary operation, rather than a projective measurement. For that purpose one would couple the transmon to the topological qubit without microwave irradiation. The transmon is initialized in the ground state |E 0 . In a time τ the coupled topological qubit evolves as α|00 + β|11 → αe i(U0−W0/2)τ / |00 (The transmon stays in the ground state during the operation, so it factors out of the wave function.) If the coupling time is chosen such that τ W 0 /2 = θ one performs a θ-phase gate operation. Eq. (12) amounts to a rotation of the logical qubit by an angle 2θ. While π/2 rotations can be performed by braiding, other rotation angles require breaking of the topological protection. For a universal quantum computation a π/4 rotation is sufficient [5], obtained from Eq. (12) by choosing θ = π/8.

B. Quantum state transfer
The operation of quantum state transfer starts from the topological qubit in the state |00 and the transmon in an arbitrary unknown superposition α|E 0 + β|E 1 of ground state and first excited state. At the end of the operation the topological qubit is in the state α|00 + β|11 , while the transmon is no longer in a superposition state. The reverse operation is also possible (transfer of an unknown state from the topological qubit to the transmon).
As we will show in this subsection, quantum state transfer can be performed by a combination of topologically protected braiding operations, plus a parityprotected cnot operation on the top-transmon. The procedure is an alternative to the quantum state transfer proposals of Refs. [13,31], which coupled the topological qubit to a flux qubit. By using a transmon rather than a flux qubit, we can offer both parity protection (no sensitivity to quasiparticle excitations) and the ability to switch the sensitivity to charge noise off in an exponential manner.
The cnot operation that we need is a conditional σ x operation on the transmon, where the condition for the σ x operation (so a switch |E 0 ↔ |E 1 ) is that the topological qubit is in the state |11 . This is a cnot gate for the top-transmon, with the topological qubit as the control and the transmon as the target.
The σ x operation on the transmon is a π-pulse of microwave radiation, at the resonant frequency ∆E/ . As before, two of the four Majorana fermions that encode the topological qubit are moved from one island to the other. The resonant frequency then depends on their fermion parity n M through Eq. (11). If the microwave radiation is resonant for n M = 1, then for n M = 0 it is detuned by an amount W 0 + W 1 . The π-pulse therefore performs the required cnot operation if the detuning is larger than the resonance width, which is the Rabi frequency Ω Rabi .
The two diagrams in Fig. 3 show, following Ref. We explain diagram (a): The topological qubit (upper line) starts out in the state |00 and is transformed by the Hadamard gate into the superposition state (|00 + |11 )/ √ 2. The transmon qubit (lower line) starts out in the (unknown) state |ψ = α|E 0 + β|E 1 . The subsequent cnot-gate entangles the two qubits, producing the state We now measure the transmon by probing the microwave resonator. If the transmon is in the ground state |E 0 , no further operation is required. If it is in the excited state |E 1 , then a final not operation on the topological qubit completes the quantum state transfer.

C. Time scales for parity protection
The operation time of a π/8 phase gate is τ = π /4W 0 , which for W 0 /h ≃ 100 MHz corresponds to τ ≃ 1.3 ns. For quantum state transfer the cnot gate needs a πpulse of duration π/Ω Rabi . The Rabi frequency should be small compared to 2W 0 /h ≃ 200 MHz. Choosing Ω Rabi /2π = 10 MHz, the operation time of the cnot gate is 50 ns. These operation times are much shorter than the coherence time of the transmon qubit, which is in the µs range [28].
The fundamental limitation to parity protection is the incoherent tunneling of unpaired electrons between the superconducting islands, a process called "quasiparticle poisoning". Since these tunnel events change the fermion parity, they break the parity protection of the operations. One would expect the density of unpaired quasiparticles to vanish exponentially as the temperature drops below the critical temperature of the superconductor, but experimentally a saturation is observed [42]. Still, the characteristic time scale for quasiparticle number fluctuations, which sets the upper limit for parity protection, becomes sufficiently large: Ref. [43] finds 2 ms in Al below 160 mK.

IV. CONCLUSION
In conclusion, we have shown how a superconducting charge qubit can be used to read out and rotate a topological qubit. These are the two operations that cannot be performed by braiding of Majorana fermions, but which are needed for a universal quantum computer [5]. Our proposal is an alternative to the read-out and rotation by means of a superconducting flux qubit [8].
In both designs, the superconducting qubit functions as a fermion parity meter. The flux qubit measures the fermion parity through the Aharonov-Casher effect, while the charge qubit relies on the 2e periodicity of the superconducting ground state energy. Both parity meters are insensitive to subgap excitations (parity protection).
The advantage we see for a charge qubit over a flux qubit is that the coupling to the topological qubit can be made exponentially small, by increasing the ratio of Josephson energy E J over charging energy E C . A superconducting charge qubit with adjustable E J /E C (a socalled transmon qubit [27]) functions as a parity meter with an exponential on-off switch.
The hybrid design of a coupled transmon and topological qubit (top-transmon) retains the full topological protection with exponential accuracy in the off-state (E J /E C ≫ 1). In the on-state (E J /E C 1) the qubit is sensitive to charge noise, but still protected from noise that preserves the fermion parity.
The experimental realization of a top-transmon is a major challenge, involving a variety of design decisions that go beyond this proposal. In Fig. 4 we give an impression of one design, to inspire further progress in this direction.

Acknowledgments
We have benefited from discussions with L. DiCarlo and V. Manucharyan. This research was supported by the Dutch Science Foundation NWO/FOM and by an ERC Advanced Investigator Grant. The parity-protected read-out and rotation of a topological qubit requires the measurement of the charge modulo 2e of a nanowire containing a pair of Majorana bound states at the end points. Most directly, this measurement of the fermion parity n M ∈ {0, 1} can be performed by bringing the nanowire in electrical contact with a Cooper pair box: a superconducting island coupled to a grounded superconductor via a Josephson junction. (See Fig. 5, left panel.) The Cooper pair box works as a parity meter because its energy spectrum E n (q ind ) is mapped onto E n (q ind + en M ) when the Majorana fermions are brought on the island. If the charge q ind induced by the gate is calibrated to zero before the operation, the spectrum E n (en M ) directly determines the fermion parity. With a single island, it is difficult to reach a large capacitance without effectively grounding the island. For that reason, the transmon uses two superconducting islands, connected to each other via a Josephson junction and disconnected from ground [27]. (See Fig. 5, right panel.) The spectrum of this differential Cooper pair box is not 2e-periodic in the total charge Q tot = Q 1 + Q 2 on both islands, so it cannot measure the fermion parity of external charges. If Majorana fermions are brought onto one of the islands from the outside, this operation changes Q tot in an unknown way -preventing the determination of the fermion parity.
To see how the 2e-periodicity is lost, we calculate the electrostatic energy of the two islands, determined by the charges Q i on the islands, the charges q i = C g,i V induced by the gates, and the elements C inv ij of the inverse of a (symmetric) capacitance matrix C.
Island i contains N i Cooper pairs plus n i unpaired electrons, so Q i = 2eN i + en i . We assume that N tot = N 1 + N 2 is even, so that δN = (N 1 −N 2 )/2 ∈ Z. (This is without loss of generality, since if N tot is odd we can make it even without changing the energy by redefining N 1 → N 1 − 1, q 1 → q 1 − 2e.) In terms of N tot and δN , the energy (A1) takes the form where the constant term is independent of δN . The induced charge q ind is given by We see that q ind depends on the total number of Cooper pairs N tot , if C 11 = C 22 . As a consequence, when external charges are brought onto the island, the value of q ind changes by an unknown amount, in general unequal to a multiple of 2e. This prevents a determination of the fermion parity of the external charges.
In order to use the differential Cooper pair box as a parity meter, the Majorana fermions should be transported from one island to the other. This changes n 1 → n 1 + n M , n 2 → n 2 − n M , hence q ind → q ind + en M -at constant N tot . That is the method described in the text.