Self-stabilization of GKP codes by parametric modulation in microwave frequency combs

Abstract

A superconducting microwave quantum circuit comprising a controllable Josephson junction element (8) connected to the energy of a linear passive circuit portion (6) exhibiting a plurality of resonant modes, where a Forster first form decomposition across the terminals of the Josephson junction element (8) comprises a target resonant mode exhibiting an impedance Z and a pulsation w higher than 13 kohm. The energy of the Josephson junction element (8) is controllable and is modulated by at least two respective pulse trains in which the pulses are separated by a duration 2 [pi] / w and have a width less than one tenth of this duration, the amplitude of the pulses within each respective pulse train being modulated by a respective sinusoidal carrier, and the amplitude of the pulses within each respective pulse train being less than one tenth of the duration 2 [pi] / w. And the pulse trains are each shifted in pairs by a duration [delta] t such that sin (w * [delta] t) equals 13 kohm / Z, such that the resonant mode of the pulse w is stabilized in one of the two GKP states of the encoded qubit.

Description

Technical Field

[0001] The present invention relates to the field of superconducting microwave resonator quantum circuits, and in particular to the realization of quantum syndrome codes using bosonic codes. Background Art

[0002] To date, the production of quantum computers has been hampered by noise, which modifies the state of quantum bits (hereinafter referred to as qubits), leading to logical errors. Despite significant progress in limiting noise sources over the past 20 years, universal fault-tolerant quantum computing is currently out of reach. Quantum error correction aims to address this problem.

[0003] The types of logical errors to be corrected have two properties: bit flips and phase flips. The main methods used aim to encode a few logical qubits in a larger information space. In this information space, the qubits are encoded into so-called "encoded" states, which are chosen so that noise does not cause a direct transition from one encoding state to another.

[0004] Therefore, the noise that causes the transition from the coded state to the non-coded state can be detected and corrected to unambiguously bring the system back to the initial coding, that is, each non-coded state is associated with a single coded state. In order not to disturb the quantum information, this error detection and correction must be done without measuring the logical qubits.

[0005] The most widely used quantum error correction method is called "surface codes". This type of solution is implemented by the largest companies in the quantum field, such as Google, IBM, Delft University, University of Zurich, etc., and constitutes the most studied solution in the world to date.

[0006] In surface codes, quantum information is carried by a 2D network of two-level systems. There are single-qubit gates and two-qubit gates that are fault-tolerant. The biggest drawback of this technology is that it requires a large number of very high-quality two-level systems to be able to perform calculations. In addition to the costs caused by this architecture, this also creates problems related to the need to control a large number of quantum systems. Summary of the invention

[0007] The applicant's work has led her to conclude that surface codes have shortcomings that require the exploration of other solutions. To this end, the applicant has investigated bosonic codes.

[0008] Bosonic codes are a family of quantum error correction codes based on storing qubits in bosonic patterns. These codes include three main subfamilies: "cat" codes, binomial codes, and GKP (Gottesman-Kitaev-Preskill) codes.

[0009] An example of a cat code is Schrödinger's two-legged cat code implemented by Yale University, the Quantic team, or companies such as Quantum Computing Inc, Alice & Bob, or Amazon Web Services.

[0010] In these codes, the coded state is a Schrodinger cat state. In contrast to GKP codes, only one of the two types of logic errors can be eliminated. Therefore, it is necessary to concatenate the code into a repetition code to be able to manage both types of logic errors.

[0011] One advantage of these codes over GKP codes is that one of the two types of errors can be arbitrarily eliminated by simply changing the size of the cat. This is not possible with GKP codes, for which only increasing the ratio between the rate of dissipation induced by the design (or the rate of measurement and correction in the case of stable measurements) and the intrinsic dissipation of the resonator allows the lifetime of the logical qubit to be extended exponentially.

[0012] Another example is the Schrödinger cat code with n>2 legs implemented by Yale University and Quantum Computing. In contrast to the Schrödinger two-legged cat code, two types of logic errors can be corrected. However, when the rate of detection and correction or the rate of dissipation caused by the design increases, the elimination does not follow an exponential law, but a polynomial law.

[0013] Binomial codes have some problems similar to those of Schrödinger's cat codes with n > 2 legs. In addition, they only protect against photon losses, not other types of noise (such as pure phase shifts).

[0014] GKP codes also have multiple implementations. The first example is a GKP code stabilized by Rabi interaction with auxiliary qubits, which was implemented by Yale University in collaboration with the Quantum Group. This method causes logical errors, such as when an auxiliary qubit experiences an error type, its state is modified according to the axis of the Bloch sphere corresponding to the operator present in the Rabi interaction during the interaction. It can be eliminated by using an auxiliary qubit that eliminates one of the errors (e.g., Schrödinger's cat), but this method is complicated to implement and has not been experimentally realized so far.

[0015] Another example is the GKP codes stabilized by modular operators. There are proposals for realizing Hamiltonian dynamics supported by modular operators, which, combined with dissipation with single photons, stabilize the GKP codes. However, the circuits proposed for realizing such dynamics are not realizable. In fact, they involve elements called "coherent phase-slip elements", or they involve "high-impedance gyrators", the realization of which has not been demonstrated so far.

[0016] Therefore, to date, no syndrome code has worked satisfactorily. The point at which state-of-the-art logical qubits have a longer coherence / lifetime than the underlying physical system has been barely reached, but never significantly exceeded.

[0017] The present invention improves this situation. To this end, it proposes a superconducting microwave quantum circuit, which includes an energy-controllable Josephson junction element connected to a linear passive circuit part exhibiting multiple resonant modes, wherein the Foster first form decomposition across the terminals of the Josephson junction element includes a target resonant mode exhibiting an impedance Z higher than 13kohm and a pulsation (pulsation) w. The energy of the Josephson junction element is controllable and modulated by at least two corresponding pulse trains, within which the pulses are separated by a duration of 2π / w and have a width less than one-tenth of the duration, the amplitude of the pulses within each corresponding pulse train is modulated by a corresponding sinusoidal carrier, and the pulse trains are respectively offset in pairs by a duration Δt, so that |sin(w*Δt)| is equal to 13kohm / Z, so that the resonant mode of the pulsation w is stabilized in one of the two GKP states of the encoded quantum bit.

[0018] The device is particularly advantageous because it allows the implementation of modular operators for realizing qubits in GKP codes, which can be reliably stabilized and can significantly improve the ratio between coherence time and gate time compared to all qubits realized to date.

[0019] According to various embodiments, the present invention may have one or more of the following features:

[0020] - the pulsation wi of the sinusoidal carrier of each corresponding pulse train is zero, so that the GKP encoded state is the lowest energy state in the interaction representation that reduces the pulsation of the target resonant mode to a pulsation below the amplitude of the effective mode potential generated for the target resonant mode divided by one fifth of the reduced Planck constant, said lowest energy state being stabilized by coupling with an environment in which the product of the Boltzmann constant and the effective temperature is below the Josephson energy in the representation,

[0021] - the pulsation wi of the sinusoidal carrier of each corresponding pulse train is a pulsation of a resonant mode of the Förster decomposition of the terminals of the Josephson junction element of the linear passive circuit part different from the pulsation w, the superconducting microwave quantum circuit also includes a homodyne linear measurement member corresponding to the resonant mode of the corresponding pulsation wi, which is arranged to measure the displacement of the resonant mode of the pulsation wi in the direction of its corresponding phase plane determined by the phase of the sinusoidal carrier of the pulsation wi, and a feedback member, which is arranged to shift the state of the target resonant mode relative to the phase reference of the target resonant mode in the direction of its phase plane determined by the phase of the pulse train corresponding to the pulsation wi in proportion to the measurement received from the measurement member,

[0022] - the pulsation wi of the carrier of each respective pulse train is a pulsation of one of the resonant modes of the Förster first form decomposition across the terminals of the Josephson junction element of the linear passive circuit portion different from the pulsation w, the linear passive circuit portion being arranged so that said resonant mode of the pulsation wi dissipates its respective excitation into an environment in which the product of the temperature and the Boltzmann constant is lower than the product of the pulsation wi and the reduced Planck constant and according to a rate higher than: √(w / 4*Z i / (6.5kohm))*e i , where Zi is the impedance of the pulsating wi mode across the terminals of the element of the Josephson junction element, and ei is the average Josephson energy integrated over the duration of the pulses, each pulse being preceded by a pre-pulse and followed by a post-pulse, the pre-pulse and post-pulse being modulated by a quadrature sinusoidal carrier with a carrier wave of the pulses and offset relative to the corresponding pulses of duration δ less than 1 / (5w) and having opposite amplitudes and absolute values ​​less than 1 / (3δw),

[0023] - the corresponding pulse trains are activated in overlapping time windows and the pulsation of the sinusoidal carrier wave that modulates their respective amplitudes is different,

[0024] - the corresponding pulse trains are activated in sequence and modulated by a sinusoidal carrier of the same pulse wi so that a pulse train ends before the next train is activated, each train comprising a number of pulses between 10 and 10^8,

[0025] - the Josephson junction element has a controllable phase, which allows to modify the position of the GKP state grid,

[0026] - the impedance of the target resonance mode is adjustable and the offset between the corresponding pulse trains can be adapted so as to enable modification of the shape of the grid of the GKP state grid,

[0027] - the impedance of the target resonance mode is equal to 13 kohm, the pulsation of the target resonance mode is modified so as to perform a π / 2 rotation of the GKP state grid, and the duration between pulses within the same train is modified in a manner adapted to this modified pulsation,

[0028] - the impedance of the target resonance mode is equal to 15 kohm, the pulsation of the target resonance mode is modified so as to perform a π / 3 rotation of the GKP state grid, and the duration between pulses within the same train is modified in a manner adapted to this modified pulsation,

[0029] - the circuit comprises two energy-controllable Josephson junction elements, each Josephson junction element being connected to the linear passive circuit portion such that the impedance of the target resonance mode presented at at least one of the two Josephson junction elements is adjustable, the energy of each Josephson junction element being controllable and modulated according to each corresponding impedance of the target resonance mode,

[0030] - the circuit comprises two circuits, at least one of which is interconnected as previously described so as to define a logic gate, and

[0031] - Both circuits present an adjustable impedance of the target resonant mode, the offset between the respective pulse trains can be adapted so as to modify the shape of the mesh of the GKP state grid, and the linear passive circuit portion of each circuit is connected to the Josephson junction element of the other circuit via an adjustable coupler so as to control the impedance of the target resonant mode of each circuit presented at the Josephson junction element of the other circuit. BRIEF DESCRIPTION OF THE DRAWINGS

[0032] Other characteristics and advantages of the invention will become more apparent on reading the following description taken by way of illustrative and non-limiting examples from the accompanying drawings, in which:

[0033] -[ Figure 1 ] is the representation of the wave function of the state of the GKP code,

[0034] -[ Figure 2 ] is a representation of a superconducting microwave resonator quantum circuit according to a first embodiment of the present invention,

[0035] -[ Figure 3 ] indicates that when the Foster first form decomposition across the terminals of the Josephson junction element includes a target resonant mode presenting an impedance Z equal to 13kohm, Figure 2 The activation time curve of the Josephson junction,

[0036] -[ Figure 4 ] indicates that when the Foster first form decomposition across the terminals of the Josephson junction element includes a resonant mode presenting an impedance Z equal to 13kohm, Figure 2The displacement of the state of the target resonant mode of the circuit in the laboratory reference frame and the rotating reference frame,

[0037] -[ Figure 5 ] indicates that when the impedance Z is higher than 13kohm Figure 3 An alternative embodiment,

[0038] -[ Figure 6 ] indicates that when the impedance Z is higher than 13kohm Figure 4 An alternative embodiment,

[0039] -[ Figure 7 ]express Figure 2 The so-called Hamiltonian implementation of the circuit,

[0040] -[ Figure 8 ]express Figure 7 The activation time curve of the Josephson junction,

[0041] -[ Fig. 9 ] means that in the rotating reference frame Figure 7 The effective potential generated by the circuit, and the probability density of the GKP encoding state,

[0042] -[ Fig.10 ]express Fig. 9 The energy level of the state,

[0043] -[ Fig.11 ]express Figure 2 The so-called feedback implementation of the circuit,

[0044] -[ Fig.12 ] indicates that the feedback Fig.11 The displacement of the state of the GKP target resonant mode of the circuit,

[0045] -[ Fig.13 ]express Figure 2 The so-called dissipative implementation of the circuit,

[0046] -[ Fig.14 ]express Fig.13 The activation time curve of the Josephson junction,

[0047] -[ Fig.15 ]express Fig.11 The state of the target resonant mode of the circuit is displaced by the dissipation, and

[0048] -[ Fig.16 ]yes Fig.13 An example of an implementation of a circuit.

[0049] The drawings and the following description mainly contain elements of certain characteristics. Therefore, they can not only serve to better understand the invention, but also help to define it if necessary. DETAILED DESCRIPTION

[0050] The present invention relates to the implementation of logical qubits by means of GKP codes. In order to better understand how to make the latter stable, Figure 1 A representation of the coded states "0" and "1" is provided, as well as the representation of their symmetric and asymmetric superpositions "+" and "-" according to square GKP codes of orthogonal q and p. The GKP code allows to ensure that orthogonal logic states have long-range support in the phase plane (q, p). Alternatively, the grid of the GKP grid can be a parallelogram, starting from the moment its area in the plane (q, p) at normalized coordinates (such as the commutator [q, p] = i) is equal to 4π. The special case when the angle of the parallelogram is equal to π / 3 is called a "hexagonal code".

[0051] In the following, the equations are presented with reference to an ideal GKP code, whose states are infinitely extended in the phase plane (infinitely compressed states), hereinafter referred to as "infinite size GKP code". The object of the invention is an implementation with a "realistic" GKP code, that is to say, whose states have a support of finite size in the phase plane (finitely compressed states), hereinafter referred to as "finite size GKP code". This implementation may involve minor changes to the equations presented below, and modifications in the stabilization scheme will be specified below as far as possible.

[0052] It can be seen from the figure that the various encoding states are separated by √(π) according to orthogonality q or according to orthogonality p.

[0053] In this figure, each scale represents the probability distribution of certain pairs of orthogonal states in bits (0; 1) or phases (+; -) according to the orthogonality q or the orthogonality p. The distance between the states 0 and 1 in q allows to obtain protection against bit flips, while the distance between the states + and - in p allows to obtain protection against phase flips. The circuit according to the invention makes it possible to obtain the probability distribution of certain pairs of orthogonal states in bits (0; 1) or phases (+; -) according to the orthogonality q or the orthogonality p. and These states are generated and stabilized by efficient coupling with the target mode, where q and p are orthogonal operators of the target mode in the rotating reference frame at the frequency of the target mode. These modular operators are stabilizers of infinite-size GKP codes.

[0054] In a so-called Hamiltonian embodiment, the invention improves the energy degradation of specific states of these stabilizers, and the encoded state has the lowest energy and is therefore stable under dissipation into an effective cryogenic container.

[0055] In so-called measurement-feedback implementations, these stabilizers are measured and the results of the measurements allow the encoding state to be stabilized by feedback.

[0056] In the so-called dissipative engineered implementation, the interaction between the target mode and the highly dissipative auxiliary mode, on the one hand via the modular operator of the target mode and on the other hand via the linear operator of the auxiliary mode, generates a modularly effective dissipation that stabilizes the GKP encoding state.

[0057] In the case of dominant errors in superconducting resonators, that is, exploiting the dissipation of single photons, the applicant believes that such encoded qubits can have lifetimes of the order of tenths of a second while remaining controllable by fault-tolerant gates and lasting on the order of microseconds, which represents an improvement of several orders of magnitude in the ratio between coherence time and gate time relative to all qubits realized to date.

[0058] The applicant has discovered a class of quantum circuits that allow the realization of the above modular operators by achieving specific control of the energy of the Josephson junction element by combining an energy-controllable Josephson junction element and a linear passive circuit part.

[0059] Figure 2 A general view of a quantum circuit of this type is shown. From this figure it can be seen that the circuit 2 comprises a transmission line 4 connected to a terminal (or "port") connected to a linear passive circuit portion 6, and a Josephson junction element 8 connected to the terminal of the linear passive circuit portion 6.

[0060] The transmission line 4 can be produced by any known means (on-chip or off-chip, 2D or 3D, etc.) in the sector, according to any known geometry of microwave transmission lines.

[0061] The passive circuit portion 6 can be produced in any known manner such that when it carries a signal in the microwave range the Förster decomposition mentioned below is obtained.

[0062] The Josephson junction element 8 has controllable Josephson energy. Figure 2 , 7 , 11 and 13, the circuit represented makes it possible to control the Josephson energy by applying a magnetic flux between two Josephson junctions forming a SQUID (superconducting quantum interference device), but it can also be controlled by other methods (more complex multi-junction circuits, semiconductor junctions with voltage bias).

[0063] According to an embodiment, the circuit 2 can be modified and arranged so that it can be implemented in a Hamiltonian implementation scheme ( Figures 7 to 10 ), feedback implementation plan ( Fig.11 and Fig.12 ) or dissipative implementation ( Figures 13 to 16 ) to stabilize the GKP code.

[0064] In general, the invention is based on two principles.

[0065] The first principle is that the linear passive circuit portion 6 exhibits one or more resonant modes, where the Foster first form decomposition across the terminals of the Josephson junction element (see, for example, the article "A Reactance Theore" by Foster, RM, Bell System Technical Journal, vol. 3, no. 2, pp. 259-267, November 1924) includes resonant modes that exhibit an impedance Z greater than or equal to 13 kohm. The equivalent circuit is Figure 2 Hereinafter, this resonant mode will also be referred to as the “target mode”.

[0066] The impedance value is particularly high, since it corresponds to twice the resistance quantum (defined by h / 4e2, i.e. 6.5 ohms), which allows the Josephson junction element 8 to realize the switching of two different modal operators and stabilizers corresponding to the GKP code at different times in the rotation reference frame of the target mode. In fact, the spatial pulsation η of the modal operator is connected to the impedance presented at the Josephson element via the following equation

[0067]

[0068] The product of the ripples of the two modular operators is equal to 4π in the case of square GKP codes and is larger for non-square GKP codes. Therefore, a value of 13 kohm is the minimum value for stabilizing GKP codes. In the case of impedance values ​​above 19.5 kohm, for example, quantum trits can be stabilized.

[0069] The second principle is the control of the Josephson energy (by penetrating Figure 2 ), which allows the states of the GKP code to be organized so that they can perform the shifts described below.

[0070] therefore, Figure 3 represents the signal used to modulate the Josephson energy. Figure 3 This control on the right is the result of the sum of two sinusoidal waves of respective pulsations w1 and w2 that each modulate the amplitude of a pulse train offset from each other by a quarter period associated with the pulsation of a resonant mode with an impedance equal to 13 kohm. Each pulse has a width less than one tenth of this quarter period.

[0071] describe Figure 3Another way is that the signal for controlling the Josephson energy of the Josephson junction element 8 is the sum of two control signals, which are used to "turn off" or "turn on" the Josephson energy in a specific way at a selected moment:

[0072] - a first control signal causes the Josephson energy to be activated at each cycle associated with the pulsation of the impedance resonance mode higher than or equal to 13 kohm and to be substantially zero at other moments, and this energy is modulated in amplitude by a sinusoid of the pulsation w1 equal to the pulsation of the first auxiliary mode, for the correction of the conjugate operator q in the Förster decomposition,

[0073] a second control signal causing the resulting Josephson energy to be activated at each cycle associated with the pulsation of the impedance resonance mode higher than or equal to 13 kohm and to be substantially zero at other moments, and modulated in amplitude by a sinusoid of the pulsation w2 equal to the pulsation of the second auxiliary mode, for the correction of the conjugate operator p, and

[0074] The first control signal and the second control signal are offset from each other by a quarter of a period associated with the pulsation w of the target resonance mode with an impedance equal to 13 kohm.

[0075] Alternatively, the control signal may be activated every half cycle instead of every cycle as described above. This may be accomplished by inverting the sign of the pulse if desired.

[0076] The result of this combination in the context of the GKP code implemented by circuit 2 is expressed as Figure 4 express.

[0077] Figure 4 Represents the various possible displacements on the GKP grid at time instants, following the pulsation w of the target mode, either in a fixed reference frame about axes q0 and p0 (at the top) or in a rotating reference frame about axes q and p (at the bottom). The concept of a rotating reference frame is also referred to as an "interaction representation", and the two expressions are interchangeable in the following.

[0078] In a fixed reference frame, the state of the resonator will rotate with the pulsation w due to the resonance of the modes of the pulsation w, and if the target mode is present at a Josephson junction element with suitable impedance, coherence and discrete displacements according to the grid of the GKP grid every quarter cycle will be possible. Figure 3activation curve, which only occurs once per quarter cycle according to one pulse train or the other. This means that the Josephson energy is zero when 2πt / w is equal to 0.125 or 0.375, which prevents any displacement outside the grid. Instead, all quarter cycles are possible according to +q0, -q0, +p0 or -p0. This can also be clearly understood by the representation in a rotating reference frame, where the state is stable (because the reference frame rotates with the pulsation w), and the displacements rotate (so displacements are possible according to +q, -q, +p or -p) and are possible every quarter cycle.

[0079] therefore, Figure 2 The circuit allows the generation and stabilization of GKP grid states because the dynamics of the patterns can be coherent and constrained according to the GKP grid.

[0080] exist Figure 3 and Figure 4 In the specific case of , the resonant mode of the pulsation w presents an impedance equal to 13 kohm, so that the unitary operators q and p cover an area equal to 4π.

[0081] However, the resonant mode of the pulsating W may present an impedance higher than 13kohm. In this case, Figure 5 and Figure 6 As shown, the pulse trains should be slightly offset from each other and the GKP grid is a diamond shape whose area remains equal to 4π.

[0082] Figures 7 to 15 Three different embodiments are presented which make it possible to stabilize the qubits encoded in the GKP code of such a circuit. Figures 7 to 10 The modulation diagrams of quantum circuits and Josephson junction elements that achieve Hamiltonian stability are described. Fig.11 and Fig.12 A quantum circuit that achieves feedback stabilization is described, and Figures 14 to 16 Quantum circuits that achieve dissipative stability are described.

[0083] Figure 7 FIG. 1 shows a Foster first form decomposition of the linear passive circuit portion 6. In this embodiment, the pulsations w1 and w2 are zero, so that the activation curve of the Josephson energy of the Josephson junction element 8 is at Figure 8 and for modes p and q Fig. 9 The energy potential shown in . Therefore, the pulse trains are each modulated by a sinusoid with zero pulsation and are therefore still equal to 1. In other words, it is to be reminded that the value 1 is a sinusoid with zero pulsation. Alternatively, the pulse train can be doubled symmetrically with respect to the period.

[0084] In a rotating reference frame, and in the absence of Josephson dynamics, all states of the oscillator are degenerate and of zero energy. The pulse train in this Hamiltonian implementation generates an egg box potential in terms of q and p, which increases the degeneracy of these energy levels and ensures that the encoded state is the lowest energy state, with a probability density at the bottom of the potential well.

[0085] This has the effect that the GKP encoded state is the lowest energy state in the interaction representation, which reduces the pulsation of the target mode to below one third of the Josephson energy divided by the reduced Planck constant.

[0086] In this embodiment, in order to allow the quantum bit to be stabilized, that is to say to return the excited state of the GKP code to the coded state, and therefore low energy, the circuit comprises a transmission line 4, which couples the circuit 2 to the environment, in which, in the interaction representation, the product of the Boltzmann constant and the effective temperature is lower than the Josephson energy in this representation.

[0087] Therefore, the high energy state tends to relax by losing energy through the transmission line 4 (e.g. Figure 7 ), so that in the event of resonator excitation, it naturally reverts to its low energy encoding state, as Fig.10 As shown, where the lines represent various GKP Hamiltonian states depending on their energy levels.

[0088] The above corresponds to a GKP code of infinite size. In the case of a finite size GKP code, the duration between pulses within a pulse train can be increased or decreased by a small amount, typically less than where E is the modulation amplitude of the effective potential (V(q) and V(p)) and h is Planck's constant.

[0089] This embodiment is particularly advantageous when the perturbation that limits the coherence of the qubits is a Hamiltonian.

[0090] Fig.11 A second embodiment of the quantum circuit 2 is shown, wherein the circuit 2 further comprises a measurement member 10 and a feedback member 12 arranged between the environment 4 and the linear passive circuit portion 6 .

[0091] The pulsation wi is here chosen to correspond to the pulsation of one of the resonance modes of the Foster decomposition at the terminals of the Josephson junction element 8 of the linear passive circuit part 6, which is different from the pulsation w of the resonance mode with an impedance higher than or equal to 13 kohm, the pulsations wi being different from each other.

[0092] The measuring member 10 is a homodyne linear corresponding to the resonance mode of the corresponding pulsation wi, arranged to measure the displacement of the resonance mode of the pulsation wi in the direction of its corresponding phase plane determined by the phase of the sinusoidal carrier of the pulsation wi, and allows the measurement of the modulus operator, that is to say the measurement of the GKP error syndrome, without the risk of error propagation, that is, the risk that the decoherence or dissipation affecting the auxiliary mode will not cause logical errors on the GKP qubit. The measuring member 10 can be implemented, for example, by a Josephson junction parametric amplifier operating close to the quantum limit.

[0093] The feedback member 12 is arranged to displace the state of the resonance mode presenting an impedance Z higher than 13 kohm in the direction of its phase plane, proportional to the measurement received by the measurement member 10, the direction of which is determined by the phase of the pulse train corresponding to the pulsation wi at the phase reference of said resonance presenting an impedance Z higher than 13 kohm. The feedback member 12 can be implemented by applying microwaves to the pulsation w, the phase and amplitude of which are proportional to the result of the measurement.

[0094] Therefore, if Fig.12 As is evident in , the GKP state is shifted by the feedback member 12 according to a vector of the phase plane, the coordinates of which are proportional to the error syndromes s1 and s2 measured by the measuring member 10. The circuit 2 may also be arranged with more than two error measurement syndromes.

[0095] This description corresponds to an implementation of infinite size GKP codes. In the case of finite size GKP codes, the measurement of the modulo operator can be replaced by linear feedback (as described above) and by low linear measurement (homodyne) with low modulo feedback.

[0096] Alternatively, the Applicant has discovered that pulse trains can be activated sequentially by modulation by a sinusoidal carrier of the same pulse wi, so that a pulse train ends before activating the next train, each train comprising a number of pulses between 10 and 10^8.

[0097] The applicant has also discovered a third embodiment in which stabilization is performed by dissipation.

[0098] In fact, stabilization may become autonomous by simple dissipation on the auxiliary mode. Then a dual-mode dissipation of the target resonator is obtained. More precisely, Fig.13 Circuit 2 makes it possible to dissipatively stabilize the GKP logical qubits via modular operators.

[0099] exist Fig.13 In the embodiment of the present invention, the Josephson junction element 8 includes two Josephson junctions 44 parallel to each other, and its energy distribution is modulated by a sinusoidal carrier to Fig.11 and Fig.12The pulse wi of the sinusoidal carrier is selected in a similar manner to the embodiment.

[0100] The linear passive circuit portion 6 is also arranged so that the resonant mode of the pulsation wi dissipates its corresponding excitation into an environment where the product of temperature and the Boltzmann constant is lower than the product of the pulsation wi and the reduced Planck constant and at a rate higher than: √(w / 4*Z i / (6.5kohm))*e i , where Zi is the impedance of the mode of the pulse wi across the terminals of the Josephson junction element, and ei is the average Josephson energy integrated over the duration of the pulse.

[0101] like Fig.14 As shown, for an implementation scheme applicable to a finite size GKP code, each pulse is preceded by a pre-pulse and followed by a post-pulse, the pre-pulse and post-pulse (dashed lines) being modulated by an orthogonal sinusoidal carrier with a carrier having pulses and being offset relative to the pulses of corresponding duration δ less than 1 / (5w) and having opposite amplitudes and absolute values ​​less than 1 / (3δw).

[0102] Preferably, the Josephson junction element 8 is coupled with four auxiliary dissipative modes so as to perform Lindblad dynamics with four dissipators. However, Lindblad dynamics may be performed with two dissipators or even six dissipators.

[0103] Alternatively, the Applicant has discovered that pulse trains can be activated sequentially by modulation by a sinusoidal carrier of the same pulse wi, so that a pulse train ends before activating the next train, each train comprising a number of pulses between 10 and 10^8.

[0104] Fig.16 Indicates that according to Fig.13 Photographic views of a practical implementation of the circuit observed from above. The photographs were obtained by optical microscopy (left) and scanning electron microscopy (right).

[0105] The figure includes several magnifications allowing a better view of the various elements of the embodiment. Thus, it is possible to see the inductance 560 generated by the long track for adjusting the impedance of the auxiliary mode present at the Josephson junction element, as well as the track for the control signal. These magnifications also make it possible to better assess the magnitudes, the Josephson junction having dimensions of the order of 500 nm, the ATS having dimensions of the order of 100 μm, and the track for the control signal having dimensions of the order of millimeters.

[0106] Fig.13The circuit can be realized with the aid of an ATS (Asymmetric Threaded SQUID) circuit. The ATS circuit comprises a symmetrical SQUID which is short-circuited at its center by a high inductance, which forms two loops.

[0107] Thus, ATS realizes an LC oscillator with added Josephson energy of controllable amplitude and phase. The common mode of the loop of the ATS circuit can have an impedance Z equal to 13kohm and a resonant frequency in the RF range for a sufficiently high inductance in the central arm of the circuit. Distributed modes of low impedance, adapted symmetry and connected to the loop can be used as auxiliary modes. In the Foster first form representation, the capacitance has a value of the order of 10fF and the inductance has a value of the order of 1 to 10μH.

[0108] In the case of Lindblad dynamics with four dissipators, it is advantageous to use an ATS instead of a SQUID. In fact, the control of the phase of each modulo operator achieved due to the Josephson junction elements is important in this case, two of the dissipators are associated with cosines and the other two are associated with sines.

[0109] This embodiment is particularly advantageous because the ATS circuit was implemented by the applicant in a practical manner, which allows particularly good control over its manufacture.

[0110] Advantageously, the impedance of the target mode presented at the Josephson junction element 8 can be adjusted, which makes it possible to generate a set of universal Clifford gates. This adjustment can be performed by introducing an adjustable coupler between the linear passive circuit part 6 and the Josephson junction element 8. The adjustable coupler can be produced in a manner known to those skilled in the art, for example in the context of superconducting circuits (or "QED circuits").

[0111] Thus, according to a first alternative embodiment, the impedance of the target resonance mode is equal to 13 kohm, which defines a square GKP code. In this case, by slightly varying the duration between two pulses within each train, the GKP state grid can be gradually pivoted until a rotation angle of π / 2 is obtained. Thus, a Hadamard gate is obtained.

[0112] In another alternative embodiment, the impedance of the target resonance mode is equal to 15 kohm, which defines a hexagonal GKP code. In this case, by slightly varying the duration between two pulses within each train, the GKP state grid can be gradually pivoted until a rotation angle of π / 3 of the GKP state grid is obtained. Thus, a gate of cyclic permutation of the three axes of the Bloch sphere of the logical qubit (X=>Y=>Z=>X...) is obtained.

[0113] A CNOT gate can be realized when two circuits present adjustable impedance and the linear passive circuit parts 6 thereof are respectively connected to the Josephson junction elements 8 of the other circuits through adjustable couplers.

[0114] In yet another alternative embodiment, adjustable impedance can be used to generate protected initialization or protected read circuits for logical qubits. In this case, the superconducting microwave quantum circuit includes two Josephson junction elements 8. At least one Josephson junction element 8 is connected to the linear passive circuit portion 6 so that the impedance of the target mode presented to it is adjustable. The grid area of ​​the GKP code is then reduced by a factor of 2, and the energy of each Josephson junction element is controlled according to the impedance of the target mode presented to each Josephson junction element. Advantageously, for a square GKP code, one of the target modes will have an impedance of 13kohm, while the other has an impedance of 6.5kohm.

Claims

1. A superconducting microwave quantum circuit comprising an energy controllable Josephson junction element (8) connected to a linear passive circuit portion (6) exhibiting a plurality of resonant modes, wherein the Foster first form decomposition across the terminals of the Josephson junction element (8) comprises a target resonant mode exhibiting an impedance Z higher than 13 kohm and a pulsation w, the energy of the Josephson junction element (8) being controllable and modulated by at least two corresponding pulse trains, within the pulse trains the pulses are separated by a duration 2π / w and have a width less than one tenth of the duration, the amplitude of the pulses within each corresponding pulse train being modulated by a corresponding sinusoidal carrier, and the pulse trains are respectively shifted in pairs by a duration Δt such that |sin(w*Δt)| is equal to 13 kohm / Z, such that the resonant mode of the pulsation w is stabilized in one of two GKP states of a coded quantum bit.

2. The superconducting microwave quantum circuit according to claim 1, wherein: The pulsation wi of the sinusoidal carrier of each corresponding pulse train is zero, so that the GKP encoded state is the lowest energy state in the interaction representation, which reduces the pulsation of the target resonant mode to a pulsation lower than the amplitude of the effective mode potential generated for the target resonant mode divided by one-fifth of the reduced Planck constant, and the lowest energy state is stabilized by coupling with an environment (4), in which the product of the Boltzmann constant and the effective temperature is lower than the Josephson energy in the representation.

3. The superconducting microwave quantum circuit according to claim 1, wherein: The pulsation wi of the sinusoidal carrier of each corresponding pulse train is a pulsation of a Foster-decomposed resonant mode of a terminal of a Josephson junction element (8) across a linear passive circuit portion (6) different from the pulsation w, and the superconducting microwave quantum circuit also includes a homodyne linear measurement member (10) corresponding to the resonant mode of the corresponding pulsation wi, the homodyne linear measurement member (10) being arranged to measure the displacement of the resonant mode of the pulsation wi in the direction of its corresponding phase plane determined by the phase of the sinusoidal carrier of the pulsation wi, and a feedback member (12) being arranged to shift the state of the target resonant mode relative to the phase reference of the target resonant mode in the direction of its phase plane determined by the phase of the pulse train corresponding to the pulsation wi in proportion to the measurement received from the measurement member (10).

4. The superconducting microwave quantum circuit according to claim 1, wherein: The pulsation wi of the carrier of each respective pulse train is a pulsation of one of the resonant modes of the Förster first form decomposition across the terminals of the Josephson junction element (8) of the linear passive circuit part (6) different from the pulsation w, the linear passive circuit part (6) being arranged so that said resonant mode of the pulsation wi dissipates its respective excitation into an environment in which the product of the temperature and the Boltzmann constant is lower than the product of the pulsation wi and the reduced Planck constant and at a rate higher than: , where Zi is the impedance of the mode of pulsation wi across the terminals of the element of the Josephson junction element, and ei is the average Josephson energy integrated over the duration of the pulses, each pulse being preceded by a pre-pulse and followed by a post-pulse, the pre-pulse and post-pulse being modulated by a quadrature sinusoidal carrier with the carrier of the pulses being offset relative to the pulses of corresponding duration δ less than 1 / (5w) and having opposite amplitudes and absolute values ​​less than 1 / (3δw).

5. The superconducting microwave quantum circuit according to any one of claims 1 to 4, wherein: The respective pulse trains are activated in overlapping time windows and the pulsations of the sinusoidal carrier waves which modulate their respective amplitudes are different.

6. The superconducting microwave quantum circuit according to any one of claims 1 to 4, wherein: Corresponding pulse trains are activated in sequence and modulated by a sinusoidal carrier of the same pulse wi so that a pulse train ends before activating the next train, each train comprising a number of pulses between 10 and 10^8.

7. A superconducting microwave quantum circuit according to any one of the preceding claims, wherein: The Josephson junction element has a controllable phase, allowing the position of the GKP state grid to be modified.

8. A superconducting microwave quantum circuit according to any one of the preceding claims, wherein: The impedance of the target resonant mode is adjustable, and wherein an offset between respective pulse trains is adaptable to allow a shape of a mesh of the GKP state grid to be modified.

9. The superconducting microwave quantum circuit according to claim 8, wherein: The impedance of the target resonance mode is equal to 13 kohm, wherein the pulsation of the target resonance mode is modified so as to perform a π / 2 rotation of the GKP state grid, and wherein the duration between pulses within the same train is modified in a manner adapted to the modified pulsation.

10. The superconducting microwave quantum circuit according to claim 8, wherein: The impedance of the target resonance mode is equal to 15 kohm, wherein the pulsation of the target resonance mode is modified so as to perform a π / 3 rotation of the GKP state grid, and wherein the duration between pulses within the same train is modified in a manner adapted to the modified pulsation.

11. The superconducting microwave quantum circuit according to claim 8, comprising two energy-controllable Josephson junction elements, each Josephson junction element is connected to the linear passive circuit portion (6), so that the impedance of the target resonance mode presented at at least one of the two Josephson junction elements is adjustable, and the energy of each Josephson junction element (8) is controllable and modulated according to each corresponding impedance of the target resonance mode.

12. A superconducting microwave quantum circuit comprising two circuits, at least one of which is interconnected according to a circuit according to one of claims 8 or 11 so as to define a logic gate.

13. The superconducting microwave quantum circuit according to claim 12, wherein: The two circuits according to claim 8, the linear passive circuit portion (6) of each circuit being connected to the Josephson junction element (8) of the other circuit through an adjustable coupler so as to control the impedance of the target resonant mode of each circuit presented at the Josephson junction element of the other circuit.

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