Autonomous stabilization of a GKP code by parametric comb modulation of microwave frequencies

Abstract

Autonomous stabilization of a GKP code by parametric comb modulation of microwave frequencies A microwave-range quantum superconducting circuit includes an energy-tunable Josephson junction element (8) connected to a portion of a linear passive circuit (6) exhibiting several resonant modes, the first-form Foster decomposition of which across the Josephson junction element (8) includes a target resonant mode having an impedance Z greater than 13 kOhms and a pulsation w.The energy of said Josephson junction element (8) is adjustable and modulated by at least two respective pulse trains in which the pulses are separated by a duration of 2π / 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 pulse trains are shifted in pairs respectively by a duration Δt such that |sin(w*Δt)| is 13 kOhms / Z, so that the resonant mode of pulsation w stabilizes in one of two GKP states encoding a qubit. Fig.2.

Description

Title of the invention: Autonomous stabilization of a GKP code by parametric comb modulation of microwave frequencies

[0001] The invention relates to the field of quantum circuits with superconducting microwave resonator, and in particular implementing a quantum correction code using a bosonic code.

[0002] The realization of a quantum computer is currently hindered by noise that alters the state of quantum bits (hereinafter qubits), causing logical errors. Despite significant progress made over the past 20 years in limiting noise sources, universal and error-tolerant quantum computing is currently out of reach. Quantum error correction aims to solve this problem.

[0003] The type of logical error to be corrected is of two kinds: bit flips and phase flips. The main methodology used aims to encode a few logical qubits in a much larger information space. In this information space, the qubits are encoded in so-called "coding" states, which are chosen so that noise does not allow a direct transition from one coding state to another.

[0004] The noise, which thus induces transitions from a coding state to a non-coding state, can be detected and corrected to return the system to the initial coding state unambiguously; that is, each non-coding state is associated with a unique coding state. To avoid disturbing the quantum information, this error detection and correction must be performed without measuring the logical qubits.

[0005] The most widespread method for correcting quantum errors is called a "surface code". This type of solution is implemented by the biggest names in the quantum field, such as Google, IBM, Delft University, the University of Zurich, etc., and is the most studied solution in the world to date.

[0006] In a surface code, quantum information is carried by a 2D network of two-level systems. Fault-tolerant one- and two-qubit gates exist. The biggest drawback of this technology is that it requires a large number of very high-quality two-level systems to perform a computation. Beyond the costs inherent in this architecture, this generates problems related to the need to control a large number of quantum systems.

[0007] The Applicant's work led it to consider that surface codes have disadvantages requiring the exploration of other solutions. For this reason, the Applicant studied bosonic codes.

[0008] Bosonic codes are a family of quantum error-correcting codes that rely on storing qubits in bosonic modes. These codes comprise three main subfamilies: 'cat' codes, binomial codes, and GKP (for Gottesman-Kitaev-Preskill) codes.

[0009] An example of cat codes are the two-legged Schrödinger's cat codes 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 encoding states are Schrödinger cat states. Unlike GKP code, only one of the two types of logical error can be suppressed. Therefore, this code must be concatenated within a repetition code to handle both types of logical error.

[0011] One advantage of these codes over GKP codes is that the suppression of one of the two types of errors can be made arbitrarily good simply by varying the size of the cat. This is not possible with GKP codes, for which only increasing the ratio between the design-induced dissipation rates (or measurement rate and correction in the case of measurement stabilization) and the intrinsic dissipation of the resonator allows for an exponential increase in the lifetime of the logic qubit.

[0012] Another example is the n>2-legged Schrödinger cat codes implemented by Yale University and Quantum Computing Inc. Unlike 2-legged Schrödinger cat codes, both types of logical error can be corrected. However, the correction does not follow an exponential but a polynomial law when the correction detection rate or the design-induced dissipation rate is increased.

[0013] Binomial codes, for their part, present problems similar to those of Schrödinger's cats with n>2 legs. Furthermore, they only protect against photon loss, and not against other types of noise such as pure phase shift.

[0014] GKP codes also have several implementations. A first example is a GKP code stabilized by Rabi interactions with an ancillary qubit, implemented by Yale University in partnership with the Quantic team. This method induces logical errors, for example, when the ancillary qubit undergoes an error that modifies its state along the axis of the Bloch sphere corresponding to the operator present in the Rabi interaction during the interaction. It is possible to eliminate these errors by using an ancillary qubit in which one of the errors is eliminated (for example, a Schrödinger's cat), but this method is complex to implement and has not yet been experimentally realized.

[0015] Another example is a GKP code stabilized by modular operators. There are proposals for implementing Hamiltonian dynamics carried by Modular operators, combined with simple one-photon dissipation, stabilize the GKP code. However, the proposed circuits to implement this dynamic are not feasible. Indeed, they either involve a so-called "coherent phase-slip element" or a "high-impedance gyrator," for which no implementation has yet been demonstrated.

[0016] No error-correcting code therefore works satisfactorily to date. For the most advanced, the point at which logical qubits have a lifetime / consistency longer than the underlying physical systems has barely been reached, but has never been significantly exceeded.

[0017] The invention improves the situation. To this end, it proposes a quantum superconducting microwave range circuit comprising an energy-tunable Josephson junction element connected to a portion of a linear passive circuit having several resonant modes, the first-form Foster decomposition of which at the terminals of the Josephson junction element includes a target resonant mode having an impedance Z greater than 13 kOhms and a pulsation w.The energy of said Josephson junction element is adjustable and modulated by at least two respective pulse trains in which the pulses are separated by a duration 2ir / 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 pulse trains being shifted two by two respectively by a duration At such that lsin(w*At)l is 13 kOhms / Z, so that the resonant mode of pulsation w stabilizes in one of two GKP states encoding a qubit.

[0018] This device is particularly advantageous because it allows the implementation of modular operators for the implementation of a qubit in a GKP code, which can be reliably stabilized and allows a considerable improvement in the ratio between coherence time and gate time compared to all qubits implemented so far.

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

[0020] - the angular frequency wi of the sinusoidal carrier of each respective pulse train is null, so that the GKP coding states are the lowest energy states in an interaction representation lowering the target resonant mode's frequency to a frequency less than one-fifth of the amplitude of the modular effective potential generated for the target resonant mode divided by the reduced Planck constant, said lowest energy states being stabilized by coupling to an environment whose product of the Boltzmann constant and the effective temperature is less than the Josephson energy in this representation,

[0021] - the angular frequency wi of the sinusoidal carrier of each respective pulse train is a pulsation of one of the resonant modes of the Foster decomposition across the Josephson junction element of the linear passive circuit portion different from the pulsation ω, the microwave-range quantum superconducting circuit further comprising a homodyne linear measuring device for the resonant modes corresponding to the respective pulsations ωi, arranged to measure the displacements of the resonant modes of pulsation ωi along a direction of their respective phase plane determined by the phase of the sinusoidal carrier of pulsation ωi, and a feedback device arranged to displace the state of the target resonant mode along a direction of its phase plane determined by the phase of the pulse train corresponding to the pulsation ωi relative to the phase reference of said target resonant mode, in a manner proportionate to a measurement received from the measuring device,

[0022] - the angular frequency wi of the carrier of each respective pulse train is a pulsation of one of the resonant modes of the Foster decomposition of the first form across the Josephson junction element of the linear passive circuit portion other than the pulsation ω, the linear passive circuit portion being arranged such that said resonant modes of pulsation ω dissipate their respective excitations in an environment whose product of the temperature by the Boltzmann constant is less than the product of the pulsation ω by the reduced Planck constant and at a rate greater than ^(^ / 4^ / (6.5 kOhms))^ where Zi is the impedance of the mode of pulsation ω across the elements of said Josephson junction element and ei is the average Josephson energy integrated over the duration of a pulse, each pulse being preceded by a pre-pulse and followed by a post-pulse,the pre-pulse and post-pulse being modulated by a sinusoidal carrier in quadrature with the pulse carrier and being shifted relative to the pulse by a respective duration ô less than l / (5w) and having an opposite amplitude and absolute value less than l / (3ôw), ,

[0023] - the respective pulse trains are activated in specific time windows vauchant, and the pulsations of the sinusoidal carriers that modulate their respective amplitudes are distinct,

[0024] - the respective pulse trains are activated one after the other and are modulated by the same sinusoidal carrier with pulsation wi, so that one pulse train terminates before the activation of the next train, each train comprising a number of pulses between 10 and 10A8,

[0025] - the Josephson junction element has an adjustable phase allowing modification the position of the GKP state grid,

[0026] - the impedance of the target resonant mode is adjustable, and the offset between the trains the respective pulses can be adapted to allow modification of the shape of the GKP state grid mesh,

[0027] - the impedance of the target resonant mode is equal to 13 kOhms, the frequency of the mode The target resonant frequency is modified to achieve a rotation of ir / 2 of the GKP state grid, and the duration between pulses within the same train is modified to suit this modified pulse rate.

[0028] - the impedance of the target resonant mode is equal to 15 kOhms, the frequency of the mode The target resonant frequency is modified to achieve a rotation of ir / 3 of the GKP state grid, and the duration between pulses within the same train is modified to suit this modified pulse rate.

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

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

[0031] - the two circuits have an adjustable target resonant mode impedance, the offset between the respective pulse trains can be adapted to allow modification of the mesh shape 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 by an adjustable coupler so as to tune the impedance of the target resonant mode of each circuit presented to the Josephson junction element of the other circuit.

[0032] Other features and advantages of the invention will become more apparent from the following description, taken from illustrative and non-limiting examples shown in the drawings:

[0033] - [Fig.1] is a representation of the wave functions of the states of a GKP code,

[0034] - [Fig.2] is a representation of a quantum circuit with a supra-microwave resonator driver according to a first embodiment of the invention,

[0035] - [Fig.3] represents the temporal activation profile of the Josephson junctions of the [Fig.2] when the first-form Foster decomposition across the Josephson junction element includes a target resonant mode exhibiting an impedance Z equal to 13 kOhms,

[0036] - [Fig.4] represents the shifts in the state of the target resonant mode of the circuit of the [Fig.2] in a laboratory frame and in a rotating frame, when the first-form Foster decomposition across the Josephson junction element includes a resonant mode with an impedance Z equal to 13 kOhms,

[0037] - [Fig.5] represents a variant of [Fig.3] when the impedance Z is greater than 13 kOhms,

[0038] - [Fig.6] represents a variant of [Fig.4] when the impedance Z is greater than 13 kOhms,

[0039] - [Fig.7] represents a so-called Hamiltonian implementation of the circuit of [Fig.2],

[0040] - [Fig.8] represents the temporal activation profile of the Josephson junctions of the [Fig.7]

[0041] - [Fig.9] represents the effective potentials generated by means of the circuit of [Fig.7] in a rotating frame of reference, as well as the probability density of the coding states GKP,

[0042] - [Fig. 10] represents the energy levels of the states of the [Fig.9],

[0043] - [Fig. 11] represents a so-called feedback implementation of the circuit of [Fig.2],

[0044] - [Fig. 12] represents the shifts in the state of the target resonant mode GKP of the circuit of the [Fig.1 1] by feedback,

[0045] - [Fig. 13] represents a so-called dissipative implementation of the circuit of [Fig.2],

[0046] - [Fig. 14] represents the temporal activation profile of the Josephson junctions of the [Fig.13]

[0047] - [Fig. 15] represents the shifts in the state of the target resonant mode of the circuit the [Fig. 11] by dissipation, and

[0048] - [Fig. 16] is an example of an implementation of the circuit of [Fig. 13].

[0049] The drawings and description below contain, essentially, elements of a definite nature. They may therefore not only serve to better understand the present invention, but also contribute to its definition, if necessary.

[0050] The invention relates to the implementation of logic qubits by means of GKP codes. To better understand how to stabilize these codes, [Fig. 1] provides a representation of the coding states "0" and "1" and their symmetric and antisymmetric superpositions "+" and "-" of a square GKP code according to quadratures q and p. The GKP code ensures that orthogonal logic states have distant supports in the phase plane (q,p). Alternatively, the mesh of the GKP grid can be a parallelogram, provided that its area in the plane (q,p) at normalized coordinates (such as the commutator [q,p]=i) is equal to 4ir. A special case where the angle of the parallelogram is jt / 3 is known as "hexagonal coding".

[0051] In what follows, the equations presented refer to idealized GKP codes, whose states extend infinitely in the phase plane ("infinitely squeezed states"), hereinafter referred to as "infinitely sized GKP codes." The invention aims at an implementation with "realistic" GKP codes, that is, whose states have finitely sized support in the phase plane ("finitely squeezed states"), hereinafter referred to as "finitely sized GKP codes." This implementation may involve minor changes to the equations presented below, and modifications to the stabilization protocol will be specified further down whenever possible.

[0052] As can be seen in this figure, the different coding states are separated by V(ji) according to quadrature q or according to quadrature p.

[0053] In this figure, each quadrant represents the probability distribution for certain pairs of orthogonal states in bit (0; 1) or in phase (+; -), according to quadrature q or quadrature p. The distance between states 0 and 1 in q provides protection against bit flips, while the distance between states + and - in p provides protection against phase flips. The circuits according to the invention allow these states to be generated and stabilized by effective coupling to the target mode via the modular operators g±2^p that it implements, where q and p are the operators of quadrature of the target mode in the rotating frame at the target mode frequency. These modular operators are the stabilizers of the infinite-size GKP code.

[0054] In a so-called Hamiltonian implementation, the invention lifts the energy degeneracy of the eigenstates of these stabilizers, and the coding states are of lower energy and therefore stabilized in the presence of dissipation to a reservoir of low effective temperature.

[0055] In a so-called feedback-by-measurement implementation, these stabilizers are measured and the result of the measurement allows the coding states to be stabilized by feedback.

[0056] In a so-called dissipation engineering implementation, the interaction between the target mode and highly dissipative ancillary modes via the modular operators of the target mode on the one hand, and the linear operators of the ancillary modes on the other hand, generates an effective modular dissipation stabilizing the GKP coding states.

[0057] In the case of the main errors for superconducting resonators, i.e. single-photon dissipation, the Applicant estimates that the qubits thus encoded can have a lifetime on the order of a tenth of a second, while remaining controllable by error-tolerant gates and of a duration on the order of a microsecond, which represents an improvement in the ratio between the coherence time and the gate time of several orders of magnitude compared to all qubits implemented to date.

[0058] The Applicant has discovered a class of quantum circuits which make it possible to implement the modular operators mentioned above by combining an energy-tunable Josephson junction element and a portion of a linear passive circuit, by implementing a particular adjustment of the energy of the Josephson junction element.

[0059] Figure 2 represents a generic view of a quantum circuit of this type. As can be seen in this figure, the circuit 2 comprises a transmission line 4 connected to a terminal (or "port") connected to a portion of linear passive circuit 6, and a Josephson junction element 8 connected to the terminals of the portion of linear passive circuit 6.

[0060] The transmission line 4 can be implemented in any known manner in the domain, on-chip or off-chip, 2D or 3D, etc., according to any geometry known for a microwave transmission line.

[0061] The passive circuit portion 6 can be made in any known way allowing the Foster decompositions mentioned below to be obtained when they carry a signal in the microwave range.

[0062] The Josephson junction element 8 has an adjustable Josephson energy. As shown in Figures 2, 7, 11 and 13, the circuits shown allow the Josephson energy to be adjusted by applying a magnetic flux between two Josephson junctions forming a SQUID (Superconducting Quantum Interference Device), but could also be adjusted by other methods (more complex multi-junction circuits, semiconductor junction with a voltage bias).

[0063] Depending on the embodiment, the circuit 2 can be modified and arranged to allow stabilization of the GKP code in a Hamiltonian implementation (figures 7 to 10), a feedback implementation (figures 11 and 12), or a dissipation implementation (figures 13 to 16).

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

[0065] The first principle is that the linear passive circuit portion 6 has one or more resonant modes whose Foster decomposition of the first form (see, for example, Foster, R.M., "A reactance theorem", Bell System Technical Journal, vol. 3, no. 2, pp. 259-267, November 1924) across the Josephson junction element includes a resonant mode having an impedance Z greater than or equal to 13 kOhms. The equivalent circuit is shown on the right in [Fig. 2]. In what follows, this resonant mode will also be referred to as the "target mode".

[0066] This impedance value is particularly important because it corresponds to twice the resistance quantum (which is defined by h / 4e², or 6.5 kΩ), allowing the Josephson junction element 8 to implement, in the rotating frame of reference of the target mode, at different times, two distinct and switching modular operators, corresponding to the stabilizers of the GKP code. Indeed, the spatial frequency d of the modular operator is related to the impedance presented to the Josephson element via

[0067] ZI 6.5 kOhms '

[0068] The product of the frequencies of the two modular operators is 4ir in the case of the square GKP code, and is greater for a non-square GKP code. The value of 13 kOhms is therefore the minimum value for stabilizing the GKP code. With impedance values ​​above 19.5 kOhms, it would be possible, for example, to stabilize qutrits.

[0069] The second principle is that the Josephson energy setting, represented by the arrow which enters the Josephson junction element 8 on [Fig.2], allows to organize the displacements which the states of the GKP code can perform as will be described below.

[0070] Thus, [Fig. 3] represents the signals used to modulate the Josephson energy. This setting, to the right of [Fig. 3], is the result of the sum of two sinusoids with respective angular frequencies w1 and w2, each modulating the amplitude of a pulse train. These trains are offset from each other by a quarter of a period associated with the angular frequency of the resonant mode, whose impedance is equal to 13 kOhms. Each pulse has a width less than one-tenth of this quarter of a period.

[0071] Another way to describe [Fig.3] is that the Josephson energy regulation signal of the Josephson junction element 8 is the sum of two control signals which act to "switch off" or "switch on" the Josephson energy in a particular way at chosen times:

[0072] - The first control signal is such that the Josephson energy is activated at each period associated with the pulsation of the resonant mode with impedance greater than or equal to 13 kOhms, and substantially zero at other times, and this energy is modulated in amplitude by a sinusoid of pulsation wl equal to the pulsation of a first ancillary correction mode of the conjugate operator q in the Foster decomposition,

[0073] - The second control signal is such that the resulting Josephson energy is activated at each period associated with the pulsation of the resonant mode with impedance greater than or equal to 13 kOhms, and substantially zero at other times, and this energy is modulated in amplitude by a sinusoid of pulsation w2 at the pulsation of a second ancillary correction mode of the conjugate operator p, and

[0074] - the first control signal and the second control signal are offset between them of a quarter period associated with the pulsation w of the target resonant mode of impedance equal to 13 kOhms.

[0075] Alternatively, the control signals could be activated every half-period instead of every period as mentioned above. If necessary, this can be done by reversing the sign of the pulse.

[0076] The result of this combination within the framework of the GKP code implemented by circuit 2 is represented with [Fig.4].

[0077] Figure 4 represents the different possible displacements on the GKP grid as a function of time, in the fixed frame of reference with axes qO and pO (top) or in a rotating frame of reference with axes q and p (bottom) at the angular frequency ω of the target mode. The concept of a rotating frame of reference is also called an "interaction representation," and the two expressions may be used interchangeably hereafter.

[0078] In the fixed frame of reference, due to the resonance of the pulsation mode w, the states of the The resonator will rotate at the angular frequency ω, and coherent and discrete displacements along the GKP grid will be possible every quarter period if the target mode is presented to the Josephson junction element with a matched impedance. However, with the activation profile of [Fig. 3], this only occurs every quarter period, depending on one pulse train or the other. This means that when 2irt / ω is 0.125 or 0.375, the Josephson energy is zero, which prevents any displacement off the grid. Conversely, every quarter period, this is possible, along +q0, -q0, +p0, or -p0. This can also be understood with the representation in the rotating frame of reference, in which the states are stable (since the frame of reference rotates at the angular frequency ω), and it is the displacements that rotate (therefore, a displacement is possible along +q, -q, +p, or -p) and are possible every quarter period.

[0079] Thus, the circuit of [Fig.2] makes it possible to generate and stabilize a GKP grid state since the dynamics of the mode can be coherent and constrained according to the GKP grid.

[0080] In the particular case of figures 3 and 4, the resonant mode of pulsation w has an impedance equal to 13 kOhms, so that the unitary operators q and p cover an area equal to 4ir.

[0081] It is nevertheless possible that the resonant mode with pulsation w has an impedance greater than 13 kOhms. In this case, as shown in Figures 5 and 6, the pulse trains should be slightly offset from each other, and the GKP grid is then a rhombus whose area remains equal to 4ir.

[0082] Figures 7 to 15 represent three distinct embodiments that allow stabilization of the qubit encoded in the GKP code of such a circuit. Thus, Figures 7 to 10 describe a quantum circuit implementing Hamiltonian stabilization and a modulation scheme of the Josephson junction element, Figures 11 and 12 describe a quantum circuit implementing feedback stabilization, and Figures 14 to 16 describe a quantum circuit implementing dissipation stabilization.

[0083] Figure 7 shows the first-form Foster decomposition of a portion of a linear passive circuit 6. In this embodiment, the pulsations ωl and ω2 are zero, so the Josephson energy activation profile of the Josephson junction element 8 is shown in Figure 8, and induces energy potentials shown in Figure 9 for the p and q modes. Thus, the pulse trains are each modulated by a sinusoid of zero pulsation, which is therefore always 1. In other words, it is recalled that the value 1 is a sinusoid of zero pulsation. Alternatively, the pulse trains could be symmetrically doubled with respect to the period.

[0084] In the rotating frame of reference, and in the absence of Josephson dynamics, all oscillator states are degenerate and have zero energy. The pulse sequence in this Hamiltonian implementation generates an egg-box potential according to q and p which lifts the degeneracy of these energy levels and ensures that the coding states are the lowest energy states, their probability density being located at the bottom of the potential wells.

[0085] This has the consequence that the GKP coding states are the lowest energy states in an interaction representation lowering the target mode frequency to a frequency less than one third of the Josephson energy divided by the reduced Planck constant.

[0086] In this embodiment, in order to stabilize the qubit, that is to say in order to bring an excited GKP code state back to a coding state, therefore of low energy, the circuit includes a transmission line 4 which couples the circuit 2 to an environment whose product of the Boltzmann constant by the effective temperature, in the interaction representation, is less than the Josephson energy in this representation.

[0087] As a result, high energy states tend to relax by losing energy through the transmission line 4 (as represented by the arrow in [Fig.7]), so that in the event of excitation of the resonator, it naturally returns to its low energy coding state as shown in [Fig. 10], in which the lines represent the different states of the GKP Hamiltonian according to their energy level.

[0088] The above corresponds to an infinite-sized GKP code. In the case of a finite-sized GKP code, the time between pulses within a pulse train can be increased or decreased by a small amount, typically less than âz4£ where E is the amplitude of the modulation of the effective potential (V(q) and V(p)), h is the Planck constant.

[0089] This embodiment is particularly advantageous when the perturbations limiting the coherence of the qubit are Hamiltonian.

[0090] Fig. 11 represents a second embodiment of the quantum circuit 2 in which the circuit 2 further comprises a measuring element 10 and a feedback element 12 arranged between the environment 4 and the portion of the linear passive circuit 6.

[0091] The pulsations wi are here chosen to correspond to a pulsation of one of the resonant modes of the Foster decomposition at the terminals of the Josephson junction element 8 of the portion of linear passive circuit 6 different from the pulsation w of the resonant mode whose impedance is greater than or equal to 13 kOhms, the pulsations wi being distinct from each other.

[0092] The measuring element 10 is a linear homodyne of the resonant modes corresponding to the respective pulsations wi, arranged to measure the displacements of the resonant modes of pulsation wi along a direction of their respective phase plane. terminated by the sinusoidal carrier phase with angular frequency wi, and allows the measurement of modular operators, that is, the measurement of GKP error syndromes, without risk of error propagation; namely, that decoherence or dissipation affecting the ancillary mode does not risk inducing a logical error on the GKP qubit. The measurement unit 10 can, for example, be implemented by a parametric amplifier with Josephson junctions operating close to the quantum limit.

[0093] The feedback member 12 is arranged to shift the state of the resonant mode having an impedance Z greater than 13 kOhms along a direction of its phase plane determined by the phase of the pulse train corresponding to the pulsation wi by the phase reference of said resonant mode having an impedance Z greater than 13 kOhms, in a manner proportionate to a measurement received from the measuring member 10. The feedback member 12 can be implemented by applying a microwave at the pulsation w whose phase and amplitude are proportional to the result of the measurement.

[0094] Thus, as shown in a [Fig.12], the GKP states are shifted by the feedback member 12 along a phase plane vector whose coordinates are proportional to the error syndromes si and s2 measured by the measuring member 10. The circuit 2 can also be arranged to have more than two error measurement syndromes.

[0095] This description corresponds to an implementation for an infinite-sized GKP code. In the case of a finite-sized GKP code, it would be possible to alternate the measurement of modular operators with linear feedback (as described above), with a weak linear (homodyne) measurement with weak modular feedback.

[0096] Alternatively, the Applicant has discovered that the pulse trains can be activated one after the other by being modulated by the same sinusoidal carrier of pulsation wi, so that one pulse train ends before the activation of the next train, each train comprising a number of pulses between 10 and IOA8.

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

[0098] Indeed, stabilization can be made autonomous with simple dissipation over ancillary modes. This results in doubly modular dissipation of the target resonator. More precisely, circuit 2 of [Fig. 13] allows the GKP logic qubit to be stabilized dissipatively using modular operators.

[0099] In the embodiment of [Fig. 13], the Josephson junction element 8 comprises two Josephson junctions 44 in parallel with each other, and its energy profile is modulated with sinusoidal carriers whose pulsations wi are chosen in a similar manner to that of the embodiment of Figures 11 and 12.

[0100] The portion of the linear passive circuit 6 is further arranged so that the modes to-\iïw / 4*Z 5.5 kOhmsH*e: °where is the impedance of the pulsation mode wi aux Resonants with pulsation wi dissipate their respective excitations in an environment where the product of the temperature and the Boltzmann constant is less than the product of the pulsation wi and the reduced Planck constant, and at a higher rate terminals of the element of said Josephson junction element and e; is the average Josephson energy integrated over the duration of a pulse.

[0101] As shown in [Fig. 14], for the implementation adapted 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 (in dashed lines) being modulated by a sinusoidal carrier in quadrature with the pulse carrier and being offset from the pulse by a respective duration ô less than l / (5w) and having an opposite amplitude and absolute value less than l / (3ôw).

[0102] Preferably, the Josephson junction element 8 is coupled to four ancillary dissipative modes to achieve a four-dissipator Lindblad dynamic. However, it would also be possible to achieve a two-dissipator or even a six-dissipator Lindblad dynamic.

[0103] Alternatively, the Applicant has discovered that the pulse trains can be activated one after the other by being modulated by the same sinusoidal carrier of pulsation wi, so that one pulse train terminates before the activation of the next train, each train comprising a number of pulses between 10 and IOA8.

[0104] Figure 16 shows a top-view photograph of an actual implementation of a circuit according to Figure 13. This photograph was obtained by optical microscopy (left photograph) and by scanning electron microscopy (right photograph).

[0105] This figure includes several enlargements to better illustrate the implementation of the various elements. For example, an inductance 560 made from a long trace used to adjust the impedance of the ancillary modes presented to the Josephson junction element can be seen, as well as the traces for the control signals. These enlargements also allow for a better appreciation of the orders of magnitude, with the Josephson junctions having dimensions on the order of 500 nm, the ATS on the order of 100 pm, and the traces for the control signals on the order of a millimeter.

[0106] The circuit of [Fig. 13] can be implemented by means of an ATS (Asymmetrically Threaded SQUID) circuit. The ATS circuit comprises a symmetrical SQUID which is short-circuited at its center by a large inductance, thus forming two loops.

[0107] Thus FATS implements an LC oscillator with added Josephson energy of controllable amplitude and phase. The common mode of the ATS circuit's ring can have an impedance Z equal to 13 kOhms and a resonant frequency in the RF range for a sufficiently large inductance in the circuit's center arm. Low-impedance, matched-symmetry distributed modes connected to the ring can serve as ancillary modes. In the first-form Foster representation, the capacitances have values ​​on the order of 10 fF, and the inductances have values ​​on the order of 1 to 10 pH.

[0108] Using an ATS instead of a SQUID is advantageous in the case of a four-dissipator Lindblad dynamics system. Indeed, phase control of each modular operator implemented by means of the Josephson junction element is important in this case, with two of the dissipators associated with a cosine wave and the other two with a sine wave.

[0109] This embodiment is particularly advantageous because the ATS circuit has been implemented practically by the Applicant, who has a particularly good grasp of its manufacture.

[0110] Advantageously, the impedance of the target mode as presented to the Josephson junction element 8 can be adjusted, thereby enabling the realization of a universal set of Clifford gates. This adjustment can be achieved by introducing an adjustable coupler between the linear passive circuit portion 6 and the Josephson junction element 8. This adjustable coupler can be implemented in a manner known to those skilled in the art, for example in the context of superconducting circuits (QED circuits).

[0111] Thus, according to a first variant, the impedance of the target resonant mode is equal to 13 kOhms, which fixes a square GKP code. In this case, by slightly varying the duration between two pulses within each train, it is possible to progressively rotate the GKP state grid until a rotation of angle ji / 2 is obtained. This yields a Hadamard gate.

[0112] In another embodiment, the impedance of the target resonant mode is 15 kOhms, which establishes a hexagonal GKP code. In this case, by slightly varying the duration between two pulses within each train, it is possible to progressively rotate the GKP state grid until a rotation angle of ir / 3 is obtained. This results in a circular permutation gate of the three axes of the Bloch sphere of the logic qubit (X=>Y=>Z=>X...).

[0113] When two circuits have an adjustable impedance and their portions of linear passive circuit 6 are respectively connected to the Josephson junction element 8 of the other circuit by an adjustable coupler, it becomes possible to implement a CNOT gate.

[0114] In yet another embodiment, the adjustable impedance can be used to achieve a protected initialization or protected readout circuit for the logic qubit. In this case, the microwave-range quantum superconducting circuit comprises two Josephson junction elements. At least one Josephson junction element is connected to the linear passive circuit portion such that the impedance of the target mode presented to it is adjustable. The area of ​​the GKP code lattice is then reduced by a factor of 2, and the energy of each Josephson junction element is tuned according to the impedance of the target mode presented to it. Advantageously, for a square GKP code, one of the target modes will have an impedance of 13 kOhms, while the other will have an impedance of 6.5 kOhms.

Claims

Demands

1. A microwave-range quantum superconducting circuit comprising an energy-tunable Josephson junction element (8) connected to a linear passive circuit portion (6) having several resonant modes, the first-form Foster decomposition across the Josephson junction element (8) of said linear passive circuit portion (6) comprising a target resonant mode having an impedance Z greater than or equal to 13 kOhms and an angular frequency ω, the energy of said Josephson junction element (8) being adjustable and modulated by at least two respective pulse trains within which the pulses are separated by a duration 2ir / ω 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 pulse trains being offset in pairs respectively by a duration At such that lsin(ω*At)l is 13 kOhms / Z,so that the target resonant mode stabilizes in one of two GKP states.

2. Microwave-range quantum superconducting circuit according to claim 1, wherein the pulsation wi of the sinusoidal carrier of each respective pulse train is zero, such that the lowest energy states in an interaction representation lowering the pulsation of the target resonant mode to a pulsation less than one-fifth of the amplitude of the modular effective potential generated for the target resonant mode divided by the reduced Planck constant, said lowest energy states being stabilized by coupling to an environment (4) whose product of the Boltzmann constant by the effective temperature is less than the Josephson energy in this representation, are GKP-coding states.

3. A microwave-range quantum superconducting circuit according to claim 1, wherein the angular frequency wi of the sinusoidal carrier of each respective pulse train is an angular frequency of one of the resonant modes of the Foster decomposition across the Josephson junction element (8) of the linear passive circuit portion (6) different from the angular frequency wi, the microwave-range quantum superconducting circuit further comprising a homodyne linear measuring element (10) for the resonant modes corresponding to the respective angular frequencies wi, arranged to measure the displacements of the resonant modes of angular frequency wi along a direction of their phase plane respective determined by the phase of the sinusoidal carrier of pulsation wi, and a feedback element (12) arranged to move the state of the target resonant mode along a direction of its phase plane determined by the phase of the pulse train corresponding to the pulsation wi with respect to the phase reference of said target resonant mode, in a manner proportionate to a measurement received from the measuring element (10).

4. Microwave-range quantum superconducting 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 Foster decomposition of the first form across the Josephson junction element (8) of the linear passive circuit portion (6) different from the pulsation w, the linear passive circuit portion (6) being arranged such that said resonant modes of pulsation wi dissipate their respective excitations in an environment whose product of the temperature by the Boltzmann constant is less than the product of the pulsation wi by the reduced Planck constant and at a rate greater than lw z, * where Z is the impedance of the mode of pulsation wi across the elements of said Josephson junction element and e;is the average Josephson energy integrated over the duration of a pulse, each pulse being preceded by a pre-pulse and followed by a post-pulse, the pre-pulse and post-pulse being modulated by a sinusoidal carrier in quadrature with the pulse carrier and being shifted relative to the pulse by a respective duration ô less than l / (5w) and having an opposite amplitude and absolute value less than l / (3ôw).;

5. Microwave-range quantum superconducting circuit according to any one of claims 1 to 4, wherein the respective pulse trains are activated in overlapping time windows, and wherein the pulsations of the sinusoidal carriers that modulate their respective amplitudes are distinct.

6. Microwave-range quantum superconducting circuit according to any one of claims 1 to 4, wherein the respective pulse trains are activated one after the other and are modulated by the same sinusoidal carrier of pulsation wi, such that one pulse train terminates before the activation of the next train, each train comprising a number of pulses between 10 and 1OA8.

7. Microwave-range quantum superconducting circuit according to one of the previous claims, wherein the Josephson junction element has an adjustable phase for changing the position of the GKP state grid.

8. Microwave-range quantum superconducting circuit one of the preceding claims, wherein the target resonant mode impedance is adjustable, and wherein the offset between the respective pulse trains can be tuned to allow modification of the mesh shape of the GKP state grid.

9. Microwave-range quantum superconducting circuit according to claim 8 wherein the target resonant mode impedance is equal to 13 kOhms, wherein the target resonant mode pulsation is modified to achieve an ir / 2 rotation of the GKP state grid, and wherein the time between pulses within a given train is modified in a manner adapted to this modified pulsation.

10. Microwave-range quantum superconducting circuit according to claim 8 wherein the target resonant mode impedance is equal to 15 kOhms, wherein the target resonant mode pulsation is modified to achieve an ir / 3 rotation of the GKP state grid, and wherein the time between pulses within a given train is modified in a manner adapted to this modified pulsation.

11. Microwave-range quantum superconducting circuit according to claim 8 comprising two energy-tunable Josephson junction elements, each connected to said linear passive circuit portion (6) such that the target resonant mode impedance presented to at least one of the two Josephson junction elements is adjustable, the energy of each Josephson junction element (8) being adjustable and modulated according to each respective target resonant mode impedance.

12. Microwave-range quantum superconducting circuit, comprising two circuits, at least one of which is, according to one of claims 8 or 11, connected together so as to define a logic gate.

13. Microwave-range quantum superconducting circuit according to claim 12, wherein said two circuits are 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 by an adjustable coupler so as to tune the target resonant mode impedance of each circuit presented to the Josephson junction element of the other circuit.

Images