Minimal superconducting quantum circuit with electrically coupled Bose-encoded circuitry

Abstract

A nonlinear superconducting quantum circuit includes at least one resonant section and an electrically connected asymmetric threaded superconducting quantum interference device. The circuit includes a first mode having a first resonant frequency and a second mode having a second resonant frequency, the ratio between the first and second resonant frequencies being different from 1 / 2. The at least one resonant section is symbolically represented and includes a linear resonant section comprising at least one inductor and at least one capacitor, and a nonlinear resonant section comprising at least one capacitor and the interference device. The linear and nonlinear resonant sections are electrically connected and arranged such that elements of one section are connected in series and elements of the other section are connected in parallel. The at least one resonant section is configured with inductance and capacitance values, which, together with the interference device, induce the first and second modes such that the zero-point fluctuation of the superconducting phase on the interference device in the first and second modes is greater than or equal to 0.05 radians.

Description

Technical Field

[0001] This invention relates to the field of superconducting quantum circuits, and more specifically, to superconducting quantum circuits containing cat qubits. Background Technology

[0002] Cat qubits are a subset of Boson-coded qubits and constitute a family of error-correcting codes for quantum applications. Generally, Boson coding relies on storing qubits in Boson mode. For cat qubits, the most common design to date is two-component cat coding.

[0003] Dissipative stabilization of two coherent states requires a nonlinear transition between two photons in the first mode (also known as the cat qubit mode) carrying the stable quantum manifold and one photon in the second mode (called the buffer mode), and vice versa. This stabilization scheme suppresses bit flips, with the degree of suppression being 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 realizations of this stabilization scheme (Leghtas et al., "Confining optical states in quantum manifolds by designed two-photon loss" published in Science, Vol. 347, p. 853, 2015, and Touzard et al., "Coherent oscillations within dissipatively stable quantum manifolds" published in Physical Review X, Vol. 8, p. 021005, 2018) failed to observe exponential suppression of bit flips because the superconducting circuit elements used to design the 2-to-1 photon conversion (i.e., the so-called "transferors") have pseudo-cross-Kerr terms that introduce additional noise processes, the escape rate of which is given by the very large transporter cat quantum ratio dispersion shift. Lescanne R. et al., in their 2020 Nature Physics article "Exponential suppression of bit flips in qubits encoded in oscillators" (hereinafter referred to as Lescanne (2020)), disclosed a significantly improved cat qubit achieved by designing a 2-to-1 photon conversion using an asymmetric threaded superconducting quantum interference device (also known as an "ATS"). The cross-Kerr term in the ATS design is much lower than that of the transporter, which allows us to observe exponential suppression of bit flips. However, the transporter 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, reaching several milliseconds. In their later work paper, "100-second bit flip time in a two-photon dissipative oscillator" (arXiv:2204.09128, https: / / arxiv.org / pdf / 2204.09128.pdf) (hereinafter referred to as Berdou (2022)), Berdou et al. successfully improved the bit flip saturation time by five orders of magnitude, reaching 100 seconds, by removing the measurement transporter and running the ATS under the assumption of dynamic stability. The significant improvement in bit flip time is possible because the ATS alone adds a spurious noise process with an extremely low escape rate. However, the confinement rate achieved in Berdou (2022) is very low.

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

[0006] Currently, no other known circuit can provide a cat qubit with good performance. The average stable lifetime of other known cat qubits is no more than a few milliseconds, which is far from sufficient for building practically applicable quantum circuits. Since the confinement rate is directly related to the 2-to-1 photon conversion rate of the design, a significant improvement in the latter's conversion rate is needed in ATS-based circuits. Summary of the Invention

[0007] This application aims to improve this situation. To this end, the applicant proposes a nonlinear superconducting quantum circuit comprising at least one resonant section and an electrically connected asymmetric threaded superconducting quantum interference device (SQFID). The nonlinear superconducting quantum 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 section is symbolically represented, comprising a linear resonant section (containing at least one inductor and at least one capacitor) and a nonlinear resonant section (containing at least one capacitor and the asymmetric threaded superconducting quantum interference device). The linear and nonlinear resonant sections are electrically connected and arranged such that elements of one are connected in series and elements of the other are connected in parallel. The at least one resonant section is configured with inductance and capacitance values ​​that, together with the asymmetric threaded superconducting quantum interference device, induce the first and second modes, such that in the first and second modes, the zero-point fluctuation of the superconducting phase on the asymmetric threaded superconducting quantum interference device is greater than or equal to 0.05 radians.

[0008] The advantage of this superconducting quantum circuit lies in its absence of coupling elements (coupling elements reduce the participation of buffered and / or stored modes in the ATS, thus reducing the 2-to-1 photon conversion rate). Theoretically, the adverse effects of coupling capacitors in the Lescanne (2020) circuit can be minimized, but large capacitors are known to introduce losses in superconducting circuits, increasing the phase-flipping rate and rendering them ineffective in practical applications. Furthermore, this design incorporates as few components as possible, thereby minimizing risks related to industrialization costs and feasibility.

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

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

[0011] - The linear resonant section includes elements arranged in series, and the nonlinear resonant section 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 parts of the nonlinear superconducting quantum circuit;

[0013] - The nonlinear superconducting quantum circuit is formed on a generally 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 section and the linear resonant section 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 harmonic of the nonlinear superconducting circuit;

[0018] - The first resonant frequency and the second resonant frequency make the difference between twice the first resonant frequency and the second resonant frequency 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-dynamic inductance material.

[0020] The present invention also relates to a quantum device comprising: a nonlinear superconducting quantum circuit according to any one of the preceding claims; a first microwave source connected to the at least one resonant portion for providing radiation at a frequency equal to the second resonant frequency; a second microwave source connected to the at least one resonant portion for providing radiation at a frequency equal to twice the difference between the first resonant frequency and the second resonant frequency; and a load coupled to the at least one resonant portion such that substantially only a second mode is coupled to the load, thereby the first mode carrying 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 comprising at least one device according to the present invention. Attached Figure Description

[0022] Other features and advantages of the invention will become apparent from the following description of the accompanying drawings, which illustrate exemplary embodiments of the invention, wherein:

[0023] - Figure 1 This demonstrates how to incorporate a current-mode cat qubit circuit into a device to stabilize quantum information;

[0024] - Figure 2 This demonstrates how to stabilize quantum information using a current-mode cat qubit circuit incorporated into a device;

[0025] - Figure 3 This diagram shows the electrical equivalent of a first embodiment of the current-mode cat qubit circuit according to the present invention.

[0026] - Figure 4 Showing respectively for Figure 3 The first and second modes of superconducting quantum circuits Values ​​and corresponding Value graph;

[0027] - Figure 5 express Figure 3 One way to implement a circuit;

[0028] - Figure 6 This diagram shows the electrical equivalent of a second embodiment of the current-mode cat qubit circuit according to the present invention.

[0029] - Figure 7 Showing respectively for Figure 6 The first and second modes of superconducting quantum circuits Values ​​and corresponding Value graph;

[0030] - Figure 8 express Figure 6 One implementation of the circuit shown;

[0031] - Figure 9 This represents a third embodiment of a current-mode cat qubit circuit;

[0032] - Figure 10 express Figure 9 The circuit shown exhibits a behavior curve as the transmission line length varies.

[0033] - Figure 11 This represents the ratio of the squares 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 This is a supplement. Detailed Implementation

[0034] The accompanying drawings and much of the following description consist of positive and well-defined features. Therefore, they not only aid in understanding the invention but can also be used to help define the invention when necessary.

[0035] In order for cat qubits 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 existing techniques belong to the family of cat qubits stabilized through parametric pump dissipation. Parametric pump dissipation is used to bridge the frequency gap between the two modes and performs 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 pump dissipation relaxes the restriction on the resonant frequency.

[0036] The first and second modes of a superconducting quantum circuit can each correspond to the circuit's inherent resonant frequencies. For example, both the first and second modes can be electromagnetic modes. The first and second modes can each have their own resonant frequencies; for example, the first mode can have... The type of resonant frequency, while the second mode can have The resonant frequency of type, where ω a and ω b This can be the angular frequency of each corresponding mode. Using the phrase "having" a first mode and a second mode means that the superconducting quantum circuit can contain components operating in the superconducting state, and these components can independently or simultaneously carry these modes. In other words, the first and second modes can be carried in different subsets of components in the superconducting circuit, or they can be carried in the same subset of components.

[0037] Superconducting quantum circuits can operate at temperatures close to absolute zero (e.g., 100 mK or lower, typically 10 mK) and be as isolated from the environment as possible to avoid energy loss and decoherence, except for some specially designed couplings. For example, only the second mode can couple to the dissipative environment, while the first mode can remain isolated from the environment.

[0038] Superconducting quantum circuits can be fabricated as one or more patterned layers of superconducting material (e.g., aluminum, tantalum, niobium, etc., as known in the art) deposited on a dielectric substrate (e.g., silicon, sapphire, etc.). Each of these one or more patterned layers can define a lumped element resonator. Capacitive elements can be formed from two adjacent superconducting material plates (on corresponding layers of the one or more patterned layers). Inductive elements can be formed from superconducting wires. Alternatively, at least one of the one or more patterned layers can define multiple transmission lines, each with a resonant frequency depending on its length. The transmission lines can be, for example, coplanar waveguides, slot lines, or microstrip lines. Alternatively, the circuit can be embedded in a three-dimensional architecture containing high-quality three-dimensional modes formed into bulk superconductors through machining or micromachining, which can be used as either mode.

[0039] The circuit can be integrated into a device that may 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, for example, an element with a given resistance (unlike a superconducting element), located outside the superconducting circuit. Load dissipation converts photon pairs from the first mode to the second mode via a 2-to-1 photon conversion. In other words, 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 microwave radiation, thereby applying microwave radiation at a frequency approximately equal to the frequency of the second mode. Thus, the first microwave source drives photons existing in the form of microwave radiation to the second mode, and the second mode in turn drives photon pairs in the first mode via a 2-to-1 photon conversion. This 2-to-1 photon conversion is bidirectional: conversely, it can be that two photons in the first mode are converted to one photon in the second mode, or one photon in the second mode is converted to two photons in the first mode. A coupler is a component that can be connected to a circuit element carrying a second mode via 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 meaning that the transmission line or waveguide terminates at a resistor at an end different from the end connected to the element carrying the second mode, and that the resistance value should be set such that most of the power flowing to 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 via 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 low-temperature environment. This allows microwave radiation to be applied without increasing thermal noise.

[0042] The second microwave source provides microwave radiation at a frequency approximately twice the resonant frequency of the first mode minus the resonant frequency of the second mode, thus achieving a 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 eventually connect to the two superconducting loops. Alternatively, two different microwave sources with the same frequency as the second microwave source can be used to directly feed the two transmission lines or waveguides with sufficient relative amplitude and phase.

[0043] Optionally, the device may include a microwave filter connected to both a first mode and a second mode 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 staggered between the load and the coupler. From a circuit perspective, 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 a first resonant frequency, or by implementing a band-pass filter at a second resonant frequency, or (if the second (or first) resonant frequency is greater than the first (or second) resonant frequency), by implementing a high-pass (or low-pass) filter with a cutoff frequency between the first and second resonant frequencies, since the photons of the second mode are the only photons that need to be dissipated in the environment. For some circuits, for example, when the two modes have different symmetries, the filter may not be necessary, and the appropriate placement of the coupler in the circuit may be sufficient to prevent the dissipation of the first mode.

[0044] Therefore, this device can stabilize two coherent states in the first mode (i.e., the quantum manifold of coherent states). For example, applying microwave radiation from a first microwave source to the second mode through a microwave filter can be viewed as two-photon driving of the first mode after a 2-to-1 photon conversion; while the load that dissipates only the photons of the second mode can be viewed as two-photon dissipation of the first mode after a 2-to-1 photon conversion. Two-photon driving and two-photon dissipation enable the stabilization of the two coherent states in the first mode.

[0045] The second mode of single-photon actuation can be formally represented by a Hamiltonian. Description, where ∈ b This is the single-photon drive rate generated by the first microwave source in the second mode. The single-photon dissipation in the second mode can be formally represented by the Lindblad operator. Description, where κ b It is the single-photon dissipation generated by the coupling of the second mode with the load.

[0046] Two-photon drive can be formally represented by the Hamiltonian. The description is given, where ∈2 is the effective two-photon drive rate. Two-photon loss can be formally represented using the Lindblad operator. The description, where κ2 is the two-photon dissipation rate. The amplitude α of the stable coherent state is ultimately determined by... Given. The confinement rate of the coherent state is κ. conf =κ2α 2 .

[0047] In κ b >> Within the range of g2α, where g2 is the 2-to-1 nonlinear transformation rate between the first and second modes, the buffer dynamics can be adiabatically eliminated, thus producing ∈2=2∈ b g2 / κ b and

[0048] In various examples, superconducting circuits can have a symbolic representation, such as consisting of a set of interconnected dipoles. The term "symbolic representation" should be understood as specifying the symbols and arrangement of lines for a set of interconnected dipoles. This set of interconnected dipoles (also called components) forms a circuit structure (or topology) that is functionally equivalent to a nonlinear superconducting circuit.

[0049] In other words, as is the classic practice in the field of superconducting circuits, the configuration of a nonlinear superconducting circuit is designed to achieve the function defined by its symbolic representation, namely, the function of the theoretical set of interconnected dipoles shown in the symbolic representation. To put it another way, while the circuit can be constructed using patterned layers of superconducting material, it should be understood that the circuit can be symbolically represented using dipoles (e.g., capacitors, inductors, and / or Josephson junctions). Although the example dipoles describe discrete elements, it will be readily understood by those skilled in the art that these elements correspond to equivalent circuits of distributed elements within a specific frequency range (e.g., low frequencies), as is well known in the art.

[0050] It is known in the art that such distributed elements can have high-frequency modes that are independent of and unimportant to the dynamic characteristics described herein. Therefore, these distributed elements can be represented symbolically. This symbolic representation can be refined by adding elements (e.g., adding a series inductor at each wire connection, or adding a parallel capacitor between any two nodes in the circuit) or by adding nodes and branches to account for other modes of the distributed element. Thus, the symbolic representation can better describe distributed elements without changing the circuit's operating principle. Therefore, as is known in the art, those skilled in the art consider the physical circuit (i.e., the circuit actually manufactured) and its symbolic representation to be equivalent. 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 the 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 junction—the only unknowns in calculating the 2-to-1 photon conversion rate for 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 injection frequency is ω. p =2ω a -ω b The external time-varying parameter. Parametric pumps are used in existing technologies to induce resonance in nonlinear interactions.

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

[0053] In Lescanne's (2020) paper, the nonlinear Hamiltonian H2 was designed using the ATS superconducting dipole. The ATS dipole possesses the following potential energy:

[0054] Where E L It is the Josephson energy of the ATS parallel inductor, E J It 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 selecting the following DC values ​​of magnetic flux (referred to as saddle points): And only for amplitudes of and frequency ωp When the σ mode is used for flux pumping, the potential energy expression becomes:

[0056]

[0057] Phase on ATS The relationship with patterns a and b is as follows:

[0058]

[0059] in and These are the zero-point fluctuations of the phase at both ends of the ATS in the first and second resonant modes, respectively. exist and Expanding to the third order and eliminating the fast rotation term, we obtain the 2-to-1 conversion Hamiltonian H2, whose 2-to-1 photon conversion rate is...

[0060] Lescanne's (2020) paper shows that bit flipping occurs with the photon α of the cat qubit encoded in the resonator. 2 The number of phases is exponentially suppressed. However, this architecture uses a transporter coupled to the cat qubit as the measurement device, causing the bit-flipping time to saturate to a few milliseconds. The applicant's research shows that this is because the confinement rate is too small to resist the dispersion shift caused by thermal excitation of the measurement device. The applicant's later research, published in the paper Berdou (2022), improved the bit-flipping saturation time by five orders of magnitude by removing the transporter and operating the ATS under the assumption of dynamic stability, despite a lower confinement rate. More precisely, in the paper Lescanne (2020), the ratio of confinement rate to phase-flipping rate is 10, while in Berdou (2022), the ratio is 0.01. As stated in the introduction 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 did not provide a potential solution to significantly improve the two-photon dissipation rate. In fact, the two-photon dissipation rate is related to the two-photon coupling rate. In the papers by Lescanne (2020) and Berdou (2022), the buffer mode is located on the ATS, while the storage mode is weakly capacitively coupled to the buffer mode and thus coupled to the ATS. This design allows for a larger [capacitive coupling]. but It is usually one to two orders of magnitude smaller. This is because the formula for g2 depends on... The square of this is so that this circuit geometry is very unfavorable for the intensity of two-photon dissipation.

[0062] The only way to solve this problem is to significantly increase the capacitance coupling the storage and buffer modes in the Lescanne (2020) and Berdou (2022) circuits, enabling the ATS to have strong participation in both buffer and storage modes. However, large capacitances are known to introduce losses in superconducting circuits.

[0063] Therefore, these existing technologies have reached a dead end: their specific geometries are crucial for achieving bit-flipping stability, but cannot be adjusted to allow for sufficiently large two-photon dissipation rates.

[0064] Examples and descriptions of circuits and devices according to the invention will now be discussed with reference to the accompanying drawings. In the following text, the terms "current-mode cat qubit circuit," "circuit," "superconducting quantum circuit," and "nonlinear superconducting circuit" are used interchangeably and refer to circuits that perform 2-to-1 photon conversions to stabilize cat qubits.

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

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

[0067] The nonlinear superconducting circuit 100 performs a 2-to-1 photon conversion between a first mode a (labeled 112) and a second mode b (labeled 114). In the following text, 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.

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

[0069] As described below, the nonlinear superconducting circuit 100 according to the invention is very special because it contains an ATS (“Asymmetric Threaded SQUID” or “Asymmetric Threaded Superconducting Quantum Interference Device”) that is electrically coupled to other elements in the nonlinear superconducting circuit 100 carrying modes a and b.

[0070] The circuit element carrying the second mode 114 is coupled to the load 106 via coupler 104. This coupling makes the second mode dissipative. A microwave source 108 is connected to a nonlinear superconducting circuit 100 for transmitting at a frequency of f. b The radiation is used to drive the second mode to operate at its resonant frequency. The microwave filter 110 is configured here with a frequency of f. b A bandpass filter. Alternatively, filter 110 can be configured with a frequency of f. a A band-stop filter is 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 f a >f b (or f) b >f a In the case of [the specific configuration], filter 110 can be configured as a low-pass (or high-pass) filter. In other embodiments, microwave filter 110 can be omitted when coupling is established between load 106 and substantially only the second mode.

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

[0072] The figure illustrates the Wigner function for the first mode with two-photon drive at a rate of ε² and two-photon dissipation at a rate of κ², the first mode having two amplitudes. The stable steady states (201, 202) are opposite in phase. Since there are two possible states, information can be encoded: state |0>202 is looped out with a solid circle, and state |1>201 is looped out with a dashed circle. Because the dynamic characteristics converging to these two states are relatively stable, this encoding is highly robust to bit-flipping errors (i.e., the system flips between states |0> and |1>). This encoding cannot correct another error path, namely phase-flipping errors. 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 a 2-to-1 photon conversion between the first and second modes.

[0073] exist Figure 3 , 6 In the diagrams of Figures 9 and 9, only 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 used for parametric pumping are not shown.

[0074] Figure 3 express Figure 1 Equivalent circuit diagram of the first embodiment of the medium current cat qubit circuit 100.

[0075] Circuit 100 includes a nonlinear resonant section 30 and a linear resonant section 32. The nonlinear section includes an ATS 34 and a capacitor element 302. The nonlinear resonant section 30 and the linear resonant section 32 are connected together by current. To avoid any misunderstanding, the term "electrical coupling" refers to the presence of a short conductive section, i.e., a short conductive track or any other means that ensures a physical continuous conductive connection, connecting the nonlinear resonant section 30 and the linear resonant section. The term "short" means that, with frequency f... a and f b Compared to the impedances of the nonlinear resonant section 30 and the linear resonant section 32, the impedance of the conductive track is negligible. If the impedance of this conductive track is not negligible, it will act as a voltage divider, reducing zero-point fluctuations in the first and second resonant modes of the ATS. and This contradicts the purpose of the present invention. In the example described herein, the arrangement of the conductive tracks also ensures that they do not shunt any nonlinear or linear resonant components.

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

[0077] This current circuit is a minimal circuit in terms of its notation, as it uses the fewest number of elements (two capacitors 302 and 322 and two inductors 34 and 320) to carry both resonant modes. This is in contrast to the prior art, where the nonlinear portion is coupled to the linear portion via a capacitive or inductive coupler, resulting in a much lower participation of one resonant mode in the ATS, thus lower zero-point fluctuations and a lower 2-to-1 photon conversion efficiency g2. Since the notation of the proposed minimal current circuit does not have such a coupler, a higher participation of both resonant modes in the ATS is expected. Although it appears that ATS 34 is capacitively coupled to the linear eigenmode via capacitor 302, in reality, due to the resonance between ATS 34 and capacitor 302, they form a nonlinear eigenmode that is electrically coupled to the linear eigenmode.

[0078] This minimized current circuit also has advantages because it allows for a direct, simple, and compact structure, as we will see later.

[0079] Furthermore, it should be noted that having one series resonant section and one parallel resonant section is the only feasible circuit topology. Therefore, there are only two possible minimal implementations of a circuit containing an ATS: one is that the ATS acts as the inductor element in the series resonant section, such as... Figure 3 As shown; another type is where the ATS acts as the inductor in the parallel resonant section, such as... Figure 6 As shown.

[0080] Let L parallel and C parallel The values ​​of L for the inductor and capacitor in the parallel resonant section are respectively. series and C series Given the values ​​of the inductor and capacitor in the series resonant section, respectively, these eigenmodes can be described as follows:

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

[0082] -through their impedance and Or equivalent

[0083] -Through their intrinsic zero-point fluctuations and Where R q =h / 4e 2 It is a superconducting resistance quantum.

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

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

[0086] Wherein, the coupling rate is The dimensionless coupling constant is

[0087] Patterns a and b above are the Hamiltonian H described above. linDiagonalized mode. The dimensionless coupling constant k can take any value between 0 and 1. Values ​​close to 0 indicate weak coupling modes, while a value of 1 indicates the maximum coupling mode. As expected, minimum current designs can easily achieve k values ​​of about 1 / 2 or higher. 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 their article "Circuits-Quantum Electrodynamics: How Strong is the Coupling Between Josephson Junction Atoms and Transmission Line Resonators?" published in Annals of Physics, Vol. 519, pp. 767-779, 2007.

[0088] like Figure 5 and Figure 8 As shown, inductive and capacitive elements constitute an LC resonator, which can be implemented using distributed elements in a patterned superconducting material layer, for example:

[0089] - Two adjacent plates connected in parallel with a superconducting wire form a capacitor; a single Josephson junction or an array of Josephson junctions forms an inductor.

[0090] A superconducting transmission line with two different boundary conditions at its ends (one end short-circuited to ground, the other open-circuited) forms a so-called λ / 4 resonator. The transmission line can be, for example, a coplanar waveguide or a microstrip line, or...

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

[0092] The implementation of the ATS 34 is similar to that known in the art, for example, as described in Lescanne's (2020) article. Its structure comprises two Josephson junctions in parallel, with an inductor connected in parallel between these two Josephson junctions. Therefore, the ATS 34 includes two connected loops, each containing a Josephson junction in parallel with a shunt inductor. Both loops of the 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 2f... a -f b Parametric pump at the location. This AC flux bias is typically chosen to drive the common mode of the two loops.

[0093] Figure 4 Inductor element 320 (L) is shown parallel ) and ATS 34 (L series When the inductance value changes Value curve results. In this figure, the first mode's Values ​​are expressed in radians using dashed lines; second mode The value is expressed in radians using a dashed line; correspondingly The level lines are represented by solid lines in MHz. These curves are constructed as follows: the first resonant frequency is fixed at 4.5 GHz, the second resonant frequency is fixed at 8.0 GHz, and the value of inductor 320 (as the east-facing coordinate) and the value of inductor 34 (as the north-facing coordinate) are changed, while the values ​​of capacitors 302 and 322 are selected to obtain the above frequencies.

[0094] Plot the ratio The reason is The amplitude of the parametric flux pump is proportional to the amplitude of the microwave source 102, which is set by the amplitude of the microwave source 102, and the amplitude of the microwave source 102 is, to some extent, arbitrary. Conversely, this ratio... It depends only on the inherent parameters of circuit 100. Figure 4 This calculation assumes the ATS 34 operates at its saddle point. In this case, w a and w b The value depends only on the inductance L of the ATS. ats And with L j Irrelevant. Due to Josephson energy E j for Therefore, it seems that by continuously decreasing L... j To increase arbitrarily However, with L ats / L j As the ratio increases, the dynamic characteristics of the ATS become more unstable (as demonstrated in Burgelman et al.'s paper "Structurally Stable Subharmonic Regions Driving Quantum Josephson Circuits," https: / / arxiv.org / abs / 2206.14631). Figure 4 In this context, the ratio is set to L. ats / L j =2, corresponding to the value in Lescanne (2020).

[0095] The figure shows that for the standard values ​​of inductors 34 and 320, frequencies exceeding 250MHz can be easily achieved. Value. In fact, the applicant's research has demonstrated that values ​​exceeding 100MHz can be guaranteed, which is nearly an order of magnitude higher than known existing technologies, and values ​​of hundreds of megahertz can be achieved. In comparison, the values ​​achieved in Lescanne (2020) The value is only 9.6 MHz. In Berdou (2022), it is at least an order of magnitude smaller than Lescanne (2020).

[0096] Figure 4 It also shows zero-point fluctuations. and Although there is only a product It has an impact on G2, but and The actual value of will directly affect the stray terms generated by ATS, which in turn may induce noisy processes that escape the constraints of stable coherent states.

[0097] Since the 2-to-1 photon conversion Hamiltonian depends on The third-order expansion of the potential energy of the ATS, as known to those skilled in the art, is that... and 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.

[0098] The term 1 / 2 in max(α,1 / 2) or max(β,1 / 2) is used to explain the minimum zero-point fluctuations. When α = 2, it remains... It is considered safe. On the other hand, β tends to be very close to zero, so max(β, 1 / 2) = 1 / 2. This means It can tolerate larger values, usually Figure 4 This indicates that, while maintaining safe parameter values, g2 can achieve an improvement of more than an order of magnitude over existing technologies.

[0099] Figure 4 It also indicates that, and More aggressive values ​​can be easily achieved, further improving g2. It should be noted that due to the relatively recent development of the cat qubit field and the lack of research in this area, the exact extent of this improvement is currently unclear. and How much can be improved? Therefore, current-mode cat qubit design has the potential for significant improvement. and Its obvious potential makes this type of research possible.

[0100] Figure 11 The solid line shows the parallel section (i.e., ...) under modes b and a. Figure 3 The ratio of the square of the zero-point fluctuation on the parallel combination of linear capacitor 322 and linear inductor 320 in the linear resonant section 32 Figure 11 The ratios of the squares of the zero-point fluctuations on capacitor 302 and ATS 34 in the nonlinear resonant section 30 are also shown in modes b and a, respectively. (Dashed line) and ratio (Dash line). If the load is coupled to the parallel sections 320 / 322, the series capacitor 302, or the ATS 34 via coupler 104, the attenuation rates of modes a and b in the load are as follows: or The ratio is proportional to the square of the value. It can be seen that, within the safety parameter range, these ratios are at most on the order of 1. That is to say, the geometry of the quantum circuit completely mixes storage and buffer modes, resulting in negligible protection of the memory during the decay of a load of 10⁶. This is precisely the price to pay for this current-mode cat qubit circuit design using a minimal number of components.

[0101] However, since the dimensionless coupling constant k is relatively large, the first resonant frequency and the second resonant frequency can be spaced far apart from each other. Figure 4 The frequency is 3.5 GHz, which provides a large space for the microwave filter 106 to strongly suppress the field strength at the first resonant mode frequency.

[0102] In addition, a first resonant frequency and a second resonant frequency can be selected to make the pump frequency 2f a -f b Slightly deviating 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 the buffer frequency f. b With memory frequency f a The difference is not much greater than twice the value, thus obtaining a smaller pump frequency f. p =2f a -f b .For example, Figure 4 f in p The value is 1GHz, which is higher than f a More than four times smaller, compared to f b Eight times smaller.

[0103] This is advantageous because the first-order terms of the ATS potential expansion indicate that the parametric pump can directly drive the circuit. As shown in the supplementary material by Lescanne (2020), this driving results in spurious dynamic AC Stark shifts and dynamic cross-Kerr terms, which are detrimental to cat qubit operations. At lower pump frequencies, the direct driving efficiency of the circuit is much lower, resulting in much smaller dynamic AC Stark shifts and dynamic cross-Kerr terms.

[0104] Another advantage is that this value 2f a -f b It can be set to be far away from f a and f bThis allows a second microwave filter to be introduced into the parametric pump circuitry to prevent the first mode from leaking into the parametric pump circuitry.

[0105] This is even more surprising because the possibility of achieving a low-loss storage mode under such strong hybridization is highly counterintuitive. The underlying principle is that the field of quantum computing is still young, especially in the area of ​​cat qubits, and changes are typically very incremental. One reason for positioning the ATS on the second mode and weakly capacitively coupling the first mode to the second mode in the paper by Lescanne (2020) and Berdou (2022) is to minimize the decay of the first mode due to the decay of the second mode in the load. In fact, traditionally, coupling the nonlinear elements of a quantum circuit to a single mode and weakly coupling other modes to that mode is considered more preferable.

[0106] The electrical coupling of nonlinear elements is clearly not a gradual change, and contrary to all prejudices.

[0107] Finally, f a and f b The larger frequency intervals between these intervals make the design more robust to uncertainties in nanofabrication, such as the well-known problem of varying Josephson junction inductance in circuits, which primarily affects f. a and f b The accuracy of the prediction.

[0108] Figure 5 Showing Figure 3 One implementation of the circuit shown.

[0109] Similar elements have used similar reference numerals. Only the first digit of the reference numeral changes from "3" to "5", for example, Figure 3 Capacitor element 322 in Figure 5 The Chinese character is marked as 522.

[0110] This diagram is a top view of the superconducting chip layout designed by the applicant, corresponding to... Figure 3 The circuitry is shown in the image. The light gray areas correspond to the metallized surfaces of tantalum or aluminum. The gray areas correspond to the sapphire substrate on which the circuitry resides. Other materials can also be used to realize superconducting circuits; for example, the metallization layer may be made of niobium, NbTi, or TiN, and the chip may be made of silicon or quartz.

[0111] The circuit includes a ground plane 50, on which circuit 55 is formed. The nonlinear resonant section includes an ATS34 and a cross-shaped capacitor element 502. The linear resonant section 30 is formed at the bottom of the figure, with a large rectangle forming a capacitor element 522, on which an array of Josephson junctions 520 forming an inductor element is connected.

[0112] Figure 5 It was not shown in the middle either, according to Figure 11 Coupler 104 can be advantageously implemented by capacitively coupling the CPW transmission line to electrode 522. Furthermore, a cat qubit can be coupled to another cat qubit or readout transmission line via a CPW bus of a different branch capacitively coupled to electrode 502. Capacitively coupling the CPW to electrodes (e.g., 502 and 522) is typically implemented using superconducting circuitry. Finally, two CPWs, respectively located near both sides of ATS 34, can be used to magnetically bias it; a DC bias is used to set its operating point, and an AC bias is used to implement a parametric pump.

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

[0114] Inductors can also be implemented in other ways, such as geometric inductors made of meandering or spiral lines.

[0115] This grounding design is more sensitive to defects in the ground plane and may generate more crosstalk than a differential design, but it is more compact and minimizes the potential for shunt ATS and thus changes. Figure 3 The symbol in the middle represents the parasitic capacitance of the precision. The simplicity and symmetry of this design demonstrate the effectiveness of a current-mode cat qubit circuit with very few components.

[0116] In another embodiment, the design can be differential, meaning that the first and second modes correspond to charge and current oscillations between electrode pairs electrically isolated from the ground plane 50. Differential designs take up more space than grounded designs, but their advantage lies in better isolation of lossy components at the ground point, such as wire bonds (not shown) or other components that may be patterned on the chip, such as other cat qubits, thereby reducing crosstalk.

[0117] Figure 6 Indicates and Figure 3 A similar second embodiment. The main difference is that the elements of the nonlinear resonant section 62 are now connected in parallel, while... Figure 3 In this embodiment, the elements of the nonlinear resonant section are connected in series. Similarly, the elements of the linear resonant section 60 are now connected in series, while... Figure 3 In the embodiment, the components of the linear resonant section are connected in parallel.

[0118] Similar elements use similar reference numerals. Only the first digit of the reference numeral changes from "3" to "6", for example, Figure 3 Capacitor element 302 in Figure 6 The Chinese character is marked as 602.

[0119] Figure 7 and Figure 4 Similar, but based on Figure 6 The circuit is described in detail below for simplicity. It is important to note that... Figure 11 and its description for Figure 6 The circuit is still effective in terms of quantity.

[0120] Figure 8 Showing Figure 6 One way to implement a circuit. (And...) Figure 5 As shown Figure 3 The circuit implementation is similar, except that the linear inductor and ATS 34 are interchanged, thus swapping the linear and nonlinear resonant sections. The advantage of this design is that the ATS 34 is now electrically coupled to the ground plane, allowing for easier and greater coupling with magnetic flux lines (not shown).

[0121] Figure 9 A third embodiment of a current-mode cat qubit circuit is shown.

[0122] exist Figure 9 In this embodiment, there is only one resonant section, which together with the ATS 34 generates the first and second modes. This embodiment is similar to... Figure 3 and Figure 6 The difference in this embodiment is that the characteristics of the first and second resonant modes cannot typically be accurately described using simplified notation involving only two LC resonators. This explains why the present invention breaks with existing biases: while the lower coupling constant k can be well understood through a simple perturbation analysis of the two eigenmodes involving only the resonant section and the ATS, this is not the case in this embodiment, where more eigenmodes of the resonant section must be considered.

[0123] As shown in the figure, ATS 34 is connected to an open transmission line 90. Since the ATS is directly terminated on a transmission line carrying a resonant mode, this circuit is a current-carrying circuit. It can be used with... Figure 5 The components shown are made of the same material. If a given... or If the required transmission line characteristic impedance is too high to be geometrically feasible, a high-dynamic-inductance material or a wide Josephson junction can be used instead of the transmission line's center conductor. Finally, transmission lines can be implemented using various geometries, such as coplanar waveguides (CPWs), microstrip lines, or slotted lines. Figure 9In the example, the CPW 90 can be connected to the microwave filter 110 and the load 106 via a capacitor or an inductor to couple to the environment.

[0124] Figure 10 and Figure 4 and Figure 7 They are somewhat similar, but the difference lies in the fact that, due to the implementation of the transmission line, multiple harmonics need to be considered. Therefore, Figure 10 It contains three graphs that describe the behavior of the 0th, 1st and 2nd harmonics of the transmission line, respectively.

[0125] Figure 10 The ATS parameters in Figure 4 and Figure 7 The same applies. Assume the characteristic impedance of the CPW is 50Ω, and, without loss of generality, take the typical effective relative permittivity ∈ [0, 1]. r =5.6.

[0126] Figure 10 The top graphic illustrates the different CPW lengths. The result. The solid line corresponds to the 2-to-1 photon rate calculated using the first (or second) resonant mode as the first (or fundamental) harmonic (or second harmonic) of the nonlinear quantum circuit. Where the first resonant mode index a = 0, and the second resonant mode index b = 1. The dashed line corresponds to the 2-to-1 photon velocity calculated by using the first resonant mode (or the second resonant mode) as the second (or fundamental) harmonic (or third harmonic) of the nonlinear quantum circuit. The first resonant mode index a = 1, and the second resonant mode index b = 2.

[0127] exist Figure 10 In the middle of the graph, the frequencies f0, f1, and f2 of the first harmonic (or fundamental wave), the second harmonic, and the third harmonic are expressed in GHz.

[0128] Figure 10 The graphic at the bottom represents the zero-point fluctuations in the phase of the first harmonic (or fundamental), second harmonic, and third harmonic of the ATS 34 in radians. and

[0129] and Figure 4 and Figure 7 Conversely, since the first and second resonant frequencies are not constant, it is necessary to show these changes. Therefore, f b -f a The detuning can reach several GHz, and The value also far exceeds that of existing technologies. and The safe value.

[0130] It is obvious to those skilled in the art that by adjusting f a f b , and The value can be adapted to specific applications, such as 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 circuit) must be included in the potential energy of the ATS, thus modifying its operating point and dynamic characteristics.

Claims

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

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

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

4. The nonlinear superconducting quantum circuit according to claim 2 or 3, wherein, The nonlinear 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 portion and the nonlinear resonant portion are implemented in physically different parts of the nonlinear superconducting quantum circuit.

5. The nonlinear superconducting quantum circuit according to claim 4, wherein, The nonlinear superconducting quantum circuit is formed on a generally flat substrate, and the width and height of the nonlinear superconducting quantum circuit are both less than a quarter wavelength corresponding to the first resonant frequency and less than a quarter wavelength corresponding to the second resonant frequency.

6. The nonlinear superconducting quantum circuit according to claim 4, wherein, The nonlinear resonant portion and the linear resonant portion are electrically connected to the common ground plane (50; 80).

7. The nonlinear superconducting quantum circuit according to claim 4, wherein, The nonlinear resonant portion and the linear resonant portion are electrically isolated from the common ground plane.

8. The nonlinear superconducting quantum circuit according to claim 1, wherein, 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 (90).

9. The nonlinear superconducting quantum circuit according to claim 8, wherein, The first mode and the second mode are respectively the fundamental wave or higher harmonic of the nonlinear superconducting quantum circuit (100).

10. The nonlinear superconducting quantum circuit according to claim 1, 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 nonlinear superconducting quantum circuit according to claim 8 or 9, wherein, The transmission line (90) is made of a Josephson junction array or a high-dynamic inductance material.

12. The nonlinear superconducting quantum circuit according to claim 1, wherein, The at least one inductor (320; 600) is made of a Josephson junction array or a high-dynamic inductance material.

13. A quantum device, comprising: The nonlinear superconducting quantum circuit according to claim 1; A first microwave source (108) connected to the at least one resonant section (30, 32; 60, 62; 90) is used to provide radiation with a frequency equal to the second resonant frequency; A second microwave source (102) connected to the at least one resonant section (30, 32; 60, 62; 90) is used to provide radiation with a frequency equal to twice the difference between the first resonant frequency and the second resonant frequency; as well as The load (106) coupled to the at least one resonant part (30, 32; 60, 62; 90) is such that only the second mode is coupled to the load (106), thereby the first mode carries the cat qubit.

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

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

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