Quantum system for stabilizing viton qubits

Abstract

The invention relates to a quantum system (1) for stabilizing Bose quantum bits, said system comprising: an instruction circuit (5) comprising three microwave sources (11, 13, 15), and a nonlinear superconducting quantum circuit (3) comprising a four-wave hybrid nonlinear element (7) and a resonant section (9), the quantum circuit (3) has a first mode having a first resonant frequency and a second mode having a second resonant frequency. The quantum circuit (3) is arranged to design a Hamiltonian H expressed as, by dissipation in the second mode, produce an effective dissipater that stabilizes the Bose quantum bits. To this end, the first microwave source (11) delivers frequency-equal radiation, the second microwave source (13) delivers frequency-equal radiation, and the third microwave source (15) delivers frequency-equal radiation to drive the second mode.

Description

TECHNICAL FIELD

[0001] The present invention relates to stabilizing a bosonic qubit. BACKGROUND

[0002] Generally, superconducting qubits can be implemented as a two-level system of a superconducting electronic circuit. Such a qubit can be stored in a bosonic mode and thus forms a special class of superconducting qubits, called bosonic qubits.

[0003] It is known from the prior art that cat qubits, i.e. bosonic qubits defined by a quantum manifold spanned by a superposition of two coherent states, can be stabilized, which are quasi-classical states of a bosonic mode. To this end, a specific dissipative stabilization mechanism can be performed and comprises a nonlinear transformation between the design of the following two modes, a first mode (also referred to as memory mode or cat qubit mode) carrying two photons of the stabilizing quantum manifold and a second mode (also referred to as buffer mode) with strong dissipative properties of one photon.

[0004] Cat qubits are advantageous in that their bit flip rate decreases exponentially, while their phase flip rate only increases linearly. Thus, a stabilized cat qubit benefits from a high noise bias, which means that the bit flip probability is much smaller than the phase flip probability.

[0005] Due to their noise structure, cat qubits can be concatenated with repetition codes, rather than using surface codes as for ordinary superconducting qubits. However, the noise bias of cat qubits seems to be insufficient for using simple repetition codes for certain large-scale algorithms and thus there is a need to find bosonic qubits with a higher noise bias.

[0006] To this end, recent research has focused on a special class of cat qubits: squeezed cat qubits. For example, Q. Xu et al. (2022) in the article “ Autonomous quantum error correction and fault-tolerant quantum computation with squeezed cat qubits ” (arXiv:2210.13406) propose to use squeezed cat codes in a continuous variable system to combat the dominant error source, excitation loss, to implement an autonomous quantum error correction (AQEC) scheme. The article announces that with only an average of four photons encoding the cat states, the noise bias is about 10 -15 For example, T. Hillmann et al. (2022) in the article “ Quantum error correction with dissipatively stabilized squeezed cat qubits ” (arXiv:2210.13359, Phys. Rev. A 107, 032423) propose and analyze the error correction performance of dissipatively stabilized squeezed cat qubits.

[0007] However, the possibility of stabilizing a compressed cat qubit in practice faces many obstacles and has therefore so far remained purely theoretical. Specifically, such stabilization requires the design of a large number of Hamiltonian terms to create the required dissipators and thus the use of too many pumps to threaten the stability of the system. It can be added that one of these terms ( where is the photon annihilation operator of the first mode and is the photon annihilation operator of the second mode) requires the injection of a large amount of energy into the system, which endangers the coherence of the system.

[0008] The present invention aims to improve this situation. SUMMARY

[0009] To this end, the Applicant proposes a quantum system for stabilizing a bosonic qubit, said system comprising: a command circuit comprising a first microwave source, a second microwave source and a third microwave source, each microwave source being arranged for delivering microwave radiation, and a nonlinear superconducting quantum circuit comprising an asymmetrically threaded superconducting quantum interference device and at least one resonant part, said asymmetrically threaded superconducting quantum interference device being connected to said resonant part, said asymmetrically threaded superconducting quantum interference device having a flux line through which radiation can be delivered to modulate a common flux and a differential flux, said nonlinear superconducting quantum circuit possessing a first mode having a first resonant frequency and a second mode having a second resonant frequency different from twice the first resonant frequency, said second mode being dissipative.

[0010] said asymmetrically threaded superconducting quantum interference device being arranged so that, when said first microwave source delivers radiation through said flux line to modulate said common flux at a frequency equal to the absolute value of the difference between twice said first resonant frequency and said second resonant frequency, said second microwave source delivers radiation through said flux line to modulate said common flux at a frequency equal to said second resonant frequency, and said third microwave source provides radiation to said at least one resonant part at a frequency equal to said second resonant frequency to drive said second mode, said nonlinear superconducting quantum circuit is designed with a Hamiltonian H expressed as wherein is proportional to the amplitude of said first microwave source, is the annihilation operator of said first mode, is a complex number whose phase and amplitude are caused by the amplitude of the second microwave source and are defined with respect to the phase and amplitude of the first microwave source, respectively, is a complex number caused by the third microwave source and defined with respect to the first microwave source, is an annihilation operator of a second mode, and is a Hermitian conjugate, the Hamiltonian H generates an effective dissipator by dissipation of the second mode , the effective dissipator stabilizes a two-dimensional manifold carrying the bosonic quantum bit in the first mode.

[0011] According to one or more embodiments, the first microwave source and the second microwave source are configured so that the respective phases of the first microwave source and the second microwave source are substantially equal.

[0012] According to one or more embodiments, the second microwave source and the third microwave source are configured so that the respective phases of the second microwave source and the third microwave source are substantially equal.

[0013] According to one or more embodiments, the first microwave source, the second microwave source and the third microwave source are configured so that the respective phases of the first microwave source, the second microwave source and the third microwave source are substantially equal.

[0014] According to one or more embodiments, the third microwave source is arranged to deliver radiation through the flux line to modulate the differential flux at a frequency equal to the second resonant frequency, thereby driving the second mode.

[0015] According to one or more embodiments, the second microwave source is configured to deliver radiation whose amplitude is such that the amplitude of is less than or equal to 1.

[0016] According to one or more embodiments, the second microwave source is configured to deliver radiation whose amplitude is such that the amplitude of is substantially equal to 1.

[0017] According to one or more embodiments, the respective phases of the first mode and the second mode in the asymmetrically-threaded superconducting quantum interference device have zero-point fluctuations , and the ratio is less than 3.

[0018] Furthermore, the Applicant also proposes a method for stabilizing a bosonic quantum bit performed by the quantum system described above, the method comprising the following operations: delivering radiation by the first microwave source through the flux line to modulate the common flux at a frequency equal to the absolute value of the difference between twice the first resonant frequency and the second resonant frequency, modulating the common flux at a frequency equal to the second resonance frequency by delivering radiation by the second microwave source through the flux line, providing radiation by the third microwave source to the at least one resonant section at a frequency equal to the second resonance frequency to drive the second mode, The implementation of the method leads to a Hamiltonian H of the nonlinear superconducting quantum circuit being designed as wherein, proportional to the amplitude of the first microwave source, is an annihilation operator of the first mode, is a complex number whose phase and amplitude are caused by the amplitude of the second microwave source and are each defined with respect to the phase and amplitude of the first microwave source, respectively, is a complex number caused by the third microwave source and is defined with respect to the first microwave source, is an annihilation operator of the second mode, and is the Hermitian conjugate, the Hamiltonian H generates an effective dissipator by dissipation of the second mode which stabilizes a two-dimensional manifold carrying the bosonic qubit in the first mode. BRIEF DESCRIPTION OF DRAWINGS

[0019] Other features and advantages of the present invention will become apparent from the following description, taken in conjunction with the accompanying drawings, which are indicative of the purposes intended to be illustrative and non-limiting purposes, in which: Figure 1 schematically illustrates a quantum system according to the present invention, the quantum system comprising a nonlinear superconducting quantum circuit and an instruction circuit.

[0020] Figure 2 illustrates a standard cat qubit and a bosonic qubit stabilized by Figure 1 the quantum system shown, Figure 3 illustrates Figure 1 a detailed schematic of the quantum system shown, Figure 4 illustrates Figure 3 a partial equivalent circuit diagram of a first current example implementation of the quantum system shown, Figure 5 illustrates Figure 3 a partial equivalent circuit diagram of a second current example implementation of the quantum system shown, Figure 6 illustrates Figure 3 a partial equivalent circuit diagram of a capacitive example implementation of the quantum system shown, Figure 7 illustrates a Hamiltonian H of the nonlinear superconducting quantum circuit being designed asFigure 1 Bit flip rate of the bosonic qubit stabilized by the quantum system as a function of the number of photons, Figure 8 is illustrated Figure 1 Non-adiabatic error of the quantum system illustrated due to the implementation of a Z gate, Figure 9 is illustrated Figure 1 Comparison between the non-adiabatic error of the bosonic qubit stabilized by the quantum system and the theoretically compressed cat qubit, Figure 10 is illustrated Figure 1 Bit flip rate of the bosonic qubit stabilized by the quantum system as a function of the ratio of the superconducting phase of the bosonic qubit mode and the corresponding zero-point fluctuations of the buffer mode. DETAILED DESCRIPTION

[0021] The drawings and the following description are mainly composed of positively and clearly defined features. They therefore contribute not only to the understanding of the invention, but they can also be used to help define the invention, if necessary.

[0022] A. Cat qubits So far, the applicant's work has been focused on the stability of cat qubits in general. In "Stabilizing a cat qubit by a nonlinear interaction", R. Lescanne et al. (Nature Physics, 2020), the possibility of implementing such a qubit was demonstrated, and they proved that this cat qubit can be stabilized by a nonlinear conversion between two photons of a first mode a (memory mode or cat qubit mode) and one photon of a second mode b (buffer mode). Exponential suppression of bit-flips in a qubit encoded in an oscillator A cat qubit is defined as a two-dimensional manifold spanned by the so-called cat states These cat states are superpositions of two coherent states

[0023] and : where: It is known that a stabilized cat qubit benefits from a high noise bias, which means that the bit flip probability is exponentially smaller than the phase flip probability. More precisely, the effective error channel (e.g. bit error or "bit flip") is suppressed exponentially with the "size" of the Schrödinger cat state of the cat qubit, i.e. the average number of photons As mentioned before, this exponential suppression of bit flip errors comes at the cost of a linear increase in phase flip errors.

[0024] ​According to the current understanding, this suppression should apply to a large class of physical noise processes that have a local impact on the phase space of the resonator. This includes, but is not limited to, photon loss, thermal excitation, photon de-coherence, and various nonlinear phenomena arising from coupling to Josephson junctions.

[0025] Recent experiments in the context of quantum superconducting circuits have observed that this exponential suppression of bit flip errors is observed with increasing average number of photons in the cat state.

[0026] A.1 Stabilization schemes Cat qubits can be stabilized or limited by the following example schemes: a) Parametric dissipation stabilization, where the transition operator is where is the two-photon dissipation rate, is the photon annihilation operator for memory mode a, and is a complex number defining the cat qubit. This can be implemented by coupling a lossy buffer mode b and a four-wave mixing device (typically a Josephson junction or an ATS) to the cat qubit mode a, and by designing the Hamiltonian to have the transition operator where is the photon annihilation operator for buffer mode b, and is the two-photon coupling rate, satisfied by applying a pump at frequency and a drive of buffer mode b at frequency .

[0027] b) Kerr Hamiltonian where is the amplitude of the Kerr Hamiltonian, is the photon annihilation operator, and is the average photon number.

[0028] c) Detuned Kerr Hamiltonian where is the amplitude of the Kerr Hamiltonian, is the photon annihilation operator, is a complex number defining the cat qubit, is a detuning factor.

[0029] d) Two-photon exchange (TPE) Hamiltonian where is a complex two-photon coupling rate, is the photon annihilation operator, is a complex number defining the cat qubit, and ​It is the decrement and elevation operator for a two-level system. This Hamiltonian can be designed in the same way as parameter dissipation stabilization a).

[0030] e) DC dissipative stability, for which the applicant has filed European patent application EP23306839.4, in which, with the previous stabilization scheme a) of the mentally stable cat qubit, the only difference is that the two-photon pump used for the nonlinear conversion between two photons in memory mode a and one photon in cache mode b is replaced by a DC voltage source biased entirely by nonlinear elements composed of one or more Josephson structures; therefore, the two-photon coupling rate is... Amplitude not pumped by two photons: , It is the Josephson energy of one or more Josephson knots. It is the zero-point fluctuation of the phase of memory mode a, and It is the phase zero-point fluctuation of cache mode b.

[0031] a) Resonant dissipation stabilization, where the transition operator is ,in, It is the two-photon dissipation rate. It is the photon annihilation operator for memory mode a, and It is the complex number that defines a cat qubit. It can be determined by the loss rate. The lossy buffer mode b and a three-wave hybrid device are coupled to the cat qubit mode a to realize the transition operator, which is designed with a stutter. ,in, It is the photon annihilation operator for cached mode b, provided that the mode frequency verification is basically true. and and add a frequency of The cache mode driver.

[0032] The following sections focus on a specific stabilization scheme: the dissipative compression stabilization scheme.

[0033] A.2 Compressed cat qubits Cat qubits possess the property of allowing the implementation of quantum gates (such as Z-gates or CNOT gates) while maintaining a noise bias. The properties used to evaluate the quality of a quantum gate are its execution time (i.e., the time required for the quantum gate to run) and the associated error probability. For cat qubits, the optimal timing of the gate's phase-flip error probability depends on the ratio... ,in, It is the single-photon loss rate (or error rate, as it causes phase-flip errors) of the cat qubit used to perform quantum gates, and It is the two-photon dissipation rate (also known as the correction rate or stability rate) of these cat qubits. The smaller, the better the gate fidelity becomes.

[0034] It is generally accepted that using single repetition codes is sufficient to correct the remaining errors, as bit flip errors are very rare; more specifically, phase flip error correction codes are sufficient to correct the remaining phase flips. For example, this can be the repetition code defined in the double basis, or any other state-of-the-art error correction code.

[0035] However, even though cat qubits have a higher noise bias, which is a particularly advantageous feature for implementing quantum error correction and designing reliable quantum computers, this noise bias does not seem to be enough for large-scale quantum algorithms with simple repetition codes. In this context, it has recently been shown that a special class of cat qubits has a higher noise bias: squeezed cat qubits.

[0036] Squeezed cat states also have fewer non-adiabatic errors during the time period of quantum gates such as Z gates or CNOT gates. Overall, reducing non-adiabatic errors can reduce the requirement for a high ratio of

[0037] Squeezed states are defined as follows: where is the displacement operator: and is the squeezing operator: where is the complex squeezing parameter with norm .

[0038] Squeezed cat states are defined as the coherent superposition of two squeezed states with opposite displacement amplitudes and the same degree of squeezing: where is a normalization constant.

[0039] Note that to obtain the correct squeezed cat state, the complex phases of and must follow: Dissipative squeezing stabilization is based on the following transition operator: where is the two-photon dissipation rate, is the photon annihilation operator, is a complex number defining the cat qubit, and are the modulus and the argument of the complex compression parameter

[0040] This transition operator can be implemented by coupling a lossy buffer mode b (with a dissipation rate ) to the cat qubit mode via a four-wave mixing device (typically a Josephson junction or an ATS) and by engineering the Hamiltonian where is the photon annihilation operator of mode b.

[0041] The dissipative form of the cat qubit is while the dissipative form of the theoretically compressed cat qubit follows a more complex form where and are assumed to be real numbers, without loss of generality. More specifically, the dissipative compression stabilization is achieved via the interaction term

[0042] Therefore, the theoretical implementation of the compressed cat qubit stabilization relies on the generation of four specific terms: i) : The first term corresponds to a nonlinear conversion between two photons of the first mode a (the memory) and one photon of the second mode b (the buffer). This term requires a pump with frequency where is the resonance frequency of the first mode a and is the resonance frequency of the second mode b.

[0043] ii) : This second term corresponds to a longitudinal coupling between the first mode a and the second mode b. This term is accessible via a pump with frequency

[0044] iii) : This third term requires a pump with frequency

[0045] iv) : This fourth term requires driving the second mode b at frequency

[0046] It must be noted that the so-called "standard cat qubit" is stabilized only via the generation of the first term and the fourth term

[0047] ​​​​​​However, one quite substantial obstacle to the application of squeezed cat qubits is the difficulty in designing such dissipative squeezing stabilization. This obstacle is first due to the large number of terms that are engineered using pumps, which can jeopardize the stability of the system, increase the number of thermal particles or reduce the coherence time. Moreover, it must be noted that the generation of the third term requires a very high frequency of about 15 gigahertz (GHz), assuming a frequency and of 5 gigahertz (GHz) in general. To bring the system to such frequencies, the amount of energy that needs to be injected into the system has an adverse effect on the coherence of the system. Moreover, the phase and amplitude of the pump light with a frequency for the generation of the third term must be precisely adjusted with respect to the phase and amplitude of the pump light with a frequency for the generation of the second term and the driving light of the second mode b with a frequency for the generation of the fourth term .

[0048] Thus, although squeezed cat qubits offer a higher noise bias in theory than cat qubits, they cannot be stabilized at present due to the many difficulties in realizing the required dissipation. On the other hand, cat qubits can be stabilized, but require very high usage conditions to achieve a sufficient noise bias and thus run large-scale quantum algorithms. This fact is the subject of the recent article by E. Gouzien et al. (2023): "A cat qubit for quantum computing" (arXiv:2502.06639). Computing 256-bit Elliptic Curve Logarithm in 9 Hours with 126133 Cat Qubits

[0049] B. Moon cat qubits Figure 1 A schematic diagram of a quantum system 1 is illustrated, the system 1 being arranged to stabilize a particular type of bosonic qubit: a moon cat qubit.

[0050] The quantum system 1 comprises a nonlinear superconducting quantum circuit 3 and an instruction circuit 5.

[0051] The nonlinear superconducting quantum circuit 3 is arranged such that four-wave mixing can take place between a first mode a and a second mode b. In the following, the first mode a carries the moon cat qubit, while the second mode b is dissipative and serves as a buffer between the moon cat qubit and the external environment.

[0052] In the remaining part of the description, the first mode a is referred to as the memory mode, while the second mode b is referred to as the buffer mode.

[0053] ​The memory mode a and the cache mode b correspond to intrinsic resonant frequencies of the nonlinear superconducting quantum circuit 3. Thus, the memory mode a and the cache mode b each have a respective resonant frequency. The resonant frequency of the memory mode a is and the resonant frequency of the cache mode b is where and are the angular frequencies of the memory mode a and the cache mode b, respectively.

[0054] By "having" a memory mode and a cache mode, it is understood that the nonlinear superconducting quantum circuit 3 comprises components operating in a superconducting state that can carry said modes independently from each other or simultaneously. In other words, the memory mode a and the cache mode b can be located in different subsets of components of the superconducting circuit or in the same subset of components.

[0055] The memory mode a has a high quality factor and the cache mode b has a low quality factor It is understood that the quality factor (Q-factor) can be determined in various ways, for example: (a) a spectral line width measurement, where the quality factor is given by where is the resonant frequency and is the spectral line width; or (b) via a time domain measurement, where a tone signal is sent and the return signal is measured after a predetermined time, where where is the characteristic decay time. Of course, the skilled person will be aware of various other ways of determining the quality factor of a particular mode.

[0056] The nonlinear superconducting quantum circuit 3 is intended to be subjected to microwave radiation provided by the instruction circuit 5 in order to design various nonlinear interactions between the memory mode a and the cache mode b. The frequency of each microwave radiation is tuned to select a particular term in the rotating wave approximation.

[0057] As detailed below, the nonlinear superconducting quantum circuit 3 is intended to interact with the instruction circuit 5 in order to design a particular Hamiltonian H that, through the dissipation of the cache mode b, creates an effective dissipator that stabilizes a two-dimensional manifold carrying the cat qubit.

[0058] Such a Hamiltonian H can be represented in the following form: Each term of the Hamiltonian H will be explained and detailed below.

[0059] The nonlinear superconducting quantum circuit 3 comprises a four-wave mixing nonlinear element 7 and at least one resonant part 9.

[0060] The four-wave mixing nonlinear element 7 is arranged to be parametrically driven by the instruction circuit 5 to make the nonlinear superconducting quantum circuit 3 generate the Hamiltonian H required for the stable cat qubit.

[0061] More specifically, the four-wave mixing nonlinear element 7 is an asymmetrically threaded superconducting quantum interference device - or ATS for short. As known to the person skilled in the art, ATSs can be used to implement 2-to-1 photon conversion, i.e. the first term of the Hamiltonian H mentioned above , thus performing dissipative stabilization, as shown by R. Lescanne et al. (2020).

[0062] Unlike the first implementation of such a stabilization scheme proposed by Z. Leghtas et al. in the article “ Confining the state of light to a quantum manifold by engineered two-photon loss ” (Science, Vol. 347, No. 6224, 2015) and S. Touzard et al. in the article “ Coherent Oscillations inside a Quantum Manifold Stabilized by Dissipation ” (Physical Review X 8, 023005, 2018), in which the superconducting circuit element used as four-wave mixer is a superconducting qubit with a single Josephson junction, the technical solution developed by R. Lescanne et al. (2020) exploits an ATS design, whose cross-Kerr term is much lower than that of a superconducting charge transmon, and therefore an exponential suppression of bit flips can be observed.

[0063] The ATS has a flux line through which radiation can be delivered to modulate the common flux and / or the differential flux.

[0064] As known to the person skilled in the art, when biased at its flux operating point (F0, F0), the Hamiltonian H of the ATS 7 or vice versa, has the following “sin-sin” form: wherein is the total superconducting phase difference across the ATS 7, is the zero-point fluctuation of the phase of the memory mode a on the ATS 7, is the zero-point fluctuation of the phase of the cache mode b on the ATS 7, is the Josephson energy of the side junction, is the inductive energy of the central inductance, corresponds to the common flux modulation of the two loops of the ATS 7,​ corresponds to the differential flux modulation of the ATS 7 two loops, and is the reduced Planck constant.

[0065] Common flux modulation can be achieved by delivering microwave radiation through the flux lines of the ATS 7 in antiphase, while the differential flux modulation can be achieved by delivering microwave radiation through the flux lines of the ATS 7 in phase.

[0066] The parametric pumping of the ATS 7 is usually achieved by pumping the common flux, as pumping the differential flux would only shift the modes coupled to the ATS 7.

[0067] For example, by pumping the common flux at a frequency , the nonlinear resonant part of the Hamiltonian can be written in the rotating frame as This is usually the 2-to-1 photon exchange Hamiltonian required to design a two-photon stabilization.

[0068] In order to design a longitudinal Hamiltonian between the memory mode a and the buffer mode b, the common flux must be pumped at a frequency : The parametric part of the Hamiltonian can be written in the rotating frame as In order to highlight the required dynamical properties, this Hamiltonian can be written as where .

[0069] The first term corresponds to the desired longitudinal coupling.

[0070] The second term is the effective drive of the buffer mode b at the resonant frequency . This term can be used to design the term of the Hamiltonian H. However, since this term needs to be adjusted independently of , a buffer driver 15 is required, or direct driving.

[0071] The third term corresponds to spurious, potentially harmful nonlinearities. Depending on the noise level the cat qubit stabilization has to deal with, this can lead to a restriction on the ratio to ensure that this last term does not have too much detrimental effect.

[0072] By parametrically pumping the ATS at a frequency a 2-to-1 photon conversion between the memory mode a and the cache mode b can be obtained. Advantageously, the pumping frequency satisfies so that this parametric pumping works as well as possible, where is the two-photon coupling rate.

[0073] The special case ( is called the resonant case and can be advantageous, as explained in the European patent application EP21306965.1 filed by the applicant. However, in this resonant case, a 2-to-1 photon conversion cannot be activated by parametrically pumping a four-wave mixing nonlinear element such as an ATS. Instead, a three-wave mixing nonlinear element should be used. As we will see later, the longitudinal term of the Hamiltonian H still requires a four-wave mixing to be designed. Thus, the resonant case, while compatible with the proposed stability of the cat qubit, requires an additional three-wave mixing nonlinear element. In other words, the resonant case requires both three-wave mixing and four-wave mixing to generate the first term and the second term respectively. The three-wave mixing can be achieved by operating the ATS at a flux point different from the usual working flux point, or by adding another nonlinear element as described in the above-mentioned European patent application EP21306965.1.

[0074] In the present invention, the applicant proposes to use an ATS to stabilize a cat qubit, so the advantage is to verify or to put it in a resonant state of 2-to-1 photon conversion.

[0075] The resonant part 9 is arranged to be coupled or connected to the ATS 7 to provide the nonlinear superconducting quantum circuit 3 with a memory mode a and a cache mode b having resonant frequencies and respectively. More specifically, the memory mode a and the cache mode b "participate" in the ATS 7, which means that the mode magnetic energy is partially or totally stored in the ATS 7. This participation can be quantified by the zero-point fluctuations of the superconducting phase on the ATS, noted for the memory mode a and for the cache mode b.

[0076] In Figure 1In the schematic diagram of the quantum system 1 illustrated in the middle, the nonlinear superconducting quantum system 3 comprises only one resonant part, namely the resonant part 9. The single resonant part can be configured to simultaneously produce the memory mode a and the buffer mode b. However, the nonlinear superconducting quantum system 3 typically comprises two resonant parts to form the memory mode a and the buffer mode b, respectively.

[0077] The instruction circuit 5 is arranged to deliver microwave radiation.

[0078] The instruction circuit 5 is arranged to parametrically drive the ATS 7 to implement a nonlinear conversion between two photons of the memory mode a and one photon of the buffer mode b, to parametrically drive the ATS 7 to implement a longitudinal coupling between the memory mode a and the buffer mode b, and to drive the buffer mode b.

[0079] To this end, as Figure 1 illustrated in the middle, the instruction circuit 5 comprises a first microwave source 11, a second microwave source 13, and a third microwave source 15.

[0080] The first microwave source 11 is configured to parametrically drive the ATS 7. More specifically, the first microwave source 11 is used to implement a parametric pumping that provides radiation to the ATS 7 at a frequency .

[0081] In the remainder of the description, the first microwave source 11 is referred to as a two- photon pumping.

[0082] The second microwave source 13 is also configured to parametrically drive the ATS 7. More specifically, the second microwave source 13 is used to implement a parametric pumping that provides radiation to the ATS 7 at a frequency .

[0083] In the remainder of the description, the second microwave source 13 is referred to as a longitudinal pumping.

[0084] Finally, the third microwave source 15 is arranged to drive the buffer mode b by providing radiation to the resonant part 9 at a frequency .

[0085] In the remainder of the description, the third microwave source 15 is referred to as a buffer driver.

[0086] As mentioned above, the combined action of the two-photon pumping 11 and the buffer driver 15 allows the nonlinear superconducting quantum circuit 3, and more specifically the ATS 7, to perform a nonlinear conversion between two photons of the memory mode a and one photon of the buffer mode b, as well as a driving of the buffer mode b.

[0087] Considering the two-dimensional manifold that stably hosts the mooncat qubit, which is the purpose of the quantum system 1, this combined action contributes to generate the term in the Hamiltonian H and in order to be designed by the nonlinear superconducting quantum circuit 3.

[0088] Moreover, the parametric pumping implemented with the longitudinal pumping 13 allows for a longitudinal coupling between the memory mode a and the buffer mode b.

[0089] As a consequence of this longitudinal coupling, the term is generated, which also contributes to the Hamiltonian H and thus helps to stabilize the two-dimensional manifold carrying the cat-in-the-moon quantum bit.

[0090] In response to the radiation delivered by the instruction circuit 5, the nonlinear superconducting quantum circuit 3 develops a Hamiltonian H whose expression is as follows: where: is the two-photon coupling rate (or interaction strength), proportional to the amplitude of the two-photon pumping 11, is the annihilation operator of the memory mode a, is a complex number whose phase and amplitude are determined by the amplitude of the radiation caused by the longitudinal pumping 13, is a complex number determined by the buffer driver 15, is the annihilation operator of the buffer mode b, denotes the Hermite conjugate.

[0091] In the expression of the Hamiltonian H, it should be noted that the parameters and are defined with respect to the two-photon pumping 11. Specifically, the phase and amplitude of the complex number are defined with respect to the phase and amplitude of the two-photon pumping 11, respectively.

[0092] In the following description, the complex number is referred to as cat-in-the-moon parameter.

[0093] It is worth noting that such a Hamiltonian H does not contain the term . As mentioned earlier, such a device requires an additional pumping, which would increase the complexity of the system and threaten its stability. Moreover, this term implies a very high frequency - corresponding to - and injects a certain amount of energy into the system, which affects the coherence of the bosonic quantum bit.

[0094] The Hamiltonian H generates, through the dissipation of the buffer mode b, an effective dissipator which stabilizes the two-dimensional manifold carrying the cat-in-the-moon quantum bit.

[0095] The desired number of photons in the memory can be obtained by adjusting the relative phases of the microwave sources.

[0096] More specifically, it is advantageous for the phase difference between the two-photon pump 11 and the longitudinal pump 13 to be substantially equal to zero. In other words, the two-photon pump 11 and the longitudinal pump 13 are configured so that their respective phases are substantially equal.

[0097] It is advantageous for the phase difference between the longitudinal pump 13 and the buffer driver 15 to be substantially equal to zero. In other words, the longitudinal pump 13 and the buffer driver 15 are configured so that their respective phases are substantially equal to each other.

[0098] Preferably, the two-photon pump 11, the longitudinal pump 13 and the buffer driver 15 are configured so that their phases are substantially equal.

[0099] By "substantially equal", it is understood that, ideally, the respective phases of the microwave sources involved are strictly equal to each other. However, in practice, such equality is difficult to achieve. Typically, the phase of one microwave source deviates by 5% from the phase of another microwave source.

[0100] Figure 2 The respective Wigner functions (or Wigner tomographies) of a standard cat qubit and of a moon cat qubit stabilized by the quantum system 1 are illustrated.

[0101] It can be noted that the standard cat qubit and the moon cat qubit stabilized by the quantum system 1 have the same number of photons but the latter is more separated in the phase space than the standard cat qubit, which means that the two binary large objects (BLOBs) are more separated. This increased separation explains the increase in noise bias in the case of the same number of photons.

[0102] It can be seen that both binary large objects take the shape of a crescent. This is why the bosonic qubit stabilized by the quantum system 1 is called a "moon cat qubit".

[0103] B.1 Example implementations of quantum systems Figure 3 The diagram is more explicitly illustrated Figure 1 .

[0104] The resonant part 9, in the form of a linear microwave network, is connected to the ATS 7 so that, when coupled to the ATS 7, which is an inductive element, via the linear coupler 17, the nonlinear superconducting quantum circuit 3 has memory modes a and buffer modes b at the resonant frequencies and which participate in the ATS 7.

[0105] The two-photon pump 11 and the longitudinal pump 13 are arranged for modulating the common flux in the ATS 7.

[0106] In Figure 3 In the illustrated example, the instruction circuit 5 further comprises a microwave source 19 arrangement for modulating the differential flux in the ATS 7.

[0107] Such a microwave source 19 can be used to drive the cache mode b by delivering to the resonant section 9 a radiation at the desired frequency . Thus, in the sense of the invention, the microwave source 19 can also be considered as a “third microwave source” or as a “cache driver”, in addition to the cache driver 15.

[0108] It is to be noted that the microwave source 19 can also be used to compensate for the parasitic driving of the cache mode b caused by the longitudinal pump 13. Although the amplitude and phase of such a compensating drive can be calculated by analytical methods, it is fine-tuned experimentally.

[0109] In order to clearly distinguish the role of each source in the Hamiltonian, Figure 3 The microwave network 21 applying the correct phase offsets is shown in Fig.

[0110] As mentioned above, the two-photon pump 11 is arranged to deliver microwave radiation to the ATS 7 at a frequency substantially equal to , so that the nonlinear superconducting quantum circuit 3 performs a 2-to-1 photon conversion between the memory mode a and the cache mode b, and thereby designs the term of the Hamiltonian H. In order to convert this 2-to-1 photon conversion into a two-photon dissipation, the cache mode b is selectively coupled to a load 23 via a linear coupler 25 and a microwave filter 27 configured as a bandpass filter having a frequency .

[0111] Alternatively, the microwave filter 27 can be configured as a bandstop filter at the frequency and can be placed on the one hand to the external environment and on the other hand between the memory mode a and the cache mode b, to isolate the memory mode a from additional losses due to unwanted coupling with the load 23.

[0112] Alternatively, if (or ), it can be configured as a low-pass (or high-pass) filter. In other embodiments, the microwave filter 27 coupling can be omitted when coupled with the load 23, and essentially only the cache mode b is established. As mentioned above, the memory mode a has a high quality factor , while the cache mode b has a low quality factor .

[0113] As mentioned above, the cache driver 15 is arranged to drive the cache mode b by delivering microwave radiation to the resonator section 9 with a frequency substantially equal to the second resonance frequency , thereby designing the term of the Hamiltonian H.

[0114] Alternatively, the cache mode b is driven by the microwave source 19 with a frequency .

[0115] In the above, the load 23 can be seen as part of the instruction circuit 5 in Figure 1 , while the linear coupler 25 and the microwave filter 27 can be seen as part of the nonlinear superconducting quantum circuit 3.

[0116] As mentioned above, the longitudinal pump 13 is arranged to deliver microwave radiation to the ATS 7 with a frequency substantially equal to the second resonance frequency , to enable longitudinal coupling between the memory mode a and the cache mode b, thereby designing the term of the Hamiltonian H.

[0117] B.1.1 Current example implementations Figure 4 and Figure 5 illustrate equivalent circuit diagrams, respectively, of embodiments of the nonlinear superconducting quantum circuit 3, in the form of a current circuit.

[0118] Such electrical equivalents are partial in the sense that they both only represent the ATS 7 and the resonator section 9, and thus neither represent the linear coupler 25 nor the microwave filter 27. More specifically, the resonator section 9 comprises a first resonator section 29 and a second resonator section 31.

[0119] The implementation of the ATS 7 is as known in the art, for example from the paper by R. Lescanne et al. (2020). The ATS 7 comprises a first Josephson junction 33 and a second Josephson junction 35 in parallel, and an inductive element 37 in parallel between them. Thus, the ATS 7 has two connected loops, each comprising a Josephson junction 33, 35 in parallel with the parallel inductive element 37. The inductive element 37 can be implemented geometrically or by a chain of junctions. Both loops of the ATS 7 are biased in DC and AC fashion. The DC bias determines the operating point of the ATS 7. It can be operated near a so-called saddle point, which is an optimal point in frequency and has a small cross-Kerr term.

[0120] Both the first resonant part 29 and the second resonant part 31 are galvanically coupled to the ATS 7. The first resonant part 29 imparts a memory mode a to the nonlinear superconducting quantum circuit 3 with a resonant frequency of , while the second resonant part 31 imparts a cache mode b to the nonlinear superconducting quantum circuit 3 with a resonant frequency of .

[0121] By “galvanically coupled”, it is understood that there is a short electrically conductive line part connecting the first resonant part 29 and the second resonant part 31 to the ATS 7, i.e. by a short electrically conductive trace or any other way of ensuring a physically continuous electrically conductive junction. The expression “short” means that the impedance of the electrically conductive trace is negligible compared to the impedance of the ATS 7, the first resonant part 29 and the second resonant part 31 at the resonant frequencies and . These short electrically conductive parts correspond to the linear coupler 17.

[0122] In embodiments where Figure 4 , the first resonant part 29 comprises a capacitive element 39 and an inductive element 41 connected in series. Similarly, the second resonant part 31 comprises a capacitive element 43 and an inductive element 45 connected in series.

[0123] In embodiments where Figure 5 , the first resonant part 29 further comprises a capacitive element 39 and an inductive element 41. However, in such embodiments, the capacitive element 39 and the inductive element 41 are connected in parallel. Similarly, the capacitive element 43 and the inductive element 45 of the second resonant part 31 are connected in parallel.

[0124] In the respective embodiments where Figure 4 and Figure 5 , the nonlinear superconducting quantum circuit 3 comprises two resonant parts. However, as mentioned before, the nonlinear superconducting quantum circuit 3 can contain only one resonant part, which is arranged for producing the memory mode a and the cache mode b.

[0125] B.1.2 Capacitive example implementations Figure 6 An equivalent circuit diagram of an embodiment of the nonlinear superconducting quantum circuit 3 in the form of a capacitive circuit is illustrated.

[0126] With Figure 4 and Figure 5 differing, the external environment is represented. The ATS 7 as a central reference point to which the other of the components are connected. The filtering of the cache mode b and the external environment are also present for this purpose. For simplicity, the microwave source is omitted, but it is arranged in the same way as in Figure 3 for driving the circuit.

[0127] In Figure 6 , the linear microwave network consists of a capacitive element and a parallel LC resonator, the capacitive element is galvanically coupled to the ATS 7 to form the cache mode b, and the parallel LC resonator is strongly capacitively coupled to the ATS 7 to form the memory mode a. The coupling to the ATS 7 corresponds to the linear coupler 17. It can be seen from the zero-point fluctuations of the phases of the two modes in the central inductance of the ATS 7 that both the memory mode a and the cache mode b are strongly coupled to the ATS 7. The cache is coupled to the external environment via the capacitive element corresponding to the linear coupler 25.

[0128] B.2 Characterization of moon cat qubits In the present invention, the nonlinear superconducting quantum circuit 3 is designed with a Hamiltonian H expressed as The Hamiltonian H generates an effective dissipator by dissipation of the cache mode b, which stabilizes a two-dimensional manifold carrying the mooncat qubit.

[0129] If the respective phases of the second microwave 13 and the cache driver 15 (or alternatively 19) are properly tuned, then and have the same complex phase, and thus can take real numbers without loss of generality. Therefore, it is assumed that and are real numbers, unless stated otherwise.

[0130] Analogous to the even cat state and the odd cat state The even mooncat state and the corresponding odd mooncat state of the mooncat qubit stabilized by the two-dimensional manifold of the quantum system 1 are superpositions of mooncat states containing only even Fock states and odd Fock states, respectively. The even mooncat state and the odd mooncat state are expanded on the Fock basis as follows (MCS stands for "Mooncat State"): where: And: Where: Where, and are normalization constants.

[0131] The performance indicator of the implementation of the quantum system 1 is the relative amplitude of the terms and In this respect, if a comparison between the moon cat qubit of the present invention and the theoretically compressed cat qubit can be established, then can be expressed as a function of the complex compression parameter as follows: As for the compressed cat, the bit flip probability decreases exponentially as the value of —or, equivalently, for the quantum cat, —increases.

[0132] B.3 Performance of moon cat qubits Figure 7 Fig. 1 graphically represents the bit flip probability as a function of the number of photons and for different values of the moon cat parameter More specifically, for each value of the moon cat parameter there is a solid line corresponding to the bit flip rate of the moon cat qubit stabilized by the quantum system 1, and there is a dashed line corresponding to the bit flip rate of the corresponding compressed cat qubit. By "comparable", here it is understood that, for a given value of the moon cat parameter we consider the compressed cat qubit obtained according to the preceding formula, with the parameter

[0133] For and a number of photons the bit flip of the moon cat qubit stabilized by the quantum system 1 exhibits an improvement of six orders of magnitude compared to the bit flip of the standard cat qubit at the same number of photons.

[0134] Figure 7 It is shown that, when the value of the moon cat parameter is less than or equal to 1, the moon cat qubit (solid line) includes the same improvement in bit flip as the compressed cat qubit (dashed line), or even a little better.​

[0135] As mentioned before, the phase and amplitude of the mooncat parameter are generated by the amplitude of the radiation caused by the longitudinal pumping 13. Advantageously, the amplitude of the radiation caused by the longitudinal pumping 13 is adjusted such that the mooncat parameter equals 1. Beyond this value, the mooncat qubit stabilized by the quantum system 1 is affected by a saturation phenomenon and thus does not perform as well as the theoretical performance of the squeezed cat qubit.

[0136] The quantum system 1 also exhibits less anharmonic errors during the implementation of the gate. The following description is mainly directed to the Z gate, but can be directly generalized to the CNOT gate.

[0137] During the implementation of the Z gate with a known cat qubit, i.e. by stabilizing the cat qubit only by the generation terms and the term two errors can occur: a single-photon loss error given by where is the number of photons in the cat state, is the single-photon loss, and is the duration of the gate, a non-adiabatic error since the gate is performed in a finite time where is the two-photon exchange rate. The non-adiabatic error prevents the gate from being performed at the fastest speed, thus avoiding increasing the single-photon loss.

[0138] A trade-off has to be found between the single-photon loss error and the non-adiabatic error, since the former is proportional to the duration of the gate and the latter is proportional to the inverse of the duration.

[0139] The non-adiabatic error of the quantum system 1 is smaller than for a known cat qubit. The non-adiabatic error of the quantum system 1 during the Z gate is given by the following formula: Figure 8 Fig. 7 shows the non-adiabatic error of the quantum system 1 during the Z gate and for a fixed number of photons as a function of the gate time in units of It can be seen that the non-adiabatic error decreases when the value of the mooncat parameter increases. The solid line shows the numerical simulation results and the dashed line shows the above formula.

[0140] Figure 9Fig. illustrates a comparison of the non-adiabatic error between the stabilized cat qubit of the moon (solid line) and the compressed cat qubit (dashed line) by the quantum system 1, and shows the execution of the quantum system 1 as well as the compressed cat qubit.

[0141] The way in which the quantum system 1 stabilizes the cat qubit of the moon means that the spurious term of the form is activated by the longitudinal pumping 13, where, is the photon annihilation operator of the buffer mode b, which is dissipative. However, if the amplitude of this spurious term is small enough, it does not harm the quantum system 1. As mentioned above, its amplitude is proportional to , where, and are the zero-point fluctuations of the superconducting phases of the memory mode a and the buffer mode b on the ATS 7, respectively, and where, is the amplitude of the sought longitudinal coupling.

[0142] Finally, Figure 10 Fig. illustrates the bit flip rate of the stabilized cat qubit of the moon by the quantum system 1 in relation to the ratio , where the photon number is approximately given by in the memory mode a. Different approximations of in this figure are due to the fact that in practice, this value is difficult to determine. This figure clearly shows that the stabilized cat qubit of the moon by the quantum system 1 has an advantage over the standard cat qubit as long as there is . More specifically, the curve corresponding to the standard cat qubit is the curve obtained for .

[0143] It should be noted that this threshold value for depends in a non-trivial way on the noise parameters of the quantum system 1 and on the ratio .

[0144] In Figure 10 , the curves correspond to the following experimentally relevant parameters: the ratio , the photon loss rate of the buffer mode b MHz, the photon loss rate of the memory mode a MHz, the dephasing rate of the memory mode a is MHz, and the thermal filling of the memory mode a .

Claims

1. A quantum system (1) for stabilizing boson qubits, comprising: The instruction circuit (5) includes a first microwave source (11), a second microwave source (13), and a third microwave source (15, 19), each microwave source being arranged to deliver microwave radiation. A nonlinear superconducting quantum circuit (3) includes an asymmetric threaded superconducting quantum interference device (7) and at least one resonant section (9), the asymmetric threaded superconducting quantum interference device (7) being connected to the at least one resonant section, the asymmetric threaded superconducting quantum interference device (7) having flux lines through which radiation can be delivered to modulate common flux and differential flux, the nonlinear superconducting quantum circuit (3) having a first mode (a) having a first resonant frequency and a second mode (b) having a second resonant frequency different from twice the first resonant frequency, the second mode (b) being dissipative. The asymmetric threaded superconducting quantum interference device (7) is arranged such that, under the following conditions, The first microwave source (11) delivers radiation through the flux line to modulate the common flux at a frequency equal to the absolute value of the difference between twice the first resonant frequency and the second resonant frequency. The second microwave source (13) delivers radiation through the flux line to modulate the common flux at a frequency equal to the second resonant frequency, and The third microwave source (15, 19) delivers radiation to the at least one resonant part (9) at a frequency equal to the second resonant frequency to drive the second mode (b). The nonlinear superconducting quantum circuit (3) is designed to be expressed as The Hamiltonian H, where, Proportional to the amplitude of the first microwave source (11), It is the annihilation operator of the first mode (a). It is a complex number whose phase and amplitude are caused by the amplitude of the second microwave source (13), and are defined relative to the phase and amplitude of the first microwave source (11), respectively. It is a complex number caused by the third microwave source (15, 19) and defined relative to the first microwave source (11). It is the annihilation operator of the second mode (b), and It is Hermitian conjugation, and the Hamiltonian H produces an effective dissipator through the dissipation of the second mode (b). The effective dissipator stabilizes the two-dimensional manifold carrying the boson qubits in the first mode (a).

2. The quantum system (1) according to claim 1, wherein, The first microwave source (11) and the second microwave source (13) are configured such that the corresponding phases of the first microwave source (11) and the second microwave source (13) are substantially equal to each other.

3. The quantum system (1) according to claim 1 or 2, wherein, The second microwave source (13) and the third microwave source (15, 19) are configured such that the corresponding phases of the second microwave source (13) and the third microwave source (15, 19) are substantially equal to each other.

4. The quantum system (1) according to any one of the preceding claims, wherein, The first microwave source (11), the second microwave source (13), and the third microwave source (15, 19) are configured such that the corresponding phases of the first microwave source (11), the second microwave source (13), and the third microwave source (15, 19) are substantially equal to each other.

5. The quantum system (1) according to any one of the preceding claims, wherein, The third microwave source (19) is arranged to deliver radiation through the flux line to modulate the differential flux at a frequency equal to the second resonant frequency to drive the second mode (b).

6. The quantum system (1) according to any one of the preceding claims, wherein, The second microwave source (13) is arranged to deliver radiation, the amplitude of which makes It has an amplitude of less than or equal to 1.

7. The quantum system (1) according to claim 6, wherein, The second microwave source (13) is arranged to deliver radiation, the amplitude of which makes It has an amplitude that is approximately equal to 1.

8. The quantum system (1) according to any one of the preceding claims, wherein, The corresponding phases of the first mode (a) and the second mode (b) across the asymmetric threaded superconducting quantum interference device (7) both exhibit zero-point fluctuations. , and among them, the ratio Less than 3.

9. A method for stabilizing boson qubits, said method being performed by a quantum system (1) according to any one of the preceding claims, and said method comprising the following operations: Radiation is delivered by the first microwave source (11) through the flux line to modulate the common flux at a frequency equal to the absolute value of the difference between twice the first resonant frequency and the second resonant frequency. Radiation is delivered by the second microwave source (13) through the flux line to modulate the common flux at a frequency equal to the second resonant frequency. The third microwave source (15, 19) delivers radiation to the at least one resonant part (9) at a frequency equal to the second resonant frequency to drive the second mode (b). The implementation of the method enables the design of the nonlinear superconducting quantum circuit (3) to be expressed as The Hamiltonian H, where, Proportional to the amplitude of the first microwave source (11), It is the annihilation operator of the first mode (a). It is a complex number whose phase and amplitude are caused by the amplitude of the second microwave source (13), and are defined relative to the phase and amplitude of the first microwave source (11), respectively. It is a complex number caused by the third microwave source (15) and defined relative to the first microwave source (11). It is the annihilation operator of the second mode (b), and It is Hermitian conjugation, and the Hamiltonian H produces an effective dissipator through the dissipation of the second mode (b). The effective dissipator stabilizes the two-dimensional manifold carrying the boson qubits in the first mode (a).

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