Superconducting quantum circuits for bosonic codes with galvanic coupling

Abstract

The nonlinear superconducting quantum circuit includes at least one resonator (30, 32) and a superconducting quantum interferometer (34) with an asymmetric thread galvanically connected to the at least one resonator (30, 32). The nonlinear superconducting circuit has a first mode having a first resonant frequency and a second mode having a second resonant frequency. The ratio between the first resonant frequency and the second resonant frequency is different from 1 / 2. The at least one resonator (30, 32) is configured with inductance and capacitance values ​​whose symbolic representations induce the first and second modes through the superconducting quantum interferometer (34) with an asymmetric thread 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 (34) with an asymmetric thread 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 present application aims to improve this situation. To this end, the applicant proposes a nonlinear superconducting quantum circuit comprising at least one resonator and a superconducting quantum interferometer with an asymmetric thread galvanically connected to the at least one resonator. The nonlinear superconducting circuit has a first mode with a first resonant frequency and a second mode with 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 is configured with inductance and capacitance values ​​whose symbolic representations induce the first and second modes through the superconducting quantum interferometer with an asymmetric thread 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 with an asymmetric thread in the first and second modes.

[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. In theory, the detrimental effects of the coupling capacitors in the circuit of Non-Patent Document 3 can be minimized, but large capacitors are known to have losses in superconducting circuits, which increase the phase inversion rate and therefore make their application in practical implementations ineffective.

[0010] In various embodiments, the method may exhibit one or more of the following features: - the at least one resonator has a symbolic representation in which the first mode is hosted in a first resonator comprising at least one inductance and at least one capacitor, and the second mode is hosted in a second resonator comprising at least one inductance and at least one capacitor, and the superconducting quantum interference device having asymmetric threads is disposed between and galvanically coupled to the first resonator and the second resonator; - the at least one inductance and the at least one capacitor of the first resonator and the second resonator are respectively in series or parallel, and the superconducting quantum interference device with asymmetric threads is respectively in parallel or series with the first resonator and the second resonator; - 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 first resonator and the second resonator 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 first resonator and the second resonator are galvanically isolated from the common ground plane; - the first resonator and the second resonator are galvanically connected to 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; and At least one or more of the inductances and / or the transmission lines 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] 4 represents a first implementation of the circuit of FIG. 3. [Figure 6] 4 illustrates a second implementation of the circuit of FIG. 3. [Figure 7] 4 shows an electrical equivalent diagram of a galvanic cat circuit of a second embodiment according to the invention; [Figure 8] FIG. 8 shows a diagram illustrating values ​​of φ for each of the first and second modes of the superconducting quantum circuit of FIG. 7, along with the corresponding g / φ values. [Figure 9] 1 shows a galvanic cat circuit of a third embodiment according to the present invention; [Figure 10] FIG. 10 shows a diagram illustrating the values ​​of φ and f for each of the first and second modes of the superconducting quantum circuit of FIG. 9, along with the corresponding g / φ values. [Figure 11] 4 shows a galvanic cat circuit of a fourth embodiment according to the present invention; [Figure 12] FIG. 12 shows a diagram illustrating the values ​​of φ and f for each of the first and second modes of the superconducting quantum circuit of FIG. 11, along with the corresponding g / φ values. [Figure 13] Complementary to FIG. 4, the ratio of the squares of the zero point fluctuations of the phases of modes b and a in the capacitor and inductor of the second resonator unit according to the first embodiment is shown together with the corresponding g2 / φp values. 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 bmay 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 the 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, 7, 9 and 11, 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 first resonant section 30, a second resonant section 32, and an ATS 34. Both the first resonant section 30 and the second resonant section 32 are galvanically connected to the ATS 34. For the avoidance of any doubt, the term "galvanically connected" means that there is a short conductive section connecting the first resonant section 30 and the second resonant section 32 to the ATS 34, i.e., a short conductive track or any other means that ensures a physically continuous conductive joint. The term "short" means that the conductive track is short enough to connect the first resonant section 30 and the second resonant section 32 to the ATS 34 at a frequency f a and f b This means that the ATS 34, the first resonator 30, and the second resonator 32 have a negligible impedance compared to the impedance of the ATS 34, the first resonator 30, and the second resonator 32. If the 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 first nor the second resonator is shunted.

[0054] This is opposite to capacitive or inductive coupling, where the coupling is ensured by the electromagnetic field without physical contact between the two parts. In these cases, the impedance of the coupling capacitor or inductor due to the self-inductance of the coupling mutual inductor is usually large and severely divides the zero point fluctuation and therefore the 2:1 photon conversion rate g2.

[0055] In the embodiment described herein, the first resonator 30 comprises a series-connected inductive element 300 and a capacitive element 302. Similarly, the second resonator 32 comprises a series-connected inductive element 320 and a capacitive element 322. If the first resonator 30 (or the second resonator 32) is isolated from the rest of the circuit 100, it will host a first bare mode (or a second bare mode).

[0056] As a result of the circuit topology described above, the ATS 34 itself is a coupling element between the first bare mode and the second bare mode. This topology naturally allows both resonant modes resulting from the interaction of the first bare mode, the second bare mode, and the ATS 34 to be strongly coupled to the ATS 34. Due to the coupling of the second mode to the cold bath (load 106) where photons are dissipated, the second mode has a larger loss and constitutes a buffer mode, while the first mode has a smaller loss and constitutes a memory mode.

[0057] The value of the inductive element 300 is L a (or L with respect to the inductive element 320) b ) and the value of the capacitive element 302 is C a (or C for capacitive element 322) b ), this first bare mode (or second bare mode) can be described by: - its angular frequency

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[0058] When the first resonator 30 and the second resonator 32 are galvanically connected to the ATS, the linear part of the Hamiltonian of the circuit is given by:

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[0059] where the dimensionless coupling constant

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[0060] The aforementioned modes a and b are expressed by the Hamiltonian H linThe 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. Values ​​close to 0 indicate weakly coupled modes, while values ​​close to 1 indicate maximally coupled modes. ats ≪L a ,L b The value of the order of 1 / 2 is obtained when L ats ~L a ,L b A value close to 1 is obtained when L ats ≫L a ,L b is obtained when k is in the order of 1 / 2. As expected, since the ATS has a mechanical superconducting inductance, the galvanic design makes it possible to easily reach values ​​of k of the order of 1 / 2 and 1. This further increases the zero-point fluctuation. The possibility of reaching such large coupling constants is unique in the field of superconducting circuits, as first shown in [5].

[0061] As shown in more detail with respect to Figures 5, 6, 9, and 11, 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.

[0062] 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.

[0063] FIG. 4 shows various values ​​of inductive elements 300 and 320, a typical value of 4 nH for the ATS inductance, and a fixed frequency f a = 4.5GHz and f b = 8.0 GHz, g2 / 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 in solid lines in MHz and are established by varying the value of inductive element 300 towards the east and the value of inductive element 320 towards the north, while selecting the values ​​of capacitive elements 302 and 322 to obtain the frequencies mentioned above.

[0064] 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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[0065] This figure shows the g2 / φ above 100 MHz for conventional values ​​of inductive elements 300 and 320. p In fact, studies by the applicant have shown that values ​​of over 50 MHz are guaranteed, and values ​​of several hundred MHz are achievable, which is close to an order of magnitude higher than known prior art. In comparison, the g2 / φ achieved in Non-Patent Document 3 is p The value of was only 9.6 MHz. In Non-Patent Document 4, it was at least an order of magnitude smaller.

[0066] Figure 4 also shows the zero point fluctuation φ a and φ b The product

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[0067] The Hamiltonian for the 2-to-1 photon conversion is φ a *(a+a † ) and 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.

[0068] The term 1 / 2 in max(α or β, 1 / 2) is used to account for the minimum zero point fluctuation. For a typical value of 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.

[0069] 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 b The galvanic cat design, with its obvious possibility of realizing this, makes it possible to carry out such research.

[0070] FIG. 13 shows the ratio of the squares of the zero point variations of modes b and a across the capacitor 322 and the inductor 320 of the second resonator 32, respectively.

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[0071] This is all the more surprising since the idea of ​​galvanically coupling the ATS to both modes would result in a strongly hybridized mode due to the large value of k. Having a low-loss memory mode and strong hybridization is highly counterintuitive. The rationale is that the field of quantum computing, especially in the area of ​​cat qubits, is very young, and very gradual changes are generally preferred. In [3] and [4], one of the reasons for having the ATS located in the second mode and the first mode weakly capacitively coupled to the second mode is to minimize the damping of the first mode due to the damping of the second mode in the load. Indeed, traditionally, it is considered much preferable to couple nonlinear elements in quantum circuits to a single mode and weakly couple other modes to it.

[0072] Galvanically coupling a nonlinear element is clearly not an incremental change and defies all preconceptions. Furthermore, applicants were surprised to find that, despite the strong hybridization of the first and second modes caused by galvanic coupling through the ATS, they were able to couple substantially only the buffer mode to the environment.

[0073] Another advantage of galvanic coupling and large values ​​of k is that the value 2f a -f b is f a and f b The advantage of this is that the first and second resonant frequencies can be selected 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 theb 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 b Therefore, 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] 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

[0077] FIG. 5 shows a first implementation of the circuit of FIG.

[0078] This figure is an optical microscope image of the surface of a superconducting chip manufactured by the applicant corresponding to the circuit of Figure 3. The white areas correspond to the metallized surface, 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.

[0079] The circuit comprises a ground plane 50 on which the circuit 52 is formed. The ground plane 50 is regularly perforated (gray dots 502) to expose the substrate. These holes are implemented to trap superconducting vortices and thereby reduce magnetic noise, but they are optional and may be omitted. Five transmission lines, 503, 504, 505, 508, and 509, are used to input and output radiation to the circuit. These transmission lines are implemented by CPW (coplanar waveguide). Line 503 allows excitation of the Cat qubit. Line 504 is a bus connected at its other end to a readout transmon used to measure the Cat qubit. Line 505 is the beginning of a filter 110 connected to a load 106 and a microwave source 108. Four wire bonds 506 are also provided to ensure equipoise across the ground plane 50. Finally, two magnetic flux lines 508 and 509 are generated by microwave source 102 to generate magnetic flux φ σ and to set the DC operating point of the ATS 34 by applying a DC current.

[0080] The circuit of Figure 5 shows a lumped and differential implementation of the resonant modes of the circuit of Figure 1. The design is lumped because the total size of circuit 100 is less than a quarter wavelength of the first and second modes. The design is differential because 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.

[0081] The pair of large rectangular electrodes labeled 550 and 552 implements the capacitor 302 in the first mode (memory mode) and 322 in the second mode (buffer mode), respectively. The small lines between the two pairs of small metal pads labeled 554 and 556 are chains of Josephson junctions, implementing the inductor 300 in the first mode and 320 in the second mode, respectively. Other implementations of inductors are also possible, for example, geometric inductors consisting of meander-shaped or spiral-shaped lines. The ATS 34 is located close to the magnetic flux lines 508 and 509 that drive it.

[0082] Although this design occupies more space than other designs, it has the advantage of providing better isolation to lossy components such as wire bonds 506 or other components that may be patterned on the chip, such as other cat qubits, thus reducing crosstalk.

[0083] FIG. 6 shows a second implementation of the circuit of FIG.

[0084] Like Figure 5, this figure is an optical microscope image of the surface of a superconducting chip manufactured by the applicant corresponding to the circuit of Figure 3. The white areas correspond to the metallized surface, 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 realize the superconducting circuit, such as niobium, NbTi, or TiN for the metallization and silicon or quartz for the chip.

[0085] Similar components have been given similar reference numerals, with only the first digit of the reference numeral changing from "5" to "6", i.e., ground plane 50 in FIG. 5 has reference numeral 60 in FIG.

[0086] This implementation differs from that of Figure 5 in that it shows a lumped and grounded implementation of the resonant modes of the circuit of Figure 3. This implementation is said to be grounded because the first and second modes correspond to the oscillation of charge and current between the electrodes and the ground plane 60 to which they are galvanically coupled via the ATS 34. The bottom electrode of the ATS is actually imprinted directly onto the ground plane 60.

[0087] The memory mode capacitor 302 has a T-shape and consists of a single electrode having the reference numeral 650. The buffer mode capacitor 322 consists of a single electrode which is a horizontal bar 652.

[0088] An important difference with respect to the circuit of Figure 5 is that in this implementation the memory mode is connected to another galvanic cat circuit (not shown) by a transmission line 620. Furthermore, the drive line 503 of the memory mode and the bus line 504 for coupling the memory mode to the readout transmon are merged into a single line labeled 603-604.

[0089] This grounded design is more sensitive to imperfections in the ground plane and can suffer from greater crosstalk than a differential design, but it is much more compact. Additionally, the coupling of the ATS to the magnetic flux lines is stronger.

[0090] Figure 7 shows a second embodiment similar to Figure 3. The difference here is that the inductances and capacitors in the first and second resonators are in parallel instead of in series. As a result, the ATS is connected in series with the first and second resonators instead of in shunt. The circuit of Figure 7 is the electronic dual of the circuit of Figure 3. All equations relating to the circuit of Figure 3 remain valid except for the following:

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[0091] Similar components have been given similar reference numerals, with only the first digit of the reference numeral changing from "3" to "7", i.e., the first resonant mode 30 in FIG. 3 has the reference numeral 70 in FIG.

[0092] Figure 8 is a diagram similar to Figure 4 but based on the circuit of Figure 7. For simplicity's sake, it will not be described further.

[0093] It will be apparent to those skilled in the art that the implementations of FIGS. 5 and 6 can be adapted to achieve the design of FIG.

[0094] Figure 9 shows a third embodiment of the galvanic cat circuit. This embodiment differs from the embodiments of Figures 3 and 7 in that the characteristics of the first and second resonant modes are usually not 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 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 and more bare modes of the resonator must be considered.

[0095] In the embodiment of FIG. 9, there is only one resonant section that, together with the ATS 34, generates both the first and second modes. As shown in this figure, the ATS 34 forms two sections 90 and 92 as shunts from an open-ended transmission line. Because the ATS is symbolically embedded in the transmission line hosting the resonant modes, the circuit is galvanic. For a small dimensionless coupling constant k, the first mode of the quantum circuit simply corresponds to the LC circuit formed by the static capacitance of the ATS and the transmission line; the second mode corresponds to the bare λ / 2 harmonic of the transmission line coupled to the ATS; the third mode corresponds to the bare λ harmonic of the transmission line coupled to the ATS; and so on. However, at larger k, the magnitudes of the symbolic inductance and capacitance values ​​associated with different bare harmonics of the transmission line are all determined to generate the first and second modes when associated with the ATS 34.

[0096] The transmission line may be implemented with the same materials as the components of Figures 5 and 6. For a given φ b or φ a If the characteristic impedance required to reach is too large to be created 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 section 90 may be capacitively or inductively connected to the microwave filter 110 and the load 106 for coupling to the environment.

[0097] FIG. 10 shows the g / φ that can be obtained for various lengths of CPW sections 90 and 92. p The resulting value curves are shown in Figure 1. In this figure, the ATS parameters are the same as in Figure 4. The CPW is assumed to have a characteristic impedance of 50 Ω, and without loss of generality, the effective dielectric constant ε for the CPW on fire is typically r= 5.6 is set. Furthermore, the first resonant mode (or the second resonant mode) is set as the first (or fundamental) harmonic (or the second harmonic) of the quantum circuit 100. In the upper graph of this figure, the value f a is shown as a dotted line in GHz, and the value φ for the second resonant frequency of the circuit b is shown by the dashed line in GHz, and the corresponding g2 / φ p The level lines are shown in solid lines in MHz. In the lower graph of this figure, the value φ for the first mode 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.

[0098] Contrary to Figures 4 and 8, 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.

[0099] 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.

[0100] The embodiment of Figure 11 is similar to that of Figure 9. Again, there is an open-ended transmission line that transmits both the first and second modes, but the ATS 34 is inserted in series with the two sections 110 and 112 of the transmission line. Figure 12 is the equivalent of Figure 8, but for the embodiment of Figure 11. For simplicity, it will not be described further.

Claims

1. A nonlinear superconducting quantum circuit (100) comprising at least one resonator (30, 32; 70, 72; 90, 92; 110, 112) and a superconducting quantum interferometer (34) having asymmetric threads galvanically connected to the at least one resonator (30, 32; 70, 72; 90, 92; 110, 112), The nonlinear superconducting circuit (100) 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; 70, 72; 90, 92; 110, 112) is configured with symbolic inductance and capacitance values ​​that induce the first mode (a) and the second mode (b) through the superconducting quantum interference device (34) with asymmetric threads, such that the nonlinear superconducting quantum circuit (100) has a zero-point variation of superconducting phase across the superconducting quantum interference device (34) with asymmetric threads in the first mode (a) and the second mode (b) of 0.05 radians or more; Nonlinear superconducting quantum circuits (100).

2. the at least one resonator (30, 32; 70, 72) has a symbolic representation in which the first mode (a) is hosted in a first resonator (30; 70) comprising at least one inductance (300; 700) and at least one capacitor (302; 702), and the second mode (b) is hosted in a second resonator (32; 72) comprising at least one inductance (320; 720) and at least one capacitor (322; 722); the asymmetric threaded superconducting quantum interference device (34) is disposed between and galvanically coupled to the first resonator (30; 70) and the second resonator (32; 72); The nonlinear superconducting quantum circuit of claim 1.

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

4. said nonlinear superconducting quantum circuit resting on a dielectric substrate and bounded to a common ground plane (50; 60) by an exposed portion of said dielectric substrate; the first resonator and the second 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 first resonator and the second resonator are galvanically isolated from the common ground plane (50); 6. The nonlinear superconducting quantum circuit according to claim 4 or 5.

7. the first resonator and the second resonator are galvanically connected to the common ground plane (60); 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, 92; 110, 112); 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. at least one or more of the inductances (300, 320; 700, 720) and / or the transmission lines (90, 92; 110, 112) 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; 70, 72; 90, 92; 110, 112), 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 resonant portion (30, 32; 70, 72; 90, 92; 110, 112) for providing radiation having a frequency equal to the difference between twice the first resonant frequency and the second resonant frequency; a load (106) coupled to the at least one resonator (30, 32; 70, 72; 90, 92; 110, 112), 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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