Majorana mode quantum computing device and method

Abstract

The invention relates to a quantum computing device comprising: - a microwave cavity (MC) having a separate or coincident input port (IP) and output port (OP); - a superconducting electronic device (SD) capacitively coupled to the microwave cavity and configured to support a chain of 2N Majorana modes (MM1 – MM4, MM2', MM3'), where N is a positive integer; and - coupling means for applying microwave excitations between each pair of adjacent Majorana modes in the chain. It also relates to a method for realizing a braiding quantum gate on a qubit using a quantum computing device. Figure for the abstract: Fig. 4

Description

Title of the invention: Majorana mode quantum computing device and method The invention lies in the field of quantum computing and more generally quantum information. Majorana fermions are hypothetical elementary particles with spin r / 2 that are their own antiparticle. More recently, this term—or expressions such as "Majorana states" and "Majorana zero modes"—has been used to refer to a particular type of quasiparticle in solids, especially in topological superconductors. A Majorana mode corresponds to a zero-energy excitation (hence the term "zero mode") that can occur, in particular, at the vortex level in a topological superconductor. By definition, a Majorana mode is represented by a self-adjoint operator: f__ 2'o ~ 70 It is understandable that an isolated Majorana mode cannot be used to define a creation or annihilation operator, as these two operators would coincide, which is contradictory. However, this difficulty can be overcome by considering two Majorana modes, identified by indices (1) and (2). We can then define operators which satisfy the usual anticommutation relation of fermions: { ei 2 ■ C12} = and can therefore be considered, respectively, a creation and annihilation operator. Since two Majorana modes are needed to define a creation or annihilation operator, such a mode is generally said to constitute "a half-fermion". The fact that a pair of separate modes is necessary to define the creation and annihilation operators of a fermion introduces a non-local quantum correlation between these modes, which drastically alters their quantum nature. Furthermore, the exchange of two Majorana modes (in real space or in an appropriate parameter space) produces a state with the same energy as the initial state, which is related to the initial state not by a simple phase factor, as with bosons or fermions, but by a unitary transformation:

[0011] = exp --7o '

[0012] If we consider three Majorana modes with indices (1), (2) and (3) and consider the exchanges (1) - (2) and (2) - (3), we can demonstrate that the operators Ui2 and U23 do not commute. We then say that the exchange process is non-abelian.

[0013] Majorana mode exchanges are also called "braiding" operations because the world lines representing these modes in spacetime form a kind of braid.

[0014] It has been shown that a set of four Majorana modes can encode a qubit and that braiding operations can be used to create quantum gates known as "Clifford gates," which are useful for performing quantum calculations. Other operations on Majorana modes also allow the manipulation of qubits, notably their "fusion" (which is equivalent to a projective measurement).

[0015] For a more detailed introduction to Majorana modes and their application to quantum computing, see (Sato 2007) and (Beenakker 2019).

[0016] Majorana modes represent a particularly promising approach for the realization of a quantum computer because braiding operations, since they perform transformations between locally indistinguishable states, are protected from local sources of decoherence ("topological protection"). However, there are considerable obstacles to overcome for the realization of a quantum processor using Majorana modes.

[0017] It is known, for example, to realize Majorana modes from a one-dimensional superconducting structure (nanofil) exhibiting strong spin-orbit coupling; see, for example, (Cottet 2013). However, the exchange of Majorana modes cannot be performed in a strictly one-dimensional geometry. More complex superconducting circuits, exhibiting a two-dimensional or lattice geometry, have been proposed—see, for example, (You 2014) and (Vijay 2016)—but the realization of such circuits faces insurmountable technological difficulties to date.

[0018] The invention aims to overcome these drawbacks and to propose a Majorana mode quantum computing device whose realization does not pose major technological difficulties.

[0019] According to the invention, this objective is achieved by coupling a superconducting electronic device, preferably one-dimensional, supporting Majorana modes to a microwave cavity and using a photonic degree of freedom of the cavity to perform the braiding. The coupling of a one-dimensional superconducting electronic device supporting Majorana modes to a microwave cavity has already described in (Cottet 2013), but not the use of this coupling to achieve braiding.

[0020] An object of the invention is therefore a quantum computing device comprising: - a microwave cavity having an input port and an output port, distinct or coincident; - a superconducting electronic device capacitively coupled to the microwave cavity and configured to support a chain of 2N Majorana modes, where N is a positive integer; and - coupling means to apply microwave excitations between each pair of adjacent Majorana modes in the chain.

[0021] According to particular embodiments:

[0022] - The device may also include a suitable electronic measuring circuit to measure at least one quadrature of a microwave field originating from the cavity's output port.

[0023] - The device may also include at least one signal generator microwave configured to generate: - microwave pulses at a first frequency resonant with the cavity; and - microwave pulses at a second frequency not resonant with the cavity and to selectively apply these pulses to said coupling means and to the cavity input port.

[0024] - N can in particular be a multiple of 3, whereby the electronic device superconducting supports at least one group consisting of four Majorana modes capable of encoding one qubit plus two auxiliary Majorana modes. In this case, the device may also include an electronic control circuit configured or programmed to drive the microwave signal generator so as to: a) apply a first microwave excitation at the first frequency between a fourth and a fifth Majorana mode from one end of the chain or a portion of the chain comprising six Majorana modes; then b) apply a second microwave excitation at the first frequency between a first and second Majorana mode from said end; then c) simultaneously apply a third microwave excitation at the second frequency between the second and third Majorana modes from said end; a fourth microwave excitation at the second frequency between the third and fourth Majorana modes, the third and fourth microwave excitations being out of phase; and a fifth microwave excitation at the first frequency at the cavity's inlet port; then d) apply a sixth microwave excitation to the first frequency between the second and third Majorana modes; then e) simultaneously apply a seventh microwave excitation at the second frequency between the fourth and fifth Majorana modes; an eighth microwave excitation at the second frequency between the fifth and sixth Majorana modes from said end; and a ninth microwave excitation at the first frequency at the cavity's inlet port; the order of operations c) and d) can be reversed.

[0025] - The electronic control circuit can be configured or programmed to drive the electronic measurement circuit to measure a component of the microwave field from the output port of the cavity in phase with the excitations at the first frequency after the application of each excitation or group of excitations.

[0026] - The electronic control circuit can also be configured or programmed to drive the microwave signal generator so as to: simultaneously apply a ninth and tenth microwave excitation at the second frequency between two adjacent pairs of adjacent Majorana modes, and to drive the electronic measurement circuit to measure two components of the microwave field from the cavity output port in phase and in quadrature with the excitations at the first frequency.

[0027] - The superconducting electronic device may include a semi-semi-nanowire conductor exhibiting spin-orbit coupling, arranged in correspondence to an antinode of the electric field of a cavity mode and on which a superconducting material is deposited, the device also comprising a magnet generating a magnetic field parallel to the nanowire.

[0028] The superconducting electronic device may also include 2N-1 electrodes allowing the application of electrostatic potentials to generate Majorana modes, said electrodes also constituting said coupling means for applying microwave excitations.

[0029] Another object of the invention is a method for realizing a quantum braiding gate on a qubit using such a quantum computing device, the method comprising the following steps: a) application of a first microwave excitation at the first frequency between a fourth and a fifth Majorana mode from one end of the chain or a portion of the chain comprising six Majorana modes; then b) application of a second microwave excitation at the first frequency between a first and a second Majorana mode from said end; then c) simultaneous application of a third microwave excitation at the second frequency between the second and a third Majorana mode from said end extremity; a fourth microwave excitation at the second frequency between the third and fourth Majorana modes; and a fifth microwave excitation at the first frequency at the cavity's entry port; then d) application of a sixth microwave excitation to the first frequency between the second and third Majorana modes; then e) simultaneous application of a seventh microwave excitation at the second frequency between the fourth and fifth Majorana modes; of an eighth microwave excitation at the second frequency between the fifth and a sixth Majorana mode from said end; and of a ninth microwave excitation at the first frequency at the cavity's inlet port; the order of steps c) and d) can be reversed.

[0030] The method may also include measuring a component of the microwave field from the cavity output port in phase with the excitations at the first frequency after the application of each excitation or group of excitations.

[0031] A method for realizing a quantum T-gate on a qubit using a quantum computing device according to claim 4 when it depends on claim 3, the method comprising the following steps: apply simultaneously two microwave excitations at the second frequency between two adjacent pairs of adjacent Majorana modes, and measure two components of the microwave field from the cavity output port in phase and in quadrature with the excitations at the first frequency.

[0032] Other features, details and advantages of the invention will become apparent from the description given with reference to the accompanying drawings provided by way of example, which represent, respectively:

[0033] [Fig.1], a four-mode Majorana system in a microwave cavity;

[0034] [Fig.2A], the cavity output signal as a function of time during an operation of fusion of two modes of Majorana of the system of the [Fig.l];

[0035] [Fig.2B], a representation in the I / Q plane of the photonic field in the cavity during of said merger operation;

[0036] [Fig.2C], the evolution of the density matrix of the superconducting device as a function time during said merger operation;

[0037] [Fig.2D] and [Fig.2E], two microwave pulse sequences to perform said fusion operation and a parity measurement;

[0038] [Fig.2F] and [Fig.2G], the states of the photonic field of the cavity during the sequences of figures [Fig.2D] and [Fig.2E], respectively;

[0039] [Fig.3A], a sequence of microwave pulses to perform an operation of braiding of the Majorana modes of the system of the [Fig.l];

[0040] [Fig.3B], the evolution of the photonic field of the cavity during the third impulse of the sequence of the figure [Fig.3A];

[0041] [Fig.3C], the evolution of the system during said braiding operation on a Bloch sphere;

[0042] [Fig.4], the schematic diagram of a device according to an embodiment of the invention, using six Majorana modes;

[0043] [Fig.5A] and [Fig.5B], two sequences of alternative pulses to perform respective braiding operations of the Majorana modes of the system of the [Fig.4];

[0044] [Fig.6] an example of a physical realization of the device in [Fig.4]; and

[0045] [Fig.7] a sequence of pulses to perform a "T-gate" type operation on the Majorana modes of the system of [Fig.4].

[0046] In the following, the term “microwave frequencies” will refer to the frequency range between 300 MHz and 300 GHz, and more particularly between 1 GHz and 100 GHz.

[0047] Figure 1 represents a linear chain of four Majorana modes MM1, MM2, MM3, MM4, which can be physically realized in a superconducting device, capacitively coupled to a microwave cavity CH that can be represented by an individual photonic mode at frequency wc. Each Majorana mode is associated with a respective self-adjoint operator y2' y^nc. The interaction between adjacent Majorana modes generates Energy shifts Cp €M, €R qUi decrease exponentially with increasing distance between Majorana modes. These energy shifts, in principle, break the topological protection, but, as will be discussed later, it is possible to make them sufficiently small to preserve the exponential scaling of the topological protection over a wide range of control parameters, thus allowing the invention to be implemented on various existing technology platforms.

[0048] The low-energy Hamiltonian H of the system can be written as the sum of three terms: a term Hcav represents the energy of the photonic field of the cavity, a term Hei represents the energy of the Majorana modes and a term Hint represents the coupling between the latter and the cavity: 100491 + / 7«,,

[0050]

[0051]

[0052]

[0053]

[0054]

[0055]

[0056]

[0057]

[0058] where g are respectively the annihilation and creation operator of the photonic field in the cavity and gL, gM and gR the coupling coefficients between pairs of adjacent Majorana modes and this photonic field. From the Majorana operators yyy$, yi 1, it is possible to define topological charges. For a chain of four Majorana modes, there are three ways to pair these modes to form two topological charges corresponding to fermionic creation operators: The electronic system can therefore be expressed in the bases {hL 3, yy ; PL y. y} where JJ € |( . / y yK yyy The parity operator associated with one of these topological charges is given by y - ny It is then considered that electron-photon coupling can be modulated at a coRF frequency in the microwave range: ÿo UJ ~ po -t- ■' with O=L, M, R and 4>0 being a phase, for example by means of grids (insulated electrodes) capacitively coupled to respective regions of the superconducting device (left, middle, right). In this way, the microwave excitation modulates the g0 coupling via a modulation of the superposition between Majorana modes, and therefore of energy shifts: y? ) — L 4- L 4- y ) where y is a constant term whose value can be varied by applying a DC voltage to the grid (i=L, M, R). An electromagnetic field CF develops in the cavity when a coupling coefficient, for example gL, is modulated at the cavity's resonant frequency: ^RF = ^c. This field is directly related to the parity of the left-hand side of the Majorana mode chain. The components Hei and Hint of the low-energy Hamiltonian can also be written as:

[0059] füa

[0060] The low-energy Hamiltonian can be rewritten in a frame of reference rotating at frequency by neglecting a static term, proportional to in accordance with the rotating wave approximation:

[0061] .. h ; _ H = / / .V 4- 4- £ C

[0062] This induces an effective coupling between the Majorana modes MMi and MM2 which can be used to measure their parity p = fvv across the CF field, as 12 illustrated by figures [Fig.2A] to [Fig.2G]. It is therefore possible to fuse the Majorana modes MMi and MM2 and to detect this fusion using photons from the cavity, extracted via a PS output port (a PE input port, coinciding with or separate from PS, is also provided to directly excite a cavity mode)

[0063] Figure [Fig. 2A] shows the evolution over time t of the output signal aout of the cavity, representative of the excited electromagnetic field within it, assuming that The modulation of the gL coupling begins at t=0. A transient is observed whose duration is of the order of magnitude of 1 / k, where k is the line width of the cavity. The amplitude of the signal after the transient is shows a representation of the output field in the I / Q plane; we note that the position of the circular spot representing the coherent mode of the field depends on the parity, which allows the measurement of the latter, and on the phase. The contrast, y, can be much greater than the width of the spots even in a strongly topological regime (g ~* 0). provided that the cavity width is sufficiently small.

[0064] The parity measurement through cavity photons, which allows the fusion of two Majorana modes, requires taking into account the two other Majorana modes, previously considered to be decoupled. By taking into account the other Majorana modes (MM3, MM4), the coupling between these modes and the cavity does not can no longer be considered purely longitudinal: there is also a transverse component of the coupling which induces a temporal evolution of the parity operator. This evolution can, however, be neglected provided that... for O=L, M, R, for example, eo, g <0.1 ûJc.

[0065] The consequences of the Majorana MMiet mode fusion operation MM2 on the additional modes MM3, MM4 are not trivial. Indeed, in the basis of the two operator charges number the fusion operation projects the two charges into a

[0066] entangled state: for Pi2=l and

[0067]

[0068]

[0069]

[0070]

[0071] for P12=-l. Entanglement can be observed by measuring the central charge, which is done in the same way as in measurement fusion, since it involves measuring the parity associated with two adjacent Majorana modes, MM2 and MM3. At the same time, the projection expressed in the basis of charges nL, hR gives a state vector; the parity measurement of modes MM3 and MM4 is therefore deterministic. [Fig.2C] represents the temporal evolution of the four components Pqq' Cf ^01' l^e 'a density matrix of the Majorana mode chain expressed in the basis L, R. We observe that we start from an entangled state and that the coherences (elements outside the diagonal of the matrix) cancel each other out during the projective measure. [Fig. 2D] and [Fig. 2F] respectively illustrate the microwave excitation sequence and the measurement sequence used to perform and measure the MM 2 ct MM fusion. First, ff is modulated at the frequency for a time greater than 1 / k (and typically less than 10 / k, for example on the order of 3 / k, since the maximum permissible measurement time is limited by the parity lifetime) to perform the fusion, and a projective measurement is performed on the cavity field. Then the parity P^ = iy y is measured by modulating M at the frequency and by performing a second measurement of the cavity field. This second, probabilistic measurement depends on the state in which the Majorana mode chain was projected, |J or I. Alternatively, it is possible to measure parity p using the sequences of [Fig.2E] and [Fig.2G]. We note that, in this case, the second measurement of the field is deterministic. If we consider k = 1 MHz, which is easily obtained for example with coplanar waveguide cavities, the measurement time is on the order of a few ps, which requires a parity lifetime of a few tens of ps. Charge-cavity couplings on the order of g = 100 MHz can be obtained, which is consistent with the aforementioned condition. Assuming that the coupling strength can being modulated by 10% it is therefore possible to perform a "one-shot" reading of the cavity without requiring electrical manipulations.

[0072] The fusion operation requires only measuring the parity operator between adjacent links in the linear chain of Majorana modes (a "link" being formed by a pair of adjacent Majorana modes) or, equivalently, activating coupling between adjacent pairs of such modes. Braiding is in principle more restrictive because it requires reading the parity corresponding to distant Majorana modes, for example MM2 and MM4. According to a widespread misconception, this would require a two-dimensional or at least lattice geometry, since it seems difficult to "jump over" a Majorana mode (MM3, in this case) in a strictly one-dimensional geometry. An idea underlying the present invention is that this is actually made possible by coupling with the microwave cavity, using two pulsed excitations to modulate the coupling coefficients gM and gR.This effectively converts the one-dimensional system into a two-dimensional system via line-to-star mapping. This allows for the creation of dynamically reconfigurable two-dimensional arrays through the use of cavity photons.

[0073] Starting from the Hamiltonian 'Aj U % / 3 “r* ïÿ We consider microwave excitations detuned with the cavity (^RF ^c) between Majorana modes MM2 and MM3 and Majorana modes MM3 and MM4. These excitations induce the following unitary transformation:

[0075] Which gives the following Hamiltonian:

[0076] _ , H • ■■■■ 4- 1 / 2)

[0077] More generally, considering a chain comprising an arbitrary number of Majorana modes, the use of a resonant cavity coupled longitudinally to said modes and non-resonant excitations makes it possible to obtain an effective Hamiltonian having the form:

[0078]

[0079] Where fnm is a linear function and 0=^-0)^ the mismatch between the excitation and the cavity. This shows that the system consisting of a chain of Majorana modes (whatever its physical realization) coupled longitudinally to a microwave cavity, with means (for example grids) allowing the application of microwave excitations between adjacent Majorana modes, is equivalent to a 2D network, which allows manipulations such as T-gates and, above all, braiding.

[0080] The protocol for performing the braiding of Majorana modes M1 and M4 is illustrated in [Fig. 3A]. First, starting from a state M1 of the system, a modulation at the frequency wc of the coupling QM followed by a reading of the cavity field allows a reading of the parity P^, after which the system is in a state Mp. For example, we consider the case where jVf = |lffi0ex) in the case P23 = 1 and Mj = Mq. Second, a modulation at the frequency of the coupling ff followed by a reading of the cavity field allows a reading of the parity P^, after which the system is in a state M2. In the example considered here, P12 = 1 and M9 = |lffi0B). Third, a measurement of the parity P24 is performed. 2 IL R. t This measurement involves two non-adjacent Majorana modes and is more complex: it requires modulation at the frequency (xJc) of the 9M coupling and simultaneously, but with a phase shift (ideally of ji / 2, in any case different from 0 or an integer multiple of ji), of the ffR coupling, as well as the application of a resonant excitation ain (at the frequency coc) to the cavity's input port. Following this measurement, the system is in a state M?. In the chosen example, = ! 0 1 ) and P24 = 1- Figure 3B shows the evolution of the field in the cavity in the two cases P24 = 1 and P24 = -1- Finally, a second measurement of the parity P^z (P23 in the example) brings the system into the state = MqG"1'4 - in other words, the system accumulates a phase of -ji / 4. Figure 3C shows the evolution of the system on a Bloch sphere for braiding and for braiding which follows the same trajectory in the opposite direction and which leads to an accumulation of phase ji / 4.In summary:

[0081] : . |1>JW UJU) ^ n ' 3 4 MW -> n “Jool c ) -U 23 / K : -W 3 jWM -M Bs .....? pU.JW k? œ^ / 4 p MW — Or p is the operator that projects the state of the system onto the parity subspace W i-

[0082] However, the protocol in [Fig. 3A] does not actually allow for the encoding and manipulation of a qubit, because the two charges 4 and μ are not independent. To overcome this difficulty, a chain of 6 Majorana modes can be used, as illustrated in [Fig. 4]. This system is obtained by extending the Majorana mode chain of [Fig. 2] on the MM4 side by adding two additional Majorana modes MM2- and MM3-, associated with the self-adjoint operators yy'. g'E denotes the coupling coefficient between MM4 and MM2- and g'M that between MM2 and MM1.

[0083] A qubit is encoded using the four Majorana modes MMb MM2, MMr and MM4 while the Majorana modes MM2 and MM3 are described as "auxiliary": they are used to prepare, braid and read the state of the qubit but cannot be used to encode it because they undergo a projection during the braiding operation.

[0084] The following fermionic operators are defined:

[0085] 1

[0086] and a base | j with To: € [AA, Lf m, £A F. m]

[0087] First, the system is initialized in a state - KUW1J ■■■ 4üa f A,^ ' V ' ..... ' '

[0088] In this way, a superposition of two different parities is created in the subspace associated with the Majorana modes MM| - MM4. As ,. ..1,, 3 and ] | ,fl | \ belong to different subspaces, they evolve independently during the braiding operation and, even if the system undergoes a projection at each step, a superposition of the evolutions of these two states is achieved, which acquire different phases.

[0089] Fig. 5A illustrates the protocol for carrying out the braiding.

[0090] Initialization is performed by modulating the coupling coefficient g'E at the frequency coc to measure the parity P42-. The braiding itself - represented by an operator jj ( involves the modulation of the coupling coefficient gL at the frequency coc; then the modulation at the frequency (^c of the coupling gM and simultaneously, but with a phase shift, ideally of ji / 2, of the coupling gR, as well as the application of a resonant excitation ain (at the frequency coc) to the input port of the cavity; then the modulation at the frequency coc of the coupling coefficient gM. Finally, the reading of the parity P3 4 completes the protocol; it requires the modulation at the frequency (^d of the coupling gM and simultaneously, but with a phase shift, ideally of ji / 2, of the coupling g'E, as well as the application of a resonant excitation ain (at the frequency coc) to the input port of the cavity.

[0091] Pulses at coc and cod frequencies exhibit constant phase relationships. For pulses of different frequencies, the phase relationship is understood to be at the beginning of the pulse; for example, the carriers at coc and cod frequencies may be in phase with each other at the initial times of the pulses. This is typically achieved by using an atomic clock to synchronize the different microwave sources.

[0092] Figure [5B] illustrates the protocol for carrying out the braiding s which differs from the previous one only by the order of two operations and which leads to a result of opposite sign of the parity measurement P3 4.

[0093] As in the previous protocol, initialization is performed by modulating the coupling coefficient g'E at the frequency coc to measure the parity P42'. The braiding Properly speaking – represented by an operator 1¾¾ – involves the modulation of the coupling coefficient gL at frequency coc; then the modulation of the coupling coefficient gM at frequency coc; then the modulation of the coupling gM at frequency ^d and simultaneously, but with a phase shift, ideally of ji / 2, of the coupling gR, as well as the application of a resonant excitation ain (at frequency coc) to the cavity input port. Finally, the reading of the P3-4 parity completes the protocol; it requires the modulation of the coupling gM at frequency ^d and simultaneously, but with a phase shift, ideally of ji / 2, of the coupling g'E, as well as the application of a resonant excitation ain (at frequency coc) to the cavity input port.

[0094] Figure 6 represents, very schematically, a device according to an embodiment of the invention, implementing the chain of 6 Majorana modes of Figure 4.

[0095] The CH cavity is of the coplanar waveguide type, obtained by etching a resonator in a superconducting metallic film (Goeppl 2008). The frequency of its fundamental mode is typically on the order of 10 GHz.

[0096] The cavity has an input port PE at one end and an output, or readout, port PS at the opposite end (in other embodiments, a single port may perform both functions). A microwave signal generator GHin is connected to the input port, enabling the excitation of a mode of the cavity, while a measurement circuit MES connected to the output port allows the measurement of at least one quadrature, and preferably both quadratures, of this mode.

[0097] A magnet AM generates a stationary magnetic field Bo in the axis of the cavity.

[0098] A superconducting electronic device DS is arranged at an antinode of the cavity mode's electric field. This device comprises a strongly spin-orbit coupled semiconductor nanowire NF, electrically isolated from the cavity's ground plane and oriented parallel to the magnetic field B. The spin-orbit coupling can be intrinsic, as for example in the case of InSb or InAs nanowires, or extrinsic, induced by a magnetic texture of a substrate on which the nanowire (e.g., a carbon nanotube) is deposited. This nanowire is coated with a superconducting film FS of bandgap width A and contacted by 5 electrodes (grids) GRi - GR5, as described in (Cottet 2013); more generally, there are 2N grids to generate 2N Majorana modes.The grids must be sufficiently spaced to ensure relatively good localization of Majorana modes (i.e., the energy of these modes must be much smaller, for example by a factor of 10 or more, than the band gap A).

[0099] The magnetic field B must be sufficiently intense to induce a topological electronic phase, at the electrochemical equilibrium potential (in a model simple, 2^2), where g is the Landé factor and iiH is the Bohr magneton), but not too intense so that the cavity can remain superconducting.

[0100] Voltage generators GT; i=l - 5 (only GT5 is shown) connected to the grids allow the chemical potential of the nanowire to be changed locally, for example to a value of U' such that U'< 2) as proposed p in (Cottet 2013). It is this modulation of chemical potential that allows the appearance of Majorana modes.

[0101] In addition, the grids are connected to microwave signal generators GH; i= 1 — 5 (only GH5 is shown) via polarization tees respective BTs. These generators are driven by a PR processor to generate a pulse sequence at the œd and coc frequencies of the type described above with reference to the [Fig.5A] and at [Fig.5B].

[0102] As explained above, the pulses have a duration of a few times 1 / K, where K is the spectral width of the cavity mode, for example 31k. Their power is typically on the order of a few milliwatts. The frequency shift between cod and coc is typically between K and 10 K; it can be, for example, 5k.

[0103] In addition to braiding, the device in [Fig. 6] (and more generally the diagram in [Fig. 4]) allows for the realization of a T-gate (also called an "ir / 8 gate"). This is achieved by simultaneously applying two off-resonance excitations (typically at the coc frequency) to two adjacent links in the Majorana mode chain and simultaneously reading the two quadratures of the field at the cavity output (which can be obtained by a single measurement along a direction forming angles of 45° with the axes of the IQ plane). For example, in the device in [Fig. 6], the excitations can be applied to the grids GR4 and GR5 to modulate the g'E and g'M couplings of the diagram in [Fig. 4]. This is illustrated in [Fig. 7].

[0104] However, it has been shown in (Karzig 2017) that braiding operations are sufficient to implement all 1-qubit Clifford gates, as well as a two-qubit entanglement gate denoted W. Together, these two allow the implementation of two-qubit gates between neighboring qubits, including the CNOT gate. Furthermore, CNOT gates and 1-qubit ir / 8 gates are sufficient to implement a universal computer (Sau 2010). A multi-qubit extension of the device in [Fig. 6] or the scheme in [Fig. 4], comprising six Majorana modes for each qubit, therefore makes it possible to implement a universal quantum computer.

[0105] To be able to perform two-qubit operations, it is sufficient to have a chain of 12 Majorana modes - that is, two subsets of 6 encoding modes Each a qubit – and to perform braiding on modes of the two subsets. The invention can therefore serve as the basis for a generic quantum computer.

[0106] Figure 6 provides an example of a physical platform for implementing the invention, but it is by no means limiting. Other embodiments are conceivable without departing from the scope of the invention; a strict one-dimensionality of the superconducting electronic device supporting the Majorana mode chain is very advantageous from a technological point of view, but not essential in principle. List of documents cited

[0107] (Sato 2017): Masatoshi Sato, Yoichi Ando “Topological superconductors: a review” arXiv:1608.03395v3, April 4, 2017.

[0108] (Beenakker 2019): CWJ Beenakker “Search for non-Abelian Majorana braiding statistics in superconductors” arXiv:1907.06497vl, July 15, 2019.

[0109] (Cottet 2013): A. Cottet, T. Kontos, B. Douçot “Squeezing light with Majorana fermions” arXiv:1307.4185v4, November 12, 2013.

[0110] (You 2014) : J. Q. You et al. « Encoding a qubit with Majorana modes in superconducting circuits » arXiv: 1108.3712v2, 20 septembre 2014.

[0111] (Vijay 2016): Sagar Vijay, Liang Fu « Braiding without Braiding: Teleportation-Based Quantum Information Processing with Majorana Zéro Modes » arXiv: 1609.00950vl, 4 septembre 2016.

[0112] (Goeppl 2008) : M. Gôppl et al. « Coplanar Waveguide Resonators for Circuit Quantum Electrodynamics », arXiv: 0807.4094vl,25 juillet 2008

[0113] (Karzig 2017) : T. Karzig et al. « Scalable Designs for Quasiparticle-Poisoning-Protected Topological Quantum Computation with Majorana Zéro Modes » arXiv: 1610.05289v4, 21 juin 2017.

[0114] (Sau 2010) : J. D. Sau et al. « Universal quantum computation in a semiconductor quantum wire network. » arXiv: 1007.4204v3, 24 novembre 2010.

Claims

Demands

1. 1. A quantum computing device comprising: - a microwave cavity (CH) having a separate or coincident input port (PE) and output port (PS); - a superconducting electronic device (DS) capacitively coupled to the microwave cavity and configured to support a chain of 2N Majorana modes (MMi - MM4, MM2-, MM3), N being a positive integer; - coupling means (GRi - GR5) for applying microwave excitations between each pair of adjacent Majorana modes in the chain; and - an electronic measurement circuit (MES) adapted to measure at least one quadrature (I, Q) of a microwave field from the output port of the cavity, characterized in that it also comprises at least one microwave signal generator (GHin, GHi - GH5) configured to generate: - microwave pulses at a first frequency resonant with the cavity;and - microwave pulses at a second non-resonant frequency with the cavity and to selectively apply these pulses to said coupling means and to the cavity input port and in that N is a multiple of 3, whereby the superconducting electronic device supports at least one group consisting of four Majorana modes capable of encoding a qubit (MMb, MM2, MM2', MM3) plus two auxiliary Majorana modes (MM2, MM3).

2. 2. A quantum computing device according to claim 1, further comprising an electronic control circuit (PR) configured or programmed to drive the microwave signal generator so as to: a) apply a first microwave excitation at the first frequency between a fourth and a fifth Majorana mode from one end of the chain or a portion of the chain comprising six Majorana modes; then b) apply a second microwave excitation at the first frequency between a first and a second Majorana mode from said end; then c) simultaneously apply a third microwave excitation at the second frequency between the second and a third Majorana mode from said end; a fourth microwave excitation at the second frequency between the third and fourth Majorana modes, the third and fourth microwave excitations exhibiting a phase shift; and a fifth microwave excitation at the first frequency at the cavity's inlet port; then d) apply a sixth microwave excitation at the first frequency between the second and third Majorana modes; then e) simultaneously apply a seventh microwave excitation at the second frequency between the fourth and fifth Majorana modes;an eighth microwave excitation at the second frequency between the fifth and sixth Majorana modes from said extremity; and a ninth microwave excitation at the first frequency at the cavity's inlet port; the order of operations c) and d) being reversible.

3. 3. Quantum computing device according to claim 2, the electronic control circuit configured or programmed to drive the electronic measurement circuit to measure a component of the microwave field from the output port of the cavity in phase with the excitations at the first frequency after the application of each excitation or group of excitations.

4. 4. Quantum computing device according to any one of claims 2 or 3 wherein the electronic control circuit (PR) is also configured or programmed to drive the microwave signal generator so as to: simultaneously apply a ninth and tenth microwave excitation at the second frequency between two adjacent pairs of adjacent Majorana modes, and to drive the electronic measurement circuit to measure two components of the microwave field from the cavity output port in phase and in quadrature with the excitations at the first frequency.

5. 5. Quantum computing device according to any one of the preceding claims, wherein the electronic device superconductor includes a semiconducting nanowire (NF) exhibiting spin-orbit coupling, arranged in correspondence to an antinode of the electric field of a mode (CF) of the cavity and on which a superconducting material (FS) is deposited, the device also including a magnet (AM) generating a magnetic field (B) parallel to the nanowire.

6. 6. Quantum computing device according to claim 4 in which the superconducting electronic device also comprises 2N-1 electrodes (GRi - GR5) enabling the application of electrostatic potentials to generate Majorana modes, said electrodes also constituting said coupling means for applying microwave excitations.

7. 7. A method for realizing a quantum braiding gate on a qubit using a quantum computing device according to any one of the preceding claims, the method comprising the following steps: a) applying a first microwave excitation at the first frequency between a fourth and a fifth Majorana mode from one end of the chain or a portion of the chain comprising six Majorana modes; then b) applying a second microwave excitation at the first frequency between a first and a second Majorana mode from said end; then c) simultaneously applying a third microwave excitation at the second frequency between the second and a third Majorana mode from said end; and a fourth microwave excitation at the second frequency between the third and fourth Majorana modes;and a fifth microwave excitation at the first frequency at the cavity's inlet port; then d) application of a sixth microwave excitation at the first frequency between the second and third Majorana modes; then e) simultaneous application of a seventh microwave excitation at the second frequency between the fourth and fifth Majorana modes; of an eighth microwave excitation at the second frequency between the fifth and a sixth Majorana mode from said end; and of a ninth microwave excitation at the first frequency at the cavity's inlet port; the order of steps c) and d) can be reversed.

8. 8. Method according to claim 7 also comprising measuring a component of the microwave field from the cavity output port in phase with the excitations at the first frequency after the application of each excitation or group of excitations.

9. 9. Method of realizing a quantum T-gate on a qubit using a quantum computing device according to any one of claims 1 to 5, the method comprising the following steps: simultaneously applying two microwave excitations at the second frequency between two adjacent pairs of adjacent Majorana modes, and measuring two components of the microwave field from the output port of the cavity in phase and in quadrature with the excitations at the first frequency.

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