Nonlinear superconducting quantum circuits

Abstract

In particular, the present disclosure relates to a nonlinear superconducting quantum circuit having a first mode and a second mode. Each of the first mode and the second mode has a respective resonant frequency. The circuit is configured such that the resonant frequency of the second mode is substantially 2N times the resonant frequency of the first mode when a predetermined current of constant magnitude is applied to the circuit. This causes the circuit to essentially perform a resonant 2N to 1 photon exchange between the first and second modes, respectively, where N is a positive integer. This forms an improved nonlinear superconducting quantum circuit.

Description

[Technical field]

[0001] The present disclosure relates to the field of quantum technologies, and more particularly to superconducting quantum circuits, devices including such circuits, quantum computing systems including such circuits, and methods including such circuits. [Background technology]

[0002] In recent years, there has been a great deal of interest in the development of quantum technologies for several applications, such as quantum computing, quantum communication, etc. In particular, superconducting quantum circuits are a promising platform for realizing quantum computing with so-called cat-qubit bits. In this particular context, superconducting quantum circuits are engineered to exhibit specific quantum dynamics, such as stabilizing a quantum manifold of coherent states. The stabilization of a quantum manifold of coherent states has been particularly studied in the following papers: ·“Exponential suppression of bit-flips in a qubit encoded in an oscillator”, Lescanne R.et.al., Nature Physics, 2020; ·“Coherent Oscillations inside a Quantum Manifold Stabilized by Dissipation”, Touzard S.et.al.; Physical Review X, 2018; and ·“Confining the state of light to a quantum manifold by engineered two-photon loss” Leghtas Z.et.al.,Science, 2015.

[0003] The aforementioned stabilization requires engineering a nonlinear transformation between two photons of a first mode that hosts the stabilized quantum manifold and one photon of a second mode, known as the buffer mode. To achieve the nonlinear transformation, existing solutions involve applying an external time-varying excitation to the superconducting quantum circuit in the form of one or more microwave tones at a specific frequency, called a parametric pump, to bridge the energy gap between the energy of the two photons of the first mode and the energy of the one photon of the second mode. These solutions are also known as parametric pumping techniques.

[0004] In this context, there remains a need for improved superconducting quantum circuits for manipulating nonlinear transformations. Summary of the Invention

[0005] Thus, a nonlinear superconducting quantum circuit is provided having a first mode and a second mode, each of the first mode and the second mode having a respective resonant frequency. The circuit is configured such that when a predetermined current of constant magnitude is applied to the circuit, the resonant frequency of the second mode is substantially 2N times the resonant frequency of the first mode. This causes the circuit to essentially perform a resonant 2N to 1 photon exchange between the first and second modes, respectively, where N is a positive integer.

[0006] The nonlinear superconducting quantum circuit may include one or more of the following: A nonlinear superconducting quantum circuit, when a given current is applied to the circuit, has a Hamiltonian that is a function of a set of parameters consisting of parameters of the circuit and parameters of the given current. -Hamiltonian is can be expanded into a sum between at least a leading term and a set of auxiliary terms of the form TIFF2025502859000001.tif6150, where g 2N is a scalar corresponding to the intrinsic coupling strength, a is the annihilation operator of the first mode, b is the annihilation operator of the second mode, TIFF2025502859000002.tif6150 is the reduced Planck constant. the circuit is configured to essentially perform a resonant two-to-one photon exchange between the first and second modes, respectively; the circuit has a symbolic representation including at least one loop including one or more Josephson junctions, the circuit being configured to perform a resonant 2N to 1 photon exchange when a predetermined current is applied to induce a phase difference across the one or more Josephson junctions. at least one loop includes a first Josephson junction arranged in parallel with a first inductive element and a first capacitive element including respective first and second extremum nodes, the symbolic representation being a second inductive element and a second capacitive element arranged in parallel and including respective first and second extreme nodes; A linear combination element, which is either a capacitive element or an inductive element; The first extreme node of each of the loops is connected to the first extreme node of each of the second inductive element and the second capacitive element arranged in parallel via a linear coupling element, and the second extreme node of each of the loops and the second extreme node of each of the second inductive element and the second capacitive element arranged in parallel are connected to a common ground. -When a given current of constant magnitude is applied to a circuit, the Hamiltonian of the circuit is a linear term describing the first and second modes of type TIFF2025502859000003.tif6150; TIFF2025502859000004.tif9150 format, or and at least one nonlinear term in the format of TIFF2025502859000005.tif9150. ω a / 2π is the resonant frequency of the first mode, and ω b / 2π is the resonant frequency of the second mode. E J is the energy of the Josephson junction, ψ DCis the phase difference across the Josephson junction induced by a given current, ψ a is the zero-point variation of the phase across the Josephson junction associated with the first mode, or ψ b is the zero-point variation of the phase across the Josephson junction associated with the second mode, ε describes the linear combination between the first and second modes. Thereby, the circuit TIFF2025502859000006.tif10150 or has TIFF2025502859000007.tif10150 The Hamiltonian can be expanded into a sum including at least the leading term, which is a nonlinear resonance term of the type TIFF2025502859000008.tif6150, and the resonance condition is 2Nω a =ω b is obtained by application of a given current. at least one loop includes a Josephson junction, a first inductive element, and a second inductive element arranged in a series topology, the series topology comprising: a first inner node connecting a pole of the Josephson junction and a pole of the first inductive element; a second inner node connecting the other pole of the Josephson junction and a pole of a second inductive element; a closed-loop node connecting the other pole of the first inductive element and the other pole of the second inductive element; At least one loop is connected to a common ground through a closed loop node; The symbolic representation also includes a first capacitor and a second capacitor; a first capacitor connected in parallel with the first inductive element between the common ground and a first inner node of the loop; a second capacitor connected in parallel with the second inductive element between the common ground and a second inner node of the loop; -When a given current of constant magnitude is applied to a circuit, the Hamiltonian of the circuit is a linear term describing the first and second modes of type TIFF2025502859000009.tif6150; and at least one nonlinear term in the format of TIFF2025502859000010.tif7150. ω a / 2π is the resonant frequency of the first mode, and ω b / 2π is the resonant frequency of the second mode. E J is the energy of the Josephson junction, ψ DC is the phase difference across the Josephson junction induced by a given current, ψ a is the zero-point variation of the phase across the Josephson junction associated with the first mode, and ψ b is the zero-point variation of the phase across the Josephson junction associated with the second mode. Thereby, the circuit has TIFF2025502859000011.tif11150 The Hamiltonian can be expanded into a sum including at least the leading term, which is the type mp nonlinear resonance term of TIFF2025502859000012.tif6150, and the resonance condition is 2Nω a =ω b is obtained by applying a predetermined current. at least one loop includes a first Josephson junction, a central inductive element, and a second Josephson junction arranged in a series topology, the series topology comprising: a first inner node connecting a pole of the first Josephson junction and a pole of the inductive element; a second inner node connecting a pole of the second Josephson junction and another pole of the inductive element; a closed loop node connecting the other pole of the first Josephson junction with the other pole of the second Josephson junction; At least one loop is connected to a common ground through a closed loop node; The symbolic representation also includes a first capacitor and a second capacitor; a first capacitor connected in parallel with the first Josephson junction between the common ground and a first inner node of the loop; a second capacitor connected in parallel with the second Josephson junction between the common ground and a second inner node of the loop; -When a given current of constant magnitude is applied to a circuit, the Hamiltonian of the circuit is a linear term describing the first and second modes of type TIFF2025502859000013.tif6150; and at least one nonlinear term in the format of TIFF2025502859000014.tif7150. ω a / 2π is the frequency of the first mode, and ω b / 2π is the frequency of the second mode. E J is the energy of the first Josephson junction and / or the second Josephson junction, ψ J0 is the phase difference across the first Josephson junction and / or the second Josephson junction induced by a given current, ψ a is the zero-point variation of the phase across the Josephson junction associated with the first mode, and ψ b is the zero-point variation of the phase across the Josephson junction associated with the second mode, Thereby, the circuit has TIFF2025502859000015.tif11150 The Hamiltonian can be expanded into a sum including at least the leading term, which is a nonlinear resonance term of the type TIFF2025502859000016.tif6150, and the resonance condition is 2Nω a =ω b is obtained by application of a given current.

[0007] Further provided is a device including the nonlinear superconducting quantum circuit, the device also including a current source configured to apply a predetermined current of constant magnitude to the nonlinear superconducting quantum circuit such that the resonant frequency of the second mode is substantially 2N times the resonant frequency of the first mode, whereby the nonlinear superconducting quantum circuit performs a uniquely resonant 2N to 1 photon exchange between the first and second modes, respectively.

[0008] In an example, the device may further include a load. The device may also include a microwave source configured to apply microwave radiation at a frequency substantially equal to the resonant frequency of the second mode or 2N times the resonant frequency of the first mode. The device may also include a coupler configured to couple the second mode of the nonlinear superconducting quantum circuit to the load and the microwave source.

[0009] Optionally, the load may be either a resistor, a matching transmission line, or a matching waveguide. Additionally or alternatively, the device may further include a bandpass or bandstop filter connected to the first and second modes of the nonlinear superconducting quantum circuit, the bandpass or bandstop filter configured to only allow coupling of the second mode to the load. Additionally or alternatively, the geometry and / or symmetry of the coupler may be such that it couples only to the second mode.

[0010] Further provided is a quantum computing system comprising at least one such nonlinear superconducting quantum circuit and / or device.

[0011] There is further provided a method comprising providing a nonlinear superconducting quantum circuit, device or quantum computing system, the method also comprising applying a predetermined current of constant magnitude to the nonlinear superconducting quantum circuit such that a resonant frequency of the second mode is substantially 2N times the resonant frequency of the first mode, the nonlinear superconducting quantum circuit implementing a uniquely resonant 2N to 1 photon exchange between the first mode and the second mode, respectively.

[0012] The method may be used in particular to stabilize a quantum manifold spanned by 2N coherent states with the same amplitude and π / N phase difference, in order to allow the encoding of quantum information in the form of cat qubits. The method may include such stabilization. The method may include providing a device as described above, comprising a load, a microwave source configured to apply microwave radiation at a frequency substantially equal to the resonant frequency of the second mode or 2N times the resonant frequency of the first mode, and a coupler configured to couple the second mode of the nonlinear superconducting quantum circuit to the load and the microwave source. The method may also include applying microwave radiation at a frequency substantially equal to the resonant frequency of the second mode or 2N times the resonant frequency of the first mode. The quantum information may then be encoded in the form of so-called cat qubits. [Brief description of the drawings]

[0013] Non-limiting examples will now be described with reference to the accompanying drawings.

[0014] [Figure 1] We show how circuits can be incorporated into devices to stabilize quantum information. [Diagram 2] 1 shows various examples of circuits used to isolate the first mode from the environment. [Diagram 3] We provide an example of how to use circuitry built into the device to stabilize quantum information. [Figure 4] 4a and 4b show how the current is adjusted to reach the frequency matching condition. [Diagram 5] 10 shows some examples of circuit symbolic representations that are alternatively used throughout this specification. [Figure 6] 1 illustrates one embodiment of a superconducting circuit. [Figure 7] 1 illustrates another embodiment of a superconducting circuit. [Figure 8] 1 illustrates another embodiment of a superconducting circuit. [Figure 9] Experimental data obtained for an example circuit is shown. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0015] A superconducting quantum circuit is provided having a first mode and a second mode. Each of the first mode and the second mode has a respective resonant frequency. The circuit is configured such that the resonant frequency of the second mode is substantially 2N times the resonant frequency of the first mode when a predetermined current of constant magnitude is applied to the circuit. Thus, the circuit essentially performs a resonant 2N to 1 photon exchange between the first and second modes, respectively. N is a positive integer (i.e., N is any positive integer equal to or greater than 1), and thus 2N is an even number, e.g., 2, 4, 6 or more. Thus, the expression "2N photons" refers to a discrete (even) quantity of photons defined by the integer 2N.

[0016] Such a superconducting quantum circuit improves the resonant 2N to 1 photon exchange between the first and second modes, respectively. Indeed, in a superconducting quantum circuit, the resonant frequency of the second mode is substantially 2N times the resonant frequency of the first mode when a given current of constant intensity is applied to the circuit. This is in contrast to prior art (such as that performed by parametric pumping techniques) in which an external time-varying excitation is used to bridge the gap between the frequencies of the two modes and perform the resonant 2N to 1 photon exchange. The external time-varying excitation used in such prior art relaxes the constraints on the mode frequencies, but causes undesirable effects such as heating the modes. This heating reduces the coherence time of the circuit and destabilizes the dynamics. This leads to the eventual destruction of any nonlinear mixing involving the resonant 2N to 1 photon exchange. In this respect, the absence of an external time-varying excitation allows the speed of the resonant 2N to 1 photon exchange to be increased by one or two orders of magnitude compared to the prior art.

[0017] The superconducting quantum circuit has a first mode and a second mode. The first mode and the second mode each correspond to a natural resonant frequency of the circuit. For example, the first mode and the second mode may be either an electromagnetic mode or a mechanical mode. The first mode and the second mode each have a respective natural resonant frequency. For example, the first mode (respectively the second mode) may be a mechanical mode and the second mode may be an electromagnetic mode (respectively the first mode is an electromagnetic mode). The first mechanical mode and the second electromagnetic mode may be coupled in the circuit via the piezoelectric effect. The first mode and the second mode each may have a respective resonant frequency, for example, the first mode may be f a =ω a / 2π type resonant frequency, and the second mode may have a resonant frequency of f b =ω b ω may have a resonant frequency of the type ω / 2π, where ω a and ω b is the angular frequency of each respective mode. By "having" a first mode and a second mode, it is meant that a superconducting quantum circuit may comprise components operating in the superconducting regime that host the modes independently of one another 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 (alternatively) in the same subset of components.

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

[0019] The superconducting quantum circuit can 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 can define a lumped element resonator. The capacitive element can be formed (on each of the one or more patterned layers) using two adjacent plates of superconducting material. The inductive element can be formed of a superconducting wire. Alternatively, at least one of the one or more patterned layers can define a portion of a transmission line that each resonates at a frequency that depends on its length. The transmission line can be, for example, a coplanar waveguide or a microstrip line. Still alternatively, the circuit can be embedded in a 3D architecture that includes high-quality 3D modes machined or micromachined into a bulk superconductor that can be used as either of two modes.

[0020] The nonlinear circuit and the predetermined current of constant magnitude are configured such that when the predetermined current is applied to the circuit, the resonant frequency of the second mode of the circuit is substantially 2N times the resonant frequency of the first mode (2Nf a =f b Also called the "frequency matching condition" of the form f a is the resonant frequency of the first mode, and f b is the resonant frequency of the second mode, or equivalently, 2Nω a =ω bIn other words, the predetermined current is an external current that induces an internal DC (direct current) bias in the circuit. The internal current is adapted so that the components / hardware forming the circuit are in a particular regime, i.e., so that the resonant frequency of each of the second modes of the circuit (also called second resonant frequency) is substantially 2N times the resonant frequency of each of the first modes (also called first resonant frequency). Thus, the nonlinearity of the circuit essentially performs a resonant 2N to 1 photon exchange that destroys 2N photons in the first mode (at the first resonant frequency) while generating one photon in the second mode (at the second resonant frequency) and conversely destroys one photon in the second mode (at the second resonant frequency) while generating 2N photons in the first mode (at the first resonant frequency). In fact, the resonant 2N to 1 photon exchange can be achieved, for example, by 2N The exchange of photons between the two modes can be mediated through an element of the circuit (e.g., a nonlinear element). A given current can be calculated as 2Nω a =ω b is applied so that a frequency matching condition of type occurs and the circuit executes resonant 2N to 1 photon exchange dynamics. The given current at which the frequency matching condition is reached is also called the bias point of the circuit. The bias point may also be called the optimal bias point if the selection of the circuit parameters causes the spurious dynamics to vanish at the frequency matching condition, as shown in the following example.

[0021] The predetermined current may be applied directly (i.e., galvanically) to the circuit such that it flows through at least a subset of the circuit's elements and splits into various possible branches. The current flowing through the branches of the circuit is called the internal current and is determined according to Kirchhoff's current law. In an example, the predetermined current may be applied directly via a current source connected to the circuit. In an example, the circuit may have a planar geometry, such that the path of the predetermined current merges with a portion of a superconducting loop of the same planar geometry, e.g., an on-chip current path.

[0022] Alternatively, the predetermined current may be passed indirectly or an internal current may be induced in the circuit by mutual inductance (for example, by means of a coil). In other words, an external inductance is inductively coupled to the circuit through a shared mutual inductance to induce a current in the circuit, which then flows through at least a subset of the circuit's elements. The internal current is therefore a current induced in the circuit through mutual inductance. Since no direct (galvanic) connection with the circuit is required, the predetermined current path may be at the same level as the circuit or may consist of an external coil above or below the circuit, its axis perpendicular to the superconducting circuit plane. In an example, when the mutual inductance is shared with a coil, the latter may be formed of several turns of a material that allows the circulation of a current to generate a magnetic field. The coil may be made of any number of coils required, which increases the mutual inductance. The coil may be made of any material that allows the circulation of a current to generate a magnetic field and then induce an internal current, for example, the coil may be made of a superconducting or non-superconducting material. Alternatively, the coil may be replaced by a permanent magnet that directly generates a constant magnetic field. However, this makes tuning the magnetic field impractical.

[0023] The application of a given current causes a given internal current I to flow through a subset of inductive elements. DC and for a given superconducting phase drop ψ occurring across each inductive element. DC Given the circuit geometry and parameters as a function of the current, the phase drop ψ DC The following example demonstrates how to experimentally tune a given current (or superconducting phase drop ψ) to reach a bias point and thereby enable resonant 2N to 1 photon exchange dynamics. DC Thus, application of a current is adapted to the configuration of the circuit to result in a unique resonant 2N to 1 photon exchange.

[0024] The provided circuit essentially performs a resonant 2N to 1 photon exchange. In other words, the circuit is configured to perform a resonant 2N to 1 photon exchange autonomously / spontaneously, i.e., without the need for an external time-varying excitation to bridge the gap between the 2N times the frequency of the first mode and the frequency of the second mode to achieve a resonant 2N to 1 photon exchange. In other words, the value of the predetermined current is set such that the components hosting the first and second modes (which perform the exchange) are in a regime that allows a resonant 2N to 1 photon exchange. That is, it does not depend on microwave devices external to the circuit, such as parametric pumps. The resonant 2N to 1 photon exchange between the first and second modes, respectively, is a quantum mechanical process that destroys 2N photons of the first mode at the respective resonant frequency of the first mode and creates one photon of the second mode at the respective resonant frequency of the second mode. The resonant 2N to 1 photon exchange is performed when the resonant frequency of the second mode is substantially 2N times the frequency of the resonant first mode by applying a constant magnitude predetermined current to the circuit. "Substantially" means that the value of the predetermined current is such that the frequency of the second mode is equal to 2N times the frequency of the first mode (also referred to in some applications as a frequency matching condition) and the resonant 2N to 1 photon exchange rate g 2N This means that the threshold is approximately equal to the magnitude of the

[0025] Since the superconducting quantum circuit inherently performs said quantum dynamics, this eliminates the need for a parametric pump since the exchange is reliably performed by a circuit including components configured to operate in the superconducting regime that allows for a unique resonant 2N to 1 photon exchange. Thus, the quality of the resonant 2N to 1 photon exchange is improved. The absence of a parametric pump eliminates the appearance of deleterious parasitic interactions that affect the quality of the resonant 2N to 1 photon exchange.

[0026] The circuit may be integrated into a device that may also include a current source configured to apply a predetermined current of constant magnitude to the circuit such that the frequency of the second mode is substantially 2N times the frequency of the first mode. The current source may be directly connected or inductively coupled to the circuit such that the induced internal current crosses at least some (e.g., all) of the elements of the circuit. In other words, the applied current may induce, in certain elements of the circuit, an internal current that travels through the surface of the circuit. The current source may be placed at room temperature and is thus a device that does not include superconducting elements. The current source may apply a predetermined current to the circuit through a conductor that is initially at room temperature, which is connected to a superconducting wire that applies the current to the superconducting circuit as the temperature decreases. The current may also be filtered using a low pass filter along its path to reduce the effects of low frequency noise.

[0027] In an example, the device may further include a load, a 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 with a given resistance, as opposed to a superconducting element) that is external to the superconducting circuit. The load dissipates the photons exchanged from the first mode to the second mode by resonant 2N to 1 photon exchange. In other words, the photons destroyed from the first mode are ejected to the environment via the second mode through the load. The microwave source may be configured to apply microwave radiation at a frequency substantially equal to the frequency of the second mode or at a frequency substantially equal to 2N times the frequency of the first mode (equivalently). In other words, the microwave source may be configured to control the amplitude and phase of the microwave radiation. Thus, the microwave source drives photons (in the form of microwave radiation) into the second mode, which drives the 2N photons of the first mode by resonant 2N to 1 photon exchange. A coupler is an element that can be connected (eg, galvanically, capacitively, or inductively) to an element of a circuit that hosts a second mode and mediates the interaction between the second mode and a load and a microwave source.

[0028] Optionally, the coupler may also be configured to couple the element of the circuit that hosts the second mode of the superconducting quantum circuit to a load and a microwave source.

[0029] The load may be a resistor, a matched transmission line or a matched waveguide. "Matched" means that the transmission line or waveguide is terminated with an appropriate resistance at an end other than the end connected to the element hosting the second mode. The load may be included within the microwave source.

[0030] In an example, this microwave source may be placed at room temperature and connected to the circuit via a coaxial cable. In an example, 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) to thermalize the microwave radiation in a cryogenic environment. This allows the microwave radiation to be applied without adding noise. This microwave radiation does not serve the same purpose as a parametric pump and is not required to obtain resonant 2N to 1 photon exchange dynamics. Here, dissipation from the load, microwave radiation, and resonant 2N to 1 photon exchange dynamics (essentially performed by the circuit) are used together to stabilize the 2N coherent state in the first mode. Indeed, without the microwave radiation, only the vacuum is stabilized in the first mode. In other words, the device allows for the stabilization of a quantum manifold of coherent states beyond the vacuum.

[0031] Optionally, the device may include a bandpass or bandstop filter connected to the first and second modes of the circuit. The bandpass or bandstop filter may be configured to only allow coupling of the second mode to the load. Optionally, the device may also include a microwave filter to protect the first mode from dissipation in the load. This microwave filter may be interleaved between the load and the coupler. From the circuit's point of view, this filter must prevent microwave photons at the first resonant frequency from escaping the circuit. This may be either a bandstop filter at the first resonant frequency or a bandpass filter at the second resonant frequency, since the photons of the second mode are the only ones that need to dissipate in the environment. In some circuits where the two modes have different symmetries, a filter may not be needed and appropriate symmetry of the coupler may be sufficient to prevent dissipation of the first mode.

[0032] Thus, the device allows for the stabilization of 2N coherent states in the first mode (i.e., quantum manifold of coherent states). For example, a microwave source applying microwave radiation through a microwave filter to the second mode can be considered as a 2N photon drive of the first mode when transformed by resonant 2N to 1 photon exchange, and a load dissipating only photons of the second mode can be considered as a 2N photon dissipation of the first mode when transformed by resonant 2N to 1 photon exchange. The 2N photon drive and 2N photon dissipation allow for the stabilization of 2N coherent states in the first mode.

[0033] A method is also provided, the method including the step of providing a superconducting quantum circuit, the method also including the step of applying a predetermined current of constant magnitude to the circuit (or a device including the circuit) such that the resonant frequency of the second mode is substantially 2N times the resonant frequency of the first mode, where N is a positive integer, which effectively operates a resonant 2N to 1 photon exchange between the two modes, respectively.

[0034] The method further includes using the device to stabilize a quantum manifold spanning 2N coherent states, each with the same amplitude and π / N phase difference from one another. This is done by combining the resonant 2N to 1 photon exchange dynamics provided by the circuit with external dissipation and microwave radiation. Quantum information can ultimately be encoded in this quantum manifold as so-called cat qubits.

[0035] A quantum computing system is further provided. The quantum computing system may include at least one of a superconducting quantum circuit and / or a device. The quantum computing system may be configured to use the superconducting quantum circuit and / or the device to execute a high-quality quantum computing protocol. In other words, the quantum system may use the superconducting quantum circuit and / or the device to execute a fault-tolerant quantum computation. This is possible thanks to the fact that the inherent resonant 2N to 1 photon exchange dynamics combined with external dissipation and microwave radiation stabilizes a quantum manifold spanning 2N coherent states of the first mode, also referred to in some applications as the "cat qubit state". In this quantum manifold, the 2N coherent states have the same amplitude and there is a phase difference of π / N between each coherent state. The quantum computing system may use these coherent states to define logical qubits, which are naturally protected, in particular against bit-flip errors, due to the stability of the quantum manifold. Thus, the quantum system may define operations (e.g., CNOT, Hadamard and / or Toffoli gates) that perform computations on the logical qubits. This opens up a complete paradigm for implementing quantum algorithms in a fault-tolerant way.

[0036] The circuit will now be described in more detail.

[0037] A circuit can have a specific Hamiltonian when a given current is applied to the circuit. As known in the art, a Hamiltonian is an operator that corresponds to the total energy of a superconducting circuit, including, for example, both kinetic and potential energy. The Hamiltonian can be used to calculate the time evolution of the circuit. The Hamiltonian can be designed to exhibit the desired quantum dynamics, in particular, a resonant 2N to 1 photon exchange between the first and second modes, respectively. The Hamiltonian of the circuit is herein a function of a specific set of parameters of the superconducting quantum circuit and parameters of the given current. By "parameters" we mean, for example, any kind of physical parameters of the circuit and / or the given current, such as capacitance, inductance, resistance, frequency, phase difference, energy level, zero point variation of phase, Josephson energy or critical current, as well as other parameters, such as voltage and / or current levels (e.g., direct current (DC) bias) from the applied given current. The set of parameters is unique in that it consists of the parameters of the circuit and the parameters of the given current. In other words, the Hamiltonian does not depend on parameters other than those specific parameters. In other words, the Hamiltonian depends only on the circuit and the given current. For the sake of explanation, the set of parameters may be represented as a set P1 U P2, where U designates the summation operator. The set of parameters P1 consists of (only) the parameters of the circuit. The set of parameters P2 consists of (only) the parameters of the given current (such as an induced DC bias).

[0038] Therefore, the total energy of the superconducting quantum circuit does not depend on any device parameters external to the superconducting quantum circuit or on a given current. Indeed, the Hamiltonian only depends on a set of parameters of the superconducting quantum circuit and / or on a given current, which can be predetermined according to quantum engineering specifications. The Hamiltonian is therefore independent of time, and in particular of any time-varying excitation, e.g. from a parametric pump.

[0039] In an example, the Hamiltonian can include linear terms that describe the presence of modes hosted by elements of the circuit, in other words, each linear term describes the presence of a respective one of the first and second modes hosted in the circuit.

[0040] The Hamiltonian may also include nonlinear terms. The nonlinear terms may describe the interaction between the first and second modes. The nonlinear terms may also be called "mixing terms", similar to the frequency mixing that occurs in classical nonlinear microwave circuits. The nonlinear terms may include constants that may act as prefactors. The constants may represent the strength of the interaction between the first and second modes. The prefactors of the nonlinear terms may be smaller or much smaller than the frequency of the system. The nonlinear terms may also be called resonant or non-resonant, depending on their compatibility with the conservation of energy.

[0041] The Hamiltonian may be expandable in a sum of terms. "Expandable in a sum of terms" means that the operator accepts a Taylor series approximation to the sum of terms (recall that it represents the energy of a superconducting quantum circuit). The total number of terms may be finite or (theoretically) infinite, but due to the Taylor series approximation, the sum of the terms always remains finite. The sum may include a leading term and a set of subsidiary terms. By "leading term" we mean a term that has a large influence on the dynamics of the system. For a term to be dominant, it must satisfy two conditions: first, the magnitude of said term (e.g. in absolute terms, or in an appropriate norm, and with respect to the other terms of the Taylor series approximation) should contribute significantly to the magnitude of the nonlinear part of the Hamiltonian. Secondly, this term must be resonant in the sense that it is compatible with the conservation of energy. A set of subsidiary terms is a term whose sum has a magnitude below a given magnitude. All this is known per se from the Taylor series approximation, so we will omit the subsidiary terms here. The Hamiltonian is of the form It can be expanded into the sum of at least the main term of TIFF2025502859000017.tif6150 and a series of auxiliary terms (omitted here).

[0042] Main terms In TIFF2025502859000018.tif6150, a is the annihilation operator of the first mode and b is the annihilation operator of the second mode. Conversely, a † is the generating operator of the first mode, and b † is the generating operator for the second mode. TIFF2025502859000019.tif6150 is a, a † , b, b † which describes a resonant 2N to 1 photon exchange. Thus, the term The leading term containing TIFF2025502859000020.tif6150 represents the annihilation of 2N photons in the first mode and the creation of one photon in the second mode. Because the Hamiltonian is a Hermitian operator, the leading term is the inverse term (also known as the Hermitian conjugate, abbreviated hc) Also includes TIFF2025502859000021.tif6150, where 2N photons of the first mode are created and 1 photon of the second mode is destroyed.

[0043] Scalar g 2N is a function of a set of parameters consisting of circuit parameters and / or parameters of a given current. The scalar g 2N is a prefactor of the leading term, so the strength of the interaction between the first and second modes (also called the "intrinsic coupling strength"), i.e., the term a 2N b † (and their Hermitian conjugates). In other words, g 2N describes the rate of resonant 2N to 1 photon exchange.

[0044] When a given current is applied to induce a resonant 2N to 1 photon exchange (i.e., the given current is adjusted at the bias point), the rate of the resonant 2N to 1 exchange term, i.e., the constant g 2N is nonzero, and the frequency matching condition ensures that the terms are resonant. Furthermore, the constant g 2Nis large (in the sense of a Taylor series approximation) compared to the other nonlinear terms (which are therefore dependent and will not be discussed further).

[0045] Therefore, since a given current is applied to induce a resonant 2N to 1 photon exchange (i.e., the frequency matching condition 2Nω a =ω b satisfies), the leading terms are resonance terms, i.e. compatible with the conservation of energy. Thus, for any N, the leading terms describe nonlinear interactions that are odd powers of the annihilation and creation operators.

[0046] Here, the Hamiltonian is in Kerr terms TIFF2025502859000022.tif27150 TIFF2025502859000023.tif6150 or cross-Kerr term TIFF2025502859000024.tif6150. The nonlinear terms can have significant magnitude and are essentially resonant despite any frequency matching conditions (this occurs only for even powers of the annihilation and creation operators). These Kerr and cross-Kerr terms can be considered detrimental to the desired engineering dynamics. However, the inventors have found that the other nonlinear terms (i.e., the constant K, among others) are not essential to the design of the system. a , K b , χ ab It is understood that the corresponding constants in ω ...

[0047] The circuit may be configured to essentially perform a resonant 2-to-1 photon exchange between the first and second modes, respectively. In other words, N is equal to 1. In this case, the leading term in the Hamiltonian is the 2-to-1 interaction Hamiltonian TIFF2025502859000025.tif6150. Alternatively, N may be greater than 1, for example N=2. The coherent states of the first mode are eigenstates of the annihilation operator a acting on the first mode, e.g., for a given coherent state |α>, it is a|α>=α|α>, where α is a complex amplitude. In the cat qubit paradigm, an intrinsic two-to-one photon exchange stabilizes the quantum manifold of the coherent states of the first mode. In the cat qubit paradigm, the cat qubit states can be defined from the coherent states, e.g., the logical qubit state |0> is defined as |α>, and the logical qubit state |1> is defined as |-α>. Both logical states belong to the quantum manifold of the coherent states stabilized by the two-to-one photon exchange. Thanks to the stability of the quantum manifold achieved by the inherent 2-to-1 photon exchange, cat qubit states are naturally protected from errors such as bit-flip errors. In fact, bit-flip errors can be suppressed autonomously and with exponential speed. This enables the implementation of quantum gates acting on logical qubit states, such as CNOT, Toffoli and / or Hadamard gates, to perform fault-tolerant computations.

[0048] In an example, a superconducting circuit may have a symbolic representation consisting of, for example, a set of interconnected dipoles.

[0049] "Symbolic representation" means an arrangement of symbols and lines that designate a set of interconnected dipoles (also called building blocks) that form a circuit structure (or topology) that is equivalent (functional) to a nonlinear superconducting circuit.

[0050] In other words, as is classical in the field of superconducting circuits, a nonlinear superconducting circuit is constructed to achieve a function defined by its symbolic representation, in other words, the function of a theoretical set of interconnected dipoles represented by the symbolic representation. In further words, although the circuit may be constructed using patterned layers of superconducting material, it should be understood that the circuit allows for symbolic representation by dipoles, e.g., capacitors, inductors, and / or Josephson junctions. Although the exemplary dipoles illustrate discrete elements, it should be understood that these elements correspond to equivalent circuits of distributed elements in certain frequency ranges (e.g., low frequencies), as known from the art.

[0051] The interconnections (of a set of interconnected dipoles of the symbolic representation) are described via a network topology, where each branch represents a dipole of the circuit, a node is a connection point between two or more branches, and a loop may be a closed path of the circuit, i.e., a path formed by starting at a given node and returning to the starting node without passing through any node more than once. Each branch may contain components (e.g., two or more) connected in parallel or in series.

[0052] For example, a pair of components including an inductor and a capacitor (i.e., an LC resonator) may be, for example, Two adjacent plates forming a capacitor in parallel with a superconducting wire forming an inductor; a section of a superconducting transmission line that is terminated with two different boundary conditions (short-circuited to ground at one end and open at the other end) forming a so-called λ / 4 resonator, the transmission line being, for example, of the coplanar waveguide or microstrip type, a section of a superconducting transmission line terminated with two identical boundary conditions (open or short circuit) forming a so-called λ / 2 resonator, the transmission line being, for example, of the coplanar waveguide or microstrip type, or A 3D cavity carved into a block of superconducting material that resonates at a given frequency that depends on its dimensions It may be implemented by distributed elements in a patterned layer of superconducting material such as

[0053] As is known in the art, such distributed elements may have higher frequency modes that are irrelevant and insignificant to the dynamics described herein. These distributed elements may therefore be represented in a symbolic representation. This symbolic representation may be refined by adding elements such as series inductors with respective wire connections or parallel capacitors between any two nodes of the circuit, or by adding nodes and branches to take into account other modes of the distributed elements. Thus, the symbolic representation better describes the distributed elements, but does not change the working principle of the circuit. Thus, as is known in the art, the physical circuit (manufactured in the real world) and the symbolic representation are usually considered equivalent. In fact, the refinement dipole of the symbolic representation only adjusts the zero-point variation of the resonant frequency or phase compared to the basic model. When designing a circuit, the final geometry can be fully and accurately simulated using a finite element solver that easily gives the frequency of the modes, the dissipation resulting from the load, and the zero-point variation of the phase across the Josephson junction, which are the only unknowns for calculating the resonant 2N to 1 photon exchange rate in any configuration.

[0054] A symbolic representation of a circuit (e.g., a set of connected dipoles or components) may include at least one superconducting loop, i.e., a series of components (represented in the symbolic representation by a dipole) connected together to form a cycle. At least one loop may include one or more Josephson junctions. Each Josephson junction may consist of a thin insulating layer separating two superconducting leads, which allows Cooper pairs to tunnel through the insulating layer. Each Josephson junction may be represented by a superconducting loop such that U(ψ)=-E J It may have a potential energy of the type cos(ψ). E J is the Josephson energy of the junction. TIFF2025502859000026.tif10150, where Φ is the integral of the voltage across the junction and Φ is the magnetic flux quantum. In an embodiment, the Josephson energy can be tuned during fabrication by selecting the surface and thickness (and therefore the room temperature resistance) of the insulating barrier.

[0055] Josephson junctions are sometimes called nonlinear inductors. In fact, E L The potential energy of an inductive element with induction energy is U(ψ)=E L ψ 2 By comparing φ / 2 with the potential energy of the junction, to first order in ψ the junction behaves as an inductor. When a given current is applied to the circuit, an internal current flows through the junction (either from a direct connection or from mutual inductance with the superconducting loop in which it is embedded), which causes a phase drop ψ across the junction. DC Thus, application of a given current modifies the potential energy of the Josephson junction as follows: TIFF2025502859000027.tif6150

[0056] The first term on the right side of the sum of the above equation is the joint E J →E J cos(ψ DC ) as a function of the effective Josephson energy. Since the inductance of a Josephson junction is inversely proportional to the Josephson energy, the DC offset ψ DC allows adjusting the inductance of the junction and therefore the frequency of the mode involved in this junction. In the example, the DC offset satisfies the frequency matching condition 2Nω a =ω b The second term represents a nonlinearity corresponding to the sinusoidal nonlinearity. The Taylor series expansion of the sinusoidal nonlinearity contains terms of odd powers of ψ that provide odd wave mixing, and thus contain terms that describe a resonant 2N to 1 photon exchange. In particular, when the phase drop is greater than the value ψ DC= π / 2 is the point where the inductance of the junction becomes infinite (so the junction behaves as an open element). As a result, the cosine nonlinearity of the potential energy vanishes (i.e., the potential energy is zero) and the sine nonlinearity is at a maximum.

[0057] Other inductive elements in the loop that are linear may maintain the same inductance when a DC current flows through the loop or when a DC phase drop occurs across the loop.

[0058] In the embodiment, the induced energy E L The energy E embedded in at least one loop of J The Josephson junction has a ratio β=E J / E L The effective magnetic flux passing through the loop can be explained by ext When biased by a given current, which is equivalently described by ext and the phase drop ψ across the junction which depends only on β DC In the example, the phase drop can be calculated numerically as the solution of the following equation: When TIFF2025502859000028.tif6150 is used, TIFF2025502859000029.tif6150

[0059] The solution of the above equation is f(ψ ext For example, the inductance E L The loop contains two junctions, each with energy E J , the phase drop across each junction is ψ DC =f(ψ ext / 2,β / 2).

[0060] The Hamiltonian of the circuit may be determined in any manner. For example, the Hamiltonian may be determined by first determining the equivalent inductance of each junction when a given supercurrent is applied to the circuit. This equivalent inductance (which depends on the DC phase drop across the junction) is L J =ψ0 / E J cos(ψ DC ), where ψ0 is the reduced flux quantum Φ0 / 2π. The frequencies of the modes can be calculated algebraically or numerically by replacing the junctions in the circuit with their equivalent inductances, and can also be done for more complex circuit layouts by performing microwave simulations with finite elements. These frequencies directly give the linear part of the Hamiltonian. The mode frequencies can be calculated together with the mode geometry, which describes what oscillatory phase difference exists between any two points in the circuit when the mode is excited. The magnitude of the oscillatory phase difference imposed on the Josephson junction is of particular interest for calculating the nonlinearity of the system. In the quantum regime, the phase difference across the junction can be written as: TIFF2025502859000030.tif6150 where ψ DC is the DC phase offset calculated earlier, and ψ a and ψ b is the zero-point variation in phase across the junction that is directly related to the shape of the mode up to normalization. The phase difference includes other terms in "..." (e.g., associated with other modes), but these terms are omitted as they are not important to the present invention.

[0061] The nonlinear term in the Hamiltonian implements a Taylor approximation to the potential energy of the junction, -E J cos(ψ) and the contribution of each junction in the circuit. To fully describe the circuit, it is necessary to find the DC phase drop across the junctions, the frequencies of the modes, and the zero variation in phase across the junctions associated with each mode of the system.

[0062] In the example, the circuit operates at each frequency ω a / 2π and ω b Since we host the first and second modes at / 2π, the linear part of the Hamiltonian can be written as: TIFF2025502859000031.tif6150, where TIFF2025502859000032.tif6150 is the reduced Planck constant.

[0063] At least one term of the interaction provided by a Josephson junction is given in the following form: TIFF2025502859000033.tif6150 where ψ DC is the DC phase drop across the junction induced by a given current, and ψ a and ψ b are the zero-point variations in phase across the junction associated with the first and second modes, respectively. The Hamiltonian can then be expanded in a Taylor series (e.g., by any desired Taylor series expansion) to give the resonant 2N to 1 photon exchange Hamiltonian The result is TIFF2025502859000034.tif6150, where TIFF2025502859000035.tif10150 is 2Nω a =ω b The resonant 2N to 1 photon exchange rate is therefore dependent only on the parameters of the circuit, such as those mentioned above, and for a given current.

[0064] In contrast, as mentioned above, prior art implementations involve the use of a parametric pump to perform the two-to-one photon exchange. In these prior art implementations, the two-to-one photon interaction Hamiltonian is TIFF2025502859000036.tif9150 format, where the coupling term g2(t) is modulated via a parametric pump, which has a frequency ω p =2ω a -ω bInject an external time-varying parameter with: Parametric pumps are used in the prior art to resonate the nonlinear interactions, but have deleterious effects.

[0065] Examples and diagrams of circuits and devices are now described with reference to the drawings.

[0066] An example of a device is shown in Figure 1.

[0067] For illustrative purposes, N=1 in the following description, but the description also applies to any N. The device incorporates a nonlinear superconducting circuit 100 that essentially performs a two-to-one photon exchange (represented by the single arrow 1200 and double arrow 1100 back and forth) between a first mode a 101 and a second mode b 102. A current source 103 is connected to the nonlinear superconducting circuit via a wire. In other words, the current source 103 is directly connected to the circuit such that a predetermined current flows through at least a subset of the elements of the nonlinear superconducting circuit 100. The current source 103 is configured to apply a predetermined current to the circuit 100. The current source 103 enables three-wave mixing interactions and achieves a frequency matching condition of 2f a =f b The component of the circuit that hosts the second mode 102 is coupled to a load 105 via a coupler 104. This coupling makes the second mode dissipative. The device also tunes the frequency f b The device incorporates a microwave source 106 of frequency f b Alternatively, the filter 107 may be configured as a bandpass filter having a frequency f a 105。 Alternatively, the first mode may be configured as a band-stop filter and may be placed between the environment and the two modes to isolate the first mode and thus prevent the first mode from suffering additional losses due to undesired coupling to the load 105.

[0068] FIG. 2 shows an example of second mode coupling of a superconducting circuit to a load, and a filter that may be integrated into the device to allow only second mode coupling to the load.

[0069] FIG. 2a illustrates a second mode 102 of the circuit 100 of FIG. 1 coupled to a load at a second resonant frequency f b 105) (first resonant frequency f a at frequency f b 2 shows a schematic diagram illustrating coupling 200 to a device via coupler 104 and filter 107 to ensure that circuit 100 is driven by microwave source 106 at frequency f b 2b and 2d) or a bandpass filter at frequency f a This allows for a band-stop filter in (also shown in Figures 2c and 2e). Alternatively, if the circuit is fabricated in a 3D architecture, another possible solution is the use of a waveguide high-pass filter to couple the second mode to the load, thanks to its high-frequency selectivity.

[0070] 2b to 2e show different possibilities for coupling 200. FIG.

[0071] FIG. 2b shows a (201) frequency f b The bandpass filter 210 (which is an LC oscillator) has an impedance of TIFF2025502859000037.tif6150 adjusts the width of the bandpass Configured to resonate with TIFF2025502859000038.tif6150.

[0072] Figure 2c shows a frequency f a At 201, a band-stop filter 220 is shown. The band-stop filter 220 (which is an LC stub) is TIFF2025502859000039.tif6150 is configured to resonate, and its impedance TIFF2025502859000040.tif6150 adjusts the width of the band rejection.

[0073] Figure 2d shows the frequency f b 2 shows a bandpass filter 210 at frequency TIFF2025502859000041.tif6150 is configured to resonate, and its impedance TIFF2025502859000042.tif6150 adjusts the width of the bandpass. This configuration is convenient because it also allows the microwave radiation (e.g., RF) input port of the device to be used to input a current bias.

[0074] FIG. 2e shows a (202) frequency f a The band-stop filter 220 (which is an LC stub) is shown at frequency TIFF2025502859000043.tif6150 is configured to resonate, and its impedance TIFF2025502859000044.tif6150 adjusts the width of the band rejection.

[0075] Figure 3 shows an example of quantum manifold stabilization of the first mode coherent state achieved by resonant 2N-to-1 photon exchange.

[0076] Figure 3a shows a linear scheme involving single-photon drive (microwave tone at the mode frequency) characterized by single-photon dissipation with amplitude ε1 and rate κ1. After a timescale 1 / κ1, the mode state converges to a single coherent state 301 of amplitude α=2ε1 / κ1. This is a stable steady state of the dynamics. However, this steady state is unique and cannot encode any information. The states are represented as blurred points in the orthogonal space of the modes due to the uncertainty principle of quantum mechanics.

[0077] Figure 3b shows the first mode, which is subjected to two-photon drive with intensity ε and two-photon dissipation rate κ. This is the amplitude TIFF2025502859000045.tif6150 and has two stable steady states (302, 303) of opposite phase. State |0> 302 is enclosed by a solid line and state |1> 303 is enclosed by a dotted line since there are two possible states and information can be encoded in them. This encoding is robust against bit-flip errors that flip the system between state |0> and state |1> thanks to the stable nature of the dynamics that converges to two states. The encoding does not correct other error channels (i.e. it does not correct phase reversal errors), although additional error correction schemes may be added to treat this separately. This stabilization is made possible by coupling to an additional mode and manipulating a two-to-one photon exchange between the first and second modes.

[0078] Figure 3c shows the N=2 case using four-photon drive and four-photon dissipation to stabilize four states in phase space. In this four-dimensional manifold, it is possible to encode a state |0> (shown as a solid line) in one superposition of two coherent states 304 and a state |1> (shown as a dashed line) in another superposition of two coherent states 305. There are two degrees of freedom remaining in the four-dimensional manifold that can serve as an error manifold used to perform first-order quantum error correction. This stabilization is made possible by coupling to an additional mode and manipulating a four-to-one photon exchange between the first and second modes.

[0079] Figure 3d shows an increase in the dimension of the stabilized manifold. The increase in dimension allows for higher order error correction compared to lower values ​​of N. With 6-photon drive and 6-photon dissipation (i.e., N=3), it is possible to stabilize a 6-dimensional manifold of coherent states that can perform second-order quantum error correction. This stabilization is made possible by coupling to an additional mode and manipulating a 6-to-1 photon exchange between the first and second modes.

[0080] FIG. 4 shows an example of how to determine the intensity of the current applied to the circuit.

[0081] FIG. 4a shows the adjustment of a given current applied to the superconducting circuit. For a given N, the bias point is set so that the resonant frequencies of the first and second modes are in the matching condition 2Nω a =ω b In other words, the equation is a point where the constant g 2N (representing the rate of resonant 2N to 1 photon exchange). The bias point can be reached by varying the current in the circuit, resulting in a hypothetical 2Nω a Frequency line 402 and ω b This corresponds to point 401, which is experimentally determined by the prevention of crossings between the spectral frequency lines 403. The parameters of the circuit elements may be selected in a range in which the frequencies of the two modes are close to the frequency matching condition.

[0082] In some circuits, there exists an optimum choice of parameters of the circuit elements. Thus, the bias point is the optimum bias point, where the spurious even terms such as Kerr and cross-Kerr terms are also cancelled.

[0083] FIG. 4b shows that the optimal bias point 404 may coincide with the point 405 where the DC phase drop across the junction is π / 2. To reach this optimal bias point, the circuit parameters can be fully targeted for manufacturing or a separate tuning knob needs to be added. This extra tuning knob (which can be realized, for example, by incorporating a SQUID in the second mode, two junctions in parallel, biased with an external current to independently control its frequency) can be used as an extra degree of freedom to reach both the frequency matching condition and the vanishing Kerr condition. It is simply a matter of implementation.

[0084] FIG. 5 shows some examples of circuit symbolic representations that are alternatively used throughout this specification.

[0085] FIG. 5a shows two exemplary implementations of current biasing: direct biasing via a galvanic connection and mutual inductance biasing represented by a transformer 503 that generates a magnetic field to induce an internal current in the circuit. The first circuit (left side of FIG. 5a) shows a current source 501 galvanically coupled to a superconducting loop. The example shows that at least one terminal of the current source is directly connected to the superconducting loop 510, which is isolated from the rest of the circuit for convenience. However, this is a matter of implementation. The current source may be connected in any way to apply a predetermined current, as shown below. Given a predetermined current I ext Applying a phase drop ψ across the junction 502 DC This is due to the internal current I DC This is because a portion of the current flows through the junction 502. The second circuit (to the right of the first circuit) shows a current source inductively coupled to the superconducting loop via mutual inductance 503. This is shown diagrammatically as a transformer (503). The current source generates a magnetic field, which in turn generates a current in the superconducting loop. This current is associated with a phase drop across the junction.

[0086] Thus, the current source may be implemented in any way to achieve a phase drop across the junction. Figure 5a illustrates this in a circuit 530 that equivalently summarizes the previous two circuits in Figure 5a. When a given current is applied (either by a standard current source or a transformer), the magnetic flux ψ ext 504 represents the effective magnetic field biasing the loop induced by a given current applied to the circuit. The magnetic flux is hereafter referred to as the external magnetic flux ψ ext The inductance 505 represents the total self-inductance of the superconducting loop embedded in the junction 502. This final notation is also used when the circuit is placed in a global magnetic field that can be generated by a magnetic coil outside the plane of the circuit whose axis is perpendicular to the plane of the circuit. The external magnetic flux is oriented at an angle ψ ext =2πΦ ext / Φ0, where Φ0 is the magnetic flux quantum. In this application, the system isext is periodic with period 2π, and ψ ext = 0. Therefore, the analysis is on the interval TIFF2025502859000046.tif6150.

[0087] 5b shows an alternative description of a transformer (as a current source). The transformer 550 may be two circuit branches that are not galvanically connected (i.e., not in direct contact) and share a mutual inductance due to their proximity. Alternatively, the transformer 550 may be part of a circuit in which two loops galvanically share a common conductor. The transformer 550 may be effectively implemented in practice.

[0088] Figure 5c shows replacing the inductance with a series junction array. Unless otherwise specified, the array can contain up to a single junction.

[0089] Figure 5d shows that a Y-Δ transformation of the inductor can be used to change the topology of the circuit without changing its behavior, producing an alternative circuit that simplifies analysis, but follows the same principles of the other embodiments.

[0090] The circuit may be configured to perform a resonant 2N to 1 photon exchange when a predetermined current is applied to induce a phase difference across one or more Josephson junctions. The following example shows how the energy of a Josephson junction (or more generally a nonlinear inductive device) can be engineered to perform a resonant 2N to 1 photon exchange. The presence of at least one loop including one or more Josephson junctions facilitates an inherent resonance of the circuit. In fact, the energy of the Josephson junction may depend on the parameters of the components hosting the first and second modes, as well as the predetermined current that induces an internal current flowing through at least one loop. In fact, the one or more Josephson junctions included in the loop provide a mixing capability that allows for a resonant 2N to 1 photon exchange. That is, the energy of the one or more Josephson junctions describes the interaction of the first and second modes that are generated by a resonant 2N to 1 photon exchange when a predetermined current is applied through at least two nodes of the circuit.

[0091] In the following examples, we will assume that the circuit is operating at the optimum bias point according to the principles above. This is to make the equations shown below easier to read. However, it should be understood that the following examples also apply in the case of non-optimum bias points.

[0092] 6a-6e show an example of a circuit 600, a symbolic representation of which includes at least one loop including one or more Josephson junctions. The circuit 600 of this example is configured to perform a resonant 2N to 1 photon exchange when a predetermined current is applied to induce a phase difference across the one or more Josephson junctions. The circuit 600 of this example is provided without an input current source. However, it should be understood that the current source may be implemented in any manner to induce a current bias in the at least one loop. This can be achieved by a standard current source (electrically connected in any manner to the circuit) or a transformer that creates a magnetic field to induce an internal current in the circuit 600. The following example shows a particularly efficient implementation of a current source in the circuit 600.

[0093] In the circuit 600, at least one loop 610 includes a first Josephson junction 601 arranged in parallel with a first inductive element 602 and a first capacitive element 603. The inductive element (such as the first inductive element 602) may be a dipole (or a collection of dipoles) designed to add inductance to the circuit. For example, the inductive element may be a single inductor (i.e., a single superconducting strip with a given inductance) or several inductors arranged in series. In the case of several inductors arranged in series, the total inductance is the sum of the respective inductances of each inductor. Alternatively and / or additionally, the inductive element 602 may also include an array of Josephson junctions, which should include at least two junctions. The capacitive element (such as the first capacitive element 603) may be a single capacitor or multiple (i.e., more than one) capacitors arranged in parallel. The loops include respective first and second extremum nodes. Each respective extremum node is therefore a common node connecting a respective node of the first Josephson junction 601 with the first inductive element 602 and the first capacitive element 603 .

[0094] Furthermore, the Josephson junction 601 has a Josephson energy E J and inductor 602 has E L and the capacitor 603 has an energy denoted as E C The energy is given as

[0095] The circuit may also include a second inductive element 605 and a second capacitive element 604 arranged in parallel to form a resonator 620. This arrangement may include a first extremum node and a second extremum node, respectively, each connecting the second inductive element 605 and the second capacitive element 604. The inductive element 605 has a capacitance, E Lr and the capacitive element 604 may have an energy denoted as E Cr603. The circuit may also include a coupled capacitive element 606 or an inductive element 607. A parallel LC resonator 620 is linearly coupled in parallel with the nonlinear resonator formed by the Josephson junction 601, the inductive element 602 and the capacitive element 603. As mentioned above, the circuit 600 is a symbolic representation of a superconducting circuit. Thus, the elements of the circuit, such as the resonator 620, may be fabricated in any manner as known in the art and may be, for example, a transmission line or part of a 3D cavity. The resonator may be a mechanical resonator that capacitively couples 606 through a piezoelectric material.

[0096] A first extremum node of each of the loops 610 may be connected to a first extremum node of each of the second inductive elements 605, and the second capacitive element 604 may be arranged in parallel via the coupling capacitive element 606 or the inductive element 607. A second extremum node of each of the loops 610 and the second extremum nodes of the second inductive element 605 and the second capacitive element 604 arranged in parallel may be connected to a common ground.

[0097] When coupled through the coupling elements (606, 607), the resonators hybridize slightly, provided their frequencies are different and the coupling elements are small (small capacitance, large inductance), forming two modes: a linear mode 608 that may be hosted (or localized) predominantly on the linear resonator 620; and a nonlinear mode 609 that is hosted (or localized) predominantly on the parallel arrangement of the loop 610 and the first capacitive element 603. Thus, the circuit is configured to essentially perform a resonant 2N to 1 photon exchange.

[0098] Figure 6b shows another circuit 630 which is a variation of circuit 600. The difference is that the coupling capacitor is replaced by an inductor 607. This is just an alternative implementation using the same principles as above.

[0099] 6c and 6d show examples of devices incorporating circuit 600. Furthermore, the examples show how the modes can be selected according to experimental constraints. In 6c, the nonlinear mode can be the first mode and the linear mode can be the second mode. 6d shows an example where the first mode is a linear mode and the second mode is a nonlinear mode.

[0100] Further, Fig. 6c shows a microwave source 640 driving the circuit, coupled to a resistive load for dissipation, and an LC resonator 650 acting as a bandpass filter. The resonant frequency of the LC resonator is tuned with the resonant frequency of the second mode to act as a bandpass filter.

[0101] 6d shows another example of a device incorporating circuit 600, in which a dc current source is coupled to a microwave source and resistive load 670. The current source is coupled to the circuit via a transformer 660 which induces a current in loop 610.

[0102] The alternatives of Figures 6b-6d follow the same principles as the circuit and device of Figure 6a, in particular the choice of which mode (of the first and second modes) may be linear depends on the trade-offs of a given experimental setup.

[0103] FIG. 6e shows an example of an implementation of a symbolic representation of the circuit 600 in a planar superconducting pattern. The superconducting circuit is fabricated in a coplanar waveguide geometry (CPW). The background 6000 of the planar superconducting pattern corresponds to the superconducting metal remaining on the dielectric substrate after fabrication. Josephson junctions are represented as black crosses. The nonlinear mode is hosted in a superconducting island 611 that has a capacitance to ground 612 and is connected to ground through a single junction 601 and a three-junction array 602 acting as a small inductive element. The superconducting loop formed is a galvanically biased magnetic flux with a DC current line that shares with the resonator a portion of the inductance to ground 613. The wire carrying the DC current is connected to a DC port 614 and to ground. The DC port 614 is connected to a current source (which may be a current source at ambient temperature) through a superconducting wire and a non-superconducting wire. This lumped resonator is capacitively coupled 606 to a coplanar waveguide λ / 2 resonator 615 that functions as the second mode. This coplanar waveguide λ / 2 resonator has multiple resonances, each of which can be modeled by just 620 LC resonators. This second mode is then capacitively 616 coupled to the external environment, which connects to the circuit via an RF input port 617. In this design, no filters are shown. The resistive load is typically a 50 Ω transmission line terminating at the 50 Ω input of the microwave generator. Inset 640 is a close-up of the junction layout of the planar superconducting pattern.

[0104] Referring again to circuit 600, the calculation of the leading terms of its Hamiltonian rate describing the resonant 2N to 1 photon exchange is shown. Again, for purposes of illustration, assume that the circuit parameters are designed such that the bias point is the optimum bias point. This means that the frequency matching condition is such that ψ DC = π / 2. In that case, the inductive energy of the loop (610) is provided by the inductance alone, and the frequency of the nonlinear resonator is therefore The frequency of the linear resonator is given by TIFF2025502859000047.tif6150. In some configurations, the frequency is given by ω l =ω b =2Nω a =2Nω nl (e.g., Fig. 6c), or ω l =ω b =ω a / 2N=ω nl / 2N (e.g., Fig. 6d). First, the calculations are performed assuming that the nonlinear mode is the first mode a and the linear mode is the second mode b, as illustrated in Fig. 6c. Regardless of how the linear coupling is performed between the two resonators, it is summarized by defining the coupling strength g. The detuning between the two modes is, in the frequency matching condition, Δ=ω b -ω a =(2N-1)ω a Since in practice g<<Δ, the detuning is in the so-called dispersion limit, which allows to simplify the calculation of the nonlinear Hamiltonian. This linear combination slightly affects the frequencies of the two modes. Full diagonalization of the two-mode system requires E C , E L , E Cr , E Lr It allows to adjust the value of the optimum bias point and limit In TIFF2025502859000049.tif6150, the interactions provided by the junction can be represented as follows: TIFF2025502859000050.tif6150, where TIFF2025502859000051.tif11150 is the zero-point variation of the phase across the Josephson junction associated with mode a, TIFF2025502859000052.tif9150 represents the linear combination, and εψ a is the zero-point variation of the phase across the Josephson junction associated with mode b. By expanding this Hamiltonian with the desired order, we obtain the resonant 2N-to-1 photon exchange Hamiltonian You can get TIFF2025502859000053.tif6150, where The file is TIFF2025502859000054.tif10150.

[0105] If the opposite selection is made, as illustrated in FIG. 6d (where the linear mode is the first mode a and the nonlinear mode is the second mode b), the interaction provided by the junction is: It is represented as TIFF2025502859000055.tif6150, where: TIFF2025502859000056.tif11150 is the zero-point variation of the phase across the Josephson junction associated with mode b, TIFF2025502859000057.tif9150 characterizes the linear combination. By expanding this Hamiltonian as a Taylor series, we obtain the resonant 2N-to-1 photon exchange Hamiltonian, You can get TIFF2025502859000058.tif6150, where The file is TIFF2025502859000059.tif10150.

[0106] 7 shows another example of a circuit, where the symbolic representation of the circuit includes at least one loop including one or more Josephson junctions. The circuit 700 according to this example is also configured to perform a resonant 2N to 1 photon exchange when a predetermined current is applied to induce a phase difference across the one or more Josephson junctions. In particular, the circuit 700 according to this example is configured to essentially perform a resonant 2N to 1 photon exchange between a first mode and a second mode, respectively, such that two modes are simultaneously hosted in the loop.

[0107] In the circuit 700, at least one loop 710 may include a first inductive element 702, a central Josephson junction element 701, and a second inductive element 703 arranged in series. Each of the first or second inductive elements (702, 703) may be an inductance, a single Josephson junction, or an array of Josephson junctions. Thus, the central Josephson junction 701 may be arranged between the first and second inductive elements as a series loop. The series configuration may include a first inner node connecting a pole of the first inductive element 702 with a pole of the Josephson junction 701. The series arrangement may also include a second inner node connecting a pole of the second inductive element 703 with the other pole of the Josephson junction 701. The series configuration may also include a closed loop node connecting the other pole of the first inductive element 702 with the other pole of the second inductive element 703.

[0108] The at least one loop may be connected to a common ground via a closed loop node. The circuit may also include a first capacitor 704 and a second capacitor 705. The first capacitor 704 may be connected in parallel with a first Josephson junction 702 between a common ground and a first internal node of the loop. The second capacitor 705 may be connected in parallel with a second Josephson junction 702 between a common ground and a second internal node of the loop. Thus, the superconducting quantum circuit 700 is configured to essentially perform a resonant 2N-to-1 photon exchange between the first and second modes, respectively, when a predetermined current is applied.

[0109] As mentioned before, the circuit 700 is therefore configured to essentially perform a resonant 2N to 1 photon exchange between the first and second modes, respectively, such that two modes are simultaneously hosted in the loop. This maximizes the participation of the central junction, and therefore increases the strength of the resonant 2N to 1 photon exchange. The circuit on the right of Figure 7a shows how the participation of the central junction can be selected to tune the nonlinearity of the circuit, and therefore the strength of the resonant 2N to 1 photon exchange. This is done by connecting the junction at a desired level along the induced portion of the mode to pick up only a portion of the zero-point fluctuations in phase.

[0110] Again, for purposes of illustration, we assume that the circuit is designed to be biased at its optimum bias point. To simplify the derivation of the Hamiltonian for this example circuit 700, we assume that the intrinsic capacitance of the junctions is zero. In practice, a full diagonalization of the system can take into account finite capacitances. In this limit, the LC resonators, a first LC resonator formed by a first inductive element 702 and a first capacitor 704, and a second LC resonator formed by a second inductive element 703 and a second capacitor 705, each host a mode of the circuit. Thus, the frequency of the first mode (e.g., the left resonator) is and the frequency of the second mode (e.g., the right resonator) is given by TIFF2025502859000060.tif6150. These components are given by TIFF2025502859000061.tif6150 and are frequency-matched under the condition 2Nω a =ω b Then, ψ DC is assumed to be π / 2, so the junction Hamiltonian is TIFF2025502859000062.tif6150, where TIFF2025502859000063.tif11150, TIFF2025502859000064.tif11150. By expanding this Hamiltonian to the desired order, we obtain the resonant 2N-to-1 photon exchange Hamiltonian You can get TIFF2025502859000065.tif6150, where The file is TIFF2025502859000066.tif10150.

[0111] FIG. 7b shows another example of a symbolic representation of a device including a circuit 700. The circuit 700 is the same as FIG. 7a, with a simple rearrangement of the capacitor 705 and the inductor 703 for the readability of the symbolic representation. Thus, in the circuit 700 included in the device, a first mode is coupled to a second mode via a Josephson junction. Both the first and second LC resonators of the circuit 700 are shorted to a superconducting ground at the other end (not coupled to the Josephson junction), allowing the formation of a superconducting loop that can be biased to reach an optimal bias point. The second LC resonator is inductively coupled (via a transformer 711) to another resonator that acts as a bandpass filter between the system and a conditioned environment 712. The environment 712 includes a load, a microwave source, and a DC current source. As before, the same input port of the device can be used to conveniently bring DC current and microwave radiation into the circuit.

[0112] Figure 7c shows an example of the fabrication of the circuit 700 as a planar superconducting pattern. The presented superconducting circuit is constructed in a coplanar waveguide geometry (CPW), with the grey areas representing the superconducting metal remaining on the dielectric substrate. The solid black crosses represent Josephson junctions. The circuit consists of two λ / 4 resonators, the one on the left (708 first mode) simply connected to ground and the one on the right (709 second mode) doubly connected to ground (711, 712) to allow for symmetry while coupling to the external environment connected via an input line 713. The junction 701 is placed between the two resonators at an antinode of the electric field to maximize the nonlinearity of the system. Alternatively, the junction can be connected at any point along each resonator transmission line to adjust the level of nonlinearity (shown in the right circuit of Figure 7a). The input line 713 shares an inductance to ground with the second mode. This allows for a second mode of inductive coupling to the environment and a DC current bias (with a current source placed at ambient temperature) to the input line 713. Thus, the current source may be connected to a standard wire at room temperature and gradually connected to a superconducting wire which is in turn connected to the input line 713 to deliver a predetermined current (not shown here). To maintain symmetry of the system, the circuit has two superconducting loops in parallel that effectively boil down to one, as shown in electrical diagram 720.

[0113] 8 shows another example of a circuit, the symbolic representation of which includes at least one loop including one or more Josephson junctions. The circuit 800 according to this example is also configured to perform a resonant 2N to 1 photon exchange when a predetermined current is applied to induce a phase difference across the one or more Josephson junctions. The circuit 800 according to this example is specifically configured to symmetrically distinguish between the first and second modes. The high symmetry of the circuit achieves an improved quality of the resonant 2N to 1 photon exchange.

[0114] With reference to FIG. 8, another example of a circuit 800 having at least one loop 810 is described. In the circuit 800, the at least one loop 810 may include a first Josephson junction 801, a central inductive element 803, and a second Josephson junction 802 arranged in series. The central inductive element 803 may be an inductance, a single Josephson junction, or an array of Josephson junctions. Thus, the central inductive element 803 may be arranged between the first and second Josephson junctions as a series loop. The series configuration may include a first inner node connecting a pole of the first Josephson junction 801 with a pole of the inductive element 803. The series arrangement may also include a second inner node connecting a pole of the second Josephson junction 802 with the other pole of the inductive element 803. The series arrangement may also include a closed loop node connecting the other pole of the first Josephson junction 801 with the other pole of the second Josephson junction 802.

[0115] The at least one loop may be connected to a common ground via a closed loop node. The circuit may also include a first capacitor 804 and a second capacitor 805. The first capacitor 804 may be connected in parallel with a first Josephson junction 801 between the common ground and a first internal node of the loop. The second capacitor 805 may be connected in parallel with a second Josephson junction 803 between the common ground and a second internal node of the loop.

[0116] Thus, the superconducting quantum circuit 800 is configured to essentially perform a resonant 2N to 1 photon exchange between the first and second modes, respectively, when a given current is applied. The Josephson junctions 801 and 802 are substantially identical, and the capacitive elements 804 and 805 are also substantially identical. The symmetry of the circuit therefore means that the actual mode of the system is in fact a superposition of the symmetric 806 and antisymmetric 807 of the two resonators. The first mode is the symmetric mode indicated by the solid arrows, and the second mode is the antisymmetric mode indicated by the dashed arrows. It can be noticed that only the antisymmetric mode has a contribution across the central inductive element 803. This contribution is advantageously used to preferentially couple the environment to the second mode.

[0117] There is no optimum bias point for this circuit. In fact, the junction is biased by ψ DC = π / 2, and ψ ext = π. However, this is not a problem, since the optimal bias point is not desired. In fact, at such a point, the Josephson junction acts as an open circuit, and thus only the parallel Josephson capacitance remains, which contradicts the fact that in this particular circuit, the junction serves as the primary inductive element for the symmetric mode. By adding one loop, the optimal bias point can be made possible. An exemplary implementation consists of a symmetrized version of the circuit of Fig. 4a and Fig. 4b, where both resonators are made identical and nonlinear by replacing the inductance 605 with a loop 610. The coupling between the two nonlinear identical resonators, as proposed in Fig. 4a and Fig. 4b, can be capacitive or inductive.

[0118] Here, circuit 800 consists of two identical resonators strongly coupled through a central inductive element. This means that the bare (before adding coupling) detuning between the two resonators is zero and the perturbative description performed for circuit 600 no longer holds. The analysis presented here is therefore different from circuit 600. Assuming that the system is perfectly symmetric (both junctions and capacitances are identical), the system can be decomposed into a symmetric mode (i.e., the first mode a) and an antisymmetric mode (i.e., the second mode b). In this eigenmode basis, the contribution of each junction to the Hamiltonian of the system can be calculated. TIFF2025502859000067.tif26150 TIFF2025502859000068.tif7150This can be expanded as follows. TIFF2025502859000069.tif6150 TIFF2025502859000070.tif6150

[0119] The two quadratic parts of the first term are The effective inductive energy of the junction at the point of application of TIFF2025502859000071.tif6150 is given. Together with the charging energy of the capacitor and the inductive energy of the central inductive element, the frequency TIFF2025502859000072.tif10150 and TIFF2025502859000073.tif10150 and phase zero point fluctuation TIFF2025502859000074.tif12150 and TIFF2025502859000075.tif12150 can be defined.

[0120] By expanding the second term in the desired order, At TIFF2025502859000076.tif10150, the resonant 2N to 1 photon exchange Hamiltonian You can get TIFF2025502859000077.tif6150.

[0121] Due to the symmetry of the system and the frequency matching conditions that must be met at the bias point of the system, the formula can be further simplified. a =ω b Therefore, TIFF2025502859000078.tif6150 and therefore From TIFF2025502859000079.tif6150 The result is TIFF2025502859000080.tif14150.

[0122] FIG. 8b shows an example of a device used to stabilize a manifold of coherent states with a circuit 800. Looking from the bottom up, at the bottom of FIG. 8b is the nonlinear superconducting circuit 800. At the top, an inductive element shares a mutual inductance 808 with another inductor that terminates in an environment 820. The environment 820 consists of a load, a microwave source, and a DC current source. As before, the same input port of the device can be used to conveniently provide DC current for the external magnetic flux and microwave radiation. In practice, as before, a portion of a transmission line is used to connect the circuit to the environment. This portion of the transmission line closest to the circuit 809 is typically a differential transmission line to maintain the symmetry of the circuit 800.

[0123] Figure 8c shows an example of a planar superconducting pattern 850 and a circuit equivalent. The presented superconducting circuit is fabricated in a coplanar waveguide geometry (CPW), the background (greyed out part) represents the superconducting metal remaining on the dielectric substrate. Each black cross represents a respective Josephson junction. In contrast to the device of Figure 8b, the inductive element is replaced by a single junction. In this circuit, to control the level of nonlinearity of the system, the central superconducting loop 810 is diluted using a tripartite of CPW transmissions called stubs (811, 812, 813). On the left and right sides (811, 812), open stubs provide the necessary capacitance with some stray inductance in series. At the bottom, the ring is not directly connected to ground, but shorted stubs 813 are interleaved to provide an inductance to ground. The mode of the circuit and its principle of operation remain the same, but the length of the transmission line added by the stubs makes it possible to limit the level of nonlinearity of the circuit. To preferentially couple to the second mode and maintain circuit symmetry, a transmission slotline 809 is used to inductively couple the mode to the environment. A transition between CPW and slotline 814 is shown above for coupling to the environment (and thus a current source) with a common geometry 815 (e.g., a CPW line or coaxial cable).

[0124] 9 shows experimental data illustrating the inherent two-to-one photon dynamics in the embodiment tested (i.e., the nonlinear reverse crossing when a given current biases the circuit). The data was taken for a device represented by a planar superconducting pattern 850 of tantalum on a sapphire substrate with Josephson junctions made of aluminum and aluminum oxide. The parameters of the symbolic representation are as follows: The Josephson energy of a pair of junctions is E J = 250 GHz, and the central inductive element has energy E L = 120 GHz. The lateral capacitance is the stray series inductance E Lc = Energy E at 400 GHz C = 40MHz. The inductance to ground is the stray capacitance E Cg= Energy E at 125MHz Lg = 150 GHz. The color map 900 shows the source frequency and bias current I ext 9 corresponds to the amplitude of the reflected microwave signal from a microwave source on the nonlinear superconducting circuit as a function of bias current. Line 901 corresponds to where the probe frequency matches the second mode frequency, and thus shows the frequency of the second mode as a function of bias current, similar to FIG. 4. A zoomed portion 910 of anti-crossing 902 shows that the two modes of the circuit meet the frequency matching condition 2ω a =ω b For this device, the bias point is approximately 7.3 mA.

Claims

1. A nonlinear superconducting quantum circuit (100, 600, 700, 800) having a first mode and a second mode, each of the first mode and the second mode has a respective resonant frequency; the circuit is configured such that when a predetermined current of constant magnitude is applied to the circuit, the resonant frequency of the second mode is substantially 2N times the resonant frequency of the first mode, whereby the circuit essentially performs a resonant 2N to 1 photon exchange between the first mode and the second mode; N is a positive integer. Nonlinear superconducting quantum circuits (100, 600, 700, 800).

2. 2. The nonlinear superconducting quantum circuit of claim 1, wherein the circuit has a Hamiltonian that is a function of a set of parameters consisting of parameters of the circuit and parameters of the predetermined current when the predetermined current is applied to the circuit.

3. The Hamiltonian has the form can be expanded into a sum between at least the leading term and a series of auxiliary terms, g ２Ｎ is a scalar corresponding to the intrinsic coupling strength, a is the annihilation operator of the first mode, b is the annihilation operator of the second mode, 3. The nonlinear superconducting quantum circuit of claim 2, wherein is a reduced Planck constant.

4. the circuit has a symbolic representation including at least one loop (610, 710, 810) including one or more Josephson junctions (601, 701, 801-802); the circuit is configured to perform the resonant 2N to 1 photon exchange when the predetermined current is applied to induce a phase difference across the one or more Josephson junctions (601, 701, 801-802); A nonlinear superconducting quantum circuit (600 to 630, 700, 800) according to any one of claims 1 to 3.

5. the at least one loop (610) includes a first Josephson junction (601) arranged in parallel with a first inductive element (602) and a first capacitive element (603) including respective first and second extremum nodes; The symbolic representation is a second inductive element (605) and a second capacitive element (604) arranged in parallel and including respective first and second extremum nodes; a linear combination element, which is either a capacitive element (606) or an inductive element (607); the first extremum nodes of the loop (610) are connected to the first extremum nodes of the second inductive element (605) and the second capacitive element (604) arranged in parallel (620) via the linear coupling elements (606-607); the respective second extremum nodes of the loop (610) and the respective second extremum nodes of the second inductive element (605) and the second capacitive element (604) arranged in parallel are connected to a common ground; 5. The nonlinear superconducting quantum circuit according to claim 4.

6. When the predetermined current of the constant magnitude is applied to the circuit, the Hamiltonian of the superconducting circuit is given by: a linear term describing the first mode and the second mode of the type in the form of, or and at least one nonlinear term of the form is the resonant frequency of the first mode frequency, is the resonant frequency of the second mode frequency, E Ｊ is the energy of the Josephson junction, ψ ＤＣ is the phase difference across the Josephson junction induced by the predetermined current, ψ ａ is the zero-point variation of the phase across the Josephson junction associated with the first mode, or ψ ｂ is the zero-point variation of the phase across the Josephson junction associated with the second mode, ε describes the linear combination between the first mode and the second mode; Thereby, the circuits (600 to 630) or have a Hamiltonian that can be expanded into a sum including at least a leading term that is a nonlinear resonance term of the type Resonance condition 2Nω ａ =ω ｂ is obtained by applying the predetermined current, A nonlinear superconducting quantum circuit (600-630) according to claim 5.

7. the at least one loop (710) includes a Josephson junction (701), a first inductive element (702), and a second inductive element (703) arranged in a series topology; The series topology is a first inner node connecting a pole of the Josephson junction (701) and a pole of the first inductive element (702); a second inner node connecting the other pole of the Josephson junction (701) and a pole of the second inductive element (703); a closed-loop node connecting the other pole of the first inductive element (702) and the other pole of the second inductive element (703); the at least one loop is connected to a common ground via the closed loop node; The symbolic representation further includes a first capacitor (704) and a second capacitor (705); the first capacitor (704) is connected in parallel with the first inductive element (702) between the common ground and the first inner node of the loop; the second capacitor (705) is connected in parallel with the second inductive element (703) between the common ground and the second inner node of the loop; 5. The nonlinear superconducting quantum circuit (700) of claim 4.

8. When the predetermined current of constant magnitude is applied to the circuit, the Hamiltonian of the circuit is: and linear terms describing the first and second modes of the type and at least one nonlinear term of the form is the resonant frequency of the first mode frequency, is the resonant frequency of the second mode frequency, E Ｊ is the energy of the Josephson junction, ψ ＤＣ is the phase difference across the Josephson junction induced by the predetermined current, ψ ａ is the zero-point variation of the phase across the Josephson junction associated with the first mode, and ψ ｂ is the zero-point variation of the phase across the Josephson junction associated with the second mode, Thereby, the circuit (700) have The Hamiltonian can be expanded into a sum including at least a leading term that is a nonlinear resonance term of the type ａ =ω ｂ is obtained by applying the predetermined current, 8. The nonlinear superconducting quantum circuit (700) of claim 7.

9. the at least one loop (810) includes a first Josephson junction (801), a central inductive element (803), and a second Josephson junction (802) arranged in a series topology; The series topology is a first inner node connecting a pole of the first Josephson junction (801) and a pole of the inductive element (803); a second inner node connecting a pole of the second Josephson junction (802) and another pole of the inductive element (803); a closed-loop node connecting the other pole of the first Josephson junction (801) and the other pole of the second Josephson junction (802); the at least one loop (810) is connected to a common ground via the closed loop node; The symbolic representation (800) further includes a first capacitor (804) and a second capacitor (805); the first capacitor (804) is connected in parallel with the first Josephson junction (801) between the common ground and the first inner node of the loop (810); the second capacitor (805) is connected in parallel with the second Josephson junction (802) between the common ground and the second inner node of the loop (810); 5. The nonlinear superconducting quantum circuit (800) of claim 4.

10. When the predetermined current of constant magnitude is applied to the circuit, the Hamiltonian of the circuit is: and linear terms describing the first and second modes of the type and at least one nonlinear term of the form is the frequency of the first mode, is the frequency of the second mode, E Ｊ is the energy of the first Josephson junction and / or the second Josephson junction, ψ Ｊ０ is the phase difference across the first Josephson junction and / or the second Josephson junction induced by the predetermined current, ψ ａ is the zero-point variation of the phase across the Josephson junction associated with the first mode, and ψ ｂ is the zero-point variation of the phase across the Josephson junction associated with the second mode, Thereby, the circuit (800) have a Hamiltonian that can be expanded into a sum including at least a leading term that is a nonlinear resonance term of the type Resonance condition 2Nω ａ =ω ｂ is obtained by applying the predetermined current, 10. The nonlinear superconducting quantum circuit (800) of claim 9.

11. The nonlinear superconducting quantum circuit according to claim 1; a current source configured to apply the predetermined current of a constant magnitude to the circuit such that the resonant frequency of the second mode is substantially 2N times the resonant frequency of the first mode, whereby the circuit performs a uniquely resonant 2N to 1 photon exchange between the first mode and the second mode, respectively. device.

12. Load and a microwave source configured to apply microwave radiation at a frequency substantially equal to the resonant frequency of the second mode or 2N times the resonant frequency of the first mode; a coupler configured to couple the second mode of the circuit to the load and the microwave source; Optionally, the load is one of a resistor, a matched transmission line, or a matched waveguide; the device further comprises a bandpass or bandstop filter connected to the first and second modes of the circuit, the bandpass or bandstop filter being configured to allow only the second mode to be coupled to the load; and / or the geometry and / or symmetry of the coupler is such that the coupler couples only to the second mode; The device of claim 11.

13. A quantum computing system comprising the nonlinear superconducting quantum circuit of claim 1 and / or the device of claim 11 or 12.

14. providing a nonlinear superconducting quantum circuit according to claim 1; applying the predetermined current of constant magnitude to the circuit such that the resonant frequency of the second mode is substantially 2N times the resonant frequency of the first mode, In this way, the circuit performs a uniquely resonant 2N to 1 photon exchange between the first mode and the second mode, respectively. method.

15. Providing a device according to claim 11 or 12; applying the predetermined current of constant magnitude to the circuit such that the resonant frequency of the second mode is substantially 2N times the resonant frequency of the first mode, In this way, the circuit performs a uniquely resonant 2N to 1 photon exchange between the first mode and the second mode, respectively. method.

16. Providing a device according to claim 12 or a quantum computing system including the device according to claim 12; applying the predetermined current of constant magnitude to the circuit such that the resonant frequency of the second mode is substantially 2N times the resonant frequency of the first mode, As such, the circuit performs a uniquely resonant 2N to 1 photon exchange between the first mode and the second mode, respectively; The method is used to stabilize a quantum manifold spanned by 2N coherent states with the same amplitude and a phase difference of π / N to enable encoding of quantum information in the form of cat qubits.

17. Providing a quantum computing system according to claim 13; applying the predetermined current of constant magnitude to the circuit such that the resonant frequency of the second mode is substantially 2N times the resonant frequency of the first mode, In this way, the circuit performs a uniquely resonant 2N to 1 photon exchange between the first mode and the second mode, respectively. method.

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