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

[0002] Cat qubits are a subset of bosonic codes, forming a family of error-correcting codes for quantum applications. Generally, bosonic codes rely on the conservation 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 proper realization of a nonlinear conversion and inverse between two photons in a first mode hosting a stabilized quantum manifold, also known as the cat qubit mode, and one photon in a second mode known as the buffer mode. Such a stabilization scheme allows for the suppression of bit flipping exponentially with respect to the number of photons in the two coherent states. However, it will only be effective if the confinement rate of the two coherent states is greater than the escape rate caused by an external noise source. The confinement rate is positively correlated with the 2:1 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 [Overview of the Initiative] [Problems that the invention aims to solve]

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

[0006] The ratio of the confinement rate to the phase inversion rate of a cat qubit is a fundamental metric for quantum error correction using cat qubits. The confinement rate, on the other hand, represents how strongly the cat qubit can be perturbed without experiencing a bit inversion, and it is positively correlated to how quickly the gate can be executed while preserving the exponential suppression of bit inversion. The phase inversion rate, on the other hand, represents how long the gate must be executed to enable error detection and correction. Theoretical analysis suggests that this ratio will be greater than 10⁴. In Non-Patent Literature 3 and Non-Patent Literature 4, this ratio is given as 10 and 0.01, respectively.

[0007] No other circuit is known that provides a well-functioning cat qubit. The average stable lifetime of other known cat qubits never exceeds a few milliseconds, which is insufficient to construct a practically usable quantum circuit. Since the confinement rate is positively correlated with a well-achieved 2:1 photon conversion rate, a significant increase in the latter is required in ATS-based circuits. [Means for solving the problem]

[0008] The purpose of this application is to improve this situation. To this end, the applicant proposes a nonlinear superconducting quantum circuit comprising at least one resonant section and a superconducting quantum interferometer having galvanically connected, asymmetric threads. The above nonlinear superconducting quantum circuit has a first mode having a first resonant frequency and a second mode having a second resonant frequency, wherein the ratio between the first resonant frequency and the second resonant frequency is different from 1 / 2, and 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 having the asymmetric thread, wherein the linear resonant section and the nonlinear resonant section are galvanically connected and 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 to have inductance and capacitance values ​​that cause the first mode and the second mode by the superconducting quantum interferometer having the asymmetric thread, such that the nonlinear superconducting quantum circuit has a zero-point fluctuation of 0.05 radians or more of the superconducting phase across the superconducting quantum interferometer having the asymmetric thread in the case of 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 ATS and therefore lacks coupling elements that reduce the 2:1 photon conversion rate. Theoretically, the detrimental effects of coupling capacitors in Lescanne2020 circuits can be minimized; however, large capacitors are known to have losses in superconducting circuits, which increases the phase inversion rate and therefore makes their application in real-world implementations ineffective. Furthermore, this design includes the minimum possible number of components, which minimizes risks related to industrialization costs and feasibility.

[0010] In various embodiments, the method may present one or more of the following features. - The linear section comprises components arranged in parallel, and the nonlinear section comprises components arranged in series. - The linear resonant section comprises components arranged in series, and the nonlinear resonant section comprises components arranged in parallel. - The above nonlinear superconducting quantum circuit exists on a dielectric substrate and has a boundary with respect to a common ground plane by the exposed portion of the dielectric substrate, and the linear resonant portion and the nonlinear resonant portion are realized in multiple physically separate parts of the above nonlinear superconducting quantum circuit. - The above nonlinear superconducting quantum circuit is formed on a substantially flat substrate and has a width and height shorter than one-quarter of the wavelength corresponding to the first and second resonant frequencies, respectively. - The linear resonant section and the nonlinear resonant section are galvanically isolated from the common ground plane. - The linear resonant section and the nonlinear resonant section are galvanically isolated from the common ground plane. - The above nonlinear superconducting quantum circuit exists on a dielectric substrate and has a boundary with respect to the common ground plane by the exposed portion of the dielectric substrate, and the at least one resonant portion is realized in a transmission line. - The first mode and the second mode described above are, respectively, the fundamental wave or higher-order harmonics of the nonlinear superconducting circuit. - The first and second resonant frequencies are set such that the difference between twice the first resonant frequency and the second resonant frequency is less than half of the first and second resonant frequencies, respectively. - The above-mentioned inductance and / or transmission line shall be composed of an array of Josephson junctions or a high-mechanical inductance material.

[0011] The present invention also, A nonlinear superconducting quantum circuit as described in one of the prior claims, A first microwave source connected to at least one of the above-mentioned resonant sections, the first microwave source for providing radiation having a frequency equal to the second resonant frequency, A second microwave source connected to at least one of the above-mentioned resonant sections, the second microwave source for providing radiation having a frequency equal to the difference between twice the first resonant frequency and the second resonant frequency, The present invention relates to a quantum device comprising a load coupled to at least one of the above-mentioned resonant parts, wherein substantially only the second mode is coupled to the load, and thereby the first mode hosts a cat qubit. This device may further include a microwave filter for coupling to the above load. The microwave filter described above is configured to allow the second resonant frequency to pass through and to block the first resonant frequency.

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

[0013] [Figure 1] This paper demonstrates a method for stabilizing quantum information by incorporating a galvanic cat qubit circuit into a device. [Figure 2] This paper demonstrates a method for stabilizing quantum information using a galvanic cat qubit circuit incorporated into the device. [Figure 3] This shows the electrical equivalent diagram of the galvanic cat circuit according to the first embodiment of the present invention. [Figure 4] Figure 3 shows the values ​​of φ for the first and second modes of the superconducting quantum circuit, along with the corresponding g2 / φp values. [Figure 5] This shows the implementation of the circuit in Figure 3. [Figure 6] This shows the electrical equivalent diagram of a galvanic cat circuit according to a second embodiment of the present invention. [Figure 7]Figure 6 shows the values ​​of φ for the first and second modes of the superconducting quantum circuit, along with the corresponding g2 / φp values. [Figure 8] This shows the implementation of the circuit in Figure 6. [Figure 9] This represents a third embodiment of the galvanic cat circuit. [Figure 10] This graph shows the behavior of the circuit in Figure 9 when the length of the transmission line changes. [Figure 11] Complementarily to Figure 4, the ratio of the squares of the zero-point phase fluctuations of modes b and a in the linear capacitor according to the first embodiment is shown. [Modes for carrying out the invention]

[0014] Other features and advantages of the present invention will be readily apparent from the description of the following drawings illustrating exemplary embodiments of the invention.

[0015] The drawings and the following description consist, for the most part, of explicitly and appropriately defined features. As a result, they are not only useful for understanding the invention, but can also be used, where necessary, to contribute to its definition.

[0016] For a cat qubit to encode useful data and be stabilized, a 2:1 photon conversion must occur between the first mode (memory) and the second mode (buffer). Most existing conventional techniques belong to a family of cat qubits stabilized by parametric pumping dissipation. Parametric pumping dissipation techniques are used to bridge the gap between the frequencies of the two modes and perform a resonant 2:1 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 the circuit's natural resonant frequency. For example, each of the first and second modes is an electromagnetic mode. Each of the first and second modes may have its own resonant frequency, for example, the first mode is of type f a =ω a The second mode may have a resonant frequency of / 2π, and the second mode is of type f b =ω b It may have a resonant frequency of / 2π, where ω a and ω b ∫

[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 couplings tailored to certain purposes, in order to avoid energy loss and decoherence. For example, the first mode may remain isolated from the environment, while only the second mode is coupled to a dissipative environment.

[0019] Superconducting quantum circuits may be fabricated as one or more patterned layers of superconducting material (e.g., aluminum, tantalum, niobium, as known in this art) deposited on a dielectric substrate (e.g., silicon, sapphire, in particular). Each of the one or more patterned layers may define a lumped-element resonator. Capacitive elements may be formed (in each of the one or more patterned layers) by two adjacent plates made of superconducting material. Inductive elements may be formed by superconducting wires. Alternatively, at least one of the one or more patterned layers may define a transmission line portion that resonates at frequencies dependent on their length. The transmission line may be, for example, a coplanar waveguide, a slotted line, or a microstrip line. Further alternative, the circuit may be embedded in a 3D architecture having high-quality 3D modes machined or micromachined to become a bulk superconductor that can also be used as any of two modes.

[0020] The circuit may be integrated as a device, which may comprise 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, for example, an element located outside the superconducting circuit and having a given resistance value opposite to that of the superconducting element. The load dissipates pairs of photons converted from the first mode to the second mode using a 2:1 photon conversion. In other words, pairs of photons destroyed from the first mode are emitted into the environment through the load via the second mode. The first microwave source may be configured to control the microwave radiation with respect to amplitude and phase so as 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 to the second mode, and then it drives pairs of photons in the first mode using a 2:1 photon conversion. This 2-to-1 photon conversion is reciprocal and may reversibly convert two photons of the first mode to one photon of the second mode, or one photon of the second mode to two photons of the first mode. A coupler is an element that can be galvanically, capacitively, or inductively connected to a circuit component hosting the second mode, and is a component that mediates the interaction between the second mode, the load, and the microwave source.

[0021] The load may be a resistor, a matched transmission line, or a matched waveguide. The expression “matched” should be interpreted as meaning 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, and the value of such a resistor is selected so that the majority of the power directed toward the load is absorbed. The load may be located inside 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 placed between the microwave source and the circuit, i.e., along the path of the microwave radiation applied by the microwave source, in order to thermally neutronize the microwave radiation in a low-temperature environment. This makes it possible to apply microwave radiation without additional thermal noise.

[0023] The second microwave source provides 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 used to obtain a 2:1 photon conversion. Since the ATS has two superconducting loops that must be processed by a flux pump with appropriate relative amplitude and phase, the radiation emitted by the second microwave source may be split to supply different transmission lines or waveguides connected at the ends to the two superconducting loops. Alternatively, two different microwave sources emitting signals at the same frequency as the second microwave source may be used to supply directly to two transmission lines or waveguides with appropriate relative amplitude and phase.

[0024] Optionally, the device may include microwave filters connected to the first and second modes of the circuit. The microwave filters may be configured to allow coupling of only the second mode to the load. These microwave filters may be interleaved between the load and the coupler. From a circuit perspective, the purpose of these filters 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, or a band-pass filter at the 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 only photons of the second mode need to dissipate into the environment. In some circuits, for example, if the two modes have different symmetries, filters may not be necessary, and the appropriate placement of the coupler in the circuit may be sufficient to prevent dissipation of the first mode.

[0025] Therefore, the apparatus enables the stabilization of the 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 may be considered a two-photon drive of the first mode once transformed by a two-to-one photon conversion, and a load that dissipates only the photons of the second mode may be considered a two-photon dissipation of the first mode once transformed by a two-to-one photon conversion. Two-photon drive and two-photon dissipation enable the stabilization of the two coherent states in the first mode.

[0026] Single-photon driving in the second mode is Hamiltonian

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[0027] Two-photon drive is Hamiltonian

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[0028] If g2 is a 2:1 nonlinear conversion ratio between the first mode and the second mode, then the condition

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[0029] In various embodiments, a superconducting circuit may have a symbolic representation consisting of, for example, a set of interconnected dipoles. The expression “symbolic representation” should be interpreted as specifying the arrangement of symbols and lines that designate the set of interconnected dipoles. The set 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, nonlinear superconducting circuits are configured to achieve a function defined by their symbolic representation, in other words, the function of a theoretical set of interconnected dipoles represented by their symbolic representation. Further in other words, the circuit may be constructed using patterned layers of superconducting material, but it should be understood that the circuit accepts symbolic representations of dipoles, such as capacitors, inductors, and / or Josephson junctions. Exemplary dipoles describe discrete elements, and as is known in the art, it will be obvious that these components correspond to equivalent circuits of distributed-parameter elements in a particular frequency range, for example, at low frequencies.

[0031] As is known in the art, such distributed-constant elements may have higher frequency modes, which are irrelevant and minor to the dynamics described in the present application. Therefore, these distributed-constant elements may be represented by symbolic expressions. This symbolic expression may be improved by adding components such as series inductors for each wire connection or parallel capacitors between any two nodes of the circuit, or by adding nodes and branches to account for other modes of the distributed-constant elements. Therefore, the symbolic expression enables a better description of the distributed-constant elements without changing the operating principle of the circuit. Therefore, as is known in the art, the physical circuit, which is the actually manufactured circuit, and its symbolic expression are regarded as equivalents by those skilled in the art. In fact, improving the dipole of the symbolic expression only adjusts the resonance frequency or the zero-point variation of the phase compared to the basic model. When designing a circuit, the final geometry may be simulated sufficiently and accurately by a finite element solver, which easily gives the frequency of each mode, the dissipation resulting from the load, and the zero-point variation of the phase across the Josephson junction, and only these are the unknowns for calculating the two-to-one photon conversion rate in any configuration.

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

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[0033] The development of cat qubit quantum circuits depends on the geometric shape of the superconducting circuit that enables the suppression of bit flipping of the encoded cat qubit in a high-Q superconducting resonator called a memory. For this purpose, two-photon dissipation of the memory is appropriately realized by coupling it to a low-Q superconducting resonator called a buffer via a nonlinear superconducting dipole.

[0034] In Non-Patent Document 3, the nonlinear Hamiltonian H2 is appropriately realized using an ATS superconducting dipole. The ATS dipole has the following potential energy.

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[0035] DC value of magnetic flux known as a saddle point |φ σ DC|=|φ δ DC|=π / 2 and amplitude φ p and frequency ω p By selecting a magnetic flux that pumps only the sigma modes having , the potential energy is given by the following equation.

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

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[0037] Here, φ a and φ bThese represent the zero-point phase fluctuations across the ATS of the first and second resonant modes, respectively.

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[0038] Non-patent document 3 states that bit inversion is the number of photons α of the cat qubit encoded in the resonator. 2 This was demonstrated to suppress exponentially. However, this architecture uses a transmon coupled to a cat qubit as the measuring device, which results in a bit flip time that saturates to several milliseconds. The applicant's research revealed that this is due to a confinement rate that is too small to resist the dispersed frequency shift caused by thermal excitation of the measuring device. Later research by the applicant disclosed in Non-Patent Document 4 increased the bit flip saturation time by five orders of magnitude by removing the transmon and operating the ATS in a manner that is considered to be dynamically stable, despite an even lower confinement rate. More precisely, in Non-Patent Document 3, the ratio of the confinement rate to the phase flip rate is 10, whereas in Non-Patent Document 4, this ratio is 0.01. As explained in the introduction of this application, such a ratio is far from the theoretically required value.

[0039] The main problem with Non-Patent Documents 3 and 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 the two-photon coupling rate.

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

[0041] Therefore, these conventional techniques are in a dead end; their specific geometric shapes are crucial for achieving bit-invert stabilization, but they cannot be fine-tuned to allow for a sufficiently large two-photon dissipation rate.

[0042] Herein, embodiments and examples of the circuits and devices according to the present invention will 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 interchangeable and refer to circuits that perform a 2-to-1 photon conversion that enables the stabilization of cat qubits.

[0043] Figure 1 shows an example of a quantum device 10 equipped with 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 2:1 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 a 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. This means that the resonant frequencies of the first and second modes are 2f. a =f b This means it is not of that type. To ensure 2-to-1 photon conversion, the parametric pump uses 2f a -f b It provides radiation at a frequency of . This is done by a microwave source 102 connected to a nonlinear superconducting circuit 100.

[0047] As will become clear from the following, the nonlinear superconducting circuit 100 according to the present invention is highly unique in that it includes 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 that host both modes a and b.

[0048] The circuit components hosting the second mode 114 are coupled to the load 106 via the coupler 104. This coupling dissipates the second mode. The microwave source 108 is connected to the nonlinear superconducting circuit 100, with frequency f b By generating radiation at its resonant frequency, it is used to drive a second mode. The microwave filter 110, in this application, has a frequency f b It is configured as a band-pass filter having a frequency f. Alternatively, filter 110 has a frequency f. aIt may be configured as a band-stop filter in and placed between the environment and the two modes to isolate the first mode and thus prevent the first mode from being affected by additional losses resulting from unwanted coupling to the load 106. Alternatively, f a >f b (or f b >f a In this case, it may be configured as a low-pass (or pass-through) 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] Figure 2 shows the stabilization of the quantum manifold in the coherent state of the first mode, achieved by the 2:1 photon conversion performed by circuit 100.

[0050] This figure shows the amplitude

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[0051] In Figures 3, 6, and 9, only circuit 100 is shown. For simplicity, the coupling to the load, the microwave source driving the buffer, and the microwave source driving the ATS for parametric pumping are omitted from the illustration.

[0052] Figure 3 shows the electrical equivalent diagram of the first embodiment of the galvanic cat circuit 100 shown in Figure 1.

[0053] The circuit 100 comprises a nonlinear resonant section 30 and a linear resonant section 32. The nonlinear section comprises an ATS 34 and a capacitive element 302. The nonlinear resonant section 30 and the linear resonant section 32 are galvanically connected together. To avoid any misunderstanding, the expression "galvanically connected" means that there is a short conductive section connecting the nonlinear resonant section 30 and the linear resonant section 32, i.e., a short conductive track or any other means that ensures a physically continuous conductive junction. The expression "short" means that the conductive track is at a frequency f a and f b This means that the impedance of this conductive track is negligible compared to the impedances of the nonlinear resonant section 30 and the linear resonant section 32 in the ATS. If this conductive track has an impedance that cannot be ignored, it means that the zero-point fluctuations φ of the first and second resonant modes in the ATS are negligible. a and φ b It operates as a voltage divider to reduce [something]. This is contrary to the purpose of the present invention. In the embodiments described herein, the conductive tracks are also arranged so that neither nonlinear nor linear resonant sections form a shunt.

[0054] In the embodiments described herein, the nonlinear resonant section 30 comprises an ATS 34 and a capacitive element 302 connected in series. The linear resonant section 32 comprises an inductive element 320 and a capacitive 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 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 is in contrast to the prior art in which the nonlinear part is coupled to the linear part via a capacitive or inductive coupler, resulting in one resonant mode having a much lower involvement in the ATS, and thus a smaller zero-point fluctuation and a smaller 2:1 photon conversion rate g2. Since the minimal galvanic circuit proposed in this application has a symbolic representation without such a coupler, both resonant modes are expected to have a significant involvement in the ATS. It could be thought that the ATS 34 is capacitively coupled to the linear bare mode via the capacitor 302, but in reality, it resonates with the capacitor 302 so that they form a nonlinear bare mode that is galvanically coupled to the linear bare mode.

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

[0057] Furthermore, it should be noted that the only possible circuit topology is one series resonant and one parallel resonant. Therefore, there are only two possible minimum implementations of the circuit including the ATS: in one case, as shown in Figure 3, the ATS acts as the inductive element of the series resonant; and in the other case, as shown in Figure 6, the ATS acts as the inductive element of the parallel resonant.

[0058] L is the value of the inductive and capacitive elements in the parallel resonant section. parallel and C parallel Using these values ​​respectively, L is set as the value of the inductive and capacitive elements in the series resonant section. series and C series When using each of these, the bare modes are described below. - 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 ) and here, the inductance L ats This is the effective inductance value of ATS34 near the global minimum of its potential energy. At the saddle point, the effective inductance L ats This is equal to the shunt inductance of the ATS34.

[0060] The linear portion of the Hamiltonian in a galvanic circuit is expressed by the following equation.

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[0061] The modes a and b mentioned above correspond to the Hamiltonian H described above. lin This is a mode that diagonalizes the θ. The dimensionless coupling constant k can take any value between 0 and 1. Values ​​close to 0 indicate a weakly coupled mode, while a value of 1 indicates a maximumly coupled mode. As expected, the smallest galvanic design allows for k values ​​greater than 1 / 2 to be easily reached. The possibility of reaching such large coupling constants is unique to the field of superconducting circuits, as first shown in Non-Patent Document 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-parameter elements in a patterned layer of superconducting material, as described below. - Two adjacent plates forming a capacitor in parallel with respect to a superconducting wire, a single Josephson junction, or an array of Josephson junctions forming an inductor. - A portion of a superconducting transmission line that forms a so-called λ / 4 resonator, terminated with two different boundary conditions (short-circuited to ground at one end and open at the other). The transmission line is, for example, a coplanar waveguide type or a microstrip type, or - A portion of a superconducting transmission line terminated with two identical boundary conditions (open-open or short-short) to form a so-called λ / 2 resonator; the transmission line is, for example, of the coplanar waveguide type or microstrip type.

[0063] In this technology, ATS34 is implemented as known, for example, in Non-Patent Document 3 of the thesis. It has a structure with two Josephson junctions in parallel and an inductive element in parallel between them. As a result, ATS34 has two connected loops, and each loop includes a Josephson junction parallel to the shunt inductive element. ATS34 generates DC and AC magnetic flux biases in both of its loops. The DC bias sets the operating point of the ATS. It may be operated near the so-called saddle point. The saddle point is a sweet spot in frequency and has a small cross-correlation term. The AC magnetic flux bias corresponds to a parametric pump at 2f a -f b This AC magnetic flux bias is usually selected to drive the common mode of the two loops.

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

[0065] The reason for plotting the ratio g2 / φ p is that φ p is proportional to the amplitude of the parametric magnetic flux pump set by the amplitude of the microwave source 102, and for some reason it is arbitrary. In contrast, the ratio

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[0066] This figure shows the conventional values ​​for inductive elements 34 and 320, specifically for g² / φ above 250 MHz. p This demonstrates that the value can be easily reached. In fact, the applicant's research has shown that values ​​exceeding 100 MHz, which are on the order of magnitude greater than known prior art, are guaranteed and that values ​​of several hundred MHz are achievable. In comparison, g2 / φ achieved with Lescanne2020 p The value was only 9.6MHz. In Berdou2022, it was at least an order of magnitude smaller.

[0067] Figure 4 also shows the zero-point variation φ a and φ b This shows the product.

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[0068] The Hamiltonian of the 2-to-1 photon conversion depends on the third-order expansion of the ATS potential energy in φ a *(a + a † ), and φ b *(b + b † ). Therefore, the rule of thumb known to those skilled in the art is to keep φ a *max(α, 1 / 2) and φ b *max(β, 1 / 2) small compared to π. Here, 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 fluctuations. When α = 2, it is considered safe to maintain φ a <0.1. On the other hand, β tends to be very close to zero, and thus max(β, 1 / 2) = 1 / 2. This means that larger values of φ b , typically φ b <0.3, are acceptable. Figure 4 shows that g2 can be increased by more than one order of magnitude compared to the prior art while maintaining safe values of the parameters.

[0070] Figure 4 also shows that it is easily possible to reach more aggressive values of φ a and φ b , resulting in a further increase in g2. It should be noted that it is unknown how far φ a and φ b can be pushed. This is because the area of the cat qubit is very recent and lacks such research. Therefore, large φ a and φ bThe galvanic cat design, with its obvious potential to achieve this, makes it possible to conduct such research.

[0071] Figure 11 shows the parallel connection of the linear capacitor 322 and linear inductor 320 of the linear resonant section 32 in the case of Figure 3, and the ratio of the squares of the zero-point fluctuations of modes b and a across the parallel section.

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

[0073] Furthermore, the pump frequency is 2f a -f b ga f a and f b It is also possible to select the first and second resonant frequencies to be somewhat distant from both of them. For example, 2f a -f b is, f a / 2 and f b It is made smaller than / 2. This means the memory frequency f a The buffer frequency f should not deviate too much from twice the current frequency. b This can be achieved by selecting a small pump frequency f p =2f a -f b This brings about f p The value of f is 1 GHz in Figure 4, which means that f a It is more than four times smaller than, and f b It is eight times smaller than that.

[0074] This is advantageous because the first-order term in the ATS potential energy expansion indicates that the parametric pump can directly drive the circuit. As shown in the appendix to Non-Patent Literature 3, this driving results in spurious dynamic AC Stark shifts and dynamic crosscar terms, which are detrimental to cat qubit operation. For low pump frequencies, direct driving of the circuit is far less efficient, which results in much smaller dynamic AC Stark shifts and dynamic crosscars.

[0075] Another advantage is the value 2f a -f b to f a and f bThis allows for separation from both sides, and 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 theoretical basis for this is that the field of quantum computing is very young, especially in the realm of cat qubits, and generally, very gradual changes are preferred. One reason why Lescanne2020 and Berdou2022 have an ATS located in the second mode and a first mode that is weakly capacitively coupled to the second mode is to minimize the decay of the first mode caused by the decay of the second mode in the load. In fact, conventionally, it has been considered much preferable to couple the nonlinear elements of a quantum circuit to a single mode and weakly couple other modes to it.

[0077] Galvanically coupling nonlinear elements is clearly not a gradual change, and goes against all preconceived notions.

[0078] Finally, f a and f b Setting a large frequency interval between them is important because of the variability of the inductance of various Josephson junctions in the circuit, mainly f a and f b This makes the design more robust to the uncertainties of nanofabrication, such as known issues that affect the accuracy of predictions.

[0079] Figure 5 shows the implementation of the circuit in Figure 3.

[0080] Similar components are given the same designation. Only the first digit of the designation changes from "3" to "5," that is, the capacitive element 322 in Figure 3 has the designation 522 in Figure 7.

[0081] This figure is a top view of a superconducting chip layout designed by the applicant in correspondence with the circuit in Figure 3. The light gray areas correspond to a metallized surface containing tantalum or aluminum. The gray areas correspond to a sapphire substrate on which the circuit is located. Other materials may be used to mount the superconducting circuit, such as niobium, NbTi, or TiN for metallization and silicon or quartz for the chip.

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

[0083] Although not shown in Figure 5, the coupler 104 has the advantage of being implemented by capacitively coupling the CPW transmission line to electrode 522, as shown in Figure 11. Furthermore, the cat qubit may be coupled to other cat qubits or readout transmons via a CPW bus 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 passing close to each side of the ATS34 may be flux-biased and used to set their operating point in DC and to realize a parametric pump in AC.

[0084] The circuit in Figure 5 shows a lumped-parameter and grounded implementation of the resonant modes of the circuit in Figure 1. Since the total size of circuit 100 is shorter than one-quarter wavelength of the first and second modes, this design can be described as lumped. Since the first and second modes correspond to the oscillations of charge and current between the electrodes and the ground plane 50 to which they are galvanically coupled via the ATS 34, this design can be described as grounded.

[0085] Other implementations of inductors, such as geometric inductors consisting of meander-shaped or spiral-shaped lines, are also possible.

[0086] This grounded design is more sensitive to defects in the ground plane and may experience greater crosstalk than the differential design, but it is much more compact and minimizes parasitic capacitance that can form a shunt in 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. This means that the first and second modes correspond to oscillations of charge and current between pairs of electrodes galvanically isolated from the ground plane 50. Although differential designs occupy more space than grounded designs, they have the advantage of providing better isolation from lossy components at the ground location, such as wire bonding (not shown in the drawings), or other components that may be patterned on the chip, such as other cat qubits, and thus reducing crosstalk.

[0088] Figure 6 shows a second embodiment similar to that in Figure 3. The main difference is that the components of the nonlinear resonant section 62 are arranged in parallel here, whereas in the embodiment of Figure 3 they were arranged in series. Similarly, the components of the linear resonant section 60 are arranged in series here, whereas in the embodiment of Figure 3 they were arranged in parallel.

[0089] Similar components are given the same designation. Only the first digit of the designation changes from "3" to "6"; that is, the capacitive element 302 in Figure 3 has the designation 602 in Figure 6.

[0090] Figure 7 is similar to Figure 4, but is a diagram based on the circuit of Figure 6. For simplicity, it is not further explained. It should be noted that Figure 11 and its explanation remain quantitatively valid even in the case of the circuit of Figure 6.

[0091] Figure 8 shows an implementation of the circuit in Figure 6. It is similar to the implementation of the circuit in Figure 3 shown in Figure 5, except that the linear inductor and ATS34 have been replaced so that the linear and nonlinear resonant sections are interchangeable. This design is advantageous in that the ATS34 is galvanically coupled to the ground plane, thereby allowing for easier and greater coupling to the flux line (not shown).

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

[0093] In the embodiment of Figure 9, there is only one resonant region that generates both the first and second modes together with the ATS34. This embodiment differs from the embodiments of Figures 3 and 6 in that the characteristics of the first and second resonant modes are not precisely described by the simplified symbolic representation that typically involves only two LC resonators. This illustrates why the present invention contradicts existing preconceptions, as 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 resonant region and the ATS, but this is not possible here, and it is necessary to consider more bare modes of the resonant region.

[0094] As shown in this figure, the ATS34 is connected to a transmission line 90 which has an open end. Since the ATS directly terminates the transmission line hosting the resonant mode, this circuit is galvanic. This means it can be implemented with the same materials as the components in Figure 5. Given φ b or φ aIf the characteristic impedance of the transmission line required to reach the target is too high to be geometrically constructed, 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 in various geometric shapes, such as a 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 to couple to the environment.

[0095] Figure 10 is somewhat similar to Figures 4 and 7, but there are differences due to the need to consider several harmonics resulting from the implementation of the transmission line. As a result, there are three graphs in Figure 10 that explain 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 is typically given for a CPW on fire. r =5.6 is set.

[0097] The upper graph of Figure 10 shows the g² / φ values ​​that can be obtained for various lengths of CPW. p The result is shown. The solid line represents the 2:1 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 This corresponds to the first resonant mode index a=0 and the second resonant mode index b=1. The dashed line represents the 2:1 photon rate g2 / φ calculated using the first resonant mode (or second resonant mode) obtained as the second harmonic (or third harmonic) of the nonlinear quantum circuit. p This corresponds to the first resonant mode index a=1 and the second resonant mode index b=2.

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

[0099] In the lower graph of Figure 10, the zero-point phase fluctuations φ0, φ1, and φ2 across the first (or fundamental), second, and third harmonics of the ATS 34 are expressed in radians.

[0100] Contrary to Figures 4 and 7, the first and second resonant frequencies are not fixed, so it is necessary to show their fluctuations. A large detuning frequency of several GHz is required. b -f a And, φ a and φ b Regarding safe values, g2 / φ significantly exceeds conventional technology. p It can be seen that the value can be reached.

[0101] For example, by fine-tuning the characteristic impedance of the transmission line, or by using other harmonics, f can be adapted to suit a given application. a ,f b , φ a , φ b , and g2 / φ p It will be obvious 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, such as an inductive short circuit, must be included in the potential energy of the ATS, and as a result, its operating point and dynamics will be modified.

Claims

1. A nonlinear superconducting quantum circuit comprising at least one resonant section (30, 32; 60, 62) and a galvanically connected superconducting quantum interferometer (34) having asymmetric threads, The above 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 described above is different from 1 / 2. The above-mentioned at least one resonant section (30, 32; 60, 62) comprises a linear resonant section (32; 60) having at least one inductance (320; 600) and at least one capacitor (322; 602), and a nonlinear resonant section (30; 62) having at least one capacitor (302; 622) and a superconducting quantum interferometer (34) having the above-mentioned asymmetric thread, The linear resonant section (32) and the nonlinear resonant section (30) are galvanically connected, and are arranged such that one side has its components connected in series and the other side has its components connected in parallel. The at least one resonant section (30, 32) is configured to have inductance and capacitance values ​​such that the nonlinear superconducting quantum circuit (100) causes the first mode (a) and the second mode (b) by the superconducting quantum interferometer (34) having asymmetric threads, such that the nonlinear superconducting quantum circuit (100) has a zero-point fluctuation of 0.05 radians or more in the superconducting phase across the superconducting quantum interferometer (34) having asymmetric threads in the cases of the first mode (a) and the second mode (b). Nonlinear superconducting quantum circuits.

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

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

4. The above nonlinear superconducting quantum circuit exists on a dielectric substrate and has a boundary with respect to the common ground plane (50;80) by the exposed portion of the dielectric substrate. The linear resonant portion and the nonlinear resonant portion are realized in multiple physically separate parts of the nonlinear superconducting quantum circuit. The nonlinear superconducting quantum circuit according to claim 2.

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

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

7. The above linear resonant section and the above nonlinear resonant section are galvanically insulated from the above common ground plane. The nonlinear superconducting quantum circuit according to claim 4.

8. The above nonlinear superconducting quantum circuit exists on a dielectric substrate and has a boundary with respect to the common ground plane due to the exposed portion of the dielectric substrate. The above-mentioned at least one resonant section is realized in the transmission line (90). The nonlinear superconducting quantum circuit according to claim 1.

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

10. The first and second resonant frequencies are set such that the difference between twice the first resonant frequency and the second resonant frequency is less than half of the first and second resonant frequencies, respectively. The nonlinear superconducting quantum circuit according to claim 1.

11. The above transmission line (90) is composed of an array of Josephson junctions or a high-dynamic-dynamic-inductance material. The nonlinear superconducting quantum circuit according to claim 8.

12. The above at least one inductance (320; 600) is composed of an array of Josephson junctions or a high-mechanical inductance material. The nonlinear superconducting quantum circuit according to claim 1.

13. A nonlinear superconducting quantum circuit according to one of claims 1 to 12, A first microwave source (108) connected to at least one of the above-mentioned resonant sections (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 at least one of the above-mentioned resonant sections (30, 32; 60, 62; 90), the second microwave source (102) 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 at least one of the above resonant parts (30, 32; 60, 62; 90), wherein only the second mode (b) is coupled to the load (106), and thereby the first mode (a) hosts the cat qubit. Quantum device.

14. The above quantum device further comprises a microwave filter (110) for coupling to the above load (106), The microwave filter (110) described above is configured to allow the second resonant frequency to pass through and to block the first resonant frequency. The quantum apparatus according to claim 13.

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

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