METHOD AND DEVICE FOR QUANTUM CALCULATION USING MAJORANA MODES

Description

[0001] The invention lies in the field of quantum computing and more generally quantum information.

[0002] Majorana fermions are hypothetical elementary particles with spin 1 / 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 correspondence of a vortex in a topological superconductor.

[0003] By definition, a Majorana mode is represented by a self-adjoint operator:

[0004] 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: 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".

[0005] 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: U ij = exp − π 4 γ 0 i γ 0 j

[0006] 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 U12 and U23 do not commute. We then say that the exchange process is non-abelian.

[0007] Majorana mode exchanges are also called "braiding" operations because the world lines representing these modes in spacetime form a kind of braid.

[0008] 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 for the manipulation of qubits, notably their "fusion" (which is equivalent to a projective measurement).

[0009] For a more detailed introduction to Majorana modes and their application to quantum computing, see (Sato 2007) and (Beenakker 2019).

[0010] 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"). Nevertheless, considerable obstacles remain to be overcome for the realization of a Majorana mode quantum processor.

[0011] For example, it is known to generate 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 two-dimensional or lattice geometries, have been proposed—see, for example, (You 2014) and (Vijay 2016)—but the realization of such circuits faces insurmountable technological difficulties to date.

[0012] (Kornich 2020) proposes to create a braid from a one-dimensional arrangement of Majorana modes by exploiting transitions between these Majorana modes and the first excited level above them. A drawback of this approach is that it compromises topological protection, as the system is subject to relaxation when in the excited state.

[0013] The invention aims to overcome these drawbacks and to propose a Majorana mode quantum computing device whose realization does not pose major technological difficulties.

[0014] According to the invention, this goal 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 been described in (Cottet 2013), but not the use of this coupling to perform braiding. The present invention is defined by independent claims 1, 7, and 9. Certain embodiments are also described in the dependent claims.

[0015] An object of the invention is therefore a quantum computing device comprising: a microwave cavity having a separate or coincident input port and output port; a superconducting electronic device capacitively coupled to the microwave cavity and configured to support a chain of 2N Majorana modes, N being a positive integer; and coupling means for applying microwave excitations between each pair of adjacent Majorana modes in the chain.

[0016] According to specific embodiments: The device may also include an electronic measurement circuit adapted to measure at least one quadrature of a microwave field from the cavity output port. The device may also include at least one microwave signal generator 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. N may, in particular, be a multiple of 3, whereby the superconducting electronic device 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)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 input 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 end; and a ninth microwave excitation at the first frequency at the cavity input port; the order of operations c) and d) being reversible. 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 cavity output port in phase with the excitations at the first frequency after the application of each excitation or group of excitations. The electronic control circuit can also be configured or programmed to drive the microwave signal generator so as to: simultaneously apply a ninth and a tenth microwave excitation to theThe second frequency is between two adjacent pairs of Majorana modes, and the electronic measurement circuit is used to measure two components of the microwave field emanating from the cavity output port in phase and in quadrature with the excitations at the first frequency. The superconducting electronic device may include a spin-orbit coupling semiconductor nanowire positioned at the antinode of the electric field of a cavity mode and onto which a superconducting material is deposited. The device also includes a magnet generating a magnetic field parallel to the nanowire.

[0017] The superconducting electronic device may also include 2N-1 electrodes enabling the application of electrostatic potentials to generate Majorana modes, said electrodes also constituting said coupling means for applying microwave excitations.

[0018] 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; of a fourth microwave excitation at the second frequency between the third and fourth Majorana modes; and of a fifth microwave excitation at the first frequency at the cavity's input 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.

[0019] 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.

[0020] Another object of the invention is a method for realizing a quantum T-gate on a qubit using such a quantum computing device, 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.

[0021] Other features, details and advantages of the invention will become apparent from the description provided with reference to the accompanying drawings given by way of example, which represent, respectively: [ Fig.1 ], a four-mode Majorana system in a microwave cavity; [ Fig. 2A ], the cavity output signal as a function of time during a fusion operation of two Majorana modes of the [ Fig. 1 ] ; ] Fig. 2B ], a representation in the I / Q plane of the photonic field in the cavity during said fusion operation; [ Fig. 2C ], the evolution of the density matrix of the superconducting device as a function of time during said fusion operation; [ Fig. 2D] et [Fig. 2E ], two microwave pulse sequences to perform said fusion operation and a parity measurement; [ Fig. 2F] et [Fig. 2G ], the states of the photonic field of the cavity during the sequences of the figures [ Fig. 2D] et [Fig. 2E ], respectively; [ Fig. 3A ], a sequence of microwave pulses to perform a braiding operation of the Majorana modes of the system of the [ Fig. 1 ] ; ] Fig. 3B ], the evolution of the photonic field of the cavity during the third pulse of the sequence in the figure [ Fig. 3A ] ; ] Fig. 3C ], the evolution of the system during said braiding operation on a Bloch sphere; [ Fig. 4 ], the schematic diagram of a device according to an embodiment of the invention, using six Majorana modes; [ Fig. 5A ] And [ Fig. 5B ], two alternating pulse sequences to perform respective braiding operations of the Majorana modes of the system of the [ Fig. 4 ] ; ] Fig. 6 ] an example of the physical implementation of the device of the [ Fig. 4 ] ; And [ Fig. 7 ] a sequence of pulses to perform a "T-gate" type operation on the Majorana modes of the system of the [ Fig. 4 ].

[0022] In the following, the term "microwave frequencies" will refer to the frequency range between 300 MHz and 300 GHz, and more specifically between 1 GHz and 100 GHz.

[0023] There [ Fig. 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 â frequency ω c . Each Majorana mode is associated with a respective self-adjoint operator. γ̂ 1, γ̂ 2, γ̂ 3, γ̂ 4. A superposition between adjacent Majorana modes generates energy shifts ε L , ε M , ε R which 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 small enough 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.

[0024] 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 Hel represents the energy of the Majorana modes, and a term Hint represents the coupling between these and the cavity: H = H el + H int + H cav H el = ℏ i ϵ L γ ^ 1 γ ^ 2 + i ϵ M γ ^ 2 γ ^ 3 + i ϵ R γ ^ 3 γ ^ 4 where, â †< are respectively the annihilation and creation operator of the photonic field in the cavity and g L , g M and g R the coupling coefficients between pairs of adjacent Majorana modes and this photonic field.

[0025] From the Majorana operators γ̂ 1 , γ̂ 2, γ̂ 3, γ̂ 4. 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: c ^ L = 1 2 γ ^ 1 + i γ ^ 2 et c ^ R = 1 2 γ ^ 3 + i γ ^ 4 c ^ m = 1 2 γ ^ 2 + i γ ^ 3 et c ^ ex = 1 2 γ ^ 1 + i γ ^ 4 c ^ o = 1 2 γ ^ 1 + i γ ^ 3 et c ^ e = 1 2 γ ^ 4 + i γ ^ 2

[0026] The electronic system can therefore be expressed in the bases 0 i , 0 j 1 i , 0 j 0 i , 1 j 1 i , 1 j où i j ∈ L R m ex o e . The parity operator associated with one of these topological charges is given by P ^ ij = iγ i γ j .

[0027] We then consider that the electron-photon coupling can be modulated at a frequency ω RF in the microwave range: g O t = g ¯ O + g ˜ O cos ω RF t + ϕ O with O=L, M, R and ϕO 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 gO coupling via a modulation of the superposition between Majorana modes, and therefore energy shifts: ε i ( t ) = ε i + ε̃ i cos( ω RF t + ϕ i ) Or ε i is a constant term whose value can be varied by applying a DC voltage to the grid (i=L, M, R).

[0028] An electromagnetic field CF develops in the cavity when a coupling coefficient, for example gL, is modulated at the resonance frequency of the cavity: ω RF = ω c . This field is directly related to the parity of the left-hand side of the Majorana mode chain. The components Hel and Hint of the low-energy Hamiltonian can also be written as: H el = iℏ ϵ L t γ ^ 1 γ ^ 2 H int = iℏg L t γ ^ 1 γ ^ 2 a ^ + a ^ †

[0029] The low-energy Hamiltonian can be rewritten in a frame of reference rotating at the frequency ω c neglecting a static term, proportional to g L in accordance with the rotating wave approximation: H ˜ = H el + ℏ 2 g ˜ L e iϕ L a ^ † + e − iϕ L a ^ i γ ^ 1 γ ^ 2

[0030] This induces an effective coupling between the Majorana modes MM 1 and MM 2 which can be used to measure their parity P̂ 12 = iγ̂ 1 γ̂ 2 across the CF field, as illustrated by the figures [ Fig. 2A] à [Fig. 2G It is therefore possible to fuse the Majorana modes MM 1 and MM 2 and to detect this fusion using photons from the cavity, extracted via an output port PS (an input port PE, coinciding with or separate from PS, is also provided to directly excite a cavity mode)

[0031] There [ Fig. 2A [ ] shows the evolution over time t of the output signal α out of the cavity, representative of the excited electromagnetic field within it, assuming that the modulation of the coupling g L begins at t=0. A transient is observed with a duration on the order of 1 / κ, where κ is the linewidth of the cavity. The amplitude of the signal after the transient is g L ˜ / κ . There [ Fig. 2B [ ] 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 its measurement, and on the phase ϕ L. The contrast, g L ˜ / κ , can be much larger than the task width even in a strongly topological regime ( g L ˜ → 0 provided that the cavity width is sufficiently small.

[0032] Measuring parity through cavity photons, which allows the merging of two Majorana modes, requires taking into account the other two Majorana modes, previously considered to be decoupled. By considering the other Majorana modes (MM3, MM4), the coupling between these modes and the cavity can no longer be considered purely longitudinal: there is also a transverse component to the coupling that induces a temporal evolution of the parity operator. This evolution can, however, be neglected provided that ε O , g O ≪ ω c for O=L, M, R, for example, ε O , g O ≤ 0.1 ω c .

[0033] The consequences of merging Majorana modes MM1 and MM2 on the additional modes MM3 and MM4 are not trivial. Indeed, in the basis of the two charge operators, there are significant changes in the number. n ^ m = 1 2 i γ ^ 2 γ ^ 3 + 1 , n ^ ex = 1 2 i γ ^ 1 γ ^ 4 + 1 The fusion operation projects the two charges into an entangled state: Ψ + = 1 L 0 R = 1 2 0 m 1 ex + 1 m 0 ex pour P 12 = 1 et Ψ − = 0 L 1 R = 1 2 0 m 1 ex − 1 m 0 ex pour P 12 = − 1 .

[0034] 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. Simultaneously, the projection expressed in the basis of charges n̂ L , n̂ R gives a state vector; the parity measure of modes MM 3 and MM 4 is therefore deterministic.

[0035] [ Fig. 2C ] represents the temporal evolution of the four components ρ 00 , ρ 10, ρ 01, ρ 11 of the 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 off the diagonal of the matrix) cancel each other out during the projective measure.

[0036] [ Fig. 2D ] And [ Fig. 2F ] respectively illustrate the microwave excitation sequence and the measurement sequence used to perform and measure the fusion of MM 1 and MM 2. First, g L is modulated at frequency ωc for a time greater than 1 / κ (and typically less than 10 / κ, for example on the order of 3 / κ, since the maximum permissible measurement time is limited by the parity lifetime) to perform fusion, and a projective measurement is performed on the cavity field. Then the parity P̂ 23 = i γ̂ 2 γ̂ 3 is measured by modulating g M at the frequency ω c 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, |Ψ + 〉 or |Ψ - 〉.

[0037] Alternatively, it is possible to measure parity P̂ 34 using the sequences of [ Fig. 2E ] And [ Fig. 2G We note that, in this case, the second measurement of the field is deterministic.

[0038] If we consider κ = 1 MHz, which is easily obtained for example with coplanar waveguide cavities, the measurement time is on the order of a few µs, which requires a parity lifetime of a few tens of µs. Charge-cavity couplings on the order of g = 100 MHz can be obtained, which is compatible with the aforementioned condition. ε O , g O ≪ ω c . Assuming that the coupling strength can be modulated by 10%, it is therefore possible to perform a "one-shot" reading of the cavity without requiring electrical manipulations.

[0039] 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 through line-to-star mapping. This allows for the creation of dynamically reconfigurable two-dimensional lattices using photons from the cavity.

[0040] Starting from the Hamilton in H el = ℏ ( iε L γ̂ 1 γ̂ 2 + iε M γ̂ 2 γ̂ 3 + iε R γ̂ 3 γ̂ 4), we consider microwave excitations that are detuned with the cavity ( ω RF ≠ ω c ) between Majorana modes MM 2 and MM 3 and Majorana modes MM 3 and MM 4. These excitations induce the following unitary transformation: U = e iω RF a ^ † a ^ t e g ¯ M ω RF − ω c e i ϕ M a ^ † + e − iϕ M a ^ i γ ^ 2 γ ^ 3 + g ¯ R ω RF − ω c e iϕ R a ^ † + e − iϕ R a ^ i γ ^ 3 γ ^ 4

[0041] This gives the following Hamiltonian: H = ℏ ω c − ω RF a ^ † a ^ + 8 i g ˜ M g ˜ R ω RF − ω c sin ϕ M − ϕ R γ ^ 2 γ ^ 4 a ^ † a ^ + 1 / 2

[0042] 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 allows us to obtain an effective Hamiltonian of the form: H eff = ∑ n , m i γ ^ n γ ^ m f nm a ^ † , a ^ , a ^ † a ^ + ℏδ a ^ † a ^

[0043] Or f nm is a linear function and δ=ω c -ω RF is 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.

[0044] The protocol for performing the braiding B 14 of the Majorana modes MM 1 and MM4 is illustrated on the [ Fig. 3A First, starting from a state M0 of the system, a modulation at the frequency ωc of the coupling g M followed by a reading of the cavity field allows for a parity reading. P 23, following which the system is in a state M1. For example, consider the case where M 0 = |1 m 0 ex 〉, in which case P 23 = 1 and M 1 = M 0. Secondly, a modulation at the frequency ω c coupling g L followed by a reading of the cavity field allows for a parity reading. P 12, following which the system is in a state M2. In the example considered here, P 12 = 1 and M 2 = |1 L 0 R 〉 . Thirdly, a measure of parity P 24 is performed. This measurement concerns two non-adjacent Majorana modes, and it is more complex: it requires modulation at the frequency ( ω d ≠ ω c ) of the coupling g M and simultaneously, but with a phase shift (ideally of π / 2, in any case different from 0 or an integer multiple of π) of the coupling g R , as well as the application of a resonant excitation αin (at frequency ωc) to the cavity's input port. Following this measurement, the system is in a state M 3. In the example chosen, M 3 = |0 o 1 e 〉 and P 24 = 1. The [ Fig. 3B ] shows the evolution of the field in the cavity in both cases P 24 = 1 and P 24 = -1. Finally, a second measure of parity P 23 ( P 23 = 1 in the example) brings the system into the state M 4 = M o e iπ / 4< - in other words, the system accumulates a phase of -π / 4. The [ Fig. 3C ] shows the evolution of the system on a Bloch sphere for braiding B̂ 14 and for braiding B̂ 41 which follows the same trajectory in the opposite direction and leads to a phase accumulation of π / 4. In summary: B̂ 14: 1 m 0 ex → Π ^ 23 1 m 0 ex → Π ^ 12 i 1 L 0 R → Π ^ 24 − e iπ / 4 0 o 1 e → Π ^ 23 e − iπ / 4 1 m 0 ex B 41: 1 m 0 ex → Π ^ 23 1 m 0 ex → Π ^ 24 − i 0 o 1 e → Π ^ 12 ie iπ / 4 1 L 0 R → Π ^ 23 e iπ / 4 1 m 0 ex Where Π̂ ij is the operator that projects the state of the system onto the parity subspace P ^ ij = 1 .

[0045] However, the protocol of the [ Fig. 3A ] does not actually allow encoding and manipulation of a qubit, because the two charges n̂ m And n̂ ex are not independent. To overcome this difficulty, one can use a chain of 6 Majorana modes, as illustrated in the [ Fig. 4 This system is obtained by extending the Majorana mode chain of the [ Fig. 2 ] on the MM 4 side by adding two additional Majorana modes MM 2', and MM 3', associated with self-adjoint operators γ̂ 2', γ̂ 3'. We denote by g' E the coupling coefficient between MM 4 and MM 2' and by g' M that between MM 2' and MM 3'.

[0046] A qubit is encoded using the four Majorana modes MM 1, MM 2, MM 3, and MM 4, while Majorana modes MM 2 and MM 3 are referred to 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 projection during the braiding operation.

[0047] The following fermionic operators are defined: c o ′ = 1 2 γ 1 + i γ 3 ′ c m ′ = 1 2 γ 2 ′ + i γ 3 ′ c e ′ = 1 2 γ 4 + i γ 2 ′ : c L ′ = 1 2 γ 1 + i γ 2 ′ : c R ′ = 1 2 γ 3 ′ + i γ 4 and a base |n i , n j , n k > with i, j, k ∈ [ e', o', m, , m', ex, m, L', R', m ]

[0048] First, the system is initialized in a state Ψ init = 0 e ′ 1 0 ′ 1 m = 1 2 0 m ′ 1 ex 1 m + 1 m ′ 0 ex 1 m .

[0049] In this way, we create a superposition of two different parities in the subspace associated with the Majorana modes MM1 - MM4. As |0 m' 1 ex 1 m 〉 and |1 m' 0 ex 1 m 〉 belong to different subspaces, they evolve independently during the braiding operation and, even if the system undergoes a projection at each stage, we end up with a superposition of the evolutions of these two states, which acquire different phases.

[0050] There [ Fig. 5A ] illustrates the protocol for carrying out the braiding B̂ 14.

[0051] Initialization is performed by modulating the coupling coefficient g'E at the frequency ωc to measure the parity P42'. The braiding itself - represented by an operator Π̂23 B̂ Step 14 involves modulating the coupling coefficient gL at frequency ωc; then modulating the coupling gM at frequency ωd ≠ ωc, and simultaneously, but with a phase shift ideally of π / 2, modulating the coupling gR, as well as applying a resonant excitation αin (at frequency ωc) to the cavity's input port; then modulating the coupling coefficient gM at frequency ωc. Finally, reading the parity P3'4 completes the protocol; it requires modulation at frequency ω d ≠ ω c of the g M coupling and simultaneously, but with a phase shift, ideally of π / 2, of the g' E coupling, as well as the application of a resonant excitation α in (at the frequency ω c ) to the cavity input port.

[0052] Pulses at frequencies ωc and ωd 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 frequencies ωc and ωd can 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.

[0053] There figure [5B ] illustrates the protocol for carrying out the braiding B̂ 41 which differs from the previous one only by the order of two operations and which leads to a result of opposite sign to the measure of parity P 3'4.

[0054] As in the previous protocol, initialization is performed by modulating the coupling coefficient g'E at the frequency ωc to measure the parity P42'. The braiding itself – represented by an operator Π̂23B̂41 – involves modulating the coupling coefficient gL at the frequency ωc; then modulating the coupling coefficient gM at the frequency ωc; and finally modulating the frequency ω d ≠ ω c of the g M coupling and simultaneously, but with a phase shift, ideally of π / 2, of the g R coupling, as well as the application of a resonant excitation α in (at the frequency ω c ) to the cavity input port. Finally, the parity reading P 3'4 completes the protocol; it requires modulation at the frequency ω d ≠ ω c of the g M coupling and simultaneously, but with a phase shift, ideally of π / 2, of the g' E coupling, as well as the application of a resonant excitation α in (at the frequency ω c ) to the cavity input port.

[0055] There [ Fig. 6 ] represents, very schematically, a device according to an embodiment of the invention, implementing the chain of 6 Majorana modes of the [ Fig. 4 ].

[0056] 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.

[0057] 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 GH in is connected to the input port, enabling the excitation of a cavity mode, while a measurement circuit MES connected to the output port allows the measurement of at least one quadrature, and preferably both quadratures, of that mode.

[0058] An AM magnet generates a stationary magnetic field B 0 in the axis of the cavity.

[0059] 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 Δ and contacted by 5 electrodes (gates) GR1–GR5, as described in (Cottet 2013); more generally, there are 2N gates to generate 2N Majorana modes. The gates must be sufficiently spaced to ensure relatively good localization of the Majorana modes (i.e., that the energy ε O of these modes must be much smaller, for example by a factor of 10 or more, than the gap width Δ).

[0060] The magnetic field B must be sufficiently intense to induce a topological electronic phase, at the electrochemical equilibrium potential µ (in a simple model, > gμ B B 2 − Δ 2 ), where g is the Landé factor and µ B is the Bohr magneton), but not too intense so that the cavity can remain superconducting.

[0061] Voltage generators GT ii=1 - 5 (only GT 5 is shown) connected to the gates allow the chemical potential of the nanowire to be changed locally, for example to a value of µ' such that μ ′ < gμ B B 2 − Δ 2 ) as proposed in (Cottet 2013). It is this modulation of chemical potential that allows the appearance of Majorana modes.

[0062] In addition, the grids are connected to microwave signal generators GH ii=1 - 5 (only GH 5 is shown) via respective BT bias tees. These generators are driven by a PR processor to generate a pulse sequence at frequencies ωd and ωc of the type described above with reference to the [ Fig. 5A ] and to the [ Fig. 5B ].

[0063] As explained above, the pulses have a duration of a few times 1 / x, κ being the spectral width of the cavity mode, for example 3 / x. Their power is typically on the order of a few milliwatts. The frequency shift between ωd and ωc is typically between κ and 10 κ, For example, it could be 5x.

[0064] In addition to braiding, the device of the [ Fig. 6 ] (and more generally of the scheme of the [ Fig. 4 ] allows for the realization of a T-gate (also called a "π / 8 gate"). This is achieved by simultaneously applying two off-resonance excitations (typically at frequency ωc) to two adjacent bonds 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 [ Fig. 6 ] excitations can be applied to the GR 4, GR 5 grids to modulate the g' E, g' M couplings of the [ Fig. 4 This is illustrated on the [ Fig. 7 ].

[0065] 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 π / 8 gates are sufficient to implement a universal computer (Sau 2010). A multi-qubit extension of the device of the [ Fig. 6 ] or the diagram of the [ Fig. 4 ], featuring six Majorana modes for each qubit, therefore makes it possible to create a universal quantum computer.

[0066] To perform two-qubit operations, one only needs a chain of 12 Majorana modes—that is, two subsets of 6 modes, each encoding one qubit—and to perform braiding on modes from both subsets. The invention can therefore serve as the basis for a generic quantum computer.

[0067] There [ Fig. 6 This constitutes an example of a physical platform for implementing the invention, but it is by no means limiting. Other implementations 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. References:

[0068] (Sato 2017) : Masatoshi Sato, Yoichi Ando « Topological superconductors: a review » arXiv:1608.03395v3, 4 avril 2017. (Beenakker 2019): C. W. J. Beenakker « Search for non-Abelian Majorana braiding statistics in superconductors » arXiv:1907.06497v1, 15 juillet 2019. (Cottet 2013): A. Cottet, T. Kontos, B. Douçot « Squeezing light with Majorana fermions » arXiv:1307.4185v4, 12 novembre 2013. (You 2014) : J. Q. You et al. « Encoding a qubit with Majorana modes in superconducting circuits » arXiv: 1108.3712v2, 20 septembre 2014. (Vijay 2016): Sagar Vijay, Liang Fu « Braiding without Braiding: Teleportation-Based Quantum Information Processing with Majorana Zero Modes » arXiv: 1609.00950v1, 4 septembre 2016. (Goeppl 2008) : M. Göppl et al. « Coplanar Waveguide Resonators for Circuit Quantum Electrodynamics », arXiv: 0807.4094v1,25 juillet 2008 (Karzig 2017) : T. Karzig et al.« Scalable Designs for Quasiparticle-Poisoning-Protected Topological Quantum Computation with Majorana Zero Modes » arXiv:1610.05289v4, 21 juin 2017. (Sau 2010) : J. D. Sau et al. « Universal quantum computation in a semiconductor quantum wire network. » arXiv:1007.4204v3, 24 novembre 2010. (Kornich 2020) : V.Kornich et al. « Braiding and all quantum operations with Majorana modes in 1D », arXiv 14 septembre 2020 et Physical Review Letters, 126(11), 117701.

Claims

1. A quantum computing device comprising: - a microwave cavity (CH) having an input port (PE) and an output port (PS) that are separate or coincident; - a superconducting electronic device (DS) capacitively coupled to the microwave cavity and configured to support a chain of 2N Majorana modes (MM1 - MM4, MM2', MM3'), N being a positive integer; and - coupling means (GR1 - GR5) for applying microwave excitations between each pair of adjacent Majorana modes of the chain and - an electronic measurement circuit (MES) suitable for measuring at least one quadrature (I, Q) of a microwave field coming from the output port of the cavity, characterized in that it also comprises at least one microwave signal generator (GHin, GH1 - GH5) 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 input port of the cavity, 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 (MM1, MM2, MM2', MM3') plus two auxiliary Majorana modes (MM2, MM3).

2. The quantum computing device as claimed in claim 1, also 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 from a portion of the chain comprising six Majorana modes; and then b) apply a second microwave excitation at the first frequency between a first and a second Majorana mode from said end; and 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 the fourth Majorana mode, the third and the fourth microwave excitations exhibiting a phase shift; and a fifth microwave excitation at the first frequency to the input port of the cavity; and then d) apply a sixth microwave excitation at the first frequency between the second and the third Majorana mode; and then e) simultaneously apply a seventh microwave excitation at the second frequency between the fourth and the fifth Majorana mode; an eighth microwave excitation at the second frequency between the fifth and a sixth Majorana mode from said end; and a ninth microwave excitation at the first frequency to the input port of the cavity; the order of operations c) and d) being able to be swapped.

3. The quantum computing device as claimed in claim 2, the electronic control circuit being configured or programmed to drive the electronic measurement circuit so as to measure a component of the microwave field coming 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. The quantum computing device as claimed in either of claims 2 and 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 a tenth microwave excitation at the second frequency between two adjacent pairs of adjacent Majorana modes, and to drive the electronic measurement circuit so as to measure two components of the microwave field coming from the output port of the cavity in phase and in quadrature with the excitations at the first frequency.

5. The quantum computing device as claimed in any one of the preceding claims, wherein the superconducting electronic device comprises a semiconductor nanowire (NF) exhibiting spin-orbit coupling, placed in correspondence with 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 comprising a magnet (AM) generating a magnetic field (B) parallel to the nanowire.

6. The quantum computing device as claimed in claim 5, wherein the superconducting electronic device also comprises 2N-1 electrodes (GR1 - GR5) for applying electrostatic potentials in order to generate the Majorana modes, said electrodes also constituting said coupling means for applying microwave excitations.

7. A method for producing a quantum braiding gate on a qubit by way of a quantum computing device as claimed in one of the preceding claims when dependent on claim 3, 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 from a portion of the chain comprising six Majorana modes; and then b) applying a second microwave excitation at the first frequency between a first and a second Majorana mode from said end; and then c) simultaneously applying 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 the fourth Majorana mode; and a fifth microwave excitation at the first frequency to the input port of the cavity; and then d) applying a sixth microwave excitation at the first frequency between the second and the third Majorana mode; and then e) simultaneously applying a seventh microwave excitation at the second frequency between the fourth and the fifth Majorana mode; an eighth microwave excitation at the second frequency between the fifth and a sixth Majorana mode from said end; and a ninth microwave excitation at the first frequency to the input port of the cavity; the order of steps c) and d) being able to be swapped.

8. The method as claimed in claim 7, also comprising measuring a component of the microwave field coming 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.

9. A method for producing a quantum T-gate on a qubit by way of a quantum computing device as claimed in 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 coming from the output port of the cavity in phase and in quadrature with the excitations at the first frequency.