Minimum superconducting quantum circuit with electrically coupled wave color coding

Abstract

A non-linear superconducting quantum circuit comprising at least one resonant portion (30, 32) and an electrically connected asymmetrically threaded superconducting quantum interference device (34), the non-linear superconducting quantum circuit comprising a first mode having a first resonant frequency and a second mode having a second resonant frequency, the ratio between the first resonant frequency and the second resonant frequency differs from 1 / 2, the at least one resonant portion (30, 32) having a symbolic representation comprising a linear resonant portion (32) comprising at least one inductance (320) and at least one capacitor (322) and a non-linear resonant portion (30) comprising at least one capacitor (302) and the asymmetrically threaded superconducting quantum interference device (34), the linear resonance part (30) and the non-linear resonance part (32) are electrically connected and respectively arranged such that one has elements connected in series and the other has elements connected in parallel, the at least one resonance part (30, 32) being configured with inductance and capacitance values, and the non-linear superconducting quantum circuit (100) induces the first mode (a) and the second mode (b) together with the asymmetric threaded superconducting quantum interference device (34), so that the zero fluctuation of a superconducting phase on the asymmetric threaded superconducting quantum interference device (34) is greater than or equal to 0.05 radian when the non-linear superconducting quantum circuit (100) is in the first mode and the second mode.

Description

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

[0002] A cat qubit is a subset of bosonic encodings and forms a family of error-correcting codes for quantum applications. Generally speaking, bosonic encodings rely on storing qubits in bosonic modes. In the case of cat qubits, the most common design to date is the two-component cat encoding.

[0003] Dissipative stabilization of two coherent states requires a non-linear conversion between two photons in a first mode (also referred to as the cat qubit mode) carrying a stable quantum manifold and one photon in a second mode (referred to as 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 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") have pseudo-cross-Kerr terms that give rise to additional noise processes whose escape rates are characterized by very large transmon-cat quantum ratio dispersive 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)) discloses a significantly improved cat qubit by using an asymmetric threaded superconducting quantum interference device (also known as an "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 allows us to observe an exponential suppression of bit flips. However, the transmon is 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" by Berdou et al. (arXiv:2204.09128, https: / / arxiv.org / pdf / 2204.09128.pdf) (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 very 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 represents 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 is 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 building a practically applicable quantum circuit. 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 connected electrically. The non-linear superconducting quantum 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 different from 1 / 2. The at least one resonant part has a symbolic representation, which includes a linear resonant part (including at least one inductor and at least one capacitor) and a non-linear resonant part (including at least one capacitor and the asymmetric threaded superconducting quantum interference device). The linear resonant part and the non-linear resonant part are electrically connected and arranged respectively such that the elements of one of them are connected in series, while the elements of the other are connected in parallel. The at least one resonant part is configured with inductance and capacitance values, which together with the asymmetric threaded superconducting quantum interference device induce the first mode and the second mode, such that the superconducting quantum circuit has a zero-point fluctuation of the superconducting phase on the asymmetric threaded superconducting quantum interference device in the first mode and the second mode that is 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 the storage mode in the ATS, thus reducing the 2-to-1 photon conversion rate). In theory, the adverse effects of the coupling capacitors in the Lescanne (2020) circuit can be minimized, but it is well known that large capacitors will generate losses in superconducting circuits, which will increase the phase flip rate and thus make it ineffective in practical applications. In addition, this design contains as few components as possible, thus minimizing the risks in terms of industrialization cost and feasibility.

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

[0010] - The linear part includes elements arranged in parallel, and the non-linear part includes elements arranged in series;

[0011] - The linear resonant part includes elements arranged in series, and the non-linear resonant part includes elements arranged in parallel;

[0012] - The nonlinear 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 linear resonant portion and the nonlinear resonant portion are implemented in physically different portions of the nonlinear superconducting quantum circuit;

[0013] - The nonlinear superconducting quantum circuit is formed on a substantially flat substrate, and its width and height are respectively less than a quarter wavelength corresponding to the first resonant frequency and the second resonant frequency;

[0014] - The nonlinear resonant portion and the linear resonant portion are electrically connected to the common ground plane;

[0015] - The nonlinear resonant portion and the linear resonant portion are electrically isolated from the common ground plane;

[0016] - The nonlinear 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 nonlinear 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;

[0019] - The at least one 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 nonlinear 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 arranged 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, respectively, for Figure 3 the first mode and the second mode of the superconducting quantum circuit values and the corresponding values of the graph;

[0027] - Figure 5 Represents Figure 3 a circuit implementation;

[0028] - Figure 6 Represents the electrical equivalent diagram of the second embodiment of the current-type cat circuit according to the present invention;

[0029] - Figure 7 Shows, respectively, for Figure 6 the first mode and the second mode of the superconducting quantum circuit values and the corresponding values of the graph;

[0030] - Figure 8 Represents Figure 6 a circuit implementation shown;

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

[0032] - Figure 10 Represents Figure 9 the behavior curve of the circuit shown with respect to the transmission line length;

[0033] - Figure 11 Represents the ratio of the square of the zero-point fluctuations of the phases of modes b and a in the linear capacitor and inductor in the first embodiment, as Figure 4 a supplement.

[0034] Most of the 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.

[0035] To enable the cat qubit to encode useful data and remain stable, a 2-to-1 photon conversion is required between a first mode (memory) and a second mode (buffer). Most of the existing technologies belong to the family of cat qubits stabilized by parametric pumping dissipation. The parametric pumping dissipation technique is used to bridge the gap between the frequencies of the two modes and perform a 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 constraints on the resonant frequencies.

[0036] 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 have their respective resonant frequencies. For example, the first mode can have type of resonant frequency, while the second mode can have type of resonant frequency, where ω a and ω b can be the angular frequencies of each corresponding mode. Adopting the first mode and the second mode "having" means that the superconducting quantum circuit can include components that operate in a superconducting state, and these components can carry these modes independently 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 in the same subset of components.

[0037] 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 a dissipative environment, while the first mode can remain isolated from the environment.

[0038] The superconducting quantum circuit can be fabricated as a patterned layer of one or more layers of 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. The capacitive element can be formed by two adjacent superconducting material plates (on the corresponding layer of the one or more patterned layers). The 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 formed by machining or microfabrication of bulk superconductors and can be used as either of the two modes.

[0039] The circuit can be integrated into a device that can include a load, a first microwave source, a second microwave source, and a coupler. The coupler can be configured to connect a second mode of the superconducting quantum circuit to the load. The load is a dissipative element, e.g., an element with a given resistance (different from superconducting elements), which is outside the superconducting circuit. The load dissipates photon pairs that are converted from the first mode to the second mode by 2-to-1 photon conversion. In other words, the photon pairs destroyed in the first mode are evacuated to the environment via the second mode through 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 whose frequency is generally equal to the frequency of the second mode. Thus, 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 by current, capacitance, or inductance, and mediates the interaction between the second mode, the load, and the microwave source.

[0040] 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 terminates at 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 is absorbed. The load can be contained within the first microwave source.

[0041] In various examples, the first microwave source can be placed at room temperature and connected to the circuit by 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 thermalize the microwave radiation using the cryogenic environment. This allows microwave radiation to be applied without increasing thermal noise.

[0042] The second microwave source is used to provide microwave radiation whose frequency is generally equal to twice the resonance frequency of the first mode minus the resonance frequency of the second mode, so as to achieve 2-to-1 photon conversion. Since the ATS has two superconducting loops and must be flux pumped with sufficient relative amplitude and phase, the radiation emitted by the second microwave source can be split for use by two different transmission lines or waveguides that 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 amplitude and phase.

[0043] Optionally, the device may include a microwave filter that is connected to the first and second modes of the circuit. The microwave filter may be configured to allow only the second mode to couple to the load. The microwave filter may 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 of the second mode are the only photons 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.

[0044] 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 the microwave filter can be regarded as a two-photon drive for the first mode after a 2-to-1 photon conversion; while the load that only dissipates the photons of the second mode can be regarded as a two-photon dissipation for the first mode after a 2-to-1 photon conversion. The two-photon drive and two-photon dissipation enable the two coherent states in the first mode to be stabilized.

[0045] 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.

[0046] The two-photon drive can be formally described by the Hamiltonian where ∈2 is the effective two-photon drive rate. The two-photon loss can be formally described by the Lindblad operator where κ2 is the two-photon dissipation rate. The amplitude α of the stabilized coherent state is finally given by . The confinement rate of the coherent state is κ conf = κ2α 2 .

[0047] In the range of κ b >> g2α, where g2 is the 2-to-1 non-linear conversion rate between the first and second modes, the buffer dynamics can be adiabatically eliminated, resulting in ∈2 = 2∈ b g2 / κ band

[0048] In various examples, a 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 referred to as components) forms a circuit structure (or topology) that is equivalent (functionally) to a non-linear superconducting circuit.

[0049] In other words, as is the classical practice in the field of superconducting circuits, the configuration of a non-linear superconducting circuit is designed to achieve the function defined by its symbolic representation, i.e., the function of the theoretical group of interconnected dipoles shown by the symbolic representation. In other words, although the circuit can be constructed using patterned superconducting material layers, 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 circuits of distributed components within a specific frequency range (such as low frequency), as is well known in the art.

[0050] It is known in the art that such distributed components can have high-frequency modes that are irrelevant and unimportant to the dynamic characteristics described herein. Therefore, these distributed components can be presented using a symbolic representation. The symbolic representation can be refined by adding components (for example, adding series inductors at each wire connection or adding parallel capacitors between any two nodes of the circuit) or adding nodes and branches to account for other modes of the distributed components. Therefore, the symbolic representation method can better describe the distributed components without changing the operating principle of the circuit. Therefore, as is known in the art, those skilled in the art consider the physical circuit (i.e., the actually manufactured circuit) and its symbolic representation to be equivalent. In fact, compared with the basic model, the refined dipoles of the symbolic representation only adjust 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 under any configuration.

[0051] The form of the 2-to-1 photon interaction Hamiltonian is In these cases, the coupling term g2(t) is modulated by a parametric pump, where the pump injects an external time-varying parameter with a frequency of ω p = 2ω a - ω b In the prior art, a parametric pump is used to make the non-linear interaction resonate.

[0052] The development of cat qubit quantum circuits relies on the geometry of superconducting circuits, which can suppress the bit flips of cat qubits encoded in high-Q superconducting resonators (referred to as memories). To this end, the two-photon dissipation of the memory is achieved by coupling the memory to a low-Q superconducting resonator (referred to as a buffer) through a nonlinear superconducting dipole.

[0053] In the article by Lescanne (2020), the ATS superconducting dipole is used to design the nonlinear Hamiltonian H2. The ATS dipole has the following potential energy:

[0054] 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 passing through the common (or differential) loop mode of the ATS.

[0055] By choosing the following DC value of the magnetic flux (referred to as the saddle point): and only performing flux pumping on the σ mode with an amplitude of and a frequency of ω p , the potential energy expression becomes:

[0056]

[0057] The phase on the ATS is related to the modes a and b as follows:

[0058]

[0059] 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 to the third order in and and eliminating the fast-rotating terms, the 2-to-1 conversion Hamiltonian H2 is obtained, and its 2-to-1 photon conversion rate is

[0060] The article by Lescanne (2020) shows that the bit flips will increase with the photon α 2The quantity is exponentially suppressed. However, this architecture uses a transmon coupled to a 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 because the confinement rate is too small to resist the dispersive frequency shift caused by the thermal excitation of the measurement device. The applicant's later research published in Berdou (2022) increased the bit flip saturation time by five orders of magnitude by removing the transmon and operating the ATS in a presumably dynamically stable state, despite a lower confinement rate. More precisely, in the paper 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.

[0061] 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 located on the ATS, and the storage mode is weakly capacitively coupled to the buffer mode and thus coupled to the ATS. This design allows for obtaining a relatively large but is typically one to two orders of magnitude smaller. Since the formula for g2 depends on squared, this circuit geometry is very unfavorable for the intensity of two-photon dissipation.

[0062] The only way to solve this problem is to sufficiently increase the capacitance coupling the storage and buffer modes in the circuits of Lescanne (2020) and Berdou (2022) so that the ATS has a strong participation in both the buffer and storage modes. However, it is known that large capacitances cause losses in superconducting circuits.

[0063] Therefore, these prior arts are in a dead end: their specific geometries are crucial for achieving bit flip stability but cannot be adjusted to allow for a sufficiently large two-photon dissipation rate.

[0064] Examples and illustrations of the 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.

[0065] Figure 1 An example of a quantum device 10 including a current-type cat qubit circuit according to the present invention is shown.

[0066] The device 10 includes a non - linear superconducting circuit 100, a microwave source 102, a coupler 104, a load 106, another microwave source 108, and a microwave filter 110.

[0067] The non - linear superconducting circuit 100 performs a 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 a cat qubit, also known as the storage mode, while the second mode b serves as a buffer between the cat qubit and the environment.

[0068] The device 10 uses parametric pumping to stabilize the cat qubit, which means that the resonance frequencies of the first and second modes do not belong to the type of 2f a = f b To ensure the 2 - to - 1 photon conversion, the parametric pump provides radiation with a frequency of 2f a - f b This is performed by the microwave source 102 connected to the non - linear superconducting circuit 100.

[0069] As described below, the non - linear 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 non - linear superconducting circuit 100 that carry modes a and b.

[0070] The circuit element carrying the second mode 114 is 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 resonance 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, thereby preventing the first mode from suffering additional losses due to unnecessary coupling with the load 106. Alternatively, in the case of f a > f b (or f b > f a ), it can be configured as a low - pass (respectively, high - pass) filter. In other embodiments, when the coupling is established only between the load 106 and generally only the second mode, the microwave filter 110 can be omitted.

[0071] Figure 2 Illustrated is the stabilization of the coherent - state quantum manifold of the first mode achieved by the 2 - to - 1 photon conversion performed by the circuit 100.

[0072] The figure shows the Wigner function of the first mode with two - photon driving at a rate of ε2 and two - photon dissipation at a rate of κ2. The first mode has two amplitudes and two stable steady - states (201, 202) with opposite phases. Since there are two possible states, information can be encoded: state |0>202 is circled with a solid line, and 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 state |0> and state |1>). This encoding cannot correct another error channel, i.e., the phase - flip error. However, an additional error - correction scheme can be added to handle this error separately. This stability is achieved by coupling to an additional mode and implementing a 2 - to - 1 photon conversion between the first mode and the second mode.

[0073] In Figure 3 、 6 and 9, only the circuit 100 will be described. For simplicity, the coupling to the load, the microwave source driving the buffer, and the microwave source driving the ATS for parametric pumping are not shown.

[0074] Figure 3 Denotes Figure 1 the equivalent circuit diagram of the first embodiment of the current - type cat circuit 100 in

[0075] The circuit 100 includes a non - linear resonant part 30 and a linear resonant part 32. The non - linear part includes an ATS 34 and a capacitive element 302. The non - linear resonant part 30 and the linear resonant part 32 are connected together in a current - conducting manner. To avoid any misunderstanding, the expression "electrically coupled" means that there is a short conducting part connecting the non - linear resonant part 30 and the linear resonant part 32, i.e., a short conducting track or any other means ensuring a physically continuous conducting connection. The expression "short" means that, compared with the impedances of the non - linear resonant part 30 and the linear resonant part 32 at frequencies f a and f b the impedance of the conducting track can be neglected. If the impedance of the conducting 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 the examples described herein, the conducting track is arranged such that it does not shunt any non - linear and linear resonant parts.

[0076] In the embodiments described herein, the nonlinear resonance section 30 includes an ATS 34 and a capacitive element 302 connected in series. The linear resonance section 32 includes an inductive element 320 and a capacitive element 322 connected in parallel. If the nonlinear resonance section 30 (or the linear resonance section 32) is isolated from the rest of the circuit 100, it will carry the first eigenmode (or the second eigenmode).

[0077] This current circuit is a minimized circuit in terms of its symbolic representation because the symbolic representation has the minimum number of elements (two capacitive elements 302 and 322 and two inductive elements 34 and 320) to carry two resonance modes. This is contrary to the prior art where the nonlinear section is coupled to the linear section through a capacitive or inductive coupler, which results in a much lower participation of one resonance mode in the ATS, and thus lower zero-point fluctuations and a lower two-to-one photon conversion rate g2. Since the symbolic representation of the minimized current circuit proposed herein does not have such a coupler, it is expected that both resonance modes will have a higher participation in the ATS. Although it may seem that the ATS 34 is capacitively coupled to the linear eigenmode through the capacitor 302, in fact, since the ATS 34 resonates with the capacitor 302, they form a nonlinear eigenmode that is electrically coupled to the linear eigenmode.

[0078] This minimized current circuit also has the advantage that it can achieve a direct, simple, and compact structure, as will be seen later.

[0079] In addition, it should be noted that having one series resonance section and one parallel resonance section is the only viable circuit topology. Therefore, there are only two possible minimum implementations for a circuit containing an ATS: one where the ATS acts as the inductive element of the series resonance section, as Figure 3 shown; and the other where the ATS acts as the inductive element of the parallel resonance section, as Figure 6 shown.

[0080] Let L parallel and C parallel be the values of the inductive element and the capacitive element of the parallel resonance section respectively, and L series and C series be the values of the inductive element and the capacitive element of the series resonance section respectively. Then these eigenmodes can be described as:

[0081] - By their angular frequencies and and L parallel,eff = L parallel and

[0082] - By their impedances and or equivalently

[0083] - via their intrinsic zero - point fluctuations and where R q = h / 4e 2 is the superconducting resistance quantum.

[0084] In the case of Figure 3 (or Figure 6 ), L series = L ats (or L parallel = L ats ), where the inductance L ats is the effective inductance value of ATS34 near the global minimum of its potential energy. At the saddle point, the effective inductance L ats equals the parallel inductance of ATS 34.

[0085] The linear part of the current - circuit Hamiltonian is:[[]]

[0086] where the coupling rate is The dimensionless coupling constant is

[0087] The above - mentioned modes a and b are the modes that diagonalize the above - mentioned Hamiltonian H lin The dimensionless coupling constant k can take any value between 0 and 1. A value close to 0 represents a weakly - coupled mode, while the value 1 represents the maximum - coupled mode. As expected, a minimum - current design can easily achieve a k value of about 1 / 2 or higher. The possibility of reaching such a large coupling constant is unique to the field of superconducting circuits, as first shown by Devoret et al. in the article "Circuit - Quantum Electrodynamics: How Strong Is the Coupling between a Josephson - Junction Atom and a Transmission - Line Resonator?" published in Annalen der Physik, Volume 519, Pages 767 - 779, 2007.

[0088] As Figure 5 and Figure 8 shown, the inductor element and the capacitor element form an LC resonator, which can be realized by distributed elements in a patterned superconducting material layer, such as:[[]]

[0089] - 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.

[0090] - A section of superconducting transmission line with two different boundary conditions at its ends (one end grounded - shorted and 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

[0091] - A superconducting transmission line with two identical boundary conditions (open - open or short - short) at its ends, thus forming a so - called λ / 2 resonator. The transmission line is, for example, of coplanar waveguide type or microstrip type.

[0092] The implementation of ATS 34 is the same as the known techniques in the art, as described, for example, in the article by Lescanne (2020). Its structure includes two parallel Josephson junctions, and an inductive element is connected in parallel between the two Josephson junctions. Thus, ATS 34 includes two connected loops, each loop containing a Josephson junction in parallel with a shunt inductive element. Both loops of ATS 34 have DC and 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 parametric pumping at 2f a –f b at. This AC flux bias is typically chosen to drive the common mode of the two loops.

[0093] Figure 4 Shows the results of the parallel value curves when the inductance values of the inductive element 320 (L series ) and ATS 34 (L ) change. In this figure, the values of the first mode are represented by dashed lines in radians; the values of the second mode are represented by dotted - dashed lines in radians; the corresponding level lines are represented by solid lines in MHz. These curves are constructed by fixing the first resonance frequency at 4.5 GHz, fixing the second resonance frequency at 8.0 GHz, changing the value of the inductive element 320 (as the east - west coordinate), changing the value of the inductive element 34 (as the north - south coordinate), and simultaneously selecting the values of the capacitive elements 302 and 322 to obtain the above frequencies.

[0094] 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 the microwave source 102, and the amplitude of the microwave source 102 is somewhat arbitrary. In contrast, this ratio depends only on the intrinsic parameters of the 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 the ATS, and are independent of L j . Due to the Josephson energy E jFor Thus, it seems that by continuously decreasing L j one can increase arbitrarily However, as L ats / L j the ratio increases, the dynamic characteristics of the ATS become more unstable (as confirmed in the article by Burgelman et al., "Structurally stable subharmonic regions in driven quantum Josephson circuits", https: / / arxiv.org / abs / 2206.14631). In Figure 4 it, this ratio is set to L ats / L j = 2, corresponding to the value in Lescanne (2020).

[0095] This figure shows that for the conventional values of the inductive elements 34 and 320, values exceeding 250 MHz can be easily achieved In fact, the applicant's research has demonstrated that values exceeding 100 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) is only 9.6 MHz. In Berdou (2022), it is at least an order of magnitude smaller than Lescanne (2020).

[0096] Figure 4 The zero-point fluctuations are also shown and Although only the product affects g2, the actual values of and will directly affect the spurious terms generated by the ATS, which in turn may induce noise processes that escape the stable coherent state limit.

[0097] Since the 2-to-1 photon conversion Hamiltonian depends on the third-order expansion of the ATS potential energy in a rule of thumb known to those skilled in the art is to make and less than π. Here, 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.

[0098] The term 1 / 2 in max(α, 1 / 2) or max(β, 1 / 2) is used to account for the minimum zero-point fluctuations. When α = 2, keeping is considered safe. On the other hand, β tends to be very close to zero, so max(β, 1 / 2) = 1 / 2. This means larger values can be tolerated, typically Figure 4 It shows that, while maintaining the safe value of the parameter, g2 can be increased by more than an order of magnitude compared to the prior art.

[0099] Figure 4 It also shows that and can easily reach more aggressive values, thus further increasing g2. It should be noted that, due to the relatively late development of the field of cat qubits and the lack of research in this area, it is currently unclear and how much can be improved. Therefore, the current-mode cat design has the obvious potential to achieve a large and , so such research can be carried out.

[0100] Figure 11 The ratio of the square of the zero-point fluctuations on the parallel part (i.e., the parallel combination of the linear capacitor 322 and the linear inductor 320 of the linear resonance part 32 in Figure 3 ) in modes b and a is shown by a solid line Figure 11 The ratios of the square of the zero-point fluctuations on the capacitor 302 and the ATS 34 of the nonlinear resonance part 30 in modes b and a are also shown respectively (dashed line) and the ratio (dash-dotted line). If the load is coupled to the parallel part 320 / 322, the series capacitor 302, or the ATS 34 through the coupler 104 respectively, the attenuation rates of modes a and b in the load are proportional to the square of or . It can be seen that within the safe parameter range, these ratios are at most of the order of 1. That is to say, the geometric structure of the quantum circuit completely mixes the storage mode and the buffer mode, resulting in a negligible degree of protection for the memory during the attenuation process of the load 106. This is exactly the price to be paid for this current-mode cat circuit design with the least number of components.

[0101] However, due to the large dimensionless coupling constant k, the first resonance frequency and the second resonance frequency can be spaced relatively far apart from each other ( Figure 4 is 3.5 GHz in), thus providing a large space for the microwave filter 106 to strongly suppress the field strength at the first resonance mode frequency.

[0102] In addition, the first resonance frequency and the second resonance frequency can also be selected such that the pump frequency 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 selecting a buffer frequency f b not very different from twice the memory frequency f a to obtain a smaller pump frequency f p = 2f a – f b . For example, Figure 4 the value of f p is 1 GHz, more than four times smaller than f a and eight times smaller than f b .

[0103] This is advantageous because the first-order term of the ATS potential energy expansion indicates that the parametric pump can directly drive the circuit. As shown in the supplementary material of Lescanne (2020), this kind of driving will lead to spurious dynamic AC Stark shifts and dynamic cross-Kerr terms, which are unfavorable for cat qubit operations. 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.

[0104] Another advantage is that the value 2f a - f b can be set far from f a and f b , so that a second microwave filter can be introduced into the parametric pump line to prevent the first mode from leaking into the parametric pump line.

[0105] This is even more surprising because in such a strongly hybridized mode, the possibility of achieving a low-loss storage mode is very counterintuitive. The basic principle is that the field of quantum computing is still young, 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 is considered more desirable to couple the nonlinear elements of a quantum circuit to a single mode and weakly couple other modes to that mode.

[0106] The electrical coupling of the nonlinear elements is obviously not a gradual change and goes against all biases.

[0107] Finally, the large frequency interval between f a and f b makes the design more robust to the uncertainties of nanofabrication, such as the well-known problem of inductance variations of various Josephson junctions in the circuit, which mainly affects the accuracy of the prediction of f a and f b .

[0108] Figure 5 shows Figure 3 an implementation of the circuit shown.

[0109] Similar elements have been given similar reference numerals. Only the first digit of the reference numeral changes from "3" to "5", for example, Figure 3 the capacitive element 322 in Figure 5 is labeled 522 in

[0110] This figure is a top view of the superconducting chip layout designed by the applicant to correspond to the circuit in Figure 3 The light gray part corresponds to the metallized surface of tantalum or aluminum. The gray part corresponds to 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 uses niobium, NbTi or TiN, and the chip uses silicon or quartz.

[0111] The circuit includes a ground plane 50 on which a circuit 55 is formed. The nonlinear resonance part includes ATS34 and a cross-shaped capacitive element 502. The linear resonance part 30 is formed by the bottom of this figure. The large rectangle forms a capacitive element 522, and an array of Josephson junctions 520 that form an inductive element is connected to this capacitive element.

[0112] Figure 5 is not shown either. According to Figure 11 , the coupler 104 can be advantageously implemented by capacitively coupling the CPW transmission line to the electrode 522. In addition, the cat qubit can be capacitively coupled to the CPW bus of different branches of the electrode 502 to be coupled to another cat qubit or the readout transmission line. Capacitively coupling the CPW to the electrodes (such as 502 and 522) is generally implemented using superconducting circuits. Finally, it can be flux-biased using two CPWs respectively close to both sides of ATS 34. The DC bias is used to set its operating point, and the AC bias is used to implement the parametric pump.

[0113] Figure 5 The circuit shown presents Figure 1 a lumped ground implementation of the circuit resonance mode. This design is called lumped because the size of the entire circuit 100 is shorter than a quarter wavelength of the first mode and the second mode. It is called ground because the first mode and the second mode correspond to the charge and current oscillations between the electrodes and the ground plane 50, and these electrodes form current coupling with the ground plane through ATS 34.

[0114] The inductor can also be implemented in other ways, such as a geometric inductor composed of a meandering wire or a spiral wire.

[0115] This grounded design is more sensitive to defects in the ground plane and may generate more crosstalk than the differential design, but it is more compact and minimizes the parasitic capacitance that may shunt the ATS and thus change Figure 3 the symbol in represents the parasitic capacitance of the accuracy. The simplicity and symmetry of this design reflect the effectiveness of the current-mode cat circuit with very few components.

[0116] In another embodiment, the design can be differential, which means that the first mode and the second mode correspond to charge and current oscillations between pairs of electrodes that are electrically isolated from the ground plane 50. The differential design takes up more space than the grounded design, but its advantage lies in being able to better isolate lossy elements at the grounding point, such as wire bonds (not shown in the figure) or other elements that may be patterned on the chip, such as other cat qubits, thus reducing crosstalk.

[0117] Figure 6 represents the second embodiment similar to Figure 3 The main difference is that the components of the nonlinear resonance part 62 are now in parallel, while in the Figure 3 embodiment, the components of the nonlinear resonance part are in series. Similarly, the components of the linear resonance part 60 are now in series, while in the Figure 3 embodiment, the components of the linear resonance part are in parallel.

[0118] Similar components are labeled with similar reference numerals. Only the first digit of the reference numeral will change from "3" to "6". For example, Figure 3 the capacitive element 302 in Figure 6 is labeled 602 in

[0119] Figure 7 is similar to Figure 4 but is based on the Figure 6 circuit. For simplicity, it will not be elaborated further. It should be noted that Figure 11 and its description are still valid in terms of quantity for the Figure 6 circuit.

[0120] Figure 8 shows Figure 6 an implementation of the circuit. Similar to the implementation of the Figure 5 circuit shown in Figure 3 only the linear inductor and the ATS 34 are interchanged, so that the linear and nonlinear resonance parts are also interchanged. The advantage of this design is that the ATS 34 is now electrically coupled to the ground plane, so that it can be more easily coupled with the magnetic flux lines (not shown).

[0121] Figure 9 shows a third embodiment of the current-mode cat circuit.

[0122] InFigure 9 In the embodiment of, there is only one resonant part, and this part generates the first mode and the second mode together through ATS 34. This embodiment is related to Figure 3 and Figure 6 The difference from the embodiment of is that the characteristics of the first resonant mode and the second resonant mode generally cannot be accurately described by a simplified symbolic representation involving only two LC resonators. This explains the reason why the present invention breaks the existing prejudice: although a lower coupling constant k can well understand the first mode and the second mode through a simple perturbation analysis of two eigenmodes including only the resonant part and ATS, this is not the case in this embodiment, and more eigenmodes of the resonant part must be considered.

[0123] As shown in the figure, ATS 34 is connected to the open transmission line 90. Since ATS directly terminates the transmission line carrying the resonant mode, this circuit is a current circuit. It can be implemented using the same materials as the Figure 5 shown components. If the characteristic impedance of the transmission line required to reach a given or is too large to be realized 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 realized in various geometries, such as coplanar waveguide (CPW), microstrip line or slot line. In the Figure 9 example, CPW 90 can be capacitively or inductively connected to the microwave filter 110 and the load 106 to couple to the environment.

[0124] Figure 10 Similar to Figure 4 and Figure 7 with slight differences. Due to the implementation of the transmission line, multiple harmonics need to be considered. Therefore, Figure 10 contains three curves, respectively describing the behaviors of the 0th, 1st and 2nd harmonics of the transmission line.

[0125] Figure 10 The ATS parameters in are the same as those in Figure 4 and Figure 7 . Assuming that the characteristic impedance of CPW is 50Ω, and, without loss of generality, the typical effective relative dielectric constant ∈ r = 5.6.

[0126] Figure 10 The top graph shows the results of for different lengths of CPW. The solid line corresponds to the 2-to-1 photon rate calculated by taking the first resonant mode (or the second resonant mode) as the first (or fundamental) harmonic (or the second harmonic) of the nonlinear quantum circuit Where the first resonance mode index a = 0 and the second resonance mode index b = 1. The dashed line corresponds to the 2-to-1 photon rate calculated for the second (or fundamental) harmonic (or third harmonic) of the first resonance mode (or second resonance mode) as a non-linear quantum circuit Where the first resonance mode index a = 1 and the second resonance mode index b = 2.

[0127] At Figure 10 In the middle figure, the frequencies f0, f1, and f2 of the first harmonic (or fundamental), second harmonic, and third harmonic are shown in GHz respectively.

[0128] Figure 10 The bottom figure shows the zero-point fluctuation values of the phases of the first harmonic (or fundamental), second harmonic, and third harmonic of the ATS 34 in radians and

[0129] With Figure 4 and Figure 7 In contrast, since the changes in the first resonance frequency and the second resonance frequency are not fixed, it is necessary to show these changes. Thus, it can be seen that the detuning of f b -f a can reach several GHz, and The value of and is also far beyond the safety values in the prior art.

[0130] 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 method of the transmission line can also be changed, although some termination methods (such as inductive short circuits) must be included in the potential energy of the ATS and will therefore modify its operating point and dynamic characteristics.

Claims

1. A non-linear superconducting quantum circuit includes at least one resonant part (30, 32; 60, 62) and an electrically connected asymmetric loop superconducting quantum interference device (34). The non-linear superconducting quantum circuit 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; 60, 62) has a symbolic representation, which includes a linear resonant part (32; 60) containing at least one inductor (320; 600) and at least one capacitor (322; 602), and a non-linear resonant part (30; 62) containing at least one capacitor (302; 622) and the asymmetric loop superconducting quantum interference device (34). The linear resonant part (30) and the non-linear resonant part (32) are electrically connected and arranged such that the elements of one of the linear resonant part and the non-linear resonant part are connected in series, and the elements of the other are connected in parallel. The at least one resonant part (30, 32) is configured with inductance and capacitance values, and the inductance and capacitance values together with the asymmetric loop superconducting quantum interference device (34) induce the first mode (a) and the second mode (b), such that the zero-point fluctuations of the superconducting phase on the asymmetric loop superconducting quantum interference device (34) of the non-linear superconducting quantum circuit (100) in the first mode (a) and the mode (b) are greater than or equal to 0.05 radians.

2. The non-linear superconducting quantum circuit according to claim 1, wherein, The linear part (32) includes elements arranged in parallel, and the non-linear part (30) includes elements arranged in series.

3. The non-linear superconducting quantum circuit according to claim 1, wherein, The linear resonant part (60) includes elements arranged in series, and the non-linear resonant part (62) includes elements arranged in parallel.

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; 80) by an exposed portion of the dielectric substrate, and the linear resonant part and the non-linear 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 width and height of the non-linear superconducting quantum circuit 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 non-linear resonant part and the linear resonant part are electrically connected to the common ground plane (50; 80).

7. The non-linear superconducting quantum circuit according to claim 4 or 5, wherein, The non-linear resonant part and the linear resonant part are electrically isolated from the common ground plane.

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 a common ground plane by an exposed portion of the dielectric substrate, and the at least one resonant part is implemented as a transmission line (90).

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 claim 1, wherein, The first resonance frequency and the second resonance frequency are such that the difference between twice the first resonance frequency and the second resonance frequency is less than half of the first resonance frequency and half of the second resonance frequency.

11. The non-linear superconducting quantum circuit according to one of the preceding claims, wherein, The at least one inductor (320; 600) and / or the transmission line (90) is 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 section (30, 32; 60, 62; 90) for providing radiation having a frequency equal to the second resonance frequency; a second microwave source (102) connected to the at least one resonant section (30, 32; 60, 62; 90) for providing radiation having a frequency equal to the difference between twice the first resonance frequency and the second resonance frequency; and a load (106) coupled to the at least one resonant section (30, 32; 60, 62; 90) 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 resonance frequency to pass through and block the first resonance frequency.

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

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