Devices and quantum measurement methods for resonant cat qubit circuits

Abstract

In particular, the present disclosure relates to a nonlinear superconducting quantum circuit having a first mode and a second mode. The first mode and the second mode each have 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. The circuit thereby essentially performs a resonant 2N-to-1 photon exchange between the first mode and the second mode, respectively, where N is a positive integer. This results in an improved nonlinear superconducting quantum circuit. When the circuit is subjected to a predetermined current and the second mode is appropriately driven, the circuit can stabilize a cat qubit. In a symmetric version of this circuit, a flux line is developed that allows substantially exclusive RF coupling to the second mode, but not the first mode, for quantum measurements. This allows quantum measurements to be performed with stabilization turned off.

Description

[Technical Field]

[0001] The present disclosure relates to the field of quantum technology, and more particularly to methods for measuring qubits in superconducting quantum circuits. [Background technology]

[0002] In recent years, the development of quantum technologies for several applications, such as quantum computing and communications, has attracted increasing attention. Superconducting quantum circuits are a promising platform for realizing quantum computers, and among them, the so-called cat qubit, stored in a superconducting resonator, is one interesting candidate.

[0003] A cat qubit is defined as a specifically chosen two-dimensional sub-manifold of a main manifold across which several coherent states are spread. As an example, a two-component cat qubit main manifold is a span of two coherent states with equal amplitudes and opposite phases, and the main manifold is two-dimensional, in which case the sub-manifold is equal to the main manifold. As another example, a four-component cat qubit main manifold is a span of four coherent states with equal amplitudes and phases shifted by 90° from each other. The two-dimensional sub-manifold is chosen as an even-photon-number parity sub-manifold. Generalizing, a 2N-component cat qubit main manifold is a span of four coherent states with equal amplitudes and phases shifted by 90° from each other. TIFF2025538425000002.tif6150 is a span of 2N coherent states with equal amplitudes and phases shifted by 150. The 2D submanifold is chosen as the 0 modulo N photon number submanifold.

[0004] In this context, superconducting quantum circuits can be engineered to exhibit specific quantum dynamics, such as stabilizing a quantum manifold of coherent states. The stabilization of a quantum manifold of two coherent states by dissipation has been investigated, among others, 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.

[0005] Stabilization of 2N coherent states requires engineering a nonlinear conversion between 2N photons in a first mode (also known as the cat qubit mode) that hosts the stabilized quantum manifold and a single photon in a second mode, known as the buffer mode, and vice versa. To achieve the nonlinear conversion, existing solutions involve applying an external time-varying excitation to the superconducting quantum circuit in the form of one or more microwave tones at specific frequencies, called parametric pumps, to bridge the energy gap between the energy of the 2N photons in the first mode and the energy of a single photon in the second mode. These solutions are also known as parametric pumping techniques.

[0006] In European Patent Application Publication No. 21306965.1, the applicant discloses a family of superconducting quantum circuits in which the resonant frequency of the second mode is substantially 2N times the resonant frequency of the first mode when a predetermined current of constant strength is applied to the circuit. This provides what are hereinafter referred to as "resonant cat qubit circuits" and "resonant cat qubits," because the 2N-to-1 photon conversion does not require parametric pumping to bridge the energy gap in the process. Once this nonlinear conversion process is enabled, the coherent state manifold is stabilized by coupling the second mode (or "buffer mode") to an environment containing a microwave source that drives it at resonance and a load that dissipates the second mode. This single-photon drive is converted into 2N-photon drive in the first mode, or cat qubit mode, by the nonlinear conversion process. Similarly, single-photon loss is converted into 2N-photon loss in the cat qubit mode.

[0007] While being able to forgo the use of parametric pumping techniques is highly advantageous, the implementation of resonant cat qubit circuits introduces new challenges to measuring cat qubits.

[0008] In previous implementations, the state of the cat qubit mode is determined by measuring the Wigner function. This measurement is most often based on the dispersive interaction between the cat qubit mode and a two-level system. The dispersive interaction embedded within the Ramsey sequence on the two-level system allows for the measurement of the field parity. Conventional Wigner function measurements in the context of superconducting circuits are described in the paper by Sun, L., Petrenko, A., Leghtas, Z. et al., "Tracking photon jumps with repeated quantum non-demolition parity measurements," Nature 511, pp. 444–448 (2014), https: / / doi.org / 10.1038 / nature13436.

[0009] Dispersive coupling is suppressed by coherent state stabilization. Therefore, the Wigner function cannot be measured while the main manifold is stabilizing. In previous implementations of stabilized Cat qubits with parametric pumping techniques, the Wigner function is measured after the stabilization process is neutralized by turning off the parametric pump, which allows for a 2N-to-1 photon conversion between the Cat qubit mode and the buffer mode. This can be done almost instantaneously (in tens of nanoseconds) thanks to the wide bandwidth of the microwave radiation.

[0010] However, this is not possible in the context of a resonant cat qubit: in fact, the stabilization is self-sustaining, so there is no parametric pump to turn off. [Prior art documents] [Patent documents]

[0011] [Patent Document 1] European Patent Application Publication No. 21306965.1 [Non-patent literature]

[0012] [Non-Patent Document 1] “Exponential suppression of bit-flips in a qubit encoded in an oscillator”, Lescanne R. et al., Nature Physics, 2020 [Non-patent document 2] “Coherent Oscillations inside a Quantum Manifold Stabilized by Dissipation”, Touzard S. et al., Physical Review X, 2018 [Non-patent document 3] “Confining the state of light to a quantum manifold by engineered two-photon loss”, Leghtas Z. et al., Science, 2015 [Non-patent document 4] Sun, L., Petrenko, A., Leghtas, Z. et al., "Tracking photon jumps with repeated quantum non-demolition parity measurements," Nature 511, pp. 444–448 (2014), https: / / doi.org / 10.1038 / nature13436 [Non-Patent Document 5] Grimm, A., Frattini, N.E., Puri, S. et al., "Stabilization and operation of a Kerr-cat qubit," Nature 584, pp. 205–209 (2020), https: / / doi.org / 10.1038 / s41586-020-2587-z Summary of the Invention

[0013] A possible solution would be to use a magnetic field to detune the resonant frequency of the resonant cat qubit circuit and deactivate the 2N-to-1 photon conversion between the cat qubit mode and the buffer mode. However, given the limitations of the resonant cat qubit circuit, it is not currently known how to provide such a magnetic field fast enough.

[0014] The present invention aims to improve the situation. To this end, the applicant provides a quantum device, which comprises: a nonlinear superconducting quantum circuit having a first mode and a second mode each having a respective resonant frequency and a central mixing component; Flux line and A quantum device comprising: The central mixing component at least one loop including a first Josephson junction, a central inductive element, and a second Josephson junction arranged in a series topology substantially symmetrical about an axis mapping a first Josephson junction to a second Josephson junction, the loop comprising a first internal node connecting a pole of the first Josephson junction to a pole of the central inductive element, a second internal node connecting a pole of the second Josephson junction to another pole of the central inductive element, and a loop closing node connecting the other pole of the first Josephson junction to the other pole of the second Josephson junction; a first circuit portion connected between a common ground and the first internal node of the loop, a second circuit portion connected between the common ground and the second internal node of the loop, and a third circuit portion connected between the loop closure node and the common ground, the third circuit portion being substantially symmetrical about the axis; Including, the first circuit portion and the second circuit portion are substantially symmetrical to each other with respect to the axis; The nonlinear superconducting quantum circuit is configured such that, when a predetermined current of constant magnitude is applied, the resonant frequency of the second mode is substantially 2N times the resonant frequency of the first mode, and a phase difference is induced across the one or more Josephson junctions, and the nonlinear superconducting quantum circuit has a form TIFF2025538425000003.tif6150, and a Hamiltonian that can be expanded into a sum between at least one dominant term and a set of auxiliary terms, where TIFF2025538425000004.tif6150 is a scalar corresponding to the intrinsic coupling strength, a is the annihilation operator of the first mode, TIFF2025538425000005.tif6150 is the annihilation operator of the second mode, TIFF2025538425000006.tif6150 is a reduced Planck constant, thereby essentially implementing a resonant 2N to 1 photon exchange between the first mode and the second mode, respectively, where N is a positive integer; the nonlinear superconducting quantum circuit is disposed on a dielectric substrate and separated from a common ground plane by an exposed portion of the dielectric substrate; the flux line is located opposite a portion of the at least one loop that includes the central inductive element and includes a first slot and a CPW portion; the CPW section is positioned to be coupled to a current source, a microwave source configured to apply microwave radiation, and a load, and separated by end wire bonds; the first slot is patterned in the ground plane exposing the dielectric substrate, extends substantially parallel to the axis, and is connected at one end to a slot in the CPW portion at the level of the end wire bond and at the other end to a portion of the exposed dielectric substrate that delimits a portion of the at least one loop that includes the central inductive element; The flux lines are configured to couple the current source to induce the predetermined current in the at least one loop, to couple the microwave radiation source to induce driving of the second mode at a frequency substantially equal to the resonant frequency of the second mode or 2N times the resonant frequency of the first mode, and to couple a load substantially exclusively to the second mode.

[0015] According to various embodiments, the quantum device may include one or more of the following features: the flux line further comprises a second slot patterned in the ground plane exposing the dielectric substrate, extending substantially parallel to the axis, connected at one end to another slot in the CPW portion at the level of the end wire bond, and terminating at its other end in the ground plane at a distance greater than 1 μm from the portion of the at least one loop including the central inductive element; the second slot includes a return portion closest to a portion of the at least one loop including the central inductive element, the return portion extending substantially perpendicular to the axis in a rotational direction away from the first slot; the length of the return portion is greater than 1 μm and less than or equal to the width of the nonlinear superconducting circuit; - said first slot is substantially symmetrical with respect to said axis; - both the first slot and the second slot are offset perpendicular to the axis and opposite to the direction of rotation; The offset amount is greater than 1 μm and is equal to or less than the width of the nonlinear superconducting circuit. the length of the first slot or the position of the wire bond is selected depending on the strength of the intended coupling between the second mode and the load; the coupling between the load and the second mode is at least 100 times greater than the coupling between the load and the first mode; further comprising a current source, a microwave source and a load coupled to said CPW portion that are operated to stabilize a coherent manifold within said device.

[0016] The present invention also relates to a quantum measurement method for the above device, said method comprising the steps of: a) varying the current output of the current source such that the predetermined current in the nonlinear superconducting circuit induces a resonant frequency of the second mode that is substantially different from twice the resonant frequency of the first mode; b) performing a quantum measurement in the first mode of the nonlinear superconducting circuit; Includes.

[0017] According to various embodiments, the method can include one or more of the following features: - modifying operation a) is carried out by applying square pulses to said current, Operation b) is a quantum non-demolition measurement and further comprises operation c) below, wherein operation c) restores the current output of said current source to obtain said predetermined current after operation b) has been performed.

[0018] The invention also relates to a quantum computing system comprising at least one device of the invention.

[0019] Non-limiting examples will now be described with reference to the accompanying drawings. [Brief explanation of the drawings]

[0020] [Figure 1] FIG. 1 illustrates how a resonant cat qubit circuit is incorporated into a device to stabilize quantum information. [Figure 2] 1A-1C illustrate various examples of circuits used to isolate the first mode from the environment. [Figure 3] FIG. 10 illustrates an example of how a resonant cat qubit circuit integrated into a device can be used to stabilize quantum information. [Figure 4] 4a and 4b show how the current is adjusted to reach the frequency matching condition. [Figure 5] 1A-1C illustrate some examples of circuit symbolic representations that are alternatively used throughout this specification. [Figure 6] FIG. 2 illustrates an embodiment of the device of FIG. 1. [Figure 7] 1 is a general circuit symbolic representation of a resonant cat qubit circuit used in a device according to the present invention; [Figure 8a] FIG. 1 shows a top view of a device according to the invention with a resonant cat qubit in the absence of flux lines. [Figure 8b] 1 shows a circuit symbolic representation of the differential mode of the resonant cat qubit circuit. [Figure 9a] 1 shows a top view of a first embodiment of a device according to the invention; [Figure 9b] 9b shows a circuit symbolic representation of the differential mode of the resonant cat qubit circuit of FIG. 9a during device operation. [Figure 10] 1 shows a top view of a second embodiment of a device according to the invention. [Figure 11] 1 shows a top view of a third embodiment of a device according to the invention. [Figure 12] 1 shows a top view of a fourth embodiment of a device according to the invention. [Figure 13]1 shows a photograph of a manufactured device according to the first embodiment. [Figure 14] 10 shows a photograph of a manufactured device according to the fourth embodiment. DETAILED DESCRIPTION OF THE INVENTION

[0021] Before describing the measurement method in accordance with the present invention, applicants will describe resonant cat qubits and the underlying principles, which will help to better understand the nature of resonant cat qubit circuits and why stabilizing those circuits is unique and poses unique challenges.

[0022] First, we explain the situation with resonant cat qubits.

[0023] To realize a resonant cat qubit, a nonlinear superconducting quantum circuit is provided having a first mode and a second mode. The first mode and the second mode each have 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. Thus, the circuit essentially performs a resonant 2N-to-1 photon exchange between the first mode and the second mode, respectively. N is a positive integer (i.e., N is any positive integer greater than or equal to 1), and therefore 2N is an even number, e.g., 2, 4, 6, or more. Thus, the phrase "2N photons" refers to a discrete, even quantity of photons defined by the integer 2N.

[0024] 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, when a given current of constant strength is applied to the circuit, the resonant frequency of the second mode is effectively 2N times that of the first mode. This contrasts with known dissipation-based cat qubit implementations, in which an external time-varying excitation, such as that implemented by parametric pumping techniques, is used to bridge the frequency gap between the two modes and achieve resonant 2N-to-1 photon exchange. The external time-varying excitation used in such prior art relaxes the constraint on the mode frequencies but causes undesirable effects, such as heating the modes. This heating shortens the coherence time of the circuit and induces dynamical instabilities, ultimately destroying any nonlinear mixing involving resonant 2N-to-1 photon exchange. In this respect, the lack of external time-varying excitation allows for an increase in the resonant 2N-to-1 photon exchange rate by one or two orders of magnitude compared to prior art.

[0025] The first and second modes of the superconducting quantum circuit can each correspond to a natural resonant frequency of the circuit. For example, the first and second modes can each be either an electromagnetic mode or a mechanical mode. The first and second modes can have respective natural resonant frequencies. For example, the first mode (or the second mode) can be a mechanical mode, and the second mode can be an electromagnetic mode (or the first mode can be an electromagnetic mode). The first mechanical mode and the second electromagnetic mode can be coupled in the circuit by the piezoelectric effect. Each of the first and second modes can have a respective resonant frequency, for example, the first mode can be a type The second mode can have a resonance frequency of type TIFF2025538425000007.tif8150. TIFF2025538425000008.tif8150, where TIFF2025538425000009.tif6150 and TIFF2025538425000010.tif6150 can be the angular frequency of each mode. "Having" a first mode and a second mode means that a superconducting quantum circuit can include 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 can be hosted in different subsets of components of the superconducting circuit, or (alternatively) in the same subset of components.

[0026] Superconducting quantum circuits, except for some tailored couplings, can operate at temperatures close to absolute zero (e.g., below 100 mK, typically 10 mK) and 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.

[0027] Superconducting quantum circuits 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 of two adjacent plates of superconducting material (on that layer of the one or more patterned layers). The inductive element can be formed of a superconducting wire. Alternatively, at least one of the one or more patterned layers can define portions of a transmission line that each resonate at a frequency that depends on the length of the patterned layer. The transmission line can be, for example, a coplanar waveguide or a microstrip line. 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.

[0028] 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 (form Also known as the "frequency matching condition" in TIFF2025538425000011.tif6150, however, TIFF2025538425000012.tif6150 is the resonance frequency of the first mode, TIFF2025538425000013.tif6150 is the resonance frequency of the second mode, or format (Also referred to as the "frequency matching condition" in TIFF2025538425000014.tif6150). In other words, the predetermined current is an external current that induces an internal DC (direct current) bias in the circuit. The internal current is such that the components / hardware forming the circuit are in a particular regime, i.e., such that the resonant frequency of the second mode (also referred to as the second resonant frequency) of the circuit is substantially 2N times the resonant frequency of the first mode (also referred to as the first resonant frequency). Thus, the nonlinearity of the circuit essentially performs a resonant 2N-to-1 photon exchange: at the first resonant frequency, 2N photons are destroyed in the first mode while generating one photon in the second mode at the second resonant frequency, and conversely, at the second resonant frequency, 2N photons are destroyed in the first mode while generating one photon in the second mode at the first resonant frequency. In fact, resonant 2N-to-1 photon exchange is hereafter referred to as The photon exchange between the two modes can be effected through a circuit element (e.g., a nonlinear element). The given current is of type The current is applied so that the frequency matching condition of TIFF2025538425000016.tif6150 occurs and the circuit executes resonant 2N-to-1 photon exchange dynamics. A given current is also called the bias point of the circuit because the frequency matching condition is reached at this current. The bias point may also be called the optimal bias point if the selection of circuit parameters eliminates spurious dynamics at the frequency matching condition, as shown in the following example:

[0029] A predetermined current can be applied directly, i.e., galvanically, to a circuit such that the current flows through at least a subset of the circuit's elements and splits into various possible branches. The current flowing within a branch of the circuit is called an internal current and is determined according to Kirchhoff's current law. In various examples, the predetermined current may be applied directly via a current source connected to the circuit. In various examples, the circuit can 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.

[0030] Alternatively, the predetermined current may be applied indirectly, i.e., an internal current may be induced in the circuit, for example, by mutual inductance with a coil. In other words, an external inductance is inductively coupled to the circuit via shared mutual inductance, inducing a current in the circuit, which in turn flows through at least a subset of the circuit's elements. Thus, the internal current is a current induced in the circuit via mutual inductance. Because a direct (galvanic) connection to the circuit is not required, the predetermined current path may be at the same level as the circuit or may be created by an external coil above or below the circuit, with the axis of the external coil perpendicular to the plane of the superconducting circuit. In various examples, when mutual inductance is shared with a coil, the coil may be formed with multiple turns of material that allow current to circulate to generate a magnetic field. The coil may be fabricated with any number of coils required, thereby increasing the mutual inductance. The coil may be fabricated from any material that allows current to circulate to generate a magnetic field and thereby induce an internal current. For example, the coil may be fabricated from a superconducting or non-superconducting material. Alternatively, the coils may be replaced with permanent magnets that directly generate a constant magnetic field, however this makes tuning the field impractical.

[0031] The application of a predetermined current is Internal currents through a subset of inductive elements TIFF2025538425000017.tif6150 and Superconducting phase drop occurring across each inductive element TIFF2025538425000018.tif6150 and result in a phase drop TIFF2025538425000019.tif6150 can be calculated taking into account the circuit geometry and the parameters as a function of a given current. The example below shows how to calculate the current (or superconducting phase drop) for a given current to reach the bias point and thereby enable resonant 2N to 1 photon exchange dynamics. TIFF2025538425000020.tif6150) can be experimentally tuned. Thus, the application of current is tailored to the configuration of the circuit to induce a unique resonant 2N-to-1 photon exchange.

[0032] The resonant Cat qubit circuit inherently performs resonant 2N-to-1 photon exchange. In other words, the circuit is configured to perform resonant 2N-to-1 photon exchange autonomously / spontaneously, i.e., without requiring external time-varying excitation to bridge the gap between 2N times the frequency of the first mode and the frequency of the second mode. In other words, a predetermined current value is set so that the components hosting the first and second modes are in a regime that allows resonant 2N-to-1 photon exchange. There is no dependency on microwave devices external to the circuit, such as parametric pumps. Resonant 2N-to-1 photon exchange between the first and second modes, respectively, is the quantum dynamics that destroys 2N photons of the first mode at the resonant frequency of the first mode and generates one photon of the second mode at the resonant frequency of the second mode, and vice versa. The resonant 2N to 1 photon exchange is achieved by applying a predetermined current of constant magnitude to the circuit when the resonant frequency of the second mode is substantially 2N times the frequency of the resonant first mode, where "substantially" means that the predetermined current value 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 is achieved. This means that it is up to a certain threshold of the same order of magnitude as TIFF2025538425000021.tif6150.

[0033] Because superconducting quantum circuits inherently implement the quantum dynamics reliably, the need for parametric pumps is suppressed, improving the quality of resonant 2N to 1 photon exchange. Indeed, the elimination of the parametric pumps eliminates the appearance of detrimental parasitic interactions that affect the quality of resonant 2N to 1 photon exchange.

[0034] The circuit may be incorporated into a device that also includes a current source configured to apply a predetermined current of a constant magnitude to the circuit so 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 so that the induced internal current traverses at least some or all of the elements of the circuit. In other words, the applied current can induce internal currents in certain elements of the circuit that travel through the surface of the circuit. The current source is a device that can be placed at room temperature and therefore does not include superconducting elements. The current source can initially apply a predetermined current to the circuit through a conductor at room temperature, which, upon cooling, is connected to a superconducting wire that applies the current to the superconducting circuit. The current may also be filtered using a low-pass filter along the current path to reduce the effects of low-frequency noise.

[0035] In various examples, 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 an element with a given resistance external to the superconducting circuit, as opposed to a dissipative element, e.g., a superconducting element. The load dissipates photons exchanged from the first mode to the second mode by resonant 2N-to-1 photon exchange. In other words, photons destroyed from the first mode are discharged to the environment through the load by the second mode. The microwave source may be configured to apply microwave radiation at a frequency substantially equal to the frequency of the second mode or substantially equal to 2N times the frequency of the first mode. 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, and the second mode drives 2N photons of the first mode by resonant 2N-to-1 photon exchange. A coupler is an element that can be galvanically, capacitively, or inductively connected to an element of the circuit that hosts the second mode and mediates the interaction between the second mode, the load, and the microwave source.

[0036] 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.

[0037] The load can be a resistor, a matched transmission line, or a matched waveguide. The term "matched" should be interpreted as meaning that the transmission line or waveguide is terminated by a resistor at an end different from the end connected to the element hosting the second mode, the value of such resistor being selected so that the power directed to the load is mostly absorbed. The load can be contained within the microwave source.

[0038] In various examples, the microwave source may be placed at room temperature and connected to the circuit via a coaxial cable. In various examples, to thermalize the microwave radiation in a cryogenic environment, 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. 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. In this case, dissipation from the load, microwave radiation, and the resonant 2N to 1 photon exchange dynamics essentially performed by the circuit are used together to stabilize 2N coherent states in the first mode. Indeed, in the absence of microwave radiation, a stable manifold is formed by the Fock state TIFF2025538425000022.tif6150 is the spreading manifold. The first mode has some residual single-photon dissipation, so the state TIFF2025538425000023.tif6150 is the status TIFF2025538425000024.tif6150, and thus only the vacuum is stabilized in the first mode over the long term. In other words, the device allows the stabilization of a quantum manifold of coherent states above the vacuum.

[0039] Optionally, the device can include a bandpass or bandstop filter connected to the first and second modes of the circuit. The bandpass or bandstop filter can be configured to only allow coupling of the second mode to the load. Optionally, the device can also include a microwave filter to protect the first mode from dissipation within the load. This microwave filter can be interleaved between the load and the coupler. From the circuit's perspective, this filter is intended to prevent microwave photons at the first resonant frequency from escaping the circuit. This can be done either by implementing a bandstop filter at the first resonant frequency or a bandpass filter at the second resonant frequency, since photons of the second mode are the only photons that need to dissipate in the environment. In some circuits where the two modes have different symmetries, a filter may not be necessary; the appropriate symmetry of the coupler may be sufficient to prevent dissipation of the first mode.

[0040] Thus, the device allows for the stabilization of 2N coherent states in the first mode, i.e., a quantum manifold of coherent states. For example, a microwave source applying microwave radiation through a microwave filter to the second mode can be viewed as a 2N-photon drive in the first mode when converted by resonant 2N-to-1 photon exchange, and a load that dissipates only photons in the second mode can be viewed as a 2N-photon dissipation in the first mode when converted 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.

[0041] The second mode of single-photon driving is given by the Hamiltonian It is formally described as TIFF2025538425000025.tif6150, however, TIFF2025538425000026.tif6150 shows the single-photon driving intensity in the second mode by the microwave source. The single-photon dissipation in the second mode is calculated by the Lindblad operator It is formally described in TIFF2025538425000027.tif6150, however, TIFF2025538425000028.tif6150 is single photon dissipation resulting from second mode coupling into a load.

[0042] The 2N photon drive is given by the Hamiltonian It is formally described in TIFF2025538425000029.tif6150, however, TIFF2025538425000030.tif6150 is an effective 2N photon drive, TIFF2025538425000031.tif6150 is the 2N to 1 nonlinear conversion ratio between the first and second modes. The 2N photon loss is due to the Lindblad operator It is formally described in TIFF2025538425000032.tif6150, however, TIFF2025538425000033.tif6150 is the 2N photon dissipation rate. The amplitude of the stabilized coherent state TIFF2025538425000034.tif6150 is finally Given as TIFF2025538425000035.tif11150.

[0043] There is also provided a method of realizing a resonant cat qubit circuit, the method comprising the steps of providing a superconducting quantum circuit as described above, and applying a predetermined current of constant magnitude to the circuit or a device comprising 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, effectively engineering a resonant 2N-to-1 photon exchange between the two modes, respectively.

[0044] This method involves the use of two equal amplitudes and The method further includes using the device with the circuit to stabilize a quantum manifold spanning 2N coherent states with phase differences. This is done by combining the resonant 2N-to-1 photon exchange dynamics provided by the resonant cat qubit circuit with external dissipation and microwave radiation. Quantum information can ultimately be encoded in this quantum manifold, such as a cat qubit.

[0045] A quantum computing system is also provided. The quantum computing system may include at least one of a superconducting quantum circuit and / or a device including the circuit. The quantum computing system may be configured to use a superconducting quantum circuit and / or a device for executing a high-quality quantum computing protocol. In other words, the quantum system may use a device including a superconducting quantum circuit and / or a circuit for executing fault-tolerant quantum computing. 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 of 2N coherent states of the first mode, also referred to in some applications as a "cat qubit state." In this quantum manifold, the 2N coherent states have the same amplitude, and there is a gap between each coherent state. TIFF2025538425000037.tif6150 phase difference. Quantum computing systems can use these coherent states to define logical qubits, which are naturally protected against errors, especially bit-flip errors, thanks to the stability of the quantum manifold. Quantum systems can therefore define operations (e.g., CNOT, Hadamard, and / or Toffoli gates) that perform computations on logical qubits. This opens up a complete paradigm for implementing quantum algorithms in a fault-tolerant manner.

[0046] We now discuss the resonant cat qubit circuit in more detail.

[0047] A circuit can have a specific Hamiltonian when a predetermined 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 engineered to exhibit desired quantum dynamics, particularly a resonant 2N-to-1 photon exchange between the first and second modes, respectively. The Hamiltonian of a circuit, as used herein, is a function of a specific set of parameters of the superconducting quantum circuit and the predetermined current. The term "parameters" refers to any kind of physical parameter of the circuit and / or the predetermined current, such as capacitance, inductance, resistance, frequency, phase difference, energy level, phase zero point shift, Josephson energy, or critical current, as well as other parameters, such as voltage and / or current level (e.g., direct current (DC) bias) from the applied predetermined current. The set of parameters is unique in that it consists of the parameters of the circuit and the parameters of the predetermined current. In other words, the Hamiltonians do not depend on parameters other than their specific parameters. In yet other words, the Hamiltonians depend on the circuit, and only the circuit, and on the given current. For purposes of illustration, the set of parameters can be represented as the set P1 U P2, where U denotes the summation operator, and the set of parameters P1 consists only of the parameters of the circuit, and the set of parameters P2 consists only of the parameters of the given current (such as an induced DC bias).

[0048] Therefore, the total energy of a superconducting quantum circuit does not depend on any device parameters or a given current external to the superconducting quantum circuit. Indeed, the Hamiltonian depends only on a set of parameters of the superconducting quantum circuit and / or a given current, which may be predetermined according to quantum engineering specifications. Therefore, the Hamiltonian is time-independent, and in particular, it does not depend on any time-varying excitation, for example from a parametric pump.

[0049] In various examples, the Hamiltonian can include linear terms that describe the existence of modes hosted by elements of the circuit. In other words, each linear term describes the existence of one of a first mode and a second mode hosted in the circuit.

[0050] The Hamiltonian can also include a nonlinear term. The nonlinear term can describe the interaction between the first and second modes. The nonlinear term is sometimes called a "mixing term," similar to the frequency mixing that occurs in classical nonlinear microwave circuits. The nonlinear term can include a constant that can act as a prefactor. The constant can describe the strength of the interaction between the first and second modes. The prefactor of the nonlinear term can be smaller or much smaller than the frequency of the system. The nonlinear term can also be called resonant or nonresonant, depending on how well the nonlinear term conforms to the conservation of energy.

[0051] The Hamiltonian may be expandable into a sum of terms. The expression "expandable into a sum of terms" should be interpreted as meaning that the operator allows a Taylor series approximation to the sum of terms, recalling the representation of the energy of a superconducting quantum circuit. The total number of terms may be finite, or even infinite, but the sum of the terms always remains finite due to the Taylor series approximation. The sum may include a dominant term and a set of auxiliary terms. The expression "dominant term" should be interpreted as designating a term that has a significant influence on the dynamics of the system. For a term to be dominant, it must satisfy two conditions: First, the magnitude of said term, whether absolute or of an appropriate norm, should contribute significantly to the magnitude of the nonlinear part of the Hamiltonian with respect to the other terms in the Taylor series approximation. Second, the term should be resonant in the sense that it is compatible with the conservation of energy. A set of auxiliary terms is a term whose sum has a magnitude below a predetermined magnitude. Since all this is known per se from the Taylor series approximation, we will omit the auxiliary terms in the following. The Hamiltonian has the form It can be expanded into a sum between at least one dominant term of TIFF2025538425000038.tif6150 and a series of auxiliary terms, which will be omitted below.

[0052] Dominating term In TIFF2025538425000039.tif6150, TIFF2025538425000040.tif6150 is the first mode annihilation operator, TIFF2025538425000041.tif6150 is the second mode annihilation operator. Conversely, TIFF2025538425000042.tif6150 is the first mode generating operator, TIFF2025538425000043.tif6150 is the generating operator of the second mode. TIFF2025538425000044.tif6150 describes a resonant 2N to 1 photon exchange, TIFF2025538425000045.tif6150 is a polynomial. Therefore, the term The dominant term containing TIFF2025538425000046.tif6150 is the first mode TIFF2025538425000047.tif6 describes the annihilation of 150 photons and the creation of one photon in the second mode. Since the Hamiltonian is a Hermitian operator, the dominant term is the inverse term (also known as the Hermitian conjugate, abbreviated as hc) TIFF2025538425000048.tif6150 is also included, where the first mode TIFF2025538425000049.tif6150 photons are generated and one photon of the second mode is destroyed.

[0053] scalar TIFF2025538425000050.tif6150 is a function of a set of parameters consisting of circuit parameters and / or parameters of a given current. TIFF2025538425000051.tif6150 is a prefactor of the dominant term, so this scalar is the strength of the interaction between the first and second modes, also called the "intrinsic coupling strength", i.e., the term TIFF2025538425000052.tif6150 (each representing its Hermitian conjugate). In other words, TIFF2025538425000053.tif6150 describes the rate of resonant 2N to 1 photon exchange.

[0054] When a given current is applied to induce a resonant 2N to 1 photon exchange, i.e., when the given current is adjusted at the bias point, the rate of the resonant 2N to 1 exchange term, i.e., the constant TIFF2025538425000054.tif6150 is non-zero, and the frequency matching condition ensures that the term is resonant. In addition, the constant TIFF2025538425000055.tif6150 is large compared to the other nonlinear terms, in the sense of Taylor series approximation, and therefore the other nonlinear terms are auxiliary and will not be discussed further.

[0055] Therefore, when a predetermined current is applied to induce a resonant 2N-to-1 photon exchange (i.e., frequency-matched condition TIFF2025538425000056.tif6150 is satisfied), the dominant terms are resonance terms, i.e., they comply with energy conservation. Thus, for any N, the dominant terms describe nonlinear interactions that are odd powers of annihilation and creation operators.

[0056] The Hamiltonian is the Kerr term TIFF2025538425000057.tif6150 TIFF2025538425000058.tif6150 or cross-Kerr term TIFF2025538425000059.tif6150. The nonlinear terms can have significant magnitudes and inherently resonate despite any frequency matching conditions (this occurs only for even powers of the annihilation and creation operators). These Kerr and cross-Kerr terms may be considered detrimental to the desired engineering dynamics. However, the applicant has determined that the corresponding constants of the other nonlinear terms (i.e., the constants TIFF2025538425000060.tif6150 TIFF2025538425000061.tif6150 TIFF2025538425000062.tif6150) can be reduced thanks to the selection of circuit parameters as shown in the following examples. In particular, the applicant has found that they vanish near the optimal bias point. Therefore, these terms are not considered further in the following examples and are only listed for completeness.

[0057] The circuit may be configured to essentially perform a resonant two-to-one photon exchange between the first and second modes, respectively. In other words, N is equal to 1. In this case, the dominant term in the Hamiltonian is the two-to-one interaction Hamiltonian TIFF2025538425000063.tif6150. Alternatively, N may be greater than 1, for example N=2. The coherent state of the first mode may be, for example, a given coherent state For TIFF2025538425000064.tif6150, the annihilation operator acting on the first mode TIFF2025538425000065.tif6150 is an eigenstate, and as a result, TIFF2025538425000066.tif6150, where TIFF2025538425000067.tif6150 is a complex amplitude. In the cat qubit paradigm, the inherent two-to-one photon exchange stabilizes a quantum manifold of coherent states in the first mode. In the cat qubit paradigm, cat qubit states can be defined from the coherent states, e.g., logical qubit states TIFF2025538425000068.tif6150 is TIFF2025538425000069.tif6150, which can be defined as the logical qubit state TIFF2025538425000070.tif6150 is TIFF2025538425000071.tif6150. Both logical states belong to a quantum manifold of coherent states stabilized by a two-to-one photon exchange. The cat qubit state is naturally protected from errors such as bit-flip errors due to the stability of the quantum manifold achieved by the inherent two-to-one photon exchange. In fact, bit-flip errors can be autonomously suppressed at exponential rates. This enables the implementation of quantum gates that operate on logical qubit states, such as CNOT gates, Toffoli gates, and / or Hadamard gates, to perform fault-tolerant computations.

[0058] In various examples, a superconducting circuit can have a symbolic representation consisting of, for example, a set of interconnected dipoles. The term "symbolic representation" should be interpreted as designating an arrangement of symbols and lines that designate the set of interconnected dipoles. The set of interconnected dipoles (also called components) forms a circuit structure (or topology) that is equivalent (functional) to a nonlinear superconducting circuit.

[0059] In other words, as is classical in the field of superconducting circuits, a nonlinear superconducting circuit is constructed to achieve a function defined by a symbolic representation of the circuit, in other words, the function of a theoretical set of interconnected dipoles represented by the symbolic representation. In other words, while a circuit may be constructed using patterned layers of superconducting material, it should be understood that the circuit also allows for symbolic representation by dipoles, e.g., capacitors, inductors, and / or Josephson junctions. While the dipole examples describe discrete elements, those skilled in the art will clearly understand that these elements correspond to equivalent circuits of distributed elements within certain frequency ranges, e.g., at low frequencies, as is known in the art.

[0060] The interconnections of a set of interconnected dipoles of a symbolic representation can be described via a network topology. In this symbolic representation network topology, each branch can represent a dipole of the circuit, a node can be a connection point between two or more branches, and a loop can 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 can include components, e.g., two or more components, connected in parallel or in series. For example, a component pair including an inductor and a capacitor, i.e., an LC resonator, can be implemented by distributed elements within a patterned layer of superconducting material, e.g., - two adjacent plates forming a capacitor in parallel with a superconducting wire forming an inductor, -So-called TIFF2025538425000072.tif6150A section of a superconducting transmission line terminated at two different boundary conditions (short-circuited to ground at one end and open at the other end) forming a resonator. The transmission line may be, for example, of the coplanar waveguide or microstrip type. -So-called TIFF2025538425000073.tif6150A section of a superconducting transmission line terminated at two identical boundary conditions (open-open or short-short) forming a 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.

[0061] As is known in the art, such distributed elements may have higher-frequency modes that are irrelevant and unimportant to the dynamics described herein. Therefore, these distributed elements may be represented symbolically. This symbolic representation can be refined by adding elements such as parallel capacitors between any two nodes of the circuit or series inductors with respective wire connections, or by adding nodes and branches to account for other modes of the distributed element. Thus, the symbolic representation allows for a better description of the distributed element without changing the circuit's operating principles. Therefore, as is known in the art, a physical circuit, i.e., a circuit actually fabricated, and its symbolic representation are considered equivalent by those skilled in the art. In fact, refinements to the symbolic representation only adjust the resonant frequency or zero-point phase variation compared to the basic model. When designing a circuit, the final geometry can be fully and accurately simulated using finite element methods, which readily provide the frequency of the modes, the dissipation resulting from the load, and the zero-point phase variation across the Josephson junction, which are the only unknowns for calculating the resonant 2N-to-1 photon exchange rate in any configuration.

[0062] A symbolic representation of a circuit, e.g., a set of connected dipoles or components, can include at least one superconducting loop, i.e., a series of cycle-forming connected components (represented by dipoles in the symbolic representation). At least one loop can include one or more Josephson junctions. Each Josephson junction can consist of a thin insulating layer separating two superconducting leads that allows Cooper pairs to tunnel through the insulating layer. Each Josephson junction can be of the type TIFF2025538425000074.tif6150, where TIFF2025538425000075.tif6150 is the Josephson energy of the junction, TIFF2025538425000076.tif10150, but TIFF2025538425000077.tif6150 is the integral of the voltage across the junction, TIFF2025538425000078.tif6150 is a magnetic flux quantum. In various examples, the Josephson energy can be tuned during fabrication by selecting the surface and thickness of the insulating barrier, and therefore the room temperature resistance.

[0063] Josephson junctions are sometimes called nonlinear inductors. TIFF2025538425000079.tif6150 is the potential energy of an inductive element, which is the inductive energy TIFF2025538425000080.tif6150 by comparing it with the potential energy of the junction. To the first order, 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, and the junction becomes embedded, which causes a phase drop across the junction. TIFF2025538425000082.tif6150. Thus, application of a given current modifies the potential energy of the Josephson junction as follows: TIFF2025538425000083.tif6150

[0064] The first term on the right side of the sum above is the change in the effective Josephson energy of the junction TIFF2025538425000084.tif6150. The inductance of a Josephson junction is inversely proportional to the Josephson energy, so the DC offset TIFF2025538425000085.tif6150 allows adjusting the inductance of the junction and therefore the frequency of the modes involved in this junction. In the example above, the DC offset is used to adjust the frequency matching condition The second term represents the nonlinearity corresponding to the sinusoidal nonlinearity. The Taylor series expansion of the sinusoidal nonlinearity provides odd wave mixing. TIFF2025538425000087.tif6150 contains odd power terms, and therefore contains terms describing a resonant 2N to 1 photon exchange. In particular, the phase drop is The point at which the inductance of the junction becomes infinite and therefore the junction behaves as an open element. As a result, the cosine nonlinearity of the potential energy vanishes, i.e., the potential energy becomes zero and the sine nonlinearity is at its maximum.

[0065] 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.

[0066] In some cases, inductive energy Energy embedded in at least one loop in TIFF2025538425000089.tif6150 The Josephson junction in TIFF2025538425000090.tif6150 is It can be written as TIFF2025538425000091.tif10150. Effective magnetic flux passing through the loop When biased by a given current, equivalently described by TIFF2025538425000092.tif6150, an internal DC current is generated within the superconducting loop, TIFF2025538425000093.tif6150 and Phase drop across the junction depends only on TIFF2025538425000094.tif6150 Generates TIFF2025538425000095.tif6150. In the examples, the phase drop is calculated using the following formula: TIFF2025538425000096.tif6150However, It can be numerically calculated as the solution to TIFF2025538425000097.tif6150.

[0067] The solution to the above equation is TIFF2025538425000098.tif6150. For example, inductance The loop in TIFF2025538425000099.tif6150 contains two junctions, each with an energy TIFF2025538425000100.tif6150, the phase drop across each junction is Represented by TIFF2025538425000101.tif6150.

[0068] The Hamiltonian of the circuit can be determined in any manner. For example, the Hamiltonian can 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 junctions) is TIFF2025538425000102.tif6150, where TIFF2025538425000103.tif6150 is a reduced flux quantum TIFF2025538425000104.tif6150. The frequencies of the modes can be calculated algebraically or numerically by replacing 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 modal frequencies can be calculated along with the modal geometry, which describes what vibrational phase difference exists between any two points in the circuit when the mode is excited.

[0069] The magnitude of the oscillatory phase difference imposed across a Josephson junction is particularly important for calculating the nonlinearity of the system. In the quantum regime, the phase difference across the junction is TIFF2025538425000105.tif6150 can be written, however, TIFF2025538425000106.tif6150 is the DC phase offset calculated previously, TIFF2025538425000107.tif6150 and TIFF2025538425000108.tif6150 is the zero-point variation in phase across the junction, which is directly related to the geometry of the modes up to normalization. The phase difference may include other terms in "..." that relate to other modes, for example. These terms are omitted below, as they are not relevant in this context.

[0070] The nonlinear term in the Hamiltonian is the potential energy of the junction It can be calculated by performing a Taylor approximation on TIFF2025538425000109.tif6150 and summing the contribution of each junction in the circuit. All that is needed to completely describe the circuit is to find the DC phase drop across the junctions, the frequencies of the modes, and the zero variation in phase between the junctions associated with each mode of the system.

[0071] In various examples, the circuitry controls the first and second modes at respective frequencies TIFF2025538425000110.tif6150 and TIFF2025538425000111.tif6150, so the linear part of the Hamiltonian is TIFF2025538425000112.tif6150 can be written, except that TIFF2025538425000113.tif6150 is the reduced Planck constant.

[0072] At least one of the interaction terms provided by the Josephson junction is of the form Assume the file is TIFF2025538425000114.tif6150, where TIFF2025538425000115.tif6150 is the DC phase drop across the junction induced by a given current, TIFF2025538425000116.tif6150 and TIFF2025538425000117.tif6150 are the zero-point variations of the phase across the junction associated with the first and second modes, respectively. The Hamiltonian is therefore the resonant 2N-to-1 photon exchange Hamiltonian TIFF2025538425000118.tif6150, where: TIFF2025538425000119.tif10150 is TIFF2025538425000120.tif6150. The resonant 2N to 1 photon exchange rate is therefore dependent only on the parameters of the circuit and the given current, such as those described above.

[0073] In contrast, as mentioned above, existing cat qubit circuits rely on the use of parametric pumps to perform two-to-one photon exchange. In these prior art realizations, the two-to-one photon interaction Hamiltonian has the form TIFF2025538425000121.tif9150. In these cases, the combined term TIFF2025538425000122.tif6150 is modulated via a parametric pump, and the pump is Injecting external time-varying parameters with TIFF2025538425000123.tif6150. Parametric pumps are used in the prior art to resonate nonlinear interactions, but have detrimental effects.

[0074] Examples and illustrative circuits and devices are now discussed with reference to the figures. In the following, the terms "resonant cat qubit circuit," "circuit," and "superconducting circuit" are used interchangeably to refer to a circuit that performs a 2N-to-1 photon exchange that allows the cat qubit to be stable when the second mode is appropriately driven and made dissipative.

[0075] FIG. 1 shows an example of a device comprising a resonant cat qubit circuit.

[0076] For purposes of explanation, hereinafter N is equal to 1, but the same applies for other integer values ​​of N. The device incorporates a nonlinear superconducting circuit 100, which has a first mode TIFF2025538425000124.tif6150101 and the second mode TIFF2025538425000125.tif6150102 essentially performs a two-to-one photon exchange represented by the single arrow 1200 and double arrow 1100 back and forth between the nonlinear superconducting circuit 100 and the nonlinear superconducting circuit 102. A current source 103 is connected to the nonlinear superconducting circuit 100 via an electric wire. In other words, the current source 103 is directly connected to the circuit 100 so 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 also satisfies frequency matching conditions. The circuit component hosting the second mode 102 is coupled to a load 105 via a coupler 104. This coupling makes the second mode dissipative. The device also The device incorporates a microwave source 106 of TIFF2025538425000127.tif6150. The microwave filter 107 may be configured as a bandpass filter of frequency 1000 kHz. TIFF2025538425000129.tif6150 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 thereby prevent the first mode from incurring additional loss due to undesired coupling to the load 105.

[0077] FIG. 2 shows an example of coupling the second mode of a superconducting circuit to a load, and a filter that may be integrated into the device to allow coupling of only the second mode to the load.

[0078] 2 a) shows a schematic diagram of a circuit 100 of FIG. 1, adapted to couple the second mode 102 to a load and to modulate the circuit 100 at a frequency 104 by a microwave source 106. TIFF2025538425000130.tif6150, showing the coupling 200 to the device via the coupler 104 and filter 107, Second Resonant Frequency In TIFF2025538425000131.tif6150, we allow only the second mode of loss, which is modeled as resistor 105, but the first resonant frequency TIFF2025538425000132.tif6150 does not allow for loss of the first mode. Figure 2 shows at least two possibilities for coupling 200, namely, frequency TIFF2025538425000133.tif6150 (also shown in schematic diagrams b) and d) in Figure 2) or frequency A band-stop filter (also shown in schematics c) and e) of Figure 2) is shown in TIFF2025538425000134.tif6150. Alternatively, when the circuit is fabricated in a 3D architecture, another possible solution is to use a waveguide wide-pass filter to couple the second mode to the load, thanks to its high-frequency selectivity.

[0079] The schematic diagrams b) to e) of FIG. 2 show different possibilities for the coupling 200.

[0080] Schematic diagram b) shows the frequency TIFF2025538425000135.tif6150 shows a bandpass filter 210 capacitively coupled (201) to the input port. The bandpass filter (which is an LC oscillator) 210 operates at a frequency TIFF2025538425000136.tif6150 is configured to resonate, and its impedance TIFF2025538425000137.tif6150 adjusts the width of the bandpass.

[0081] Schematic diagram c) shows the frequency TIFF2025538425000138.tif6150 shows a band-stop filter 220 capacitively coupled 201 to the input port. The band-stop filter 220 is an LC stub, and TIFF2025538425000139.tif6150 is configured to resonate, and its impedance TIFF2025538425000140.tif6150 adjusts the width of the band rejection.

[0082] Schematic diagram d) shows the frequency TIFF2025538425000141.tif6150 shows a bandpass filter 210 inductively coupled 202 to an input port. The bandpass filter 210 is an LC oscillator with a frequency TIFF2025538425000142.tif6150 is configured to resonate, and its impedance TIFF2025538425000143.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.

[0083] Schematic diagram e) shows the frequency TIFF2025538425000144.tif6150 shows a band-stop filter 220 inductively coupled (202) to the input port. The band-stop filter is an LC stub with a frequency TIFF2025538425000145.tif6150 is configured to resonate, and its impedance TIFF2025538425000146.tif6150 adjusts the width of the band rejection.

[0084] Figure 3 shows an example of quantum manifold stabilization of first-mode coherent states achieved by resonant 2N-to-1 photon exchange.

[0085] The schematic diagram in Figure 3a) shows the amplitude TIFF2025538425000147.tif6150 and rate Shown is a linear scheme with a single-photon driver (microwave tone at the mode frequency) characterized by single-photon dissipation according to TIFF2025538425000148.tif6150. Time scale After TIFF2025538425000149.tif6150, the mode state is Amplitude The system converges to a single coherent state 301 at TIFF2025538425000150.tif6150. This is a stable steady state of the dynamics. However, this steady state is unique, and no information can be encoded into it. This state is represented as a blurred point in the Cartesian space of modes due to the uncertainty principle of quantum mechanics.

[0086] The schematic diagram in Figure 3 b) shows Strength TIFF2025538425000151.tif6150 Two-photon drive and amplitude Rate with two stable steady states (302, 303) in TIFF2025538425000152.tif6150 TIFF2025538425000153.tif6150 shows the first mode undergoing two-photon dissipation with opposite phases. Since there are two possible states, information can be encoded, i.e., state TIFF2025538425000154.tif6150302 is surrounded by a solid line, TIFF2025538425000155.tif6150303 is enclosed by a dotted line. This encoding is based on the stable nature of the dynamics that converges to two states. TIFF2025538425000156.tif6150 and status The system is robust to bit-flip errors that cause the system to flip between the first and second modes. The coding does not correct the other error channel, namely, the phase-flip error. However, an additional error correction scheme may be added to handle this separately. This stabilization is made possible by coupling into an extra mode and engineering a two-to-one photon exchange between the first and second modes.

[0087] Schematic diagram c) of Figure 3 shows the case of N=2, where four-photon drive and four-photon dissipation are used to stabilize four states in phase space. In this four-dimensional manifold, the state TIFF2025538425000158.tif6150, and states in another superposition of two coherent states 305 shown by dashed lines. TIFF2025538425000159.tif6150 can be encoded. Two degrees of freedom remain in the four-dimensional manifold, which can act as an error manifold used to perform first-order quantum error correction. This stabilization is made possible by coupling to an extra mode and engineering a four-to-one photon exchange between the first and second modes.

[0088] Schematic diagram d) of Figure 3 shows the increased dimensionality of the stabilized manifold. The increased dimensionality allows for higher-order error correction compared to lower values ​​of N. With six-photon drive and six-photon dissipation (i.e., N=3), we can stabilize a six-dimensional manifold of coherent states capable of performing second-order quantum error correction. This stabilization is made possible by coupling into an extra mode and engineering a six-to-one photon exchange between the first and second modes.

[0089] FIG. 4 shows an example of how to determine the magnitude of the current applied to the circuit to achieve resonant 2N to 1 photon exchange.

[0090] Figure 4a shows the adjustment of a given current to be 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 a matched condition. In other words, the equivalence is the constant that represents the rate of resonant 2N-to-1 photon exchange: The bias point can be reached by varying the current in the circuit, and the bias point is determined by the virtual TIFF2025538425000162.tif6150 Frequency line 402 and TIFF2025538425000163.tif6150 corresponds to point 401 experimentally determined by the anti-crossing between spectral frequency line 403. The parameters of the circuit elements can be selected within a range where the frequencies of each of the two modes are close to the frequency matching condition.

[0091] For some circuits, there is an optimum choice of parameters for the circuit's elements. Therefore, the bias point is the optimum bias point, where spurious even terms such as Kerr and cross-Kerr terms are also cancelled as described above.

[0092] FIG. 4b shows that the optimum bias point 404 is at a point where the DC phase drop across the junction is TIFF2025538425000164.tif6150. To reach this optimal bias point, circuit parameters can be adjusted during manufacturing. Alternatively, a separate tuning knob can be added. This extra tuning knob, which can be realized, for example, by incorporating a SQUID (two junctions in parallel biased with external currents to independently control their frequencies) in the second mode, can be used as an extra degree of freedom to reach both the frequency-matching condition and the vanishing Kerr condition. Those skilled in the art will recognize that this is simply a matter of implementation.

[0093] FIG. 5 shows some examples of circuit symbolic representations that are alternatively used throughout this specification.

[0094] Schematic diagram a) of Figure 5 shows two example 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 schematic diagram a)) shows a current source 501 galvanically coupled to a superconducting loop. The example shows at least one terminal of the current source connected directly to the superconducting loop 510, which is conveniently isolated from the rest of the circuit. However, this is a matter of implementation. The current source may be connected in any way to apply a predetermined current, as shown below. Predetermined Current When TIFF2025538425000165.tif6150 is applied, a phase drop occurs. TIFF2025538425000166.tif6150 flows across the junction 502. This is the internal current TIFF2025538425000167.tif6150 passes through junction 502. The second circuit (to the right of the first circuit) shows a current source inductively coupled to the superconducting loop through 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.

[0095] Thus, the current source may be implemented in any way to achieve a phase drop across the junction. Schematic diagram a) of Figure 5 illustrates this with circuit 530, which equivalently summarizes the two circuits 510 and 520 of schematic diagram a). When a given current is applied, either by a standard current source or a transformer, the magnetic flux TIFF2025538425000168.tif6150504 represents the effective magnetic field biasing a loop induced by a given current applied to the circuit. The magnetic flux, hereafter referred to as the external magnetic flux when integrated on the surface of the loop, It is sometimes called TIFF2025538425000169.tif6150. The inductance 505 represents the total self-inductance of the superconducting loop embedded in the junction 502. This last 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, the axis of the magnetic coil being perpendicular to the plane of the circuit. The external magnetic flux is oriented at an angle TIFF2025538425000170.tif6150, where TIFF2025538425000171.tif6150 is a magnetic flux quantum. In this application, the system period TIFF2025538425000172.tif6150 It is periodic in TIFF2025538425000173.tif6150, It can be assumed that the image is symmetric about TIFF2025538425000174.tif6150. Therefore, the analysis is performed on the interval It can be limited to TIFF2025538425000175.tif6150.

[0096] Schematic diagram b) of Figure 5 shows an alternative description of a transformer as a current source. The transformer, referred to here as 550, can be two circuit branches that are not galvanically connected (i.e., not in direct contact), but share a mutual inductance due to their proximity. Alternatively, transformer 550 can be part of a circuit in which two loops galvanically share a common conductor. Transformer 550 can be effectively implemented in practice.

[0097] 5 c) shows replacing the inductance with an array of junctions in series. Unless otherwise specified, this array can comprise as few as a single junction.

[0098] Schematic diagram d) of FIG. 5 shows a Y-inductor that can be used to generate alternative circuits that follow the same principles of the other embodiments, but change the topology of the circuit without changing its behavior, simplifying the analysis. TIFF2025538425000176.tif6150 conversion shown.

[0099] A circuit can be configured to perform 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 illustrates how the energy of a Josephson junction, or more generally, the energy of a nonlinear inductive device, can be engineered to perform resonant 2N to 1 photon exchange. The presence of at least one loop containing one or more Josephson junctions facilitates the inherent resonance of the circuit. In fact, the energy of the Josephson junctions can depend on the parameters of the components hosting the first and second modes, as well as the predetermined current that induces the internal current through at least one loop. In fact, the one or more Josephson junctions included in the loop provide the mixing capability that enables 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 is generated by resonant 2N to 1 photon exchange when a predetermined current is applied through at least two nodes of the circuit.

[0100] The following examples are presented under the assumption that the circuit operates at the optimal bias point according to the principles described above. This allows for improved readability of the equations provided below. However, it should be noted that the following examples also apply to non-optimal bias points.

[0101] 6 illustrates another embodiment in which a symbolic representation of a circuit includes at least one loop containing one or more Josephson junctions. Circuit 60600 according to this example is also configured to perform resonant 2N to 1 photon exchange when a predetermined current is applied to induce a phase difference across one or more Josephson junctions. Circuit 60600 according to this example is specifically configured to symmetrically distinguish between the first and second modes. The high symmetry of the circuit achieves improved quality of the resonant 2N to 1 photon exchange.

[0102] Next, an example of a circuit 60600 having at least one loop 610 will be discussed with reference to schematic diagram a) of FIG. 6 . In the circuit 600, the at least one loop 610 can include a first Josephson junction 601, a central inductive element 603, and a second Josephson junction 602 arranged in series. The central inductive element 603 can be an inductance, a single Josephson junction, or an array of Josephson junctions. Thus, the central inductive element 603 can be arranged between the first and second Josephson junctions as a series loop. The series arrangement can include a first inner node connecting a pole of the first Josephson junction 601 with a pole of the inductive element 603. The series arrangement can also include a second inner node connecting a pole of the second Josephson junction 602 with another pole of the inductive element 603. The series arrangement may also include a closed loop node connecting the other pole of the first Josephson junction 601 with the other pole of the second Josephson junction 602 .

[0103] The at least one loop 610 may be connected to a common ground through a closed-loop node. The circuit may also include a first capacitor 604 and a second capacitor 605. The first capacitor 604 may be connected in parallel with a first Josephson junction 601 between the common ground and a first internal node of the loop. The second capacitor 605 may be connected in parallel with a second Josephson junction 603 between the common ground and a second internal node of the loop.

[0104] Thus, superconducting quantum circuit 600 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. Josephson junctions 601 and 602 are substantially identical, and capacitive elements 604 and 605 are also substantially identical. Thus, the symmetry of the circuit implies that the actual modes of the system are a symmetric superposition 606 of two resonators (indicated by a full arrow) and an antisymmetric superposition 607 of two resonators (indicated by a dashed arrow). The first mode is a symmetric superposition 606, and the second mode is an antisymmetric superposition 607. It can be noted that only the second mode has a contribution across central inductive element 603, which is advantageously used to preferentially couple the environment to this second mode while isolating the first mode from the environment.

[0105] There is no optimum bias point for this circuit. In fact, the junction must have a non-zero phase drop across the inductor, so It may be biased towards TIFF2025538425000177.tif6150 The resonator cannot be biased to the maximum budget of TIFF2025538425000178.tif6150. However, this is not a problem, since an optimal bias point is not desired. In fact, at such a point, the Josephson junction acts as an open circuit, thus leaving only the parallel Josephson capacitance. This contradicts the fact that, in this particular embodiment, the junction functions as the primary inductive element for the symmetric mode. Adding a loop can enable an optimal bias point. An exemplary implementation is a symmetrized version of the circuit of Figures 6a and 6b, where both resonators are identical and nonlinear by replacing inductance 605 with loop 610. As proposed in Figures 6a and 6b, the coupling between the two nonlinear identical resonators can be capacitive or inductive.

[0106] Here, circuit 600 consists of two identical resonators that are strongly coupled via a central inductive element. This implies that the bare detuning (before adding coupling) between the two resonators is zero, and the perturbative description performed for circuit 600 no longer holds. Therefore, the analysis presented here differs from that of circuit 600. If we assume that the system is perfectly symmetric, i.e., both junctions and both capacitances are identical, the system will operate in a symmetric mode (i.e., the first mode TIFF2025538425000179.tif6150) and antisymmetric mode (i.e., the second mode TIFF2025538425000180.tif6150). In this eigenmode basis, the contribution of each junction to the Hamiltonian of the system can be calculated, TIFF2025538425000181.tif7150 TIFF2025538425000182.tif7150This can be factorized as follows: TIFF2025538425000183.tif6150 TIFF2025538425000184.tif6150

[0107] The two quadratic parts of the first term are The effective inductive energy of the junction at the point of application of TIFF2025538425000185.tif6150 is given. Along with the charging energy of the capacitor and the inductive energy of the central inductive element, the frequency TIFF2025538425000186.tif10150 and TIFF2025538425000187.tif10150 and phase zero point fluctuation TIFF2025538425000188.tif12150 and It becomes possible to define TIFF2025538425000189.tif12150.

[0108] By expanding the second term to the desired order, we obtain the resonant 2N-to-1 photon exchange Hamiltonian TIFF2025538425000190.tif6150, but You can get TIFF2025538425000191.tif10150.

[0109] Due to the symmetry of the system and the frequency matching conditions that must be met at the bias point of the system, the equation can be further simplified. Since it is TIFF2025538425000192.tif6150, TIFF2025538425000193.tif6150= TIFF2025538425000194.tif6150, therefore, TIFF2025538425000195.tif6150, It can be shown that it leads to TIFF2025538425000196.tif14150.

[0110] Schematic diagram b) of Figure 6 shows an example of a device used to stabilize a manifold of coherent states using circuit 600. Viewed from the bottom up, the nonlinear superconducting circuit 600 is at the bottom, with an inductive element sharing mutual inductance 608 with another inductor terminating in environment 620 above. Environment 620 consists of a load, a microwave source, and a DC current source. In the same manner as described above, the same input ports of the device can be conveniently used to carry DC current for external magnetic flux and microwave radiation. In practice, a portion of a transmission line is used to connect the circuit to the environment as described above. This portion of the transmission line closest to circuit 609 is typically a differential transmission line to preserve the symmetry of circuit 600.

[0111] Schematic diagram c) of Figure 6 shows an example of a planar superconducting pattern 650 and its circuit equivalent. The presented superconducting circuit is fabricated in a coplanar waveguide (or CPW) geometry, with the background (grayed out) representing the remaining superconducting metal on the dielectric substrate. Each black cross represents a Josephson junction. In contrast to the device in schematic diagram b), the inductive element is replaced with a single junction. In this embodiment, the central superconducting loop 610 is diluted by using three sections of CPW transmission line, called stubs (611, 612, 613), to control the level of nonlinearity in the system. On the left and right sides, open stubs 611 and 612 provide some stray inductance in series with the required capacitance. At the bottom, the ring is not directly connected to ground; instead, a shorted stub 613 is interleaved to provide inductance to ground. While the circuit mode and its operating principle remain the same, the length of transmission line added by the stubs allows for limiting the level of nonlinearity in the circuit. To preferentially couple to the second mode and maintain the symmetry of the circuit, a transmission slotline 609 is used to inductively couple the mode to the environment. A transition between the CPW and the slotline 614 is shown above for coupling to the environment (and therefore a current source) with a common geometry 615 (e.g., a CPW line or coaxial cable).

[0112] A flux line design according to the invention is described, which offers superior properties in stabilizing the manifold of coherent states compared to the embodiment of schematic diagram c) of Figure 6. As a precaution, in order to be able to stabilize the manifold of coherent states using a resonant cat circuit, Inducing a bias current in the central loop to activate the nonlinear transformation; coupling to and driving a second mode or buffer mode and allowing it to dissipate; Importantly, we do not couple the first or cat qubit modes, This is what is required.

[0113] Compared to the implementation proposed in European Patent Application No. 21306965.1, the present invention aims to propose a solution that fully meets these requirements with a single flux line. Specifically, the circuit design 650 has been improved to provide better coupling to the central loop at DC and better coupling to the buffer modes at RF.

[0114] Figure 7 shows a general circuit diagram of a resonant cat qubit circuit to recall important properties of the circuit proposed in Figure 6. Specifically, the circuit is constructed around a central loop 710 containing two substantially equal Josephson junctions 711, 712 and one central inductive element 713. The loop is arranged so that the two substantially equal Josephson junctions define an axis of symmetry A. Details of the circuit portions 721, 722 on the left and right of the central loop and the circuit portion 723 below the central loop are not important to the present invention, which is focused locally around the central loop, except that they are symmetric about the axis of symmetry A. These circuit portions are omitted from subsequent figures. Due to the symmetry of the circuit, all modes supported by the circuit are symmetric or antisymmetric about the axis of symmetry. Symmetric modes are also referred to as common modes, and antisymmetric modes are also referred to as differential modes. Below, we focus on two modes of the circuit: one with a common geometry to store the cat qubit, and the other with a differential geometry used as a buffer mode in the context of the cat qubit. The mode geometry is emphasized by showing the voltage configuration around the central loop. In what follows, the common mode is shown as a solid line, the differential mode as a dashed line, currents as thick arrows, and potential differences as thin arrows.

[0115] FIG. 8 illustrates an embodiment of a quantum device with a symmetric resonant cat qubit circuit that has no flux lines and is not connected to the environment.

[0116] As shown in FIG. 8a, device 800 includes a ground plane 820 and a symmetric resonant Cat qubit circuit 840 for performing 2N-to-1 photon exchange. The gray areas represent superconducting material disposed on a dielectric substrate and patterned to form the circuit, separated by exposed substrate portions (white areas). The majority of the superconducting material is within the ground plane 820 that surrounds the circuit. To explain the coupling strategy, it is useful to represent common-mode currents circulating within the ground plane next to a central loop, circled and labeled 844. It is important to note that the common-mode current vanishes at the circuit's axis of symmetry A, while the differential-mode current has a nonzero current at the same location.

[0117] Figure 8b shows a generalized differential-mode equivalent circuit 850 for the configuration shown in Figure 8a, which is useful for characterizing coupling. The equivalent circuit includes an inductive element 851 and a capacitive element 852, which are oriented horizontally so that the + / - polarity directions of the circuit match those of the differential mode in Figure 8a.

[0118] FIG. 9 shows a first embodiment of a quantum device including a symmetric resonant cat qubit circuit and flux lines in accordance with the present invention.

[0119] As shown in FIG. 9a, device 900 comprises an environment 910, a ground plane 920, a flux line 930, and a resonant cat qubit circuit 940 for performing a 2N-to-1 photon exchange.

[0120] Environment 910 comprises a current source (not shown in FIG. 9), a microwave source 914, and a load 916. In this embodiment, the current source is not shown because the focus is on the RF characteristics of the induced second mode (differential mode or buffer mode) and first mode (common mode or cat qubit mode).

[0121] Ground plane 920 is connected to a common ground, as indicated by the triangles connected to the dots. Flux lines 930 are formed on common ground 920 by exposing a portion of the dielectric substrate on which common ground 920 is located.

[0122] Flux line 930 comprises a coplanar waveguide (CPW) section 932, a first slot 934, and a second slot 936. More precisely, CPW section 932 comprises slot 9320 and slot 9322 that delimit a centerline that is further connected to a current source, a microwave source 914, and a load 916. CPW section 932 comprises multiple wire bonds that ensure that the ground planes on either side of CPW section 932 are at the same potential. For simplicity, only wire bonds 938 closest to resonant cat qubit circuit 940 are shown in FIGS. 9a and 10-12, separating CPW section 932 of flux line 930 from first slot 934 and second slot 936.

[0123] In the example shown herein, resonant cat qubit circuit 940 is similar to the circuit shown in schematic diagram a) of Figure 6 and includes two Josephson junctions 941 and 942 and an inductance, central inductive element 943. In other embodiments, central inductive element 943 may be a Josephson junction or an array of Josephson junctions.

[0124] As mentioned above, the main focus of this diagram is to clarify the common and differential modes induced by this embodiment. To better describe these modes, the following notation is used: The solid arrows refer to the common mode. The dashed arrows refer to the differential mode. The thick arrows represent the circulation of current, The thin arrows relate to the potential difference. This is what is used.

[0125] Thus, in Figure 9a, arrows, whether solid or dashed, thick or thin, indicate common-mode and differential-mode current paths. For example, Figure 9a shows that differential-mode current flows from the environment into CPW section 932, and return current flows along first slot 934.

[0126] 9a, second slot 936 is much shorter than first slot 934, which is connected to the gap between resonant cat qubit circuit 940 and the rest of ground plane 920. In other words, first slot 934 divides ground plane 920 into two unconnected portions 9210 and 9220.

[0127] In this embodiment, the second slot 936 is optional. Both slots extend substantially parallel to axis A. Axis A constitutes an axis of symmetry for the resonant cat qubit circuit 940 and for the first slot 934. This exploitation of the symmetry of the resonant cat qubit circuit 940 is important because, since the common mode has currents flowing in opposite directions in the ground plane portions 9210 and 9220, the common mode current is zero at the level of axis A, while the differential mode current is non-zero, resulting in differential mode coupling. This results in microwave radiation emitted from the environment 910 coupling almost exclusively with the differential mode rather than the common mode, which is essential for stabilizing the cat qubit. The terms "almost exclusively coupled" or "substantially exclusively coupled" should be interpreted to mean that differential mode (or common mode) coupling is much greater than common mode (or differential mode) coupling. In a preferred manner, this ratio exceeds 100. It is important to note that slot 934 extends to the central loop of the circuit, completely cutting the ground plane into two halves, allowing for strong differential-mode coupling. This contrasts sharply with the implementation in schematic diagram c) of Figure 6, where the slot does not completely cut the ground plane. This prior art cut does not extend completely, leaving a residual galvanic path between the two halves of the ground plane that can couple DC currents but at least partially shunts differential-mode RF coupling. The coupling ratio can be further improved by interleaving a microwave filter that passes both DC and microwave radiation at differential-mode frequencies, as described in Figure 2.

[0128] Figure 9b shows the differential-mode equivalent circuit of resonant cat qubit circuit 940 when the device is operated to guarantee 2N-to-1 photon exchange and the microwave source (or load) and differential-mode coupling are substantially exclusively to this differential mode. In this case, the presence of flux lines affects the equivalent circuit of Figure 8b. Specifically, the disconnection induced by slot 934 is represented by capacitor 952, and the remaining inductive path from left ground plane 9210 to right ground plane 9220 through wire bond 938 is represented by inductor 950. Inductance 851 of Figure 8b is disconnected into two parts, 954 and 956, while main capacitor 958 remains largely unchanged. This equivalent circuit shows that the coupling to the environment can be tuned by selecting the location of wire bond 938. Indeed, moving the wire bond away from the resonant cat qubit circuit increases the impedance of the reactive path (including inductor 950 and capacitor 952), increasing the tendency for differential-mode current to pass through the resistor, i.e., increasing the coupling to the environment.

[0129] Applicant's testing has revealed that this embodiment is very interesting in that the microwave source and load are substantially coupled in differential mode and not in common mode, but at the expense of insufficient coupling of the current from the current source (hereinafter DC coupling) to supply a given current.

[0130] Intuitively, the applicant reasoned that while placing the first slot 934 at the center of axis A would allow for substantially exclusive coupling between the differential modes of the microwave source 914 and the load 916, the second slot 936 would need to be very short (if not non-existent) to respect the symmetry of the device, resulting in insufficient DC coupling.

[0131] Therefore, applicant has prepared a second embodiment shown in Figure 10, in which similar elements to Figure 9a share the same reference numerals except for the first two digits, i.e., wire bond 938 in Figure 9a has reference numeral 1038 in Figure 10. The same is true for Figures 11 and 12.

[0132] For brevity, only the differences from Figure 9a will be described, as will Figures 11 and 12. In this figure, the emphasis is on showing DC coupling, so only current source 1012 is shown.

[0133] In this embodiment, the length of the second slot 1036 is approximately the same as that of the first slot 1034, except that the second slot 1036 does not extend all the way to the exposed dielectric substrate portion corresponding to the boundary of the portion of the resonant cat qubit circuit 1040 including the central inductive element 1043. For example, the distance between the tip of the second slot 1036 and the aforementioned exposed dielectric substrate portion can be as small as 1 μm or as large as tens of μm. In this embodiment, as shown by the thick dashed line, DC current flows from the CPW portion 1032 along the first slot 1034 and the second slot 1036, and spreads within the ground plane portion 10220 approximately perpendicular to axis A in a direction away from the first slot 1034, hereinafter referred to as the "direction of rotation."

[0134] Applicant's performance analysis of this embodiment shows that while DC coupling is significantly improved, it comes at the cost of degraded differential mode coupling and increased undesired common mode coupling. Applicant has discovered that this is due to the effect that second slot 1036 has on the resonant cat qubit circuit 1040, breaking the symmetry of the device. Note that second slot 1036 can be implemented with mirror symmetry about axis A in FIG. 10 .

[0135] To overcome the newly created deficiency, applicants came up with the idea of ​​offsetting the flux lines perpendicular to axis A to re-establish common mode isolation.

[0136] This effort resulted in the embodiment illustrated in Figure 11. In this embodiment, the first slot 1134 and the second slot 1136 are offset in the opposite direction of rotation. This offset can be as small as 1 μm or as large as the width of the device, meaning that the first slot 1134 and the second slot 1136 are located at the edges of the device.

[0137] Applicant's performance analysis of this embodiment shows that the common mode coupling exhibits a sweet spot where coupling to the environment disappears while maintaining differential mode coupling, but that the differential mode coupling could be improved.

[0138] To overcome the newly arising defects, the applicant came up with the idea of ​​adding a return portion to the second slot 1236 substantially along the rotational direction.

[0139] This effort resulted in the embodiment illustrated in Figure 12. In this embodiment, the first slot 1234 and second slot 1236 are still axially offset, but the second slot 1236 has an additional return portion 12360. The return portion can be as small as 1 μm or as large as the width of the device, i.e., it can extend to the edge of the device.

[0140] Applicant's performance analysis of this embodiment has shown that this embodiment exhibits the best performance, as it has high DC and differential mode coupling, but very low common mode coupling.

[0141] Applicant has also discovered that the return portion can be used in the second and third embodiments with some modification.

[0142] Figures 13 and 14 show photographs of fabricated devices corresponding to the embodiment of Figure 9 and the embodiment of Figure 12, respectively. In each figure, the enlarged portion allows a better view of how the first slot divides the ground plane and how the first and second slots are positioned relative to the central mixing component.

[0143] In tests conducted by the applicant, the circuit of Figure 13 achieved differential target coupling up to 100 MHz. It has been shown that a magnetic flux density of approximately 0.1 flux quanta (FQ) per mA of current can be achieved, depending on the exact wirebond configuration. TIFF2025538425000198.tif6150), which allows for improved DC coupling to the center loop. This is also achieved in the circuit of Figure 14, which achieves the target differential mode coupling up to 50 MHz, i.e. TIFF2025538425000199.tif6150, and achieve DC coupling to the central loop on the order of one flux quantum per mA, a typical target parameter in this field.

[0144] Next, the measurement method according to the present invention will be discussed.

[0145] The cat qubit belongs to a family of bosonic qubits that are encoded in harmonic oscillators, which we also call cat qubit modes. In contrast to two-level systems, harmonic oscillators have infinitely more levels than can be used to encode information.

[0146] Tomography is an operation that allows for complete knowledge of the state of a quantum system. This operation requires measurements of different observables. For a two-level system, measurements of observables X, Y, and Z are sufficient to fully characterize the system's state. For a harmonic oscillator with infinitely many energy levels, the system must be assumed to allow tomography to be performed with a finite number of measurements. Generally, the system is assumed to be in a low-energy Fock state subspace. Some forms of measurement can be performed experimentally using harmonic oscillators in superconducting circuits, such as the Husimi-Q function or the Wigner function and its corresponding characteristic functions. These functions are defined over the oscillator's phase space, with quadrature phases designated I and Q. By sampling a finite portion of the phase space, these functions can be measured directly. From this finite sampling and the assumption that the system involves a low-energy subspace, a maximum likelihood algorithm can be used to reconstruct the system's state upon completion of tomography.

[0147] One can make more specific assumptions about the subspace in which the system exists. For example, in the cat qubit paradigm, coherent state manifold stabilization restricts the possible states to within a span of 2N coherent states. Therefore, with this assumption, fewer measurements are needed to perform tomography of the system. For example, For a two-component CAT qubit, defined within the span of {TIFF2025538425000200.tif6150}, one can perform a full Wigner function measurement of the state, or measure the effective X, Y, or Z, similar to a two-level physical system. The former is primarily used to tune and characterize the operation of superconducting circuits, while the latter is used during calculations. In that particular case, the measurement of X is also the parity of the number of photons in that state, and Z is a measure of whether the population is in one or the other coherent state. Another example is the four coherent states in the so-called four-component CAT paradigm { The goal is to define qubits in an even manifold of spans {TIFF2025538425000201.tif6150}. It is clear that measuring the complete Wigner function of a system is more complete than measuring a few observables while guessing at possible states. As a result, it is generally possible to reconstruct the mean values ​​of these observables using the complete Wigner function.

[0148] In the context of cat qubits, the Wigner function or its characteristic function is usually easier to use. In fact, the Husimi-Q function also contains all the information in principle, but the Husimi-Q function is sensitive to noise in the context of cat qubits. The Wigner function in TIFF2025538425000202.tif6150 is a function that modulates the field This can be determined by measuring the parity of the field after displacing it by 6150 (as shown in Sun, L., Petrenko, A., Leghtas, Z. et al., "Tracking photon jumps with repeated quantum non-demolition parity measurements," Nature 511, 444-448 (2014), https: / / doi.org / 10.1038 / nature13436). The characteristic function in TIFF2025538425000204.tif6150 is the displacement operator The mean value of TIFF2025538425000205.tif6150 can be determined by measuring the mean value of TIFF2025538425000205.tif6150 (as shown in Campagne-Ibarcq, P., Eickbusch, A., Touzard, S. et al., "Quantum error correction of a qubit encoded in grid states of an oscillator," Nature 584, pp. 368–372 (2020), https: / / doi.org / 10.1038 / s41586-020-2603-3). In what follows, the Wigner function is used because it contains more readily available information about the cat qubit. For example, in a two-component cat qubit, measuring X is equivalent to measuring the Wigner function at 0. Note, however, that the Husimi-Q function or the characteristic function of the Wigner function can also be used.

[0149] To measure the Wigner function, it is first necessary to displace the state of the system and then measure the parity. Displacing the state corresponds to sending a finite duration pulse with a frequency close to the mode frequency so that the mode frequency is within the pulse frequency spectrum. This pulse is created using a microwave source connected to the mode through a transmission line coupled to the mode. This coupling can be capacitive, inductive, or galvanic. The amplitude and phase of this pulse define the amplitude and phase of the displacement. Measuring the parity can be done indirectly by coupling the mode to a two-level system and mapping the parity of the field of the mode to the state of the two-level system. This mapping is then used to measure the parity of the field of the mode. The photon number operator of TIFF2025538425000206.tif6150 is used for the two-level system TIFF2025538425000207.tif8150 or This can be achieved by realizing a Hamiltonian that couples to either the Z or X operator of TIFF2025538425000208.tif8150. The former can be achieved using so-called dispersive interactions (Sun, L., Petrenko, A., Leghtas, Z. et al., "Tracking photon jumps with repeated quantum non-demolition parity measurements," Nature 511, 444-448 (2014), https: / / doi.org / 10.1038 / nature13436), while the latter can be achieved using so-called longitudinal interactions (see, for example, S. Touzard, A. Kou, N.E. Frattini et al., "Gated Conditional Displacement Readout of Superconducting Qubits," Phys. Rev. Lett. 122, 080502), which can be operated with a parametric pump at the two-level frequency. The states of the two-level system are reconstructed at a rate that depends on the number of photons in the mode. TIFF2025538425000209.tif6150 Rotate around the axis (X axis respectively). By adjusting the tif to TIFF2025538425000210.tif9150, we can ensure that for even photon numbers in the oscillator, the two-level system accumulates an integer number of rotations, and for odd photon numbers, the two-level system accumulates a half-integer number of rotations. In the case of dispersive interactions, the two-level system accumulates a state Starting with TIFF2025538425000211.tif6150, the two-level system is If there is an even number of photons in the mode, the state TIFF2025538425000212.tif6150, and if there is an odd number of photons, the state TIFF2025538425000213.tif6150. Measuring X changes the qubit to state TIFF2025538425000214.tif6150 or TIFF2025538425000215.tif6150 and therefore determine the photon number parity. In the case of longitudinal interactions, the two-level system is in the state Starting with TIFF2025538425000216.tif6150, the two-level system is If there is an even number of photons in the mode, the state TIFF2025538425000217.tif6150, and if there is an odd number of photons, the state TIFF2025538425000218.tif6150. By measuring Z, the quantum bit TIFF2025538425000219.tif6150 or Determine whether the photon number is in TIFF2025538425000220.tif6150 and therefore determine the photon number parity.

[0150] Unfortunately, the fundamental interaction required to perform Wigner tomography, namely the photon number operator Coupling to TIFF2025538425000221.tif6150 is incompatible with the coherent state stabilization mentioned above. This is Merge to TIFF2025538425000222.tif6150 This is also true for its characteristic function, which depends on either the coupling to TIFF2025538425000223.tif6150. This is not coincidental; in fact, it is the desired effect of the stabilization mechanism, which aims to suppress spurious coupling. As a result, the Wigner function cannot be measured while the cat qubit stabilization is on.

[0151] There are several measurements that are compatible with stabilization; for example, it is possible to determine which stabilized coherent state the system is in by coupling the mode to a heterodyne detector (or a homodyne detector if there are only two coherent states).

[0152] For a two-component cat qubit, this corresponds to measuring Z. However, although theoretically one could also measure X, Y of the two-component cat state while stabilization is on, measuring X and Y requires engineering a nonlocal Hamiltonian, which is only at the theoretical proposal stage. As explained earlier, experimentally measuring X results in a parity measurement that relies on interactions that are protected by stabilization, just as in Wigner tomography.

[0153] In the case of a four-component cat qubit, measuring which stabilized coherent state the system is in does not measure the observables of the qubit, but rather projects the system out of code space.

[0154] Therefore, only a few measurements can be performed while stabilization is on, and this measurement set does not cover the need for calibration or quantum computation.

[0155] In the prior art, this problem is solved for stabilized two-component Cat qubits that rely on parametric pumping by simply turning off the parametric pump, which enables the two-to-one photon exchange—a key element of stabilization. In the case of the resonant Cat qubit circuit described above, the inherently resonant nature of the stabilization mechanism precludes the use of this solution, since the parametric pump is not required. Energy conservation is built in because the buffer mode is tuned to have twice the frequency of the Cat qubit mode. As a result, the two-to-one photon exchange dynamics is always on, and this description also applies to the more general case of 2N-to-one photon exchange.

[0156] Compared to previously proposed implementations, the fact that the resonant Cat qubit device has a flux line that is well coupled to the buffer (or differential mode, or second mode) at both DC and RF allows new possibilities for readout. In particular, this flux line allows for a rapid (tens of nanoseconds) change of a given current. By suddenly changing the given current value, the resonant frequencies of both the first and second modes are changed, and the system shifts from the resonant frequency matching condition, which is one condition that allows 2N to 1 photon conversion. TIFF2025538425000224.tif6150 and TIFF2025538425000225.tif6150, and their frequencies under resonant frequency matching conditions TIFF2025538425000226.tif6150 and Detuning from TIFF2025538425000227.tif6150 TIFF2025538425000228.tif6150 and In TIFF2025538425000229.tif6150, the 2N to 1 photon exchange is The condition for TIFF2025538425000230.tif6150 is off, where TIFF2025538425000231.tif6150 is the strength of the nonlinear conversion. This is typically achieved by modifying the magnetic flux in the device's central loop by a fraction of a flux quantum (a common unit of magnetic flux known in the field). This is achieved by modifying the output current of the current source by less than 1 mA, depending on the DC coupling of the flux lines. Hereafter, we refer to the new predetermined current value as the tomographic bias point, rather than the stabilization bias point, at which 2N-to-1 photon conversion becomes active. The current can be modified in any appropriate way, such as by applying a square pulse, ramp, or other signal-modifying shape, taking into account signal distortion between the current source and the device.

[0157] In some embodiments, another method exists for detuning the frequency of the mode from the resonant frequency matching condition by exploiting the fact that the buffer mode has a Kerr nonlinearity. These embodiments are preferred in which the central inductive element is a single Josephson junction. In this case, by suddenly applying a strong parametric pump to the buffer mode via a flux line or another RF coupling line at a frequency different from the nonlinear transitions in the device, the buffer frequency is AC Stark-shifted, detuning the effective resonant frequency of the buffer mode (and, to a lesser extent, the cat qubit mode) from the frequency that satisfies the frequency matching condition. While this solution does not require rapid DC current changes, it has been found to have several adverse effects on the device, including degrading the coherence time of modes and two-level systems used for tomography and generally heating the modes in the device.

[0158] When the system is detuned from the resonant frequency matching condition, coherent state stabilization is no longer active, and the cat qubit mode dynamics are free to perform arbitrary quantum measurements on the cat qubit mode. Special care must be taken when performing measurements because the cat qubit mode frequency at the tomography bias point differs from the frequency at the stabilization bias point. However, the accumulated phase of the cat qubit state due to the frequency difference between the two points can be calculated and corrected while the cat qubit is operating.

[0159] In the preferred embodiment, not only is the predetermined current suddenly changed to turn off the 2N to 1 photon conversion, but the buffer mode drive is also turned off since it serves no purpose.

[0160] If the quantum measurement is non-destructive (QND), stabilization can be conveniently resumed by returning the current source output to the stabilizing DC bias point. This also occurs if the buffer mode drive was turned off. Again, the accumulated phase due to the detuning of the cat qubit mode between the two bias points must be calculated and corrected.

Claims

1. A nonlinear superconducting quantum circuit having a first mode and a second mode, each having its own resonant frequency, and a central mixed component (840, 940, 1040, 1140, 1240), Flux lines (930, 1030, 1130, 1230) and A quantum device equipped with, The aforementioned central mixing component is The first Josephson junctions (841, 941, 1041, 1141, 1241) are arranged in a series topology substantially symmetric with respect to axis (A) that maps the first Josephson junctions (841, 941, 1041, 1141, 1241) to the second Josephson junctions (842, 942, 1042, 1142, 1242), and the central inductor (843, 943, 1043, 1143, 1243) and the second Josephson junctions (842, 942, 1042, 1142, 1242) are arranged in a series topology substantially symmetric with respect to axis (A) that maps the first Josephson junctions (841, 941, 1041, 1141, At least one loop (710) comprising: a first internal node connecting the pole of 1241) to the pole of the central inductor (843, 943, 1043, 1143, 1243); a second internal node connecting the pole of the second Josephson junction (842, 942, 1042, 1142, 1242) to another pole of the central inductor (843, 943, 1043, 1143, 1243); and a loop closing node connecting another pole of the first Josephson junction (801) to another pole of the second Josephson junction (842, 942, 1042, 1142, 1242), A first circuit portion (721) connected between the common ground and the first internal node of the loop (710), a second circuit portion (722) connected between the common ground and the second internal node of the loop (710), and a third circuit portion (723) connected between the loop closing node and the common ground and substantially symmetrical with respect to the axis (A), Includes, The first circuit portion (721) and the second circuit portion (722) are substantially symmetrical with respect to the axis (A), The nonlinear superconducting quantum circuit is configured such that, when a predetermined current of a constant strength is applied, the resonant frequency of the second mode becomes substantially 2N times the resonant frequency of the first mode, and a phase difference is induced between the ends of one or more Josephson junctions (841, 941, 1041, 1141, 1241; 842, 942, 1042, 1142, 1242), and the nonlinear superconducting quantum circuit is formal It has a Hamiltonian that can be expanded into a sum between at least one governing term and a set of auxiliary terms, where, is a scalar corresponding to the intrinsic bond strength, is the annihilation operator for the first mode, is the annihilation operator for the second mode, is the reduced Planck constant, which essentially performs a resonant 2N-to-1 photon exchange between the first and second modes, respectively, where N is a positive integer. The nonlinear superconducting quantum circuit is arranged on a dielectric substrate and is separated from the common ground plane (820, 920, 1020, 1120, 1220) by the exposed portion of the dielectric substrate. The flux lines (930, 1030, 1130, 1230) are located opposite to the portion of the at least one loop (710) that includes the central inductor (843, 943, 1043, 1143, 1243), and include the first slots (934, 1034, 1134, 1234) and the CPW portion (932, 1032, 1132, 1232), The CPW sections (932, 1032, 1132, 1232) are arranged to be coupled with a current source (1012), microwave sources (914, 1114, 1214) configured to apply microwave radiation, and loads (916, 1116, 1216), and are separated by end wire bonds (938, 1038, 1138, 1238), The first slots (934, 1034, 1134, 1234) are patterned in the ground plane with the dielectric substrate exposed, extending substantially parallel to the axis (A), and connected at one end to the slots (9320, 10320, 11320, 12320) of the CPW portion (932, 1032, 1132, 12320) at the level of the end wire bonds (938, 1038, 1138, 1238), and at the other end to the exposed dielectric substrate portion that demarcates the portion of the at least one loop (710) that includes the central inductor (843, 943, 1043, 1143, 1243). The flux lines (930, 1030, 1130, 1230) are configured such that coupling with the current source (1012) induces a predetermined current in the at least one loop (710), coupling with the microwave sources (914, 1114, 1214) induces driving of the second mode at a frequency substantially equal to the resonant frequency of the second mode or 2N times the resonant frequency of the first mode, and substantially exclusively couples the loads (916, 1116, 1216) to the second mode. Quantum devices.

2. The flux lines (930, 1030, 1130, 1230) are patterned in the ground plane, exposing the dielectric substrate, extending substantially parallel to the axis (A), and at the level of the end wire bonds (938, 1038, 1138, 1238) to another slot (9322, 10322, 11322, 12322) of the CPW portion (932, 10322, 11322, 12322). The quantum device according to claim 1, further comprising a second slot (936, 1036, 1136, 1236) connected at one end to the at least one loop (710) and having its other end terminated in the ground plane (820, 920, 1020, 1120, 1220) at a distance of more than 1 μm from the portion of the at least one loop (710) containing the central inductor (843, 943, 1043, 1143, 1243).

3. The quantum device according to claim 2, wherein the second slot (936, 1036, 1136, 1236) includes a return portion (12360) closest to the portion of the at least one loop (710) containing the central inductor (843, 943, 1043, 1143, 1243), and the return portion (12360) extends substantially perpendicular to the axis (A) in a rotational direction away from the first slot (934, 1034, 1134, 1234).

4. The quantum device according to claim 3, wherein the length of the return portion (12360) is greater than 1 μm and less than or equal to the width of the nonlinear superconducting quantum circuit.

5. The quantum device according to claim 1, wherein the first slots (934, 1034, 1134, 1234) are substantially symmetric with respect to axis (A).

6. The quantum device according to claim 1, wherein both the first slots (934, 1034, 1134, 1234) and the second slots (936, 1036, 1136, 1236) are perpendicular to the axis (A) and offset in the direction opposite to the direction of rotation.

7. The quantum device according to claim 6, wherein the offset amount is greater than 1 μm and less than or equal to the width of the nonlinear superconducting quantum circuit.

8. The quantum device according to claim 1, wherein the length of the first slots (934, 1034, 1134, 1234) or the position of the end wire bonds (938, 1038, 1138, 1238) is selected according to the strength of the intended bond between the second mode and the load (916, 1116, 1216).

9. The quantum device according to claim 1, wherein the coupling between the load and the second mode is 100 times greater than the coupling between the load and the first mode.

10. The quantum device according to claim 1, further comprising a current source (1012), microwave sources (914, 1114, 1214), and loads (916, 1116, 1216) coupled to the CPW portion (932, 1032, 1132, 1232) which operates to stabilize a coherent manifold within the quantum device.

11. A quantum measurement method for the device described in claim 10, a) Modifying the current output of the current source such that the predetermined current in the nonlinear superconducting quantum circuit induces a resonant frequency of the second mode that is substantially different from twice the resonant frequency of the first mode, b) Performing a quantum measurement in the first mode of the nonlinear superconducting quantum circuit, A quantum measurement method that includes this.

12. The quantum measurement method according to claim 11, wherein the modification of operation a) is performed by applying a rectangular pulse to the current.

13. Operation b) is a quantum nondestructive measurement, and further includes operation c) below: Operation c) restores the current output of the current source to obtain the predetermined current after operation b) has been performed. The quantum measurement method according to claim 11 or claim 12.

14. A quantum computing system comprising at least one device according to any one of claims 1 to 10.

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