Minimal superconducting quantum circuits for bosonic codes with galvanic coupling

Abstract

The nonlinear superconducting quantum circuit includes a resonator and a galvanically connected asymmetric thread superconducting quantum interferometer. The nonlinear superconducting quantum circuit has a first mode having a first resonant frequency and a second mode having a second resonant frequency. The ratio of the first and second resonant frequencies is different from 1 / 2. The resonator has a symbolic representation including a linear resonator including an inductance and a capacitor, and a nonlinear resonator including a capacitor and a superconducting quantum interferometer. The linear resonator and the nonlinear resonator are galvanically connected and arranged such that one has components connected in series and the other has components connected in parallel. The resonator is configured with an inductance and a capacitance that induces the first and second modes through the superconducting quantum interferometer such that the nonlinear superconducting quantum circuit has a zero-point variation of 0.05 radians or more in the superconducting phase across the superconducting quantum interferometer in the first and second modes.

Description

[Technical Field]

[0001] The present invention relates to the field of superconducting quantum circuits, and more particularly to the field of superconducting quantum circuits including cat qubits. [Background technology]

[0002] Cat qubits are a subset of bosonic codes, which form a family of error-correcting codes for quantum applications. Generally, bosonic codes rely on the storage of qubits in bosonic modes. For cat qubits, two-component cat codes have been the most common design so far.

[0003] Dissipative stabilization of two coherent states requires the appropriate realization of a nonlinear conversion between two photons in a first mode, also known as the cat qubit mode, hosting a stabilized quantum manifold, and one photon in a second mode, known as the buffer mode, and vice versa. Such a stabilization scheme allows for exponential suppression of bit flips with respect to the number of photons in the two coherent states. However, it will only be effective if the confinement ratio of the two coherent states is greater than the escape rate caused by external noise sources. The confinement ratio is positively related to the 2-to-1 photon conversion rate. [Prior art documents] [Non-patent literature]

[0004] [Non-Patent Document 1] Leghtas et al., "Confining the state of light to a quantum manifold by engineered two-photon loss", Science 347, 853 (2015) [Non-patent document 2] Touzard et al., "Coherent Oscillations inside a Quantum Manifold Stabilized by Dissipation", Physical Review X 8, 021005 (2018) [Non-patent document 3] Lescanne R. et Al., "Exponential suppression of bit-flips in a qubit encoded in an oscillator", Nature Physics, 2020 [Non-patent document 4] Berdou et. al., "One hundred second bit-flip time in a two-photon dissipative oscillator", arXiv:2204.09128, https: / / arxiv.org / pdf / 2204.09128.pdf [Non-patent document 5] Devoret et al., "Circuit-QED: How strong can the coupling between a Josephson junction atom and a transmission line resonator be?", Ann. Phys. 519, 767-779 (2007) [Non-patent document 6] Burgelman et al., "Structurally stable subharmonic regime of a driven quantum Josephson circuit", https: / / arxiv.org / abs / 2206.14631 Summary of the Invention [Problem to be solved by the invention]

[0005] In the first implementations of this stabilization scheme (Non-Patent Documents 1 and 2), exponential suppression of bit flips could not be observed. The reason is that the superconducting circuit elements used to properly realize the two-to-one photon conversion (so-called transmons) have spurious cross-Kerr terms, which cause additional noise processing in the escape rate given by the very large transmon cat qubit dispersion shift. Non-Patent Document 3 disclosed a significantly improved cat qubit by properly realizing the two-to-one photon conversion using an asymmetrical threaded superconducting quantum interference device (also called "ATS"). The ATS design has a much lower cross-Kerr term than transmons, which made it possible to observe exponential suppression of bit flips. However, transmons were also used to measure the cat qubit state. Although less detrimental to the cat qubit in this position, it still leads to saturation of the bit flip time, up to several milliseconds. Subsequent work [4] succeeded in increasing the bit-flip saturation time by five orders of magnitude, up to 100 seconds, by eliminating measurement transmons and operating the ATS in a manner that was considered to be dynamically stable. This dramatic increase in bit-flip time was possible because the ATS only adds spurious noise processing with a very low escape rate. However, the confinement rate achieved in [4] was very low.

[0006] 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. The confinement rate, on the other hand, describes how strongly the cat qubit can be perturbed without experiencing a bit flip, which is positively related to how fast a gate can be executed while preserving exponential suppression of bit flips. The phase flip rate, on the other hand, describes how long a gate must be executed to allow for error detection and correction. Theoretical analysis suggests that this ratio should be greater than 10. In [3] and [4], the ratios are 10 and 0.01, respectively.

[0007] No other circuits are known that provide well-behaved cat qubits. The average stable lifetime of other known cat qubits is no more than a few milliseconds, which is insufficient for building practically usable quantum circuits. Because the confinement ratio is positively related to a properly realized 2-to-1 photon conversion rate, a significant increase of the latter is required for circuits based on ATS. [Means for solving the problem]

[0008] The aim of the present application is to improve this situation. To this end, the applicant proposes a nonlinear superconducting quantum circuit comprising at least one resonator and a galvanically connected superconducting quantum interferometer with asymmetric threads. the nonlinear superconducting quantum circuit has a first mode having a first resonant frequency and a second mode having a second resonant frequency, wherein a ratio between the first resonant frequency and the second resonant frequency is different from 1 / 2; the at least one resonant section has a symbolic representation comprising: a linear resonant section comprising at least one inductance and at least one capacitor; and a nonlinear resonant section comprising at least one capacitor and the superconducting quantum interferometer with the asymmetric thread, the linear resonant section and the nonlinear resonant section being galvanically connected and respectively arranged such that one has its components connected in series and the other has its components connected in parallel, and the at least one resonant section is configured with inductance and capacitance values ​​that induce the first mode and the second mode by the superconducting quantum interferometer with the asymmetric thread such that the nonlinear superconducting quantum circuit has a zero point variation of 0.05 radians or more of superconducting phase across the superconducting quantum interferometer with the asymmetric thread in the first mode and the second mode.

[0009] This superconducting quantum circuit is advantageous because it reduces the involvement of buffer and / or memory modes in the ATS and therefore does not have coupling elements that reduce the 2 to 1 photon conversion rate. Although theoretically it may be possible to minimize the detrimental effects of coupling capacitors in the Lescanne 2020 circuit, large capacitors are known to have losses in superconducting circuits, which increases the rate of phase reversal and therefore makes their application in practical implementations ineffective. Furthermore, the design includes the smallest possible components, which minimizes industrialization costs and feasibility risks.

[0010] In various embodiments, the method may exhibit one or more of the following features: - the linear section comprises components arranged in parallel and the non-linear section comprises components arranged in series; - the linear resonator comprises components arranged in series and the nonlinear resonator comprises components arranged in parallel; - the nonlinear superconducting quantum circuit resides on a dielectric substrate and is bounded to a common ground plane by an exposed portion of the dielectric substrate, and the linear resonator portion and the nonlinear resonator portion are realized in physically separate portions of the nonlinear superconducting quantum circuit; - the nonlinear superconducting quantum circuit is formed on a substantially planar substrate and has a width and a height that are less than one-quarter of a wavelength corresponding to the first resonant frequency and the second resonant frequency, respectively; - the linear resonator portion and the nonlinear resonator portion are galvanically isolated from the common ground plane; - the linear resonator portion and the nonlinear resonator portion are galvanically isolated from the common ground plane; - the nonlinear superconducting quantum circuit resides on a dielectric substrate and is bounded to a common ground plane by an exposed portion of the dielectric substrate, and the at least one resonator is realized in a transmission line; - the first mode and the second mode are each a fundamental or higher harmonic of the nonlinear superconducting circuit; - the first resonant frequency and the second resonant frequency are set such that a difference between twice the first resonant frequency and the second resonant frequency is less than half the first resonant frequency and half the second resonant frequency; - the at least one inductance and / or the transmission line are constituted by an array of Josephson junctions or by a high mechanical inductance material.

[0011] The present invention also provides A nonlinear superconducting quantum circuit according to one of the preceding claims; a first microwave source connected to the at least one resonator for providing radiation having a frequency equal to the second resonant frequency; a second microwave source connected to the at least one resonator for providing radiation having a frequency equal to the difference between twice the first resonant frequency and the second resonant frequency; a load coupled to the at least one resonator, such that substantially only the second mode is coupled to the load, whereby the first mode hosts a cat qubit. The apparatus may further comprise a microwave filter for coupling to the load; The microwave filter is configured to pass the second resonant frequency and block the first resonant frequency.

[0012] The invention also relates to a quantum computing system comprising at least one device according to the invention. [Brief explanation of the drawings]

[0013] [Figure 1] We demonstrate how galvanic cat qubit circuits can be incorporated into devices to stabilize quantum information. [Figure 2] We demonstrate how quantum information can be stabilized using a galvanic cat qubit circuit built into the device. [Figure 3] 1 shows an electrical equivalent diagram of a galvanic cat circuit of a first embodiment according to the invention; [Figure 4] FIG. 4 shows a diagram illustrating values ​​of φ for each of the first and second modes of the superconducting quantum circuit of FIG. 3, along with the corresponding g / φ values. [Figure 5] This represents an implementation of the circuit in Figure 3. [Figure 6] 4 shows an electrical equivalent diagram of a galvanic cat circuit of a second embodiment according to the invention; [Figure 7]FIG. 7 shows a diagram illustrating values ​​of φ for each of the first and second modes of the superconducting quantum circuit of FIG. 6, along with the corresponding g / φ values. [Figure 8] This represents an implementation of the circuit in Figure 6. [Figure 9] 1 represents a third embodiment of a galvanic cat circuit. [Figure 10] 10 illustrates a graph showing the behavior of the circuit of FIG. 9 as the length of the transmission line is changed. [Figure 11] 4 shows the ratio of the squares of the zero-point fluctuations of the phases of modes b and a in the linear capacitor according to the first embodiment. DETAILED DESCRIPTION OF THE INVENTION

[0014] Other features and advantages of the present invention will become readily apparent from the following description of the drawings, which show illustrative embodiments of the invention.

[0015] The drawings and the following description are, for the most part, composed of explicit and well-defined features, so that they are not only useful for understanding the invention, but can also be used, if necessary, to contribute to its definition.

[0016] In order for a cat qubit to encode useful data and be stabilized, a two-to-one photon conversion must occur between the first mode (memory) and the second mode (buffer). Most existing prior art belongs 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 resonant two-to-one photon conversion when the second mode does not have a resonant frequency that is a multiple of two of the resonant frequency of the first mode. In other words, the external time-varying excitation used in parametric pumping dissipation relaxes the constraint on the resonant frequency.

[0017] The first and second modes of the superconducting quantum circuit may each correspond to a natural resonant frequency of the circuit. For example, the first and second modes may each be electromagnetic modes. Each of the first and second modes may have a respective resonant frequency, for example, the first mode may be a type f a =ω a / 2π, and the second mode may have a resonant frequency of type f b =ω b ω may have a resonant frequency of ω / 2π, where ω a and ω b may be the angular frequency of each mode. By "having" a first mode and a second mode, it is meant that a superconducting quantum circuit may comprise components operating in a superconducting manner that host multiple modes independently of each other or simultaneously. In other words, the first mode and the second mode may be hosted in different subsets of the components of the superconducting circuit, or in the same subset of the components.

[0018] Superconducting quantum circuits may be operated at temperatures close to absolute zero (e.g., below 100 mK, typically 10 mK) and may be isolated as much as possible from the environment, except for some tailored coupling, to avoid energy loss and decoherence. For example, a first mode may be kept isolated from the environment, while only a second mode is coupled to a dissipative environment.

[0019] A superconducting quantum circuit may be fabricated as one or more patterned layers of superconducting material (e.g., aluminum, tantalum, niobium, among others, as known in the art) deposited on a dielectric substrate (e.g., silicon, sapphire, among others). Each of the one or more patterned layers may define a lumped-element resonator. Two adjacent plates of superconducting material may form a capacitive element (in each of the one or more patterned layers). A superconducting wire may form an inductive element. Alternatively, at least one of the one or more patterned layers may define sections of transmission lines, each resonating at a frequency dependent on its length. The transmission lines may be, for example, coplanar waveguides, slot lines, or microstrip lines. As a further alternative, the circuit may be embedded in a 3D architecture with high-quality 3D modes machined or micromachined into bulk superconductors that can be used as any of two modes.

[0020] The circuit may be integrated as a device, and the device may include a load, a first microwave source, a second microwave source, and a coupler. The coupler may be configured to connect the second mode of the superconducting quantum circuit to the load. The load is a dissipative element, e.g., an element having a given resistance value external to the superconducting circuit, as opposed to a superconducting element. The load dissipates photon pairs converted from the first mode to the second mode using two-to-one photon conversion. In other words, photon pairs destroyed from the first mode are discharged to the environment through the load via the second mode. The first microwave source may be configured to control the microwave radiation in terms of amplitude and phase to apply microwave radiation at a frequency substantially 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, which then drives photon pairs in the first mode using two-to-one photon conversion. This two-to-one photon conversion is reciprocal, and may reversibly convert two photons in the first mode to one photon in the second mode, or one photon in the second mode to two photons in the first mode. A coupler is an element that may be galvanically, capacitively, or inductively connected to a component in a circuit hosting the second mode, and mediates the interaction between the second mode, a load, and a microwave source.

[0021] The load may be a resistor, a matched transmission line, or a matched waveguide. The term "matched" should be interpreted to mean that the transmission line or waveguide is terminated by a resistor at an end different from the end connected to the component hosting the second mode, the value of such resistor being selected so that most of the power destined for the load is absorbed. The load may be provided internal to the first microwave source.

[0022] In various embodiments, the first microwave source may be located at room temperature and connected to the circuit via a coaxial cable. In various embodiments, an attenuator may be located 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 due to the low temperature environment. This allows the microwave radiation to be applied without additional thermal noise.

[0023] The second microwave source is used to provide microwave radiation at a frequency substantially equal to twice the resonant frequency of the first mode minus the resonant frequency of the second mode, thereby obtaining two-to-one photon conversion. Because the ATS has two superconducting loops that must be flux-pumped with the appropriate relative amplitude and phase, the radiation emitted by the second microwave source may be split to feed different transmission lines or waveguides connected at their ends to the two superconducting loops. Alternatively, two different microwave sources radiating signals at the same frequency as the second microwave source may be used to directly feed the two transmission lines or waveguides with the appropriate relative amplitude and phase.

[0024] Optionally, the device may include a microwave filter connected to the first and second modes of the circuit. The microwave filter may be configured to only allow coupling of the second mode to the load. This microwave filter may be interleaved between the load and the coupler. From the circuit's perspective, the filter's purpose is 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 resonant frequency, since only photons in the second mode need to be dissipated into the environment, or by implementing a band-pass filter at the second resonant frequency, or by implementing a high-pass (or low-pass) filter with a cutoff frequency between the first and second resonant frequencies if the second (or first) resonant frequency is greater than the first (or second) resonant frequency. For some circuits, for example, if the two modes have different symmetries, a filter may not be necessary, and appropriate placement of the coupler in the circuit may be sufficient to prevent dissipation of the first mode.

[0025] Thus, the device enables stabilization of two coherent states in the first mode, i.e., a quantum manifold of coherent states. For example, a first microwave source that applies microwave radiation to the second mode via a microwave filter can be considered a two-photon drive of the first mode once converted by two-to-one photon conversion, and a load that dissipates only photons in the second mode can be considered a two-photon dissipation of the first mode once converted by two-to-one photon conversion. The two-photon drive and two-photon dissipation enable stabilization of two coherent states in the first mode.

[0026] The single-photon drive in the second mode is given by the Hamiltonian

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[0027] Two-photon drive is given by the Hamiltonian

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[0028] If g2 is the 2-to-1 nonlinear conversion ratio between the first and second modes, then the condition

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[0029] In various embodiments, a superconducting circuit may have a symbolic representation consisting of, for example, a collection of interconnected dipoles. The term "symbolic representation" should be interpreted as specifying an arrangement of symbols and lines that designate the collection of interconnected dipoles. The collection of interconnected dipoles (also called components) forms a circuit structure (or topology) that is (functionally) equivalent to a nonlinear superconducting circuit.

[0030] In other words, and as is classical in the field of superconducting circuits, a nonlinear superconducting circuit is constructed to achieve a function defined by its symbolic representation, i.e., the function of a collection of theoretical interconnected dipoles represented by the symbolic representation. In further terms, while a circuit may be constructed using patterned layers of superconducting material, it should be understood that the circuit also admits symbolic representation by dipoles, e.g., capacitors, inductors, and / or Josephson junctions. While the exemplary dipoles describe discrete elements, those skilled in the art will clearly understand that these components correspond to equivalent circuits of distributed elements at particular frequency ranges, e.g., at low frequencies, as is known in the art.

[0031] As is known in the art, such distributed elements may have higher frequency modes that are irrelevant and insignificant to the dynamics described herein. Therefore, these distributed elements may be represented by a symbolic representation. This symbolic representation can be improved by adding components such as a series inductor for each wire connection or a parallel capacitor between any two nodes of the circuit, or by adding nodes and branches to account for other modes of the distributed element. Thus, the symbolic representation allows for a better description of the distributed element without changing the circuit's operating principles. Therefore, as is known in the art, the physical circuit, which is the actual fabricated circuit, and its symbolic representation are considered equivalent by those skilled in the art. In practice, improving the dipoles of the symbolic representation only adjusts the zero-point shift in resonant frequency or phase compared to the basic model. When designing a circuit, the final geometry may be fully and accurately simulated by a finite element solver, which easily gives the frequency of each mode, the dissipation due to the load, and the zero-point variation of phase across the Josephson junction, which are the only unknowns for calculating the 2-to-1 photon conversion rate in any configuration.

[0032] The Hamiltonian for the two-to-one photon interaction is of the form

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[0033] The development of cat qubit quantum circuits relies on a superconducting circuit geometry that allows for the suppression of bit flips of cat qubits encoded in a high-Q superconducting resonator called the memory. To this end, the two-photon dissipation of the memory is conveniently realized by coupling it to a low-Q superconducting resonator called the buffer via a nonlinear superconducting dipole.

[0034] In [3], the nonlinear Hamiltonian H2 is suitably realized using an ATS superconducting dipole. The ATS dipole has the potential energy

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[0035] DC value of magnetic flux |φ, known as the saddle point σ ,DC|=|φ δ ,DC|=π / 2 and amplitude φ p and frequency ω p By choosing a flux that pumps only sigma modes with

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[0036] The phase φ across the ATS is related to modes a and b by:

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[0037] where φ a and φ bare the zero-point variations of phase across the ATS of the first and second resonant modes, respectively.

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[0038] Non-Patent Document 3 states that bit flipping is performed by increasing the number of photons of the cat qubit encoded in the resonator, α 2 We demonstrated that the ATS is exponentially suppressed by . However, this architecture uses transmons coupled to the cat qubit as the measurement device, which leads to bit-flip times that saturate down to a few milliseconds. Our work revealed that this is due to a confinement factor that is too small to resist the dispersive frequency shifts caused by thermal excitation of the measurement device. Subsequent work by us, disclosed in [4], despite an even lower confinement factor, increased the bit-flip saturation time by five orders of magnitude by eliminating transmons and operating the ATS in a manner that is considered dynamically stable. More precisely, in [3], the ratio of the confinement factor to the phase-flip rate is 10, whereas in [4], this ratio is 0.01. As explained in the introduction, such a ratio is very far from the theoretically required value.

[0039] The main problem with Non-Patent Document 3 and Non-Patent Document 4 is that they do not provide a potential solution for significantly increasing the two-photon dissipation rate. In fact, the two-photon dissipation rate is

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[0040] The only way to avoid this problem is to increase the capacitance coupling the memory and buffer modes of the circuits of Non-Patent Document 3 and Non-Patent Document 4 sufficiently so that the ATS is strongly involved in both buffer and memory modes. However, large capacitors are known to have losses in superconducting circuits.

[0041] Thus, these prior art techniques are at an impasse: their specific geometry, which is crucial for achieving bit-flip stabilization, cannot be fine-tuned to allow for a sufficiently large two-photon dissipation rate.

[0042] Embodiments and examples of circuits and devices according to the present invention will now be described with reference to the drawings. In the following, the terms "galvanic Cat qubit circuit," "circuit," "superconducting quantum circuit," and "nonlinear superconducting circuit" are used interchangeably to refer to a circuit that performs a two-to-one photon conversion that allows for stabilizing a Cat qubit.

[0043] FIG. 1 shows an example of a quantum device 10 comprising a galvanic cat qubit circuit according to the present invention.

[0044] The apparatus 10 comprises a nonlinear superconducting circuit 100 , a microwave source 102 , a coupler 104 , a load 106 , another microwave source 108 , and a microwave filter 110 .

[0045] The nonlinear superconducting circuit 100 performs a two-to-one photon conversion between a first mode a having code 112 and a second mode b having code 114. In the following, the first mode a hosts the cat qubit and is also known as the memory mode, while the second mode b is used as a buffer between the cat qubit and the environment.

[0046] The device 10 stabilizes the cat qubit using a parametric pump, which means that the resonant frequencies of the first and second modes are 2f a =f b To guarantee two-to-one photon conversion, the parametric pump is a -f b This is done by a microwave source 102 connected to a nonlinear superconducting circuit 100.

[0047] As will become apparent below, the nonlinear superconducting circuit 100 of the present invention is highly unique in that it comprises an ATS ("Asymmetrical threaded SQUID" or "Asymmetrical threaded Superconducting quantum interference device") that is galvanically coupled to other components of the nonlinear superconducting circuit 100 hosting both modes a and b.

[0048] The circuit component hosting the second mode 114 is coupled to a load 106 via a coupler 104. This coupling makes the second mode dissipative. A microwave source 108 is connected to the nonlinear superconducting circuit 100 and generates a microwave signal at a frequency f b The microwave filter 110 is used to drive the second mode at its resonant frequency by causing radiation at frequency f b Alternatively, the filter 110 may be configured as a bandpass filter having a frequency f a, and may be placed between the environment and the two modes to isolate the first mode and thus prevent it from suffering additional losses resulting from unwanted coupling to the load 106. Alternatively, f a >f b (or f b >f a ), it may be configured as a low-pass (or band-pass) filter. In other embodiments, the microwave filter 110 may be omitted if coupling can be established between the load 106 and substantially only the second mode.

[0049] FIG. 2 illustrates the stabilization of a quantum manifold of coherent states of the first mode achieved by the two-to-one photon conversion performed by circuit 100.

[0050] This figure shows the amplitude

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[0051] In the drawings of Figures 3, 6 and 9, only the circuit 100 is illustrated: the coupling to the load, the microwave source driving the buffer, and the microwave source driving the ATS for parametric pumping are not shown for simplicity.

[0052] FIG. 3 represents an electrical equivalent diagram of a first embodiment of the galvanic cat circuit 100 of FIG.

[0053] The circuit 100 comprises a nonlinear resonant portion 30 and a linear resonant portion 32. The nonlinear portion comprises an ATS 34 and a capacitive element 302. The nonlinear resonant portion 30 and the linear resonant portion 32 are galvanically connected together. For the avoidance of any doubt, the expression "galvanically connected" means that there is a short conductive portion connecting the nonlinear resonant portion 30 and the linear resonant portion 32, i.e., a short conductive track or any other means that ensures a physically continuous conductive joint. The expression "short" means that the conductive track is short enough to connect the nonlinear resonant portion 30 and the linear resonant portion 32 at a frequency f a and f b This means that the conductive track has a negligible impedance compared to the impedance of the nonlinear resonant part 30 and the linear resonant part 32 in the ATS. If this conductive track has a non-negligible impedance, it will cause a zero point shift φ of the first and second resonant modes in the ATS. a and φ b This would be contrary to the object of the present invention. In the embodiment described herein, the conductive tracks are also arranged so that neither the nonlinear nor the linear resonant part is shunted.

[0054] In the embodiment described herein, the nonlinear resonator 30 comprises an ATS 34 and a capacitive element 302 connected in series. The linear resonator 32 comprises an inductive element 320 and a capacitive element 322 connected in parallel. If the nonlinear resonator 30 (or the linear resonator 32) is isolated from the rest of the circuit 100, it will host a first bare mode (or a second bare mode).

[0055] This galvanic circuit is minimal in the sense that its symbolic representation has the minimum number of components possible to host two resonant modes (two capacitive elements 302 and 322, and two inductive elements 34 and 320). This contrasts with prior art in which a nonlinear section is coupled to a linear section via a capacitive or inductive coupler, resulting in a much lower contribution of one resonant mode in the ATS, thus resulting in a smaller zero-point shift and a smaller 2-to-1 photon conversion rate g2. The minimal galvanic circuit proposed herein has a symbolic representation without such a coupler, so both resonant modes are expected to have a significant contribution in the ATS. While one might think of the ATS 34 as capacitively coupled to the linear bear mode via capacitor 302, in reality, because it resonates with capacitor 302, they form a nonlinear bear mode that is galvanically coupled to the linear bear mode.

[0056] This minimal galvanic circuit is also advantageous because, as will be seen later, it allows for a straightforward, simple, and compact implementation.

[0057] Furthermore, it should be noted that having one series resonant section and one parallel resonant section is the only possible circuit topology. Therefore, there are only two possible minimal implementations of a circuit including an ATS: one in which the ATS plays the role of the inductive element in the series resonant section, as shown in Figure 3, and one in which the ATS plays the role of the inductive element in the parallel resonant section, as shown in Figure 6.

[0058] The values ​​of the inductive and capacitive elements of the parallel resonant section are L parallel and C parallel and L are used as the values ​​of the inductive and capacitive elements of the series resonant section. series and C series These bare modes are described below, using respectively: - their angular frequencies

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[0059] 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 the ATS34 near the global minimum of its potential energy. At the saddle point, the effective inductance L ats is equal to the shunt inductance of the ATS34.

[0060] The linear part of the Hamiltonian of the galvanic circuit is given by:

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[0061] The aforementioned modes a and b are expressed by the above Hamiltonian H lin The dimensionless coupling constant k can take any value between 0 and 1. Values ​​close to 0 indicate weakly coupled modes, while values ​​of 1 indicate maximally coupled modes. As expected, minimal galvanic designs allow values ​​of k greater than 1 / 2 to be easily reached. The possibility of reaching such large coupling constants is unique in the field of superconducting circuits, as first shown in [5].

[0062] As shown in more detail with respect to Figures 5 and 8, the inductive and capacitive elements form an LC resonator, which may be implemented by distributed elements in a patterned layer of superconducting material, as described below. - two adjacent plates forming a capacitor in parallel with a superconducting wire, a single Josephson junction, or an array of Josephson junctions forming an inductor; - a section of a superconducting transmission line terminated at two different boundary conditions (short-circuited to ground at one end and open at the other), forming a so-called λ / 4 resonator. The transmission line may be, for example, of the coplanar waveguide or microstrip type, or - a section of a superconducting transmission line terminated with two identical boundary conditions (open-open or short-circuited), forming a so-called λ / 2 resonator, the transmission line being, for example, of the coplanar waveguide or microstrip type.

[0063] The ATS 34 is implemented as known in the art, for example in the article "Analysis of the ATS 34: A Study of the ATS 34", Vol. 1, No. 1, pp. 111-114, 2003. It is a structure with two Josephson junctions in parallel with a parallel inductive element between them. As a result, the ATS 34 comprises two connected loops, each loop comprising a Josephson junction in parallel with a shunt inductive element. The ATS 34 produces DC and AC flux biases in both of its loops. The DC bias sets the operating point of the ATS, which may be operated near the so-called saddle point. The saddle point is a sweet spot in frequency and has a small cross-Kerr term. The AC flux bias is 2f a -f b This AC flux bias is typically chosen to drive the common mode of the two loops.

[0064] FIG. 4 shows the inductive element 320 (L parallel ) and the inductance of the ATS34 (L series ) can be obtained for various values ​​of g / φ p The resulting value curve is shown in Figure 1. In this figure, the value φ for the first mode is a is shown in radians as a dotted line, and the second mode φ b The value of is shown in radians by the dashed-dotted line, and the corresponding g2 / φ p The level lines are shown as solid lines in MHz. These curves are established by fixing the first resonant frequency at 4.5 GHz and the second resonant frequency at 8.0 GHz, varying the value of inductive element 320 towards the east and varying the value of inductive element 34 towards the north, while selecting the values ​​of capacitive elements 302 and 322 to obtain the frequencies mentioned above.

[0065] Ratio g2 / φ p The reason for plotting φ p is proportional to the amplitude of the parametric flux pump, which is set by the amplitude of the microwave source 102, and is somehow arbitrary.

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[0066] This figure shows the g2 / φ above 250 MHz for conventional values ​​of inductive elements 34 and 320. p In fact, work by the applicant has shown that values ​​of over 100 MHz are guaranteed, and values ​​of several hundred MHz are achievable, which is an order of magnitude greater than known prior art. In comparison, the g2 / φ achieved by Lescanne 2020 is p The value of was only 9.6MHz. Berdou2022 was at least an order of magnitude smaller.

[0067] Figure 4 also shows the zero point fluctuation φ a and φ b The product

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[0068] The Hamiltonian for the 2-to-1 photon conversion is φ a *(a+a † )+φ b *(b+b † ), a rule of thumb known to those skilled in the art is that the ATS potential energy depends on the third order expansion of φ a *max(α,1 / 2) and φ b The goal is to keep *max(β, 1 / 2) small compared to π, where α is the amplitude of the stabilized coherent state in the cat qubit and β is the amplitude of the residual electromagnetic field in the buffer.

[0069] The term 1 / 2 in max(α or β, 1 / 2) is used to account for the minimum zero point fluctuation. When α=2, φ a <0.1 is considered safe. On the other hand, β tends to be very close to zero, so max(β, 1 / 2) = 1 / 2. This means that φ b Larger values ​​of , typically φ b This means that g<0.3 is acceptable. Figure 4 shows that g can be increased by more than an order of magnitude compared to the prior art while maintaining safe values ​​of the parameters.

[0070] Figure 4 also shows the relationship between φ a and φ b We show that more aggressive values ​​of φ are easily achievable, resulting in a further increase in g2. a and φ b It should be noted that it is unknown how far we can push φ, since the cat qubit field is very recent and lacks such research. a and φ bThe galvanic cat design, with its obvious possibility of realizing this, makes it possible to carry out such research.

[0071] FIG. 11 shows the ratio of the squares of the zero point fluctuations of modes b and a across the parallel section, which is the parallel combination of the linear capacitor 322 and the linear inductor 320 of the linear resonant section 32 in the case of FIG.

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[0072] However, thanks to the large value of the dimensionless coupling constant k, the first and second resonant frequencies may be widely separated from each other (3.5 GHz in FIG. 4), thus providing the microwave filter 106 with a large chamber for strong blocking of the electromagnetic field at the frequency of the first resonant mode.

[0073] Furthermore, the pump frequency 2f a -f b is f a and f b It is also possible to select the first and second resonant frequencies to be somewhat far from both 2f a -f b is f a / 2 and f b This means that the memory frequency f a The buffer frequency f should be set so that it is not too far from twice the b This can be achieved by choosing a small pump frequency f p =2f a -f b For example, f p The value of is 1 GHz in Figure 4, which means that f a is more than four times smaller than f b is eight times smaller than

[0074] This is advantageous because the first-order terms in the ATS potential energy expansion indicate that the parametric pump can directly drive the circuit. As shown in the appendix of [3], this driving leads to spurious dynamical AC Stark shifts and dynamical cross-Kerr terms, which are detrimental to cat qubit operation. For low pump frequencies, direct operation of the circuit is much less efficient, which leads to much smaller dynamical AC Stark shifts and dynamical cross-Kerr terms.

[0075] Another advantage is that the value 2f a -f b f a and f bTherefore, by introducing a second microwave filter into the parametric pump line, it is possible to prevent the first mode from being emitted into the parametric pump line.

[0076] This is all the more surprising because the possibility of having a low-loss memory mode with such strongly hybridized modes is highly counterintuitive. The rationale is that the field of quantum computing, particularly in the area of ​​cat qubits, is very young, and very gradual changes are generally favored. One of the reasons for having the ATS located in the second mode and the first mode weakly capacitively coupled to the second mode in Lescanne 2020 and Berdou 2022 is to minimize the damping of the first mode due to the damping of the second mode in the load. Indeed, it is traditionally considered much preferable to couple nonlinear elements in quantum circuits to a single mode and weakly couple other modes to it.

[0077] Galvanically coupling nonlinear elements is clearly not a gradual change and defies all preconceptions.

[0078] Finally, f a and f b The large frequency spacing between f is due to the variability of the inductance of the various Josephson junctions in the circuit, primarily f a and f b This makes the design more robust to the uncertainties of nano-manufacturing, such as the known challenges that affect the accuracy of predictions of

[0079] FIG. 5 shows an implementation of the circuit of FIG.

[0080] Similar components have been given similar reference numerals, with only the first digit of the reference numeral changing from "3" to "5", i.e., capacitive element 322 in FIG. 3 has reference numeral 522 in FIG.

[0081] This figure shows a top view of the layout of a superconducting chip designed by the applicant to correspond to the circuit of Figure 3. The light gray areas correspond to the metallized surfaces, which may include tantalum or aluminum. The gray areas correspond to the substrate, made of sapphire, on which the circuit sits. Other materials may be used to implement the superconducting circuit, such as niobium, NbTi, or TiN for the metallization and silicon or quartz for the chip.

[0082] The circuit comprises a ground plane 50 on which circuit 55 is formed. The nonlinear resonator comprises an ATS 34 and a cross-shaped capacitive element 502. The linear resonator 30 is formed at the bottom of the figure by a large rectangle forming the capacitive element 522, to which is connected an array of Josephson junctions 520 forming the inductive element.

[0083] Although not shown in Figure 5, coupler 104 may advantageously be implemented by capacitively coupling a CPW transmission line to electrode 522, according to Figure 11. Furthermore, cat qubits may be coupled to other cat qubits or readout transmons via CPW buses capacitively coupled to different branches of electrode 502. Capacitive coupling of CPWs to electrodes such as 502 and 522 is conventionally done in superconducting circuits. Finally, two CPWs running closely to each side of ATS 34 may be used to flux bias it to set its operating point at DC and to realize a parametric pump at AC.

[0084] The circuit of Figure 5 shows a lumped and grounded implementation of the resonant modes of the circuit of Figure 1. The design is lumped because the total size of the circuit 100 is less than a quarter wavelength of the first and second modes. The design is grounded because the first and second modes correspond to oscillations of charge and current between the electrodes and the ground plane 50 to which they are galvanically coupled via the ATS 34.

[0085] Other implementations of inductors may also be possible, for example geometric inductors consisting of meander-shaped or spiral-shaped lines.

[0086] This grounded design is more sensitive to imperfections in the ground plane and may suffer from greater crosstalk than a differential design, but it is much more compact and minimizes parasitic capacitances that can shunt the ATS and thus alter the accuracy of the symbolic representation in Figure 3. The simplicity and symmetry of this design demonstrate the effectiveness of a galvanic cat circuit with a minimum number of components.

[0087] In another embodiment, the design may be differential, meaning that the first and second modes correspond to charge and current oscillations between a pair of electrodes that are galvanically isolated from the ground plane 50. Differential designs occupy more space than grounded designs, but they have the advantage of providing better isolation from lossy components of the ground location, such as wire bonds (not shown in the drawings), or other components that may be patterned on the chip, such as other cat qubits, thus reducing crosstalk.

[0088] Figure 6 shows a second embodiment similar to Figure 3. The main difference is that the components of the nonlinear resonator 62 are now in parallel, as opposed to being in series in the embodiment of Figure 3. Similarly, the components of the linear resonator 60 are now in series, as opposed to being in parallel in the embodiment of Figure 3.

[0089] Similar components have been given similar reference numerals, with only the first digit of the reference numeral changing from "3" to "6", i.e., capacitive element 302 in FIG. 3 has reference numeral 602 in FIG.

[0090] Figure 7 is a diagram similar to Figure 4, but based on the circuit of Figure 6. For simplicity's sake, it will not be further described. It should be noted that Figure 11 and its description remain quantitatively valid for the circuit of Figure 6.

[0091] Figure 8 shows an implementation of the circuit of Figure 6. It is similar to the implementation shown in Figure 5 of the circuit of Figure 3, except that the linear inductance and ATS 34 are swapped so that the linear and nonlinear resonant sections are swapped. This design is advantageous in that the ATS 34 is now galvanically coupled to the ground plane, thereby allowing easier and greater coupling to magnetic flux lines (not shown).

[0092] FIG. 9 shows a third embodiment of the galvanic cat circuit.

[0093] In the embodiment of Figure 9, there is only one resonator that, together with the ATS 34, generates both the first and second modes. This embodiment differs from the embodiments of Figures 3 and 6 in that the characteristics of the first and second resonating modes are not typically accurately described by a simplified symbolic representation involving only two LC resonators. This illustrates why the present invention goes against existing preconceptions: while a low coupling constant k may allow for a good understanding of the first and second modes from a simple perturbation analysis involving only the two bare modes of the resonator and the ATS, this is not possible here; more bare modes of the resonator must be considered.

[0094] As shown in this figure, the ATS 34 is connected to a transmission line 90 having an open end. Since the ATS directly terminates the transmission line hosting the resonant mode, the circuit is galvanic. This can be implemented with the same materials as the components in Figure 5. For a given φ b or φ aIf the characteristic impedance of the transmission line required to reach is too large to be fabricated geometrically, the center conductor of the transmission line may be replaced by a high mechanical inductance material or by a chain of wide Josephson junctions. Finally, the transmission line may be implemented with various geometries, e.g., coplanar waveguide (CPW), microstrip, or stripline. In the embodiment of Figure 9, the CPW 90 may be capacitively or inductively connected to the microwave filter 110 and load 106 for coupling to the environment.

[0095] Figure 10 is somewhat similar to Figures 4 and 7, but there are differences in that due to the implementation of the transmission line, some harmonics must be taken into account. As a result, there are three graphs in Figure 10 that illustrate the behavior of harmonics 0, 1, and 2 of the transmission line.

[0096] In Figure 10, the ATS parameters are the same as in Figures 4 and 7. The CPW is assumed to have a characteristic impedance of 50 Ω, and without loss of generality, the effective dielectric constant ε in the case of a CPW on a fire is typically r =5.6 is set.

[0097] The top graph of Fig. 10 shows the g2 / φ that can be obtained for various lengths of the CPW. p The solid line represents the 2-to-1 photon rate g2 / φ calculated by the first resonant mode (or second resonant mode) obtained as the first (or fundamental) harmonic (or second harmonic) of the nonlinear quantum circuit. p where the first resonant mode index a = 0 and the second resonant mode index b = 1. The dashed line represents the 2-to-1 photon rate g2 / φ calculated with the first resonant mode (or the second resonant mode) obtained as the second harmonic (or the third harmonic) of the nonlinear quantum circuit. p where the first resonant mode index a=1 and the second resonant mode index b=2.

[0098] In the middle graph of FIG. 10, the frequency values ​​f0, f1, and f2 of the first (or fundamental), second, and third harmonic are expressed in units of GHz.

[0099] In the bottom graph of FIG. 10, the values ​​φ0, φ1, and φ2 of the respective zero point variations of phase across the ATS 34 of the first (or fundamental), second, and third harmonics are expressed in units of radians.

[0100] Contrary to Figures 4 and 7, the first and second resonant frequencies are not fixed, so their fluctuations must be shown. b -f a and φ a and φ b The safe value of g2 / φ is significantly higher than that of the conventional technology. p It can be seen that values ​​of and are achievable.

[0101] f to suit a given application, for example by fine-tuning the characteristic impedance of the transmission line or by using other harmonics. a , f b , φ a , φ b , and g2 / φ p It will be clear to those skilled in the art that the value of can be adjusted. It is also possible to change the termination of the transmission line, but any termination, e.g., an inductive short circuit, will need to be included in the potential energy of the ATS and will consequently modify its operating point and dynamics.

Claims

1. A nonlinear superconducting quantum circuit comprising at least one resonator (30, 32; 60, 62) and a galvanically connected superconducting quantum interferometer (34) having asymmetric threads, the nonlinear superconducting quantum circuit has a first mode (a) having a first resonant frequency and a second mode (b) having a second resonant frequency; the ratio between the first resonant frequency and the second resonant frequency is different from 1 / 2; the at least one resonator (30, 32; 60, 62) has a symbolic representation comprising a linear resonator (32; 60) comprising at least one inductance (320; 600) and at least one capacitor (322; 602), and a nonlinear resonator (30; 62) comprising at least one capacitor (302; 622) and the superconducting quantum interference device (34) with the asymmetric thread; the linear resonator (30) and the nonlinear resonator (32) are galvanically connected and arranged such that one has its components connected in series and the other has its components connected in parallel; the at least one resonator (30, 32) is configured with inductance and capacitance values ​​that induce the first mode (a) and the second mode (b) with the superconducting quantum interference meter (34) having the asymmetric thread such that the nonlinear superconducting quantum circuit (100) has a zero point variation of superconducting phase across the superconducting quantum interference meter (34) having the asymmetric thread in the first mode (a) and the second mode (b) of 0.05 radians or more. Nonlinear superconducting quantum circuits.

2. The linear section (32) comprises components arranged in parallel and the non-linear section (30) comprises components arranged in series. The nonlinear superconducting quantum circuit of claim 1.

3. The linear resonator (60) comprises components arranged in series, and the nonlinear resonator (62) comprises components arranged in parallel. The nonlinear superconducting quantum circuit of claim 1.

4. the nonlinear superconducting quantum circuit resides on a dielectric substrate and is bounded to a common ground plane (50; 80) by an exposed portion of the dielectric substrate; the linear resonator and the nonlinear resonator are realized in physically separate portions of the nonlinear superconducting quantum circuit; 4. The nonlinear superconducting quantum circuit according to claim 2 or 3.

5. the nonlinear superconducting quantum circuit is formed on a substantially flat substrate and has a width and a height that are shorter than a quarter of a wavelength corresponding to the first resonant frequency and the second resonant frequency, respectively; 5. The nonlinear superconducting quantum circuit according to claim 4.

6. the linear resonator and the nonlinear resonator are galvanically connected to the common ground plane (50; 80); 6. The nonlinear superconducting quantum circuit according to claim 4 or 5.

7. the linear resonator portion and the nonlinear resonator portion are galvanically isolated from the common ground plane.

6. The nonlinear superconducting quantum circuit according to claim 4 or 5.

8. the nonlinear superconducting quantum circuit resides on a dielectric substrate and is bounded to a common ground plane by an exposed portion of the dielectric substrate; The at least one resonator is realized in a transmission line (90). The nonlinear superconducting quantum circuit of claim 1.

9. The first mode (a) and the second mode (b) are each a fundamental wave or a higher harmonic of the nonlinear superconducting circuit (100).

9. The nonlinear superconducting quantum circuit of claim 8.

10. the first resonant frequency and the second resonant frequency are set so that a difference between twice the first resonant frequency and the second resonant frequency is smaller than half the first resonant frequency and half the second resonant frequency; A nonlinear superconducting quantum circuit according to any one of claims 1 to 9.

11. the at least one inductance (320; 600) and / or the transmission line (90) are constituted by an array of Josephson junctions or by a high mechanical inductance material; A nonlinear superconducting quantum circuit according to any one of claims 1 to 10.

12. A nonlinear superconducting quantum circuit according to any one of claims 1 to 11; a first microwave source (108) connected to the at least one resonator (30, 32; 60, 62; 90), the first microwave source (108) for providing radiation having a frequency equal to the second resonant frequency; a second microwave source (102) connected to the at least one resonator (30, 32; 60, 62; 90), for providing radiation having a frequency equal to the difference between twice the first resonant frequency and the second resonant frequency; a load (106) coupled to the at least one resonator (30, 32; 60, 62; 90), such that substantially only the second mode (b) is coupled to the load (106), whereby the first mode (a) hosts a cat qubit; Quantum device.

13. The quantum device further comprises a microwave filter (110) for coupling to the load (106); the microwave filter (110) is configured to pass the second resonant frequency and block the first resonant frequency; 13. The quantum device of claim 12.

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

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