Superconducting quantum circuit with electrically coupled glass color coding

Abstract

A non-linear superconducting quantum circuit includes at least one resonant portion (30, 32) and an asymmetrically threaded superconducting quantum interference device (34) electrically connected to the at least one resonant portion (30, 32). The nonlinear superconducting circuit includes a first mode having a first resonant frequency and a second mode having a second resonant frequency, the ratio of the first resonant frequency to the second resonant frequency being different from 1 / 2. The at least one resonant portion (30, 32) is configured with inductance and capacitance values represented by its symbols that, together with the asymmetric threaded superconducting quantum interference device (34), induce the first mode and the second mode such that the non-linear superconducting quantum circuit has, in the first mode and the second mode, at least one of the resonant portions (30, 32) is coupled to the non-symmetric threaded superconducting quantum interference device (34), and the at least one resonant portion (30, 32) is coupled to the non-symmetric threaded superconducting quantum interference device (34). And the zero fluctuation of the superconducting phase on the asymmetric thread superconducting quantum interference device (34) is greater than or equal to 0.05 radian.

Description

[0001] The present invention relates to the field of superconducting quantum circuits, and more particularly, to a superconducting quantum circuit comprising cat qubits.

[0002] Cat qubits are a subset of bosonic encodings, which form a family of error-correcting codes for quantum applications. Generally speaking, bosonic encodings rely on storing qubits in bosonic modes. In the implementation of cat qubits, the two-component cat code is by far the most common design scheme.

[0003] Dissipative stabilization of two coherent states requires a non-linear conversion between two photons in a first mode (also called the cat qubit mode) carrying a stable quantum manifold and one photon in a second mode (called the buffer mode), and vice versa. This stabilization scheme can suppress bit flips, and the degree of suppression is exponentially related to the number of photons in the two coherent states. However, this method is only effective when the confinement rate of the above two coherent states is greater than the escape rate caused by external noise sources. The confinement rate is directly related to the 2-to-1 photon conversion rate.

[0004] The first implementations of this stabilization scheme (the articles "Trapping of light states in a quantum manifold by engineered two-photon losses" by Leghtas et al. in *Science* volume 347, page 853 in 2015 and "Coherent oscillations within a quantum manifold stabilized by dissipation" by Touzard et al. in *Physical Review X* volume 8, 021005 in 2018) failed to observe an exponential suppression of bit flips because the superconducting circuit elements used to engineer the 2-to-1 photon conversion (so-called "transmons") had pseudo-cross-Kerr terms that gave rise to additional noise processes whose escape rates were given by very large transmon-cat quantum ratio characteristic dispersion shifts. The article "Exponential suppression of bit flips in qubits encoded in an oscillator" by Lescanne R. et al. in *Nature Physics* in 2020 (hereinafter referred to as Lescanne (2020)) disclosed a significantly improved cat qubit by using an asymmetric threaded superconducting quantum interference device (also known as "ATS") to engineer the 2-to-1 photon conversion. The cross-Kerr terms of the ATS design are much lower than those of the transmon, which enabled us to observe an exponential suppression of bit flips. However, the transmon was also used to measure the state of the cat qubit. While this design is less harmful to the cat qubit in this position, it still causes the bit flip time to saturate at several milliseconds. In the later work paper "Achieving hundred-second bit flip times in two-photon dissipative oscillators" (arXiv:2204.09128, https: / / arxiv.org / pdf / 2204.09128.pdf) by Berdou et al. (hereinafter referred to as Berdou (2022)), by removing the measurement transmon and operating the ATS in a putatively dynamically stable state, the bit flip saturation time was successfully increased by five orders of magnitude to 100 seconds. The large increase in the bit flip time was made possible because the ATS alone adds spurious noise processes with extremely low escape rates. However, the confinement rate achieved in Berdou (2022) is very low.

[0005] The ratio of the confinement rate to the phase flip rate of the cat qubit is a fundamental metric for quantum error correction using cat qubits. On the one hand, the confinement rate indicates to what extent the cat qubit can be perturbed without a bit flip occurring, which is directly related to the speed at which gates can be executed while maintaining an exponential suppression of bit flips. On the other hand, the phase flip rate indicates how much time is required to execute a gate to detect and correct an error. Theoretical analysis shows that this ratio should be greater than 10 4 . In Lescanne (2020) and Berdou (2022), this ratio was 10 and 0.01 respectively.

[0006] Currently, there is no other known circuit that can provide well-performing cat qubits. The average stable lifetime of other known cat qubits does not exceed a few milliseconds, which is far from sufficient for constructing practical quantum circuits. Since the limiting rate is directly related to the designed 2-to-1 photon conversion rate, in the ATS-based circuit, it is necessary to significantly increase the latter's conversion rate.

[0007] This application aims to improve this situation. To this end, the applicant proposes a non-linear superconducting quantum circuit, which includes at least one resonant part and an asymmetric threaded superconducting quantum interference device that is current-connected to the at least one resonant part. The non-linear superconducting circuit includes a first mode with a first resonant frequency and a second mode with a second resonant frequency, and the ratio of the first resonant frequency to the second resonant frequency is not equal to 1 / 2. The at least one resonant part is configured with inductance values and capacitance values represented by their symbols, and these inductance values and capacitance values, together with the asymmetric threaded superconducting quantum interference device, induce the first mode and the second mode, such that the non-linear superconducting quantum circuit has zero-point fluctuations of the superconducting phase on the asymmetric threaded superconducting quantum interference device in the first mode and the second mode that are greater than or equal to 0.05 radians.

[0008] The advantage of this superconducting quantum circuit is that it has no coupling elements (coupling elements will reduce the participation of the buffer mode and / or storage mode in the ATS, thereby reducing the 2-to-1 photon conversion rate). In theory, the adverse effects of the coupling capacitor in the Lescanne (2020) circuit can be minimized, but it is well known that large capacitors will cause losses in superconducting circuits, which will increase the phase flip rate and make it ineffective in practical applications.

[0009] In various embodiments, the method may have one or more of the following features:

[0010] - The at least one resonant part has a symbol representation, where the first mode is carried in the first resonant part including at least one inductor and at least one capacitor, and the second mode is carried in the second resonant part including at least one inductor and at least one capacitor. The asymmetric threaded superconducting quantum interference device is disposed between the first resonant part and the second resonant part and is electrically coupled to the first resonant part and the second resonant part;

[0011] - The at least one inductor and the at least one capacitor of the first resonant part and the second resonant part are respectively connected in series or in parallel, and the asymmetric threaded superconducting quantum interference device is respectively connected in parallel or in series with the first resonant part and the second resonant part;

[0012] - The non-linear superconducting quantum circuit is located on a dielectric substrate and is separated from a common ground plane by an exposed portion of the dielectric substrate, and the first resonant portion and the second resonant portion are implemented in physically different portions of the non-linear superconducting quantum circuit.

[0013] - The non-linear superconducting quantum circuit is formed on a substantially planar substrate, the width and height of which are respectively less than one quarter of the wavelength corresponding to the first resonant frequency and the second resonant frequency.

[0014] - The first resonant portion and the second resonant portion are electrically isolated from the common ground plane.

[0015] - The first resonant portion and the second resonant portion are electrically connected to the common ground plane.

[0016] - The non-linear superconducting quantum circuit is located on a dielectric substrate and is separated from a common ground plane by an exposed portion of the dielectric substrate, and the at least one resonant portion is implemented as a transmission line.

[0017] - The first mode and the second mode are respectively the fundamental wave or higher-order harmonics of the non-linear superconducting circuit.

[0018] - The first resonant frequency and the second resonant frequency are such that the difference between twice the first resonant frequency and the second resonant frequency is less than half of the first resonant frequency and half of the second resonant frequency, and

[0019] - At least one of the inductor and / or the transmission line is made of a Josephson junction array or a high kinetic inductance material.

[0020] The present invention also relates to a quantum device, which includes: a non-linear superconducting quantum circuit according to one of the foregoing claims; a first microwave source connected to the at least one resonant portion for providing radiation having a frequency equal to the second resonant frequency; a second microwave source connected to the at least one resonant portion for providing radiation having a frequency equal to the difference between twice the first resonant frequency and the second resonant frequency; and a load coupled to the at least one resonant portion such that generally only the second mode is coupled to the load, so that the first mode carries a cat qubit. The device may further include a microwave filter for coupling to the load, the microwave filter being configured to allow the second resonant frequency to pass through and block the first resonant frequency.

[0021] The present invention also relates to a quantum computing system, which includes at least one device according to the present invention.

[0022] Other features and advantages of the present invention will be apparent from the following description of the drawings, which show exemplary embodiments of the present invention, wherein:

[0023] - Figure 1 Shows how a current-type cat qubit circuit can be incorporated into a device to stabilize quantum information;

[0024] - Figure 2 Shows how to use the current-type cat qubit circuit incorporated into the device to stabilize quantum information;

[0025] - Figure 3 Represents the electrical equivalent diagram of the first embodiment of the current-type cat circuit according to the present invention;

[0026] - Figure 4 Shows for Figure 3 the first mode and the second mode of the superconducting quantum circuit values and the corresponding value graph;

[0027] - Figure 5 Shows Figure 3 the first implementation of the circuit;

[0028] - Figure 6 Shows Figure 3 the second implementation of the circuit;

[0029] - Figure 7 Shows the electrical equivalent diagram of the second embodiment of the current-type cat circuit according to the present invention;

[0030] - Figure 8 Shows for Figure 7 the first mode and the second mode of the superconducting quantum circuit values and the corresponding value graph;

[0031] - Figure 9 Represents the third embodiment of the current-type cat circuit described in the present invention;

[0032] - Figure 10 Shows for Figure 9 the first mode and the second mode of the superconducting quantum circuit shown and f values and the corresponding value graph;

[0033] - Figure 11 Represents the fourth embodiment of the current-type cat circuit described in the present invention;

[0034] - Figure 12 Shows for Figure 11 the first mode and the second mode of the superconducting quantum circuit shown and f values and the corresponding value graph;

[0035] - Figure 13 Shows the ratio of the square of the zero-point fluctuations of the phases of modes b and a in the capacitor and inductor of the second resonant part in the first embodiment, and the corresponding value, as Figure 4 a supplement.

[0036] Most of the accompanying drawings and the following description consist of positive and well-defined features. Therefore, they not only help to understand the present invention, but can also be used to assist in defining the present invention when needed.

[0037] In order for the cat qubit to encode useful data and remain stable, 2-to-1 photon conversion is required between the first mode (memory) and the second mode (buffer). Most of the existing technologies belong to the family of stable cat qubits achieved through parametric pumping dissipation. The parametric pumping dissipation technique is used to bridge the gap between the two mode frequencies and perform resonant 2-to-1 photon conversion when the resonant frequency of the second mode is not a multiple of 2 of the resonant frequency of the first mode. In other words, the external time-varying excitation used in parametric pumping dissipation relaxes the restrictions on the resonant frequencies.

[0038] The first mode and the second mode of the superconducting quantum circuit can correspond to the natural resonant frequencies of the circuit respectively. For example, both the first mode and the second mode are electromagnetic modes. The first mode and the second mode can each have their own resonant frequencies. For example, the first mode can have a type of resonant frequency, while the second mode can have a type of resonant frequency, where ω a and ω b can be the angular frequencies of their respective modes. Employing "having" the first mode and the second mode means that the superconducting quantum circuit can include components that operate in a superconducting state, and these components carry these modes independently of each other or simultaneously. In other words, the first mode and the second mode can be carried in different subsets of components of the superconducting circuit, or can be carried in the same subset of components.

[0039] The superconducting quantum circuit can operate at a temperature close to absolute zero (e.g., 100 mK or lower, typically 10 mK), and be isolated from the environment as much as possible to avoid energy loss and decoherence (except for some special couplings). For example, only the second mode can be coupled to the dissipative environment, while the first mode can remain isolated from the environment.

[0040] A superconducting quantum circuit can be fabricated as a patterned layer of one or more superconducting materials (such as aluminum, tantalum, niobium, etc. known in the art) deposited on a dielectric substrate (such as silicon, sapphire, etc.). Each of the one or more patterned layers can define a lumped element resonator. A capacitive element can be formed by two adjacent superconducting material plates (on the corresponding layer of the one or more patterned layers). An inductive element can be formed by a superconducting wire. Alternatively, at least one of the one or more patterned layers can define a multi-segment transmission line, and the resonant frequency of each segment of the transmission line depends on its length. The transmission line can be, for example, a coplanar waveguide, a slot line, or a microstrip line. Alternatively, the circuit can be embedded in a three-dimensional architecture that contains high-quality three-dimensional modes, which are formed by machining or microfabrication of a bulk superconductor and can be used as either of the two modes.

[0041] The circuit can be integrated into a device, which can include a load, a first microwave source, a second microwave source, and a coupler. The coupler can be configured to connect the second mode of the superconducting quantum circuit to the load. The load is a dissipative element, for example, an element with a given resistance (different from the superconducting element), which is located outside the superconducting circuit. The load dissipates photon pairs that are converted from the first mode to the second mode through 2-to-1 photon conversion. In other words, the photon pairs destroyed in the first mode are evacuated to the environment through the second mode via the load. The first microwave source can be configured to control the amplitude and phase of the microwave radiation, so as to apply microwave radiation with a frequency substantially equal to the frequency of the second mode. Therefore, the first microwave source drives photons in the form of microwave radiation into the second mode, and the second mode in turn drives photon pairs in the first mode through 2-to-1 photon conversion. This 2-to-1 photon conversion is bidirectional: conversely, it can be 2 photons in the first mode converted into 1 photon in the second mode, or 1 photon in the second mode converted into 2 photons in the first mode. The coupler is an element that can be connected to the circuit element carrying the second mode through current, capacitance, or inductance, and mediates the interaction between the second mode, the load, and the microwave source.

[0042] The load can be a resistor, a matched transmission line, or a matched waveguide. The term "matched" should be understood as the transmission line or waveguide being terminated by a resistor at one end different from the end connected to the element carrying the second mode, and the resistance value of the resistor should be set so that most of the power flowing into the load can be absorbed. The load can be included within the first microwave source.

[0043] In various examples, the first microwave source can be placed at room temperature and connected to the circuit through a coaxial cable. In various examples, an attenuator can be placed between the microwave source and the circuit (i.e., along the path of the microwave radiation applied by the microwave source) to thermally equilibrate the microwave radiation using the cryogenic environment. This allows microwave radiation to be applied without increasing thermal noise.

[0044] The second microwave source is used to provide microwave radiation with a frequency approximately equal to twice the first-mode resonance frequency minus the second-mode resonance frequency, thereby achieving a 2-to-1 photon conversion. Since the ATS has two superconducting loops and must be flux-pumped with sufficient relative amplitudes and phases, the radiation emitted by the second microwave source can be split for use in two different transmission lines or waveguides, which will ultimately be connected to the two superconducting loops. Alternatively, two different microwave sources with the same frequency as the second microwave source can be used and fed directly into the two transmission lines or waveguides with sufficient relative amplitudes and phases.

[0045] Optionally, the device may include a microwave filter connected to the first and second modes of the circuit. The microwave filter can be configured to allow only the second mode to couple to the load. The microwave filter can be interleaved between the load and the coupler. From the perspective of the circuit, the filter is designed to prevent microwave photons in the first mode from escaping the circuit. This can be achieved by implementing a band-stop filter at the first resonance frequency, or by implementing a band-pass filter at the second resonance frequency, or (if the second (or first) resonance frequency is greater than the first (or second) resonance frequency), by implementing a high-pass (or low-pass) filter with a cut-off frequency between the first and second resonance frequencies, since the photons in the second mode are the only ones that need to dissipate in the environment. For some circuits, for example, when the two modes have different symmetries, the filter may not be necessary, and the proper placement of the coupler in the circuit may be sufficient to prevent the dissipation of the first mode.

[0046] Therefore, the device is capable of stabilizing two coherent states in the first mode (i.e., the quantum manifold of the coherent state). For example, the first microwave source applying microwave radiation to the second mode through a microwave filter can be regarded as a two-photon drive for the first mode after the 2-to-1 photon conversion; while the load that only dissipates the photons in the second mode can be regarded as a two-photon dissipation for the first mode after the 2-to-1 photon conversion. The two-photon drive and two-photon dissipation enable the stabilization of the two coherent states in the first mode.

[0047] The single-photon drive of the second mode can be formally described by the Hamiltonian where ∈ b is the single-photon drive rate generated by the first microwave source on the second mode. The single-photon dissipation of the second mode can be formally described by the Lindblad operator where κ b is the single-photon dissipation generated by the coupling of the second mode to the load.

[0048] The two-photon drive can be formally described by the Hamiltonian Description, where ∈2 is the effective two-photon drive rate. The two-photon loss can be formally described by the Lindblad operator Description, where κ2 is the two-photon dissipation rate. The amplitude α of the stable coherent state is finally given by Given. The confinement rate of the coherent state is κ conf = κ2α 2 .

[0049] In the range of κ b >> g2α, where g2 is the 2-to-1 nonlinear conversion rate between the first and second modes, the buffer dynamics can be adiabatically eliminated, resulting in ∈2 = 2∈ b g2 / κ b and

[0050] In various examples, the superconducting circuit can have a symbolic representation, for example, consisting of a set of interconnected dipoles. The term "symbolic representation" should be understood to specify an arrangement of a set of interconnected dipole symbols and lines. This set of interconnected dipoles (also called components) forms a circuit structure (or topology) that is (functionally) equivalent to the nonlinear superconducting circuit.

[0051] In other words, as is the classical practice in the field of superconducting circuits, the configuration of the nonlinear superconducting circuit is designed to achieve the function defined by its symbolic representation, i.e., the function of the theoretical set of interconnected dipoles shown in the symbolic representation. In other words, although the circuit can be constructed with a patterned layer of superconducting material, it should be understood that the circuit can be symbolically represented by dipoles (such as capacitors, inductors, and / or Josephson junctions). Although the example dipoles describe discrete components, those skilled in the art clearly understand that these components correspond to the equivalent circuit of distributed components within a specific frequency range (such as low frequency), as is well known in the art.

[0052] As is known in the art, such distributed elements can have high-frequency modes that are irrelevant and unimportant to the dynamic characteristics described herein. Therefore, these distributed elements can be presented using symbolic representations. The symbolic representation can be refined by adding elements (e.g., inductors in series at each wire connection, or capacitors in parallel between any two nodes of the circuit) or adding nodes and branches to account for other modes of the distributed elements. Thus, the symbolic representation can better describe the distributed elements without changing the operating principle of the circuit. Therefore, as is known in the art, the physically manufactured circuit (i.e., the actually manufactured circuit) and its symbolic representation are recognized as equivalent in the art. In fact, compared to the basic model, the refined dipole of the symbolic representation only adjusts the resonant frequency or the zero-point fluctuations of the phase. When designing a circuit, the final geometry can be fully and accurately simulated using a finite element solver, which can easily give the frequencies of the modes, the dissipation from the load, and the zero-point fluctuations of the phase across the Josephson junctions, which are the only unknowns for calculating the 2-to-1 photon conversion rate for any configuration.

[0053] The Hamiltonian for the 2-to-1 photon interaction has the form In these cases, the coupling term g2(t) is modulated by a parametric pump, where the pump injects at a frequency ω p = 2ω a -ω b of an external time-varying parameter. Parametric pumps have been used in the prior art to make the non-linear interaction resonate.

[0054] The development of the cat qubit quantum circuit relies on the geometry of the superconducting circuit, which can suppress bit flips of the cat qubit encoded in a high-Q superconducting resonator (referred to as the memory). To this end, two-photon dissipation of the memory can be achieved by coupling the memory to a low-Q superconducting resonator (referred to as the buffer) via a non-linear superconducting dipole.

[0055] In the article by Lescanne (2020), the non-linear Hamiltonian H2 is designed using an ATS superconducting dipole. The ATS dipole has the following potential energy:

[0056] where E L is the Josephson energy of the ATS parallel inductor, and E J is the Josephson energy of the ATS SQUID junction, (or ) is the magnetic flux through the common (or differential) loop mode of the ATS.

[0057] By choosing the following DC value of the magnetic flux (referred to as the saddle point): and only for the σ mode (where the amplitude is and a frequency of ω p ) for flux pumping, the potential energy expression becomes:

[0058]

[0059] The phase at both ends of the ATS The relationship with modes a and b is:

[0060]

[0061] where and are the zero-point fluctuations of the phase at both ends of the ATS in the first and second resonant modes respectively. Expanding at and to the third order and eliminating the fast-rotating terms, the 2-to-1 conversion Hamiltonian H2 is obtained, where the 2-to-1 photon conversion rate is

[0062] The article by Lescanne (2020) shows that bit flips are exponentially suppressed with the number of photons a of the cat qubit encoded in the resonator. However, this architecture uses a transmon coupled to the cat qubit as a measurement device, resulting in the bit flip time saturating to a few milliseconds. The applicant's research shows that this is due to the confinement rate being too small to resist the dispersive frequency shift caused by the thermal excitation of the measurement device. As later disclosed by the applicant in Berdou (2022), by removing the transmon and operating the ATS in the expected dynamically stable state (although with a lower confinement rate), the bit flip saturation time was increased by 5 orders of magnitude. More precisely, in the paper by Lescanne (2020), the ratio of the confinement rate to the phase flip rate was 10, while in Berdou (2022), this ratio was 0.01. As mentioned in the introduction section of this application, such a ratio is far from the theoretically required value. 2

[0063] The main problem with Lescanne (2020) and Berdou (2022) is that they do not provide a potential solution to significantly increase the two-photon dissipation rate. In fact, the two-photon dissipation rate is related to the two-photon coupling rate In both Lescanne (2020) and Berdou (2022), the buffer mode is on the ATS, and the storage mode is weakly capacitively coupled to the buffer mode and thus to the ATS. This design allows for obtaining a large but is usually one to two orders of magnitude smaller. Since the formula for g2 depends on ​The square, so this circuit geometry is very unfavorable for the intensity of two-photon dissipation.

[0064] The only way to solve this problem is to sufficiently increase the capacitance coupling the storage mode and the buffer mode in the circuits of Lescanne (2020) and Berdou (2022) so that the ATS has a strong participation in both the buffer mode and the storage mode. However, it is known that large capacitances cause losses in superconducting circuits.

[0065] Therefore, these prior arts reach a dead end: their specific geometry is crucial for achieving bit-flip stability, but it cannot be adjusted to allow a large enough two-photon dissipation rate.

[0066] Examples and descriptions of circuits and devices according to the present invention will now be discussed with reference to the accompanying drawings. Hereinafter, expressions such as "current-type cat qubit circuit", "circuit", "superconducting quantum circuit", and "nonlinear superconducting circuit" may be used interchangeably to refer to a circuit that performs 2-to-1 photon conversion to stabilize a cat qubit.

[0067] Figure 1 An example of a quantum device 10 including a current-type cat qubit circuit according to the present invention is shown.

[0068] The device 10 includes a nonlinear superconducting circuit 100, a microwave source 102, a coupler 104, a load 106, and another microwave source 108 and a microwave filter 110.

[0069] The nonlinear superconducting circuit 100 performs 2-to-1 photon conversion between a first mode a (labeled 112) and a second mode b (labeled 114). Hereinafter, the first mode a carries the cat qubit, also referred to as the storage mode, while the second mode b serves as a buffer between the cat qubit and the environment.

[0070] The device 10 uses parametric pumping to stabilize the cat qubit, which means that the resonance frequencies of the first mode and the second mode do not belong to the 2f a = f b type. To ensure 2-to-1 photon conversion, the parametric pump provides radiation at a frequency of 2f b - f b This is performed by the microwave source 102 connected to the nonlinear superconducting circuit 100.

[0071] As described below, the nonlinear superconducting circuit 100 according to the present invention is very special because it includes an ATS ("asymmetric threaded SQUID" or "asymmetric threaded superconducting quantum interference device"), which is electrically coupled to other elements in the nonlinear superconducting circuit 100 that carry modes a and b.

[0072] The components of the circuit carrying the second mode 114 are coupled to the load 106 through the coupler 104. This coupling makes the second mode dissipative. The microwave source 108 is connected to the non-linear superconducting circuit 100 to drive the second mode to operate at its resonant frequency by emitting radiation with a frequency of f b The microwave filter 110 is configured here as a band-pass filter with a frequency of f b . Alternatively, the filter 110 can be configured as a band-stop filter with a frequency of f a and placed between the environment and the two modes to isolate the first mode, thus preventing the first mode from suffering additional losses due to unnecessary coupling with the load 106. Alternatively, when f a > f b (or f b > f a ), it can be configured as a low-pass (or high-pass) filter. In other embodiments, when coupling is established only between the load 106 and generally only the second mode, the microwave filter 110 can be omitted.

[0073] Figure 2 Illustrated is the stabilization of the quantum manifold of the coherent state of the first mode achieved through 2-to-1 photon conversion performed by the circuit 100.

[0074] This figure shows the Wigner function of the first mode when it is driven by two-photon at a rate of ε2 and dissipated by two-photon at a rate of κ2. The first mode has two stable steady states (201, 202) with amplitudes of and opposite phases. Since there are two possible states, information can be encoded: the state |0> 202 is circled with a solid line, and the state |1> 201 is circled with a dashed line. Due to the relatively stable dynamics of convergence to these two states, this encoding is highly robust against bit-flip errors (i.e., the system flips between the state |0> and the state |1>). This encoding cannot correct another error channel, i.e., the phase-flip error. However, additional error correction schemes can be added to handle this error separately. This stability is achieved by coupling to an additional mode and implementing 2-to-1 photon conversion between the first mode and the second mode.

[0075] In Figure 3 , 7 , 9, and 11, only the circuit 100 will be described. For simplicity, the coupling to the load, the microwave source of the drive buffer, and the microwave source driving the ATS for parametric pumping are not shown.

[0076] Figure 3 Represents Figure 1 the equivalent circuit diagram of the first embodiment of the current-type cat circuit 100 in

[0077] The circuit 100 includes a first resonant section 30, a second resonant section 32, and an ATS 34. Both the first resonant section 30 and the second resonant section 32 are electrically connected to the ATS 34. To avoid any misunderstanding, the expression "electrically coupled" means that there are short conductive parts connecting the first resonant section 30 and the second resonant section 32 to the ATS 34, i.e., a short conductive track or any other means ensuring a physically continuous conductive connection. The expression "short" means that, compared with the impedances of the ATS 34, the first resonant section 30, and the second resonant section 32 at frequencies f a and f b the impedance of the conductive track can be neglected. If the impedance of the conductive track cannot be neglected, it will act as a voltage divider, reducing the zero-point fluctuations of the first and second resonant modes in the ATS and which is contrary to the purpose of the present invention. In this example, the conductive track is arranged in such a way that it also ensures that it does not shunt either the first resonant section or the second resonant section.

[0078] This is contrary to capacitive coupling or inductive coupling, in which the coupling is ensured by an electromagnetic field and there is no need for physical contact between the two parts. In these cases, due to the self-inductance effect of the coupling transformer, the impedance of the coupling capacitance or inductance is usually large, which will seriously divide the zero-point fluctuations, thus affecting the 2-to-1 photon conversion rate g2.

[0079] In the embodiment described herein, the first resonant section 30 includes an inductive element 300 and a capacitive element 302 connected in series. Similarly, the second resonant section 32 includes an inductive element 320 and a capacitive element 322 connected in series. If the first resonant section 30 (or the second resonant section 32) is separated from the rest of the circuit 100, the first resonant section 30 (or the second resonant section 32) will carry the first eigenmode (or the second eigenmode).

[0080] Due to the above circuit topology, the ATS 34 itself is a coupling element between the first eigenmode and the second eigenmode. This topology naturally allows the two resonant modes generated by the interaction of the first eigenmode, the second eigenmode, and the ATS 34 to strongly participate in the ATS 34. By coupling the second mode to the cold bath (load 106) where photons are dissipated, the second mode has greater loss and constitutes a buffer mode, while the first mode has less loss and constitutes a storage mode.

[0081] Let L a be the inductance value of the inductive element 300 (L b be the inductance value of the inductive element 320), C a be the inductance value of the capacitive element 302 (C bIf it is the capacitance value of the capacitance element 322), then the first eigenmode (second eigenmode) can be described as:

[0082] - By its angular frequency (Or ), where (Or )

[0083] - By its impedance (Or ), or equivalently

[0084] - By its intrinsic zero - point fluctuation (Or ), where R q = h / 4e 2 is the superconducting resistance quantum.

[0085] The inductance L ats is the effective inductance value of the ATS 34 near the global minimum of its potential energy. At the saddle point, the effective inductance L ats is equal to the parallel inductance of the ATS 34.

[0086] When the first resonant part 30 and the second resonant part 32 are electrically connected to the ATS, the linear part of the circuit Hamiltonian is:

[0087]

[0088] where the coupling rate is The dimensionless coupling constant is

[0089] The above - mentioned modes a and b are the modes that diagonalize the Hamiltonian H lin . The dimensionless coupling constant k can take any value between 0 and 1. A value close to 0 represents a weak - coupling mode, while the value 1 represents the maximum - coupling mode. When L ats << L a , L b , a value close to 0 is obtained. When L ats ~L a , L b , a value of the order of 1 / 2 is obtained. When L ats >> L a , L bWhen, a value close to 1 is obtained. As expected, due to the dynamic superconducting inductance of the ATS, the current design can easily achieve k values of the order of 1 / 2 and 1. This will further increase the zero-point fluctuations. The possibility of achieving such a large coupling constant is unique to the field of superconducting circuits, as first demonstrated by Devoret et al. in the article "Circuit-QED: How Strong Is the Coupling between a Josephson Junction Atom and a Transmission-Line Resonator?" published in the Annals of Physics, Vol. 519, pp. 767-779, 2007.

[0090] As Figure 5 , 6 , 9, and 11 show that the inductive element and the capacitive element form an LC resonator, which can be realized by distributed elements in a patterned superconducting material layer, for example:

[0091] - Two adjacent plates in parallel with a superconducting wire form a capacitor, and a single Josephson junction or an array of Josephson junctions forms an inductor.

[0092] - A section of superconducting transmission line with two different boundary conditions at its ends (one end grounded and short-circuited, the other end open-circuited) forms a so-called λ / 4 resonator. The transmission line is, for example, of coplanar waveguide type or microstrip type, or

[0093] - A section of superconducting transmission line with two identical boundary conditions at its ends (open-open or short-short) forms a so-called λ / 2 resonator, and the transmission line is, for example, of coplanar waveguide type or microstrip type.

[0094] The implementation of the ATS 34 is the same as the prior art known in the art, as described, for example, in the article by Lescanne (2020). Its structure has two parallel Josephson junctions, and an inductive element is connected in parallel between the two Josephson junctions. Therefore, the ATS 34 includes two connected loops, each loop containing a Josephson junction in parallel with a shunt inductive element. Both loops of the ATS 34 have direct current (DC) and alternating current (AC) flux biases. The DC bias sets the operating point of the ATS. It can operate near the so-called saddle point, which is the optimal point in terms of frequency and has a small cross-Kerr term. The AC flux bias corresponds to a parametric pump with a frequency at 2f a –f b . Usually, this AC flux bias is selected to drive the common mode of the two loops.

[0095] Figure 4 Shows the results of the value curve, which can be obtained at different values of the inductive elements 300 and 320, where the typical value of the ATS inductance is 4 nH, the frequency f a = 4.5 GHz and f b= 8.0 GHz. In the figure, the value of the first mode is represented by a dashed line in radians, and the value of the second mode is represented by a dotted line in radians, while the corresponding level line is represented by a solid line in MHz. These curves are constructed in such a way that by varying the value of inductive element 300 (as the eastward coordinate) and the value of inductive element 320 (as the northward coordinate), while selecting the values of capacitive elements 302 and 322, the above frequencies are obtained.

[0096] The reason for plotting this ratio is that is proportional to the amplitude of the parametric flux pump, and the amplitude of the parametric flux pump is set by the amplitude of microwave source 102, and the amplitude of microwave source 102 is somewhat arbitrary. In contrast, this ratio depends only on the intrinsic parameters of circuit 100. Figure 4 is calculated assuming that ATS 34 operates at its saddle point. In this case, w a and w b values depend only on the inductance L ats of ATS, and are independent of L j . Since the Josephson energy E j is it thus seems possible to make j arbitrarily large by continuously reducing L . However, as the ratio of L ats / L j increases, the dynamic characteristics of ATS become more unstable (as confirmed by Burgelman et al. in the article "Structurally Stable Subharmonic Regions in Driven Quantum Josephson Circuits", https: / / arxiv.org / abs / 2206.14631). In Figure 4 , this ratio is set to L ats / L j = 2, corresponding to the value in Lescanne (2020).

[0097] The figure shows that for conventional values of inductive elements 300 and 320, it is possible to easily achieve values exceeding 100 MHz. In fact, the applicant's research has shown that values exceeding 50 MHz can be guaranteed, which is nearly an order of magnitude higher than the known prior art, and values in the hundreds of megahertz can be achieved. In contrast, the value achieved in Lescanne (2020) was only 9.6 MHz. In Berdou (2022), it is at least an order of magnitude smaller than Lescanne (2020).

[0098] Figure 4 The zero-point fluctuations are also shown. and Although only the product affects g2, and the actual values of

[0099] will directly affect the spurious terms generated by the ATS, which in turn may induce noise processes that escape the stable coherent state limit. and Since the 2-to-1 photon conversion Hamiltonian depends on the third-order expansion of the ATS potential energy in and The rule of thumb known to those skilled in the art is to make

[0100] less than π. Where α is the amplitude of the stable coherent state in the cat qubit, and β is the amplitude of the residual electromagnetic field in the buffer. is considered safe. On the other hand, β tends to be very close to zero, so max(β, 1 / 2) = 1 / 2. This means can tolerate larger values, typically Figure 4 shows that, while maintaining the safe values of the parameters, g2 can be increased by more than an order of magnitude compared to the prior art.

[0101] Figure 4 It also shows that and can easily reach more aggressive values, thereby further increasing g2. It should be noted that, due to the relatively late development of the cat qubit field and the lack of research in this area, it is currently not clear and how much can be improved. Therefore, the current-mode cat design has the obvious potential to achieve larger and and will thus be able to conduct such research.

[0102] Figure 13 The ratio of the square of the zero-point fluctuations of modes b and a across the capacitor 322 and the inductor 320 of the second resonant section 32 is shown respectively (shown as a dashed line) and the ratio (shown as a dotted line). If the load is coupled to the capacitor 322 of the circuit 100 through the coupler 104, the attenuation rates of modes a and b in the load are respectively related to and is proportional to the square of; if the load is coupled to the inductor 320, the attenuation rates of modes a and b in the load are respectively proportional to and the square of. Therefore, these ratios indicate to what extent mode a is immune to load attenuation depending on how the load is coupled to the circuit. When the load is coupled to the capacitor 322 as is common practice in the field of superconducting circuits, the ratio value is about an order of magnitude of 10 within the safe parameter region. Although this level of protection is not significant, it provides room for achieving strong and efficient filtering thanks to the relatively wide frequency interval between fa and fb allowed by the relatively large k value of the current design. It is worth noting that when the load is connected to the inductor 320 of the second resonant section 32 (for example, an inductive coupler can be used), the ratio increases by a factor of 10, which is quite good, and can be easily increased to the necessary level through the general implementation of the filter 110.

[0103] This is even more surprising because the idea of electrically coupling the ATS to the two modes leads to strong hybrid modes due to the relatively large k value. Having a low-loss storage mode and strong hybridization is highly counterintuitive. The basic principle is that the field of quantum computing is still in its infancy, especially in the field of cat qubits, and usually tends to make very gradual changes. One of the reasons for positioning the ATS on the second mode and weakly capacitively coupling the first mode to the second mode in the papers of Lescanne (2020) and Berdou (2022) is to minimize the attenuation of the first mode due to the attenuation of the second mode in the load. In fact, traditionally, it has been considered more preferable to couple the nonlinear element of the quantum circuit to a single mode and weakly couple other modes to that mode.

[0104] The electrical coupling of the nonlinear element is clearly not a gradual change and goes against all biases. In addition, the applicant surprisingly found that although the electrical coupling through the ATS induces strong mixing of the first mode and the second mode, generally only the buffer mode can be coupled to the environment.

[0105] Another advantage of the electrical coupling and the relatively large k value is that the first resonant frequency and the second resonant frequency can be selected such that the value 2f a -f b slightly deviates from f a and f b . For example, make 2f a -f b less than f a / 2 and f b / 2. This can be achieved by choosing the buffer frequency f b not too far from twice the storage frequency f a , resulting in a relatively small pump frequency f p= 2f a –f b For example, Figure 4 the f in p has a value of 1 GHz, which is more than four times smaller than f a and eight times smaller than f b .

[0106] This is advantageous because the first-order term of the ATS potential energy expansion indicates that parametric pumping can directly drive the circuit. As shown in the supplementary materials of Lescanne (2020), this kind of driving will lead to spurious dynamic AC Stark shifts and dynamic cross-Kerr terms, which are detrimental to the operation of cat qubits. At lower pump frequencies, the direct driving efficiency of the circuit is much lower, which results in much smaller dynamic AC Stark shifts and dynamic cross-Kerr terms.

[0107] Another advantage is that the value 2f a - f b can be set far away from f a and f b , so that a second microwave filter can be introduced into the parametric pumping line to prevent the first mode from leaking into the parametric pumping line.

[0108] Finally, the large frequency interval between f a and f b makes the design more robust to uncertainties in nanofabrication, such as the well-known problem of variations in the inductance of various Josephson junctions in the circuit, which mainly affects the accuracy of the prediction of f a and f b .

[0109] Figure 5 Shows Figure 3 the first implementation of the circuit shown.

[0110] This figure is an optical microscope image of the surface of a superconducting chip fabricated by the applicant to correspond to the circuit shown in Figure 3 . The white part represents the metallized surface of tantalum or aluminum. The gray part represents the sapphire substrate on which the circuit is placed. Other materials can also be used to implement the superconducting circuit. For example, the metallization layer can use niobium, NbTi or TiN, and the chip can use silicon or quartz.

[0111] The circuit includes a ground plane 50 on which a circuit 52 is formed. The ground plane 50 has regular perforations to expose the substrate (the gray dots marked as 502). These perforations are for reducing magnetic noise by capturing superconducting vortices, but these perforations are optional and can be omitted. Five transmission lines, respectively marked as 503, 504, 505, 508, and 509, are used to introduce and extract radiation into and out of the circuit. These transmission lines are implemented using CPW (coplanar waveguide). Line 503 can be used to excite the cat qubit. Line 504 is a bus line whose other end is connected to a readout transmitter for measuring the cat qubit. Line 505 is the starting end of a filter 110, connected to a load 106 and a microwave source 108. In addition, four wire bonds 506 are provided to ensure equipotentiality on the ground plane 50. Finally, two flux lines 508 and 509 are used to pump the flux of ATS 34 through a microwave source 102 and set the DC operating point of ATS 34 by applying a DC current. and set the DC operating point of ATS 34 by applying a DC current.

[0112] Figure 5 The circuit shown illustrates Figure 1 lumped and differential implementations of the resonant modes of the circuit shown. This design is called lumped because the total size of circuit 100 is less than a quarter wavelength of the first and second modes. This design is called differential because the first and second modes correspond to charge and current oscillations between pairs of electrodes electrically isolated from the ground plane 50.

[0113] A pair of large rectangular electrodes marked as 550 and 552 respectively implement capacitor 302 of the first mode (storage mode) and capacitor 322 of the second mode (buffer mode). The thin lines between two pairs of small metal pads marked as 554 and 556 are Josephson junction chains, respectively implementing inductor 300 of the first mode and inductor 320 of the second mode. The inductor can also be implemented in other ways, such as a geometric inductor made of a meandering or spiral wire. ATS 34 is located near the flux lines 508 and 509 that drive it.

[0114] This design occupies more space than other designs, but its advantage is that it can better isolate lossy elements at the ground end (such as wire bond 506) or other elements that may be patterned on the chip (such as other cat qubits), thereby reducing crosstalk.

[0115] Figure 6 Shows Figure 3 a second implementation of the circuit shown.

[0116] Similar to Figure 5 this figure is an optical microscope image of the surface of a superconducting chip manufactured by the applicant to correspond to Figure 3The circuit shown. The white part represents the metallized surface of tantalum or aluminum. The gray part represents the sapphire substrate on which the circuit is located. Other materials can also be used to implement the superconducting circuit. For example, the metallization layer can use niobium, NbTi or TiN, and the chip can use silicon or quartz.

[0117] Similar components have been given the same reference numerals. Only the first digit of the reference numeral will change from "5" to "6". For example, Figure 5 the ground plane 50 in Figure 6 is labeled 60 in

[0118] This implementation is different from Figure 5 the implementation in that it shows a lumped ground implementation of the resonant modes of the Figure 3 circuit. This implementation is called grounded because the first and second modes correspond to the oscillation of charge and current between the electrode and the ground plane 60, and these oscillations of charge and current are electrically coupled to the ground plane 60 through ATS 34. The bottom electrode of the ATS is actually directly imprinted on the ground plane 60.

[0119] The capacitor 302 for the storage mode is made of a single electrode (which is T-shaped and labeled 650). The capacitor 322 for the buffer mode is made of a single electrode (which is a horizontal bar) 652.

[0120] The significant difference from the Figure 5 circuit is that in this implementation, the storage mode is connected to another electrically coupled cat circuit (not shown) through the transmission line 620. In addition, the drive line 503 for the storage mode and the bus 504 for coupling the storage mode to the readout transmission line are combined into one line, labeled 603 - 604.

[0121] This grounded design is more sensitive to defects in the ground plane and may produce more crosstalk than the differential design, but it is more compact. In addition, the coupling between the ATS and the flux line is stronger.

[0122] Figure 7 represents a second embodiment similar to Figure 3 The difference is that the inductance and capacitance in the first resonant part and the second resonant part are in parallel instead of in series. Therefore, the ATS is connected in series with the first and second resonant parts instead of in parallel. Figure 7 The circuit of Figure 3 is the electrical dual of the Figure 3 circuit. All the equations of the

[0123]

[0124] Similar elements are denoted by the same reference numerals. Only the first digit of the reference numerals will change from "3" to "7". For example, Figure 3 the first resonance mode 30 in Figure 7 is denoted as 70 in

[0125] Figure 8 is similar to Figure 4 but is based on the circuit of Figure 7 For simplicity, it will not be elaborated further.

[0126] It is obvious to those skilled in the art that Figure 5 and Figure 6 can be implemented to put the design of Figure 7 into practice.

[0127] Figure 9 represents a third embodiment of a current-mode cat circuit. This embodiment is different from the embodiments of Figure 3 and Figure 7 in that the characteristics of the first and second resonance modes generally cannot be accurately described by a simplified symbolic representation involving only two LC resonators. This explains the reason for the present invention to break the existing prejudice: although the lower coupling constant k can be well understood for the first and second modes through a simple perturbation analysis of only two eigenmodes and ATS including the resonant part, this is not the case here, and more eigenmodes of the resonant part must be considered.

[0128] In the embodiment of Figure 9 there is only one resonant part, which together with ATS 34 generates the first and second modes. As shown, ATS 34 shunts the open transmission line into two parts 90 and 92. Since ATS is symbolically embedded in the transmission line carrying the resonant mode, this circuit is a current circuit. If the dimensionless coupling constant k is small, the first mode of the quantum circuit will simply correspond to the LC circuit formed by ATS and the static capacitance of the transmission line. The second mode will correspond to the eigen λ / 2 harmonic of the coupling between the transmission line and ATS; the third mode will correspond to the eigen λ harmonic of the coupling between the transmission line and ATS; and so on. However, when k is large, the inductance and capacitance values associated with different eigenharmonics of the transmission line are determined simultaneously to generate the first and second modes when associated with ATS 34.

[0129] The transmission line can be implemented with the same material as the elements in Figure 5 and Figure 6 If a given or If the required characteristic impedance is too large to be achieved geometrically, the central conductor of the transmission line can be replaced with a high kinetic inductance material or a wide Josephson junction chain. Finally, the transmission line can be implemented in various geometries, such as coplanar waveguide (CPW), microstrip line, or slot line. In Figure 9 the example of

[0130] Figure 10 the results of the curves of Figure 4 values obtained for different lengths of CPW sections 90 and 92 are shown. In this figure, the ATS parameters are the same as in r . It is assumed that the CPW has a characteristic impedance of 50 Ω, and without loss of generality, the typical effective relative permittivity ∈ a of the CPW on sapphire is taken as 5.6. In addition, the first resonant mode (or the second resonant mode) is taken as the first (or fundamental) harmonic (or the second harmonic) of the quantum circuit 100. In the top graph of this figure, the value of the first resonant frequency of the circuit f b is represented by a dashed line in GHz, the value of the second resonant frequency of the circuit f is represented by a dotted line in GHz, and the corresponding level line is represented by a solid line in MHz. In the bottom graph of this figure, the value of the first mode is represented by a dashed line in radians, the value of the second mode is represented by a dotted line in radians, and the corresponding

[0131] level line is represented by a solid line in MHz. Figure 4 and Figure 8 In contrast to b - f a , since the variations of the first resonant frequency and the second resonant frequency are not fixed, it is necessary to display them. It seems that a large detuning f of several GHz can be achieved, and the and values far exceed the safety values in the prior art.

[0132] Those skilled in the art can obviously adapt to specific applications by adjusting the values of f a , f b , and , for example, by adjusting the characteristic impedance of the transmission line or using other harmonics. The termination of the transmission line can also be changed, but some terminations (such as inductive short circuits) must be included in the potential energy of the ATS, thus changing its operating point and dynamic characteristics.

[0133] Figure 11 The embodiments of Figure 9 are similar. There is also an open transmission line in the figure that carries both the first mode and the second mode at the same time, but ATS 34 is connected in series with the two parts 110 and 112 of the transmission line. Figure 12 Equivalent to Figure 8 , but the embodiment of Figure 11 is adopted. For simplicity, it will not be elaborated here.

Claims

1. A non-linear superconducting quantum circuit (100) includes at least one resonant part (30, 32; 70, 72; 90, 92; 110, 112) and an asymmetric threaded superconducting quantum interference device (34) electrically connected to the at least one resonant part (30, 32; 70, 72; 90, 92; 110, 112). The non-linear superconducting circuit (100) includes a first mode (a) having a first resonant frequency and a second mode (b) having a second resonant frequency, and the ratio of the first resonant frequency to the second resonant frequency is different from 1 / 2. The at least one resonant part (30, 32; 70, 72; 90, 92; 110, 112) is configured with inductance and capacitance values represented by their symbols, and the inductance and capacitance values, together with the asymmetric threaded superconducting quantum interference device (34), induce the first mode (a) and the second mode (b), such that the non-linear superconducting quantum circuit (100) has zero-point fluctuations of the superconducting phase on the asymmetric threaded superconducting quantum interference device (34) in the first mode (a) and the second mode (b) that are greater than or equal to 0.05 radians.

2. The non-linear superconducting quantum circuit according to claim 1, wherein The at least one resonant part (30, 32; 70, 72) has the following symbol representation: the first mode (a) is carried in a first resonant part (30; 70) including at least one inductor (300; 700) and at least one capacitor (302; 702); the second mode (b) is carried in a second resonant part (32; 72) including at least one inductor (320; 720) and at least one capacitor (322; 722); the asymmetric threaded superconducting quantum interference device (34) is disposed between the first resonant part (30; 70) and the second resonant part (32; 72) and is electrically coupled to the first resonant part (30; 70) and the second resonant part (32; 72).

3. The non-linear superconducting quantum circuit according to claim 2, wherein, The at least one inductor (300, 320; 700, 720) and the at least one capacitor (302, 322; 702, 722) of the first resonant part (30; 70) and the second resonant part (32; 72) are respectively in series or in parallel, and the asymmetric threaded superconducting quantum interference device (34) is respectively in parallel or in series with the first resonant part (30; 70) and the second resonant part (32; 72).

4. The non-linear superconducting quantum circuit according to claim 2 or 3, wherein, The non-linear superconducting quantum circuit is located on a dielectric substrate and is separated from a common ground plane (50; 60) by an exposed part of the dielectric substrate, and the first resonant part and the second resonant part are implemented in physically different parts of the non-linear superconducting quantum circuit.

5. The non-linear superconducting quantum circuit according to claim 4, wherein, The non-linear superconducting quantum circuit is formed on a substantially flat substrate, and the non-linear superconducting quantum circuit has a width and a height that are respectively less than a quarter wavelength corresponding to the first resonant frequency and the second resonant frequency.

6. The non-linear superconducting quantum circuit according to claim 4 or 5, wherein, The first resonant part and the second resonant part are electrically isolated from the common ground plane (50).

7. The non-linear superconducting quantum circuit according to claim 4 or 5, wherein The first resonant portion and the second resonant portion are electrically connected to the common ground plane (60).

8. The non-linear superconducting quantum circuit according to claim 1, wherein, The non-linear superconducting quantum circuit is located on a dielectric substrate and is separated from the common ground plane by an exposed portion of the dielectric substrate, and the at least one resonant portion is implemented as a transmission line (90, 92; 110, 112).

9. The non-linear superconducting quantum circuit according to claim 8, wherein, The first mode (a) and the second mode (b) are respectively the fundamental wave or higher-order harmonics of the non-linear superconducting circuit (100).

10. The non-linear superconducting quantum circuit according to one of the foregoing claims, wherein, The first resonant frequency and the second resonant frequency are such that the difference between twice the first resonant frequency and the second resonant frequency is less than half of the first resonant frequency and half of the second resonant frequency.

11. The non-linear superconducting quantum circuit according to one of the preceding claims, wherein, At least one of the inductors (300, 320; 700, 720) and / or one or more of the transmission lines (90, 92; 110, 112) are made of a Josephson junction array or a high kinetic inductance material.

12. A quantum device, comprising: A non-linear superconducting quantum circuit according to one of the preceding claims; A first microwave source (108) connected to the at least one resonant portion (30, 32; 70, 72; 90, 92; 110, 112) for providing radiation having a frequency equal to the second resonant frequency; a second microwave source (102) connected to the at least one resonant portion (30, 32; 70, 72; 90, 92; 110, 112) for providing radiation having a frequency equal to the difference between twice the first resonant frequency and the second resonant frequency; and a load (106) coupled to the at least one resonant portion (30, 32; 70, 72; 90, 92; 110, 112) such that generally only the second mode (b) is coupled to the load (106), whereby the first mode (a) carries the cat qubit.

13. The quantum device according to claim 12, further comprising a microwave filter (110) for coupling to the load (106), the microwave filter (110) being arranged to allow the second resonant frequency to pass through and block the first resonant frequency.

14. A quantum computing system comprising at least one device according to claim 12 or 13.

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