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. To perform a quantum measurement with the circuit, a method is proposed for switching off the drive of the second mode and switching to an undriven manifold capable of performing the quantum measurement.

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. TIFF2025538423000002.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 resonant cat qubit situation. Indeed, the stabilization is self-sustaining, so there is no parametric pump to turn off. The only comparable prior art situation is the case of the Kerr cat qubit, discussed in 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. However, in this situation, the nature of the Kerr cat qubit confinement introduces significant differences, as discussed further below. [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] The present invention aims to improve the situation, and to that end, the Applicant has disclosed a quantum measurement method for a device comprising a nonlinear superconducting quantum circuit having a first mode and a second mode, each having a respective resonant frequency, and a symbolic representation comprising at least one loop including one or more Josephson junctions, the method comprising: 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 one or more Josephson junctions, and the nonlinear superconducting quantum circuit is of the form TIFF2025538423000003.tif6150 and a Hamiltonian that can be expanded into a sum between at least one dominant term and a set of auxiliary terms, where TIFF2025538423000004.tif6150 is a scalar corresponding to the intrinsic bond strength, TIFF2025538423000005.tif6150 is the first mode annihilation operator, TIFF2025538423000006.tif6150 is the second mode annihilation operator, TIFF2025538423000007.tif6150 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 device further comprises a current source configured to provide the predetermined current of a constant magnitude; a microwave source configured to apply microwave radiation at a frequency substantially equal to a resonant frequency of a second mode or 2N times the resonant frequency of a first mode, the microwave source coupled to the nonlinear superconducting quantum circuit to drive the second mode; and a load coupled substantially only to the second mode. The method comprises the following operations: 1) turning off the second mode of driving; 2) pausing for a duration comprised between 10,150 seconds; TIFF2025538423000009.tif6150 is the rate (radians / second) of 2N photon dissipation in the nonlinear superconducting circuit; 3) performing a quantum measurement to determine a characteristic of the photon number distribution on the first mode of the circuit; Includes.

[0014] According to various embodiments, the method can include one or more of the following features: - operation 1) includes applying a square pulse shape to reduce the amplitude drive from its nominal stabilization value to 0; - operation 1) comprises applying two successive square pulse shapes, a first square pulse shape causing said amplitude drive to go from its nominal regulated value to a negative value substantially equal to or greater than the nominal regulated absolute value, and a second square pulse shape causing the current to go from this value to zero; Operation 3) is a quantum non-demolition measurement and further comprises the following operation 4), in which operation 4) restores the second mode of operation after operation 3) is performed; - operation 4) includes applying a square pulse shape to bring the amplitude drive from 0 to a nominal regulated value; - operation 4) comprises applying two successive square pulse shapes, a first square pulse shape for bringing the amplitude drive to a value substantially comprised between 0 and 1-10 times the nominal stabilization value, and a second square pulse shape for bringing the amplitude drive from this value to the nominal stabilization value; The method further includes the following operation 0), in which operation 0) applies to the first mode an electromagnetic pulse having a frequency substantially equal to a resonant frequency of the first mode, the electromagnetic pulse comprising: TIFF2025538423000010.tif6150 and TIFF2025538423000011.tif6150, which has a duration smaller than - Action 0) is executed before action 1), Actions 0) and 1) are performed simultaneously.

[0015] The invention also relates to a quantum tomography method, said method comprising the following operations: a) obtaining a set of complex displacements defining displacement amplitudes and displacement phases that scan the phase space of cat qubit modes; b) preparing a state in a cat qubit mode based on a selected set of parameters for operation of a device comprising a nonlinear superconducting quantum circuit having a first mode and a second mode each having a respective resonant frequency and a symbolic representation comprising at least one loop including one or more Josephson junctions; 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 one or more Josephson junctions; Nonlinear superconducting quantum circuits are of the form TIFF2025538423000012.tif6150 and a Hamiltonian that can be expanded into a sum between at least one dominant term and a set of auxiliary terms, where TIFF2025538423000013.tif6150 is a scalar corresponding to the intrinsic bond strength, TIFF2025538423000014.tif6150 is the first mode annihilation operator, TIFF2025538423000015.tif6150 is the second mode annihilation operator, TIFF2025538423000016.tif6150 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 device further comprising: a current source configured to provide a predetermined current of constant magnitude; a microwave source configured to apply microwave radiation at a frequency substantially equal to a resonant frequency of a second mode or 2N times the resonant frequency of a first mode, the microwave source coupled to the nonlinear superconducting quantum circuit to drive the second mode; and a load coupled substantially only to the second mode. c) applying the method according to any one of claims 7 to 9, for a given complex displacement in the set of complex displacements, the method defining the amplitude, duration and phase of the electromagnetic pulse of step 0) using the associated displacement amplitude and displacement phase; d) repeating operations b) and c) using different complex displacements; Includes.

[0016] The present invention also provides a method for adjusting operation of a quantum device, comprising the steps of: a) obtaining a plurality of sets of parameters for operation of a quantum device; b) for each set of parameters of operation a) performing the method of the present invention; c) deriving an adjusted parameter set based on the results of operation b); The present invention relates to a method for adjusting the operation of a quantum device, including:

[0017] The present invention also relates to a quantum computing device comprising a nonlinear superconducting quantum circuit having a first mode and a second mode, each having a respective resonant frequency, and a symbolic representation comprising at least one loop including one or more Josephson junctions, 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 one or more Josephson junctions, and the nonlinear superconducting quantum circuit is of the form TIFF2025538423000017.tif6150 and a Hamiltonian that can be expanded into a sum between at least one dominant term and a set of auxiliary terms, where TIFF2025538423000018.tif6150 is a scalar corresponding to the intrinsic bond strength, TIFF2025538423000019.tif6150 is the first mode annihilation operator, TIFF2025538423000020.tif6150 is the second mode annihilation operator, TIFF2025538423000021.tif6150 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 device further comprises a current source configured to provide the predetermined current of a constant magnitude; a microwave source configured to apply microwave radiation at a frequency substantially equal to a resonant frequency of a second mode or 2N times the resonant frequency of a first mode, the microwave source coupled to the nonlinear superconducting quantum circuit to drive the second mode; and a load coupled substantially only to the second mode; The device operates using an adjusted parameter set determined by the method of the present invention.

[0018] The present invention also relates to a quantum computing device comprising a nonlinear superconducting quantum circuit having a first mode and a second mode, each having a respective resonant frequency, and a symbolic representation comprising at least one loop including one or more Josephson junctions, 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 one or more Josephson junctions, and the nonlinear superconducting quantum circuit is of the form TIFF2025538423000022.tif6150 and a Hamiltonian that can be expanded into a sum between at least one dominant term and a set of auxiliary terms, where TIFF2025538423000023.tif6150 is a scalar corresponding to the intrinsic bond strength, TIFF2025538423000024.tif6150 is the first mode annihilation operator, TIFF2025538423000025.tif6150 is the second mode annihilation operator, TIFF2025538423000026.tif6150 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 device further comprises a current source configured to provide the predetermined current of a constant magnitude; a microwave source configured to apply microwave radiation at a frequency substantially equal to a resonant frequency of a second mode or 2N times the resonant frequency of a first mode, the microwave source coupled to the nonlinear superconducting quantum circuit to drive the second mode; and a load coupled substantially only to the second mode; The device performs the method of the present 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 shows a first embodiment of the device of FIG. 1. [Figure 7] FIG. 2 shows another embodiment of the device of FIG. [Figure 8] FIG. 2 shows another embodiment of the device of FIG. [Figure 9] FIG. 9 shows experimental data obtained using the example resonant cat qubit circuit of FIG. 8. [Figure 10] 1 is a diagram of a quantum measurement method according to the present invention; [Figure 11]10A-10C illustrate various microwave source ramp-up and ramp-down embodiments for turning second mode operation off or on. [Figure 12] FIG. 10 shows a non-drive manifold. [Figure 13] FIG. 1 illustrates an alternative embodiment of a quantum measurement method according to the present invention. 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 TIFF2025538423000027.tif8150. TIFF2025538423000028.tif8150, provided that TIFF2025538423000029.tif6150 and TIFF2025538423000030.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 TIFF2025538423000031.tif6150, however, TIFF2025538423000032.tif6150 is the resonance frequency of the first mode, TIFF2025538423000033.tif6150 is the resonance frequency of the second mode, or format (Also referred to as the "frequency matching condition" in TIFF2025538423000034.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 TIFF2025538423000036.tif6150 occurs and the circuit performs 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 TIFF2025538423000037.tif6150 and Superconducting phase drop occurring across each inductive element TIFF2025538423000038.tif6150 and result in a phase drop TIFF2025538423000039.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. TIFF2025538423000040.tif6150) can be experimentally tuned. Thus, the application of current is tailored to the circuit configuration 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 TIFF2025538423000041.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 TIFF2025538423000042.tif6150 is the spreading manifold. The first mode has some residual single-photon dissipation, so the state TIFF2025538423000043.tif6150 is in the state TIFF2025538423000044.tif6150, and therefore only the vacuum is stable in the first mode in 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 in TIFF2025538423000045.tif6150, however, TIFF2025538423000046.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 TIFF2025538423000047.tif6150, however, TIFF2025538423000048.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 TIFF2025538423000049.tif6150, however, TIFF2025538423000050.tif6150 is an effective 2N photon drive, TIFF2025538423000051.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 TIFF2025538423000052.tif6150, however, TIFF2025538423000053.tif6150 is the 2N photon dissipation rate. The amplitude of the stabilized coherent state TIFF2025538423000054.tif6150 is finally Given as TIFF2025538423000055.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. TIFF2025538423000057.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 TIFF2025538423000058.tif6150 and a series of auxiliary terms, which will be omitted below.

[0052] Dominating term In TIFF2025538423000059.tif6150, TIFF2025538423000060.tif6150 is the first mode annihilation operator, TIFF2025538423000061.tif6150 is the second mode annihilation operator. Conversely, TIFF2025538423000062.tif6150 is the first mode generating operator, TIFF2025538423000063.tif6150 is the generating operator of the second mode. TIFF2025538423000064.tif6150 describes a resonant 2N to 1 photon exchange, TIFF2025538423000065.tif6150 is a polynomial. Therefore, the term The dominant term containing TIFF2025538423000066.tif6150 is the first mode TIFF2025538423000067.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) TIFF2025538423000068.tif6150 is also included, where the first mode TIFF2025538423000069.tif6150 photons are generated and one photon of the second mode is destroyed.

[0053] scalar TIFF2025538423000070.tif6150 is a function of a set of parameters consisting of circuit parameters and / or parameters of a given current. TIFF2025538423000071.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 TIFF2025538423000072.tif6150 (each representing its Hermitian conjugate). In other words, TIFF2025538423000073.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 TIFF2025538423000074.tif6150 is non-zero, and the frequency matching condition ensures that the term is resonant. In addition, the constant TIFF2025538423000075.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 TIFF2025538423000076.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 TIFF2025538423000077.tif6150 TIFF2025538423000078.tif6150 or cross-Kerr term TIFF2025538423000079.tif6150. The nonlinear terms can have significant magnitude 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 TIFF2025538423000080.tif6150 TIFF2025538423000081.tif6150 TIFF2025538423000082.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 TIFF2025538423000083.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 TIFF2025538423000084.tif6150, the annihilation operator acting on the first mode TIFF2025538423000085.tif6150 is an eigenstate, and as a result, TIFF2025538423000086.tif6150, where TIFF2025538423000087.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., the logical qubit states TIFF2025538423000088.tif6150 is TIFF2025538423000089.tif6150, which can be defined as the logical qubit state TIFF2025538423000090.tif6150 is TIFF2025538423000091.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 TIFF2025538423000092.tif6150A section of a superconducting transmission line terminated at two different boundary conditions (short-circuited to ground at one end and open-circuited at the other end) forming a resonator. The transmission line may be, for example, of the coplanar waveguide or microstrip type. -So-called TIFF2025538423000093.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 TIFF2025538423000094.tif6150, where TIFF2025538423000095.tif6150 is the Josephson energy of the junction, TIFF2025538423000096.tif10150, but TIFF2025538423000097.tif6150 is the integral of the voltage across the junction, TIFF2025538423000098.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. TIFF2025538423000099.tif6150 is the potential energy of an inductive element, which is the inductive energy TIFF2025538423000100.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. TIFF2025538423000102.tif6150. Thus, application of a given current modifies the potential energy of the Josephson junction as follows: TIFF2025538423000103.tif6150

[0064] The first term on the right side of the sum above is the change in the effective Josephson energy of the junction TIFF2025538423000104.tif6150. The inductance of a Josephson junction is inversely proportional to the Josephson energy, so the DC offset TIFF2025538423000105.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. TIFF2025538423000107.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 TIFF2025538423000109.tif6150 The Josephson junction in TIFF2025538423000110.tif6150 is It can be written as TIFF2025538423000111.tif10150. Effective magnetic flux passing through the loop When biased by a given current, equivalently described by TIFF2025538423000112.tif6150, an internal DC current is generated within the superconducting loop, TIFF2025538423000113.tif6150 and Phase drop across the junction depends only on TIFF2025538423000114.tif6150 Generates TIFF2025538423000115.tif6150. In the examples, the phase drop is calculated using the following formula: TIFF2025538423000116.tif6150However, It can be numerically calculated as the solution to TIFF2025538423000117.tif6150.

[0067] The solution to the above equation is TIFF2025538423000118.tif6150. For example, inductance The loop in TIFF2025538423000119.tif6150 contains two junctions, each with an energy TIFF2025538423000120.tif6150, the phase drop across each junction is Represented by TIFF2025538423000121.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 TIFF2025538423000122.tif6150, where TIFF2025538423000123.tif6150 is a reduced flux quantum TIFF2025538423000124.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 TIFF2025538423000125.tif6150 can be written, except that TIFF2025538423000126.tif6150 is the DC phase offset calculated previously, TIFF2025538423000127.tif6150 and TIFF2025538423000128.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 TIFF2025538423000129.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 TIFF2025538423000130.tif6150 and TIFF2025538423000131.tif6150, so the linear part of the Hamiltonian is TIFF2025538423000132.tif6150 can be written, except that TIFF2025538423000133.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 TIFF2025538423000134.tif6150, where TIFF2025538423000135.tif6150 is the DC phase drop across the junction induced by a given current, TIFF2025538423000136.tif6150 and TIFF2025538423000137.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 TIFF2025538423000138.tif6150, where: TIFF2025538423000139.tif10150 is TIFF2025538423000140.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 TIFF2025538423000141.tif9150. In these cases, the combined term TIFF2025538423000142.tif6150 is modulated via a parametric pump, and the pump is Injecting external time-varying parameters with TIFF2025538423000143.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 TIFF2025538423000144.tif6150101 and the second mode TIFF2025538423000145.tif6150102, essentially performing 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 100. 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 TIFF2025538423000147.tif6150. The microwave filter 107 may be configured as a bandpass filter of frequency TIFF2025538423000148.tif6150. Alternatively, the filter 107 may be configured as a bandpass filter of frequency TIFF2025538423000148.tif6150. TIFF2025538423000149.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. TIFF2025538423000150.tif6150 shows the coupling 200 to the device via the coupler 104 and filter 107 to ensure that the Second Resonant Frequency In TIFF2025538423000151.tif6150, we allow only the second mode of loss, which is modeled as resistor 105, but the first resonant frequency TIFF2025538423000152.tif6150 does not allow for loss of the first mode. Figure 2 shows at least two possibilities for coupling 200, namely, frequency The bandpass filter in TIFF2025538423000153.tif6150 (also shown in schematic diagrams b) and d) of Figure 2) or frequency A band-stop filter (also shown in schematics c) and e) of Figure 2) is shown in TIFF2025538423000154.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 TIFF2025538423000155.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 TIFF2025538423000156.tif6150 is configured to resonate, and its impedance TIFF2025538423000157.tif6150 adjusts the width of the bandpass.

[0081] Schematic diagram c) shows the frequency TIFF2025538423000158.tif6150 shows a band-stop filter 220 capacitively coupled 201 to the input port. The band-stop filter 220 is an LC stub, and TIFF2025538423000159.tif6150 is configured to resonate, and its impedance TIFF2025538423000160.tif6150 adjusts the width of the band rejection.

[0082] Schematic diagram d) shows the frequency TIFF2025538423000161.tif6150 shows a bandpass filter 210 inductively coupled 202 to the input port. The bandpass filter 210 is an LC oscillator, with a frequency TIFF2025538423000162.tif6150 is configured to resonate, and its impedance TIFF2025538423000163.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 TIFF2025538423000164.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 TIFF2025538423000165.tif6150 is configured to resonate, and its impedance TIFF2025538423000166.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 TIFF2025538423000167.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 TIFF2025538423000168.tif6150. Timescale After TIFF2025538423000169.tif6150, the mode state is Amplitude The system converges to a single coherent state 301 at TIFF2025538423000170.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 TIFF2025538423000171.tif6150 Two-photon drive and amplitude Rate with two stable steady states (302, 303) in TIFF2025538423000172.tif6150 TIFF2025538423000173.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 TIFF2025538423000174.tif6150302 is surrounded by a solid line, TIFF2025538423000175.tif6150303 is enclosed by a dotted line. This encoding is based on the stable nature of the dynamics that converges to two states. TIFF2025538423000176.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 TIFF2025538423000178.tif6150, and states in another superposition of two coherent states 305 shown by dashed lines. TIFF2025538423000179.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 TIFF2025538423000182.tif6150 Frequency line 402 and TIFF2025538423000183.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 TIFF2025538423000184.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 TIFF2025538423000185.tif6150 is applied, a phase drop occurs. TIFF2025538423000186.tif6150 flows across the junction 502. This is the internal current TIFF2025538423000187.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 TIFF2025538423000188.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 TIFF2025538423000189.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 radiated by an angle TIFF2025538423000190.tif6150, where TIFF2025538423000191.tif6150 is a magnetic flux quantum. In this application, the system period TIFF2025538423000192.tif6150 It is periodic in TIFF2025538423000193.tif6150, It can be assumed that the image is symmetric about TIFF2025538423000194.tif6150. Therefore, the analysis is performed on the interval It can be limited to TIFF2025538423000195.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. TIFF2025538423000196.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] Schematic diagrams a)-e) of FIG. 6 illustrate an embodiment of a circuit 600 in which the symbolic representation of the circuit includes at least one loop containing one or more Josephson junctions. Circuit 600 in this example is 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 600 in this example does not include an input current source. However, it should be understood that the current source may be implemented in any manner, so long as it induces an appropriate current bias in at least one loop. This can be accomplished with a standard current source (galvanically connected to the circuit in any manner) or a transformer that generates a magnetic field to induce an internal current in circuit 600. The following example illustrates a particularly efficient implementation of a current source in circuit 600.

[0102] In the circuit 600, at least one loop 610 includes a first Josephson junction 601 arranged in parallel with a first inductive element 602 and a first capacitive element 603. An inductive element, such as the first inductive element 602, may be a dipole or a collection of dipoles designed to add inductance to the circuit. For example, the inductive element may be a single inductor, i.e., a single superconducting strip with a given inductance, or multiple inductors arranged in series. In the case of multiple inductors arranged in series, the total inductance is the sum of the respective inductances of each inductor. Alternatively and / or additionally, the inductive element 602 may also include an array of Josephson junctions, including at least two junctions. A capacitive element, such as the first capacitive element 603, may be a single capacitor or multiple capacitors arranged in parallel. The loops include respective first and second extremum nodes. Each such extremum node is therefore a common node connecting a respective node of the first Josephson junction 601 with the first inductive element 602 and the first capacitive element 603 .

[0103] Furthermore, the Josephson junction 601 has the Josephson energy TIFF2025538423000197.tif6150, and inductor 602 has TIFF2025538423000198.tif6150, and capacitor 603 has With energy shown as TIFF2025538423000199.tif6150.

[0104] The circuit may also include a second inductive element 605 and a second capacitive element 604 arranged in parallel to form a resonator 620. This arrangement may include a first extremum node and a second extremum node, respectively, connecting the second inductive element 605 and the second capacitive element 604. The inductive element 605 may be TIFF2025538423000200.tif6150, and the capacitive element 604 may have an energy The circuit 600 may have an energy represented as TIFF2025538423000201.tif6150. The circuit may also include a coupled capacitive element 606 or an inductive element 607. A parallel LC resonator 620 is linearly coupled in parallel to the nonlinear resonator formed by the Josephson junction 601, the inductive element 602, and the capacitive element 603. As noted above, the circuit 600 is a symbolic representation of a superconducting circuit. Therefore, the elements of the circuit, such as the resonator 620, may be fabricated by any method known in the art. For example, the elements of the circuit may be a transmission line or part of a 3D cavity. The resonator may also be a mechanical resonator capacitively coupled 606 via a piezoelectric material.

[0105] A first extremum node of the loop 610 may be connected to a first extremum node of the second inductive element 605, and the second capacitive element 604 may be arranged in parallel via the coupling capacitive element 606 or the inductive element 607. A second extremum node of the loop 610 and the second extremum nodes of the parallel arranged second inductive element 605 and second capacitive element 604 may be connected to a common ground.

[0106] When coupled via coupling element 606 (607, respectively, for schematic b), the resonators hybridize slightly if their frequencies differ and the coupling element is small (small capacitance, large inductance), thus forming two modes. A linear mode 608 may be hosted or localized predominantly on the linear resonator 620. A nonlinear mode 609 is hosted or localized predominantly on the parallel arrangement of loop 610 and first capacitive element 603. Thus, the circuit is arranged to essentially perform a resonant 2N-to-1 photon exchange.

[0107] Schematic diagram 6b shows circuit 630, a variation of circuit 600. The difference between the two circuits is that coupling capacitor 606 has been replaced with inductor 607. This alternative implementation uses the same principles as those described above.

[0108] Schematics c) and d) show examples of devices incorporating circuit 600. Furthermore, these examples show how modes can be selected according to experimental constraints. In schematic c), the nonlinear mode is the first mode and the linear mode is the second mode. In schematic d), the linear mode is the first mode and the nonlinear mode is the second mode.

[0109] Additionally, schematic c) shows a microwave source 640 driving the circuit, coupled to a resistive load for dissipation, and an LC resonator 650 acting as a bandpass filter tuned at the resonant frequency of the second mode.

[0110] Schematic diagram d) shows another example of a device incorporating circuit 600, where a DC current source is coupled to a microwave source and a resistive load 670. The current source is coupled to the circuit via a transformer 660, which induces a current in loop 610.

[0111] The alternatives in schematics b)-d) follow the same principles as the circuit and device in schematic a). In particular, the choice of whether the first or second mode can be linear depends on trade-offs that depend on the experimental setup.

[0112] Schematic diagram e) shows an example of a symbolic implementation of circuit 600 in a planar superconducting pattern. The superconducting circuit is fabricated in a coplanar waveguide (CPW) geometry. The background 6000 of the planar superconducting pattern corresponds to the superconducting metal remaining on the dielectric substrate after fabrication. Josephson junctions are represented as black crosses. The nonlinear mode is hosted within a superconducting island 611, which has a capacitance to ground 612 and is connected to ground via a single junction 601 and a three-junction array 602 acting as a small inductive element. The formed superconducting loop is a galvanically biased magnetic flux with a DC current line that shares a portion of the inductance to ground 613 with the resonator. The DC current-carrying wire is connected to DC port 614 and ground. DC port 614 is connected to a current source, which can be a current source at ambient temperature, via superconducting and non-superconducting wires. This lumped resonator is capacitively coupled 606 to a coplanar waveguide λ / 2 resonator 615, which serves as the second mode. This coplanar waveguide λ / 2 resonator has multiple resonances, each of which can be modeled by an LC resonator, similar to resonator 620. This second mode is then capacitively coupled 616 to the external environment, which connects to the circuit via RF input port 617. No filters are shown in this design. The resistive load is typically a 50 Ω transmission line terminating at the 50 Ω input of the microwave generator. Inset 640 is a close-up of the junction layout of the planar superconducting pattern.

[0113] Referring again to circuit 600, the calculation of the governing terms of the Hamiltonian rate describing the resonant 2N to 1 photon exchange will now be discussed. As mentioned above, for simplicity, we will assume that the circuit parameters are designed so that the bias point is the optimum bias point. This means that the frequency matching condition is This means that the loop 610's inductive energy is provided by the inductance alone, and the frequency of the nonlinear resonator is therefore The frequency of the linear resonator is given in TIFF2025538423000203.tif6150. Given in TIFF2025538423000204.tif6150. In some configurations, the frequency is TIFF2025538423000205.tif6150 (e.g., schematic diagram c)) or TIFF2025538423000206.tif6150 (e.g., schematic diagram d)). First, as illustrated in schematic diagram c), the nonlinear mode can be selected as the first mode. TIFF2025538423000207.tif6150, linear mode is the second mode TIFF2025538423000208.tif6150. Regardless of how the linear coupling between the two resonators is done, this is the coupling strength The detuning between the two modes can be summarized by defining It can be defined as TIFF2025538423000210.tif6150. Since the detuning is in the so-called dispersion limit, which allows us to simplify the calculation of the nonlinear Hamiltonian, this linear combination only slightly affects the frequencies of the two modes. Full diagonalization of the two-mode system is then required to obtain the target frequency. It allows you to adjust the value of TIFF2025538423000212.tif6150. The optimum bias point and limit In TIFF2025538423000213.tif6150, the interactions provided by the junction are TIFF2025538423000214.tif6150, where TIFF2025538423000215.tif11150 is in mode is the zero-point variation of the phase across the Josephson junction associated with TIFF2025538423000216.tif6150, TIFF2025538423000217.tif9150 describes the linear combination, TIFF2025538423000218.tif6150 is in mode The zero-point fluctuation of the phase across the Josephson junction associated with TIFF2025538423000219.tif6150. By expanding this Hamiltonian to the desired order, we obtain the resonant 2N-to-1 photon exchange Hamiltonian TIFF2025538423000220.tif6150However, You can get TIFF2025538423000221.tif10150.

[0114] Linear mode is the first mode TIFF2025538423000222.tif6150, and the nonlinear mode is the second mode When reverse selection is performed, the interactions provided by the conjugation are: Write TIFF2025538423000224.tif6150, but TIFF2025538423000225.tif11150 is in mode is the zero-point variation of the phase across the Josephson junction associated with TIFF2025538423000226.tif6150, TIFF2025538423000227.tif9150 characterizes the linear combination. By expanding this Hamiltonian as a Taylor series, we obtain the resonant 2N-to-1 photon exchange Hamiltonian TIFF2025538423000228.tif6150, but You can get TIFF2025538423000229.tif10150.

[0115] 7 illustrates another embodiment of a circuit in which a symbolic representation of the circuit includes at least one loop containing one or more Josephson junctions. Circuit 700 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 700 according to this example is configured to essentially perform resonant 2N to 1 photon exchange between a first mode and a second mode such that two modes are simultaneously hosted within the loop.

[0116] In the circuit 700, at least one loop 710 can include a first inductive element 702, a central Josephson junction element 701, and a second inductive element 703 arranged in series. The first inductive element 702 and the second inductive element 703 can each be an inductance, a single Josephson junction, or an array of Josephson junctions. Thus, the central Josephson junction 701 can be arranged between the first inductive element and the second inductive element as a series loop. The series arrangement can include a first inner node connecting a pole of the first inductive element 702 to a pole of the Josephson junction 701. The series arrangement can also include a second inner node connecting a pole of the second inductive element 703 to another pole of the Josephson junction 701. The series arrangement can also include a closed-loop node connecting another pole of the first inductive element 702 to another pole of the second inductive element 703.

[0117] The at least one loop 710 may be connected to a common ground through a closed loop node. The circuit may also include a first capacitor 704 and a second capacitor 705. The first capacitor 704 may be connected in parallel with the first Josephson junction 702 between the common ground and a first internal node of the loop. The second capacitor 705 may be connected in parallel with the second Josephson junction 702 between the common ground and a second internal node of the loop.

[0118] Thus, superconducting quantum circuit 700 is configured to essentially perform a resonant 2N to 1 photon exchange between the first and second modes, respectively, when a predetermined current is applied.

[0119] As mentioned above, two modes are simultaneously hosted within loop 710. This maximizes their participation at the central junction, thus increasing the strength of the resonant 2N to 1 photon exchange. The circuit on the right of schematic diagram a) in Figure 7 shows how the participation of the central junction can be selected to tune the circuit's nonlinearity and therefore the strength of the resonant 2N to 1 photon exchange. This is done by connecting the junction at a desired level along the induced portion of the mode to pick up only a portion of the zero-point fluctuations in phase.

[0120] Again, for simplicity, we assume that the circuit is designed to be biased at its optimum bias point. To simplify the derivation of the Hamiltonian for this example circuit 700, we assume that the intrinsic capacitance of the junctions is zero. In practice, a full diagonalization of the system can take finite capacitances into account. In this example, the modes of the circuit are hosted by respective LC resonators, with the first LC resonator formed by the first inductive element 702 and the first capacitor 704, and the second LC resonator formed by the second inductive element 703 and the second capacitor 705, respectively. Thus, the frequency of the first mode (e.g., hosted by the left resonator) is: TIFF2025538423000230.tif6150 and the frequency of the second mode (e.g. hosted in the right resonator) is given by TIFF2025538423000231.tif6150, and these components are frequency-matched. It is adjusted to reach TIFF2025538423000232.tif6150. Then TIFF2025538423000233.tif6150 is TIFF2025538423000234.tif6150, so the joint Hamiltonian is TIFF2025538423000235.tif6150, however, TIFF2025538423000236.tif11150 and Write TIFF2025538423000237.tif11150. By expanding this Hamiltonian to the desired order, we obtain the resonant 2N-to-1 photon exchange Hamiltonian: TIFF2025538423000238.tif6150, however, You can get TIFF2025538423000239.tif10150.

[0121] Schematic diagram b) of FIG. 7 shows another example of a symbolic representation of a device including circuit 700. Circuit 700 is the same as schematic diagram a), except that capacitor 705 and inductor 703 have been rearranged for easier symbolic representation. Thus, in circuit 700 included in the device, a first mode is coupled to a second mode via a Josephson junction. Both the first and second LC resonators of circuit 700 are shorted to superconducting ground at their other ends (not coupled to the Josephson junction), allowing for the formation of a superconducting loop that can be biased to reach an optimal bias point. The second LC resonator is inductively coupled via transformer 711 to another resonator that acts as a bandpass filter between the system and environment 712. Environment 712 includes a load, a microwave source, and a DC current source. As before, the same input port of the device can be conveniently used to introduce DC current and microwave radiation into the circuit.

[0122] Schematic diagram c) of Figure 7 shows an example of fabricating circuit 700 as a planar superconducting pattern. The superconducting circuit shown is constructed in a coplanar waveguide (CPW) geometry, with the gray area representing the remaining superconducting metal on the dielectric substrate. The black crosses represent Josephson junctions. The circuit consists of two λ / 4 resonators: the left λ / 4 resonator (referenced 708 and hosting the first mode) is connected to ground, and the right λ / 4 resonator (referenced 709 and hosting the second mode) is doubly connected to ground at reference numerals 711 and 712 to allow for symmetry maintenance while coupling to the external environment connected via input line 713. Junction 701 is placed between the two resonators at an antinode of the electric field to maximize the system's nonlinearity. Alternatively, the junction can be connected at any point along each resonator's transmission line to adjust the level of nonlinearity (as shown in the right-hand circuit of schematic diagram a). Input line 713 shares an inductance to ground with the second mode. This allows for inductive coupling of the second mode to the environment and a DC current bias on input line 713 from a current source placed at ambient temperature. Thus, the current source may be connected to a standard electrical wire at room temperature and gradually connected to a superconducting wire which in turn is connected to input line 713 to carry a predetermined current (not shown here). To maintain symmetry in the system, the circuit has two superconducting loops in parallel that effectively combine into one, as shown in electrical diagram 720.

[0123] 8 illustrates another embodiment in which the symbolic representation of the circuit includes at least one loop containing one or more Josephson junctions. Circuit 800 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 800 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.

[0124] Next, an example of a circuit 800 having at least one loop 810 will be discussed with reference to schematic diagram a) of FIG. 8 . In the circuit 800, the at least one loop 810 can include a first Josephson junction 801, a central inductive element 803, and a second Josephson junction 802 arranged in series. The central inductive element 803 can be an inductance, a single Josephson junction, or an array of Josephson junctions. Thus, the central inductive element 803 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 801 with a pole of the inductive element 803. The series arrangement can also include a second inner node connecting a pole of the second Josephson junction 802 with another pole of the inductive element 803. The series arrangement may also include a closed loop node connecting the other pole of the first Josephson junction 801 with the other pole of the second Josephson junction 802 .

[0125] The at least one loop 810 may be connected to a common ground through a closed-loop node. The circuit may also include a first capacitor 804 and a second capacitor 805. The first capacitor 804 may be connected in parallel with a first Josephson junction 801 between the common ground and a first internal node of the loop. The second capacitor 805 may be connected in parallel with a second Josephson junction 803 between the common ground and a second internal node of the loop.

[0126] Thus, superconducting quantum circuit 800 is configured to essentially perform a resonant 2N-to-1 photon exchange between the first and second modes, respectively, when a predetermined current is applied. Josephson junctions 801 and 802 are substantially identical, and capacitive elements 804 and 805 are also substantially identical. Thus, the symmetry of the circuit implies that the actual modes of the system are a symmetric superposition 806 of two resonators (indicated by a full arrow) and an antisymmetric superposition 807 of two resonators (indicated by a dashed arrow). The first mode is a symmetric superposition 806, and the second mode is an antisymmetric superposition 807. It can be noted that only the second mode has a contribution across central inductive element 803, which is advantageously used to preferentially couple the environment to this second mode while isolating the first mode from the environment.

[0127] 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 TIFF2025538423000240.tif6150 The resonator cannot be biased to the maximum budget of TIFF2025538423000241.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.

[0128] Here, circuit 800 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. The analysis presented here therefore differs from that of circuit 600. Assuming 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 TIFF2025538423000242.tif6150) and antisymmetric mode (i.e., the second mode TIFF2025538423000243.tif6150). In this eigenmode basis, the contribution of each junction to the Hamiltonian of the system can be calculated, TIFF2025538423000244.tif7150 TIFF2025538423000245.tif7150This can be factorized as follows: TIFF2025538423000246.tif6150 TIFF2025538423000247.tif6150

[0129] The two quadratic parts of the first term are The effective inductive energy of the junction at the point of application of TIFF2025538423000248.tif6150 is given. Along with the charging energy of the capacitor and the inductive energy of the central inductive element, the frequency TIFF2025538423000249.tif10150 and TIFF2025538423000250.tif10150 and phase zero point fluctuation TIFF2025538423000251.tif12150 and It becomes possible to define TIFF2025538423000252.tif12150.

[0130] By expanding the second term to the desired order, we obtain the resonant 2N-to-1 photon exchange Hamiltonian TIFF2025538423000253.tif6150, but You can get TIFF2025538423000254.tif10150.

[0131] 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 TIFF2025538423000255.tif6150, TIFF2025538423000256.tif6150= TIFF2025538423000257.tif6150, therefore, TIFF2025538423000258.tif6150, It can be shown that it leads to TIFF2025538423000259.tif14150.

[0132] Schematic diagram b) of Figure 8 shows an example of a device used to stabilize a manifold of coherent states using circuit 800. Viewed from the bottom up, the nonlinear superconducting circuit 800 is at the bottom, with an inductive element sharing mutual inductance 808 with another inductor terminating in environment 820 above. Environment 820 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 809 is typically a differential transmission line to preserve the symmetry of circuit 800.

[0133] Schematic diagram c) of Figure 8 shows an example of a planar superconducting pattern 850 and its circuit equivalent. The presented superconducting circuit is fabricated in a coplanar waveguide (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 810 is diluted by using three sections of CPW transmission line, called stubs (811, 812, 813), to control the level of nonlinearity in the system. On the left and right sides, open stubs 811 and 812 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 813 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 809 is used to inductively couple the mode to the environment. A transition between the CPW and the slotline 814 is shown above for coupling to the environment (and therefore a current source) with a common geometry 815 (e.g., a CPW line or coaxial cable).

[0134] Figure 9 shows experimental data demonstrating the inherent two-to-one photon dynamics in a tested implementation, i.e., nonlinear anticrossing when a given current biases the circuit. The data is measured on a device represented by a planar superconducting pattern 850 made of tantalum on a sapphire substrate with Josephson junctions made of aluminum and aluminum oxide. The parameters in symbolic representation are as follows: The Josephson energy of a pair of junctions is: TIFF2025538423000260.tif6150GHz, and the central inductive element is TIFF2025538423000261.tif61 It is made of a single Josephson junction with 50 GHz. The lateral capacitance is the stray series inductance TIFF2025538423000262.tif6150GHz energy TIFF2025538423000263.tif6150MHz. Inductance to ground is stray capacitance TIFF2025538423000264.tif6150MHz Energy TIFF2025538423000265.tif6150GHz. Colormap 900 shows the source frequency and bias current TIFF2025538423000266.tif6150 corresponds to the amplitude of the reflected microwave signal from a microwave source on the nonlinear superconducting circuit as a function of bias current. Line 901 corresponds to where the probe frequency matches the second mode frequency, and therefore shows the frequency of the second mode as a function of bias current, similar to Figure 4. An enlarged portion 910 of anticrossing 902 shows where the two modes of the circuit meet the frequency matching condition. This occurs when TIFF2025538423000267.tif6150 is met. For this device, the bias point is approximately 7.3mA.

[0135] Next, the measurement method according to the present invention will be discussed.

[0136] 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.

[0137] 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.

[0138] 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 {TIFF2025538423000268.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 {TIFF2025538423000269.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.

[0139] 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 TIFF2025538423000270.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 TIFF2025538423000272.tif6150 is the displacement operator The mean value of TIFF2025538423000273.tif6150 can be determined by measuring the mean value of TIFF2025538423000273.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.

[0140] 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 TIFF2025538423000274.tif6150 is used for the two-level system TIFF2025538423000275.tif8150 or This can be achieved by realizing a Hamiltonian that couples to either the Z or X operator of TIFF2025538423000276.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. TIFF2025538423000277.tif6150 Rotate around the axis (X axis respectively). By adjusting the tif to TIFF2025538423000278.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 TIFF2025538423000279.tif6150, the two-level system is If there is an even number of photons in the mode, the state TIFF2025538423000280.tif6150, and if there is an odd number of photons, the state TIFF2025538423000281.tif6150. Measuring X changes the qubit to state TIFF2025538423000282.tif6150 or TIFF2025538423000283.tif6150 and therefore determine the photon number parity. In the case of longitudinal interactions, the two-level system is in the state Starting with TIFF2025538423000284.tif6150, the two-level system is If there is an even number of photons in the mode, the state TIFF2025538423000285.tif6150, and if there is an odd number of photons, the state TIFF2025538423000286.tif6150. By measuring Z, the qubit changes to state TIFF2025538423000287.tif6150 or Determine whether the photon number is in TIFF2025538423000288.tif6150 and therefore determine the photon number parity.

[0141] Unfortunately, the fundamental interaction required to perform Wigner tomography, namely the photon number operator Coupling to TIFF2025538423000289.tif6150 is incompatible with the coherent state stabilization mentioned above. This is Merge to TIFF2025538423000290.tif6150 This is also true for its characteristic function, which depends on either the coupling to TIFF2025538423000291.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.

[0142] 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).

[0143] 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.

[0144] 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.

[0145] 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.

[0146] 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 are always on.

[0147] Using a resonant cat qubit circuit, 2N coherent state manifold stabilization is enabled by single-photon drive on the buffer and single-photon loss in the buffer. Single-photon drive can be easily controlled using a microwave source, but single-photon loss is built in. This entails that in the context of a resonant cat qubit circuit, 2N-photon drive can be easily controlled, but 2N-photon loss cannot be easily controlled.

[0148] Therefore, in the context of a two-component cat qubit, applicants have experimented with two-photon drive control. In doing so, applicants have demonstrated that when only two-photon losses are active, a stable manifold of cat qubit modes is TIFF2025538423000292.tif6150 Fock state and TIFF2025538423000293.tif6150 We have discovered that this manifold is a span of Fock states. In what follows, we will refer to this manifold as a coherent state for the two-component cat qubit. We call this the undriven manifold in contrast to the driven manifold containing TIFF2025538423000294.tif6150. This means that whatever state the cat qubit starts in, the cat qubit will This means that the photon number operator is projected by dissipation onto the undriven manifold across TIFF2025538423000295.tif6150. Advantageously, this projection preserves parity. In fact, since the cat qubit modes lose photons two at a time, parity does not change during this process. Furthermore, when only two-photon loss is on, there is no preferred phase in the cat qubit mode, and the state is then free to rotate, so the photon number operator Coupling to TIFF2025538423000296.tif6150 is again possible. This means that we can measure which Fock state the mode will be in. This can be generalized to many more states. If only 2N photon dissipation is effective, the initial state of the cat qubit mode will be dissipated by dissipation. The photon count is then projected onto a new stable manifold within a span of 6150. This projection preserves the photon number modulo 2N, and ultimately, the number of photons in the newly stabilized undriven manifold can be measured. At first glance, this appears similar to the Kerr-cat qubit situation described in 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. However, there are two important differences that would prevent those skilled in the art from considering this approach: For Kerr Cat, mapping requires turning off the parametric pump, This pump turn-off must be adiabatic compared to the confinement velocity and is ineffective in this case.

[0149] Several techniques known in the art can be used to characterize the photon number distribution in the undriven manifold.

[0150] The following shows the non-drive manifold TIFF2025538423000298.tif6150 spans a list of examples in the two-component cat qubit paradigm. - Measuring the parity of the photon number distribution, a two-level system coupled as described above can be used. Stabilized by two-photon dissipation Note that for the TIFF2025538423000299.tif6150 manifold, the parity is directly related to the photon number, Measuring whether the -cat qubit mode is in a particular photon number state, by sending a frequency-selective pulse on the two-level system, can use a dispersively coupled two-level system as known in the art (see, for example, Hofheinz, M., Weig, E., Ansmann, M. et al., "Generation of Fock states in a superconducting quantum circuit," Nature 454, 310-314 (2008), https: / / doi.org / 10.1038 / nature07136). Since there are only two Fock states in this manifold, the result of the measurement is a complete representation of the photon number). - Measuring the cat qubit modes in the Fock state by engineering dispersive coupling to linear modes coupled to heterodyne or homodyne detectors (see, for example, the paper by S. Touzard, A. Kou, N.E. Frattini et al., "Gated Conditional Displacement Readout of Superconducting Qubits," Phys. Rev. Lett. 122, 080502), - Measuring that the cat qubit mode is due to dispersive coupling with its linear counterpart in the Fock state, provided that the cat qubit mode has a slight anharmonicity (see, for example, Wallraff, A., Schuster, D., Blais, A. et al., "Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics," Nature 431, pp. 162-167 (2004), https: / / doi.org / 10.1038 / nature02851).

[0151] Thus, for a resonant two-component Cat qubit circuit, methods are available to measure Z and X, and Y is easily mapped to X. Measuring these three observables is sufficient to know the complete state of the system, if it is assumed to be in a two-component Cat qubit manifold. As a reminder, Z can be measured by heterodyne or homodyne detection while stabilization is on, and X, which is also the photon number parity, is the Fock state TIFF2025538423000300.tif6150 and It can be mapped to the measurements in TIFF2025538423000301.tif6150.

[0152] For a four-component cat qubit, the undriven manifold is a Fock state The initial state of the cat qubit mode is projected onto this manifold while maintaining the photon number modulo 4. In this manifold, we can measure parity to know whether the cat qubit mode state is in the cat qubit (even) or error (odd) manifold. Assuming the state is within the cat qubit manifold, we can measure whether it has 0 photons modulo 4 or 2 photons modulo 4. This measurement is equivalent to the Z measurement of a four-component cat qubit (Mirrahimi M. et al., "Dynamically protected cat-qubits: a new paradigm for universal quantum computation," 2014, New J.Phys.16 045014).

[0153] If one desires more information about the system, or wants to rely on fewer assumptions, a full Wigner measurement should be performed as described above. As a reminder, this requires the measurement of parity after a displacement, where the displacement is scanned over the phase space of the modes. The displacement is then measured over a period of time. TIFF2025538423000303.tif6150 Applied Hamiltonian If the displacement is sudden and strong enough, i.e. TIFF2025538423000305.tif6150 and TIFF2025538423000306.tif6150 overcomes the 2N photon stabilization and displaces the state. The resulting displacement The complex amplitude of TIFF2025538423000307.tif6150 is the complex amplitude of the microwave source used for displacement. TIFF2025538423000308.tif6150 and Displacement pulse duration TIFF2025538423000309.tif6150. Then, when the 2N photon drive is turned off, the system operates in the undriven manifold. TIFF2025538423000310.tif6150, where the photon number operator TIFF2025538423000311.tif6150, so parity measurements can be performed.

[0154] Note that the reverse operation is possible, i.e., by turning on the single-photon drive again in buffer mode, we map the undriven manifold into a driven manifold while maintaining the photon number modulo 2N. For example, in the two-component cat qubit paradigm, the Fock state If you start with TIFF2025538423000312.tif6150, TIFF2025538423000313.tif6150 Cat qubit state or even state, Fock state If you start with TIFF2025538423000314.tif6150, TIFF2025538423000315.tif6150 cat qubit state or odd state. When performing a QND measurement on the undriven manifold, this guarantees a QND measurement of the X operator of the two-component cat qubit.

[0155] As a result, applicants have devised a method, shown in FIG. 10, for performing quantum non-demolition (QND) measurements on a device comprising a resonant cat qubit circuit hosting a cat qubit.

[0156] The QND measurement begins with operation 1000, which turns off the drive on the buffer. As a result, the two-photon drive on the resonant Cat qubit is turned off, and only the two-photon loss is active. Operation 1000 can be performed by turning off the microwave source that provides the radiation to drive the second mode. Turning off the drive can be done by applying a specific microwave radiation sequence, illustrated in FIG. 11.

[0157] In the schematic diagram a) of Figure 11, according to a first embodiment, the microwave radiation is suddenly brought to zero, which corresponds to switching off the microwave source.

[0158] In schematic diagram b) of FIG. 11, according to another embodiment, the microwave source is ramped down to zero due to the finite bandwidth of both the microwave source and the microwave path for driving.

[0159] In schematic diagram c) of FIG. 11, according to a preferred embodiment, the microwave radiation is driven to a negative level before being driven to zero. This allows for the fastest convergence to the undriven manifold. The negative level of drive is of a similar absolute value to the stabilized drive, although larger values ​​are clearly available. When properly adjusted, this embodiment allows for faster arrival at the undriven manifold than the embodiments of schematic diagrams a) and b), as shown in schematic diagram d) described below.

[0160] The schematic diagram in Figure 11 (d) shows the amplitude starting from the center of the Bloch sphere of the driven manifold. TIFF2025538423000316.tif6150 shows a simulation of the performance of the embodiment of schematic c) compared to the two-component cat qubit embodiment of schematic a) with TIFF2025538423000316.tif6150. As the transition to the undriven manifold progresses, the remaining population outside the undriven manifold is driven by the stabilized drive amplitude. Shown together with the corresponding time-dependent buffer drive amplitude for TIFF2025538423000317.tif6150. This schematic shows an embodiment of schematic c) where the buffer drive amplitude undershoot quickly converges to the undriven manifold.

[0161] In yet another embodiment, operation 1000 can be performed by turning off the coupling between the microwave source and the resonant cat qubit circuit, rather than the microwave source itself. In this way, the drive is again turned off, allowing tomography to be performed. In a preferred method, such decoupling is performed while maintaining the coupling of the resonant cat qubit circuit to the current source and load.

[0162] After operation 1000 is performed, a pause occurs in operation 1010. This pause occurs when: TIFF2025538423000318.tif10150. This allows the encoding of the stabilized cat qubit to converge to an undriven manifold, as shown in Figure 12 for a two-component cat qubit.

[0163] Finally, a property of the photon number distribution of the cat qubit mode restricted to the undriven manifold can be measured in operation 1020, for example, by measuring the photon number parity. This measurement can be performed in a number of known ways, as demonstrated in the articles above. Alternatively, operation 1020 can measure whether the number of photons is zero. This measurement can be performed in a number of known ways, as demonstrated in the articles above.

[0164] For two-component cat qubits, parity is the undriven manifold As preserved by the dissipative projection onto TIFF2025538423000319.tif6150, operation 1010 maps the X operator of the cat qubit to the photon number operator in the undriven manifold. As a result, the measurement of operation 1020 is effectively a measurement of the operator X of the cat qubit.

[0165] Once the measurement of operation 1020 has been performed, the drive in the second mode can be restored in operation 1030, thereby restabilizing the Cat qubit. Operation 1030 is particularly advantageous when the photon number measurement of operation 1020 is QND, because the entire process can be QND by remapping the undriven manifold to the Cat qubit manifold. Schematics e)-g) of FIG. 11 illustrate various embodiments for performing the microwave source ramp-up of operation 1030. These schematics correspond to the ramp-down embodiments shown in schematics a)-c) of FIG. 11, respectively.

[0166] This operation may be omitted, especially if operation 1020 is not QND.

[0167] The measurement method of Figure 10 is particularly useful when using the device in a quantum algorithm. However, this measurement method is thus limited to measuring the X operator in the context of a two-component Cat qubit. Figure 13 shows another embodiment of the method according to the invention, which allows for more complete tomography to be performed.

[0168] The method of Figure 13 is very similar to the method of Figure 10. Therefore, only the differences will be described and the operations of Figures 10 and 13 having the same last two digits will be considered identical.

[0169] The main difference between Figure 13 and Figure 10 is the application of operation 1340. In operation 1340, the field in the cat qubit host circuit is displaced by a short, intense pulse that can overcome the stabilization mechanism of the resonant cat qubit circuit. In the embodiments described herein, such a pulse can be achieved in the manner described above. The displacement induced by this operation is expressed as the complex amplitude of the displacement, Features TIFF2025538423000320.tif6150.

[0170] In a preferred embodiment, operation 1340 is performed before operation 1300. In an alternative embodiment, operations 1340 and 1300 are performed simultaneously. After operations 1340 and 1300 are performed, pause operation 1310 and measure operation 1320 are performed.

[0171] The displacement of operation 1340 and operations 1300-1320 combined effectively measure the parity of the displacement field of the cat qubit mode. By definition, this is the value of the Wigner function Measure TIFF2025538423000321.tif6150.

[0172] In this case, the method is not QND due to the displacement, and act 1030 of restoring the drive to its original state is not performed. To perform a full tomography, the cat qubit must be re-prepared, and acts 1340 and 1300 through 1320 must be performed. TIFF2025538423000322.tif6150 It is necessary to repeat with other values. In this way, by fixing the parameters of the resonant cat qubit circuit's operation, a full tomography can be performed. This full tomography can then be used to modify the parameters of the resonant cat qubit circuit's operation, thereby adjusting its operation, until the appropriate characteristics are achieved.

Claims

1. A quantum measurement method for a device comprising a nonlinear superconducting quantum circuit (100, 600, 700, 800) having a symbolic representation comprising a first mode and a second mode, each having its own resonant frequency, and at least one loop (610, 710, 810) including one or more Josephson junctions (601, 701, 801-802), 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 (601, 701, 801-802), 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 device further comprises: a current source configured to provide the predetermined current of constant strength; a microwave source configured to apply microwave radiation at a frequency substantially equal to the resonant frequency of the second mode or 2N times the resonant frequency of the first mode, and coupled to the nonlinear superconducting quantum circuit to drive the second mode; and a load substantially coupled to the second mode only. The above method performs the following actions 1) The step of turning off the drive of the second mode, ２） A step of pausing for a duration that is included within seconds, The step is the rate of 2N photon dissipation (radians / second) of the aforementioned nonlinear superconducting quantum circuit, and 3) A step of performing a quantum measurement to determine the characteristics of the photon number distribution on the first mode of the circuit, A quantum measurement method, including

2. The quantum measurement method according to claim 1, wherein operation 1) includes applying a rectangular pulse shape to reduce the amplitude drive from its nominal stabilized value to 0.

3. The quantum measurement method according to claim 1, wherein operation 1) comprises applying two consecutive rectangular pulse shapes, the first rectangular pulse shape causing the amplitude drive to a negative value substantially equal to or greater than the nominal stabilized absolute value, and the second rectangular pulse shape causing the current to decrease from this value to zero.

4. Operation 3) is a quantum non-destructive measurement, and further includes operation 4) below, The quantum measurement method according to claim 1, wherein operation 4) restores the driving of the second mode after operation 3) has been performed.

5. The quantum measurement method according to claim 4, wherein operation 4) includes applying a rectangular pulse shape to change the amplitude drive from 0 to the nominal stabilized value.

6. The quantum measurement method according to claim 4), wherein operation 4) includes applying two consecutive rectangular pulse shapes, the first rectangular pulse shape causing the amplitude drive to a value substantially between 0 and 1 to 10 times the nominal stabilization value, and the second rectangular pulse shape causing the amplitude drive to this value and the nominal stabilization value.

7. The following action 0) is further included: Operation 0) applies an electromagnetic pulse to the first mode having a frequency substantially equal to the resonant frequency of the first mode, the electromagnetic pulse is Larger amplitude and The method according to claim 1, having a shorter duration than [a shorter duration].

8. The method according to claim 7, wherein operation 0) is performed before operation 1).

9. The method according to claim 7, wherein operation 0) and operation 1) are performed simultaneously.

10. A quantum tomography method, which operates as follows: a) A step of obtaining a set of complex displacements that define the displacement amplitude and displacement phase for scanning the phase space of the cat qubit mode, b) Preparing a state in the cat qubit mode based on a selected set of parameters for the operation of a device comprising a nonlinear superconducting quantum circuit (100, 600, 700, 800) having a symbolic representation comprising a first mode and a second mode, each having its own resonant frequency, and at least one loop (610, 710, 810) including one or more Josephson junctions (601, 701, 801-802), the step of 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 (601, 701, 801-802). The aforementioned 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 device further comprises: a current source configured to provide a predetermined current of constant strength; a microwave source configured to apply microwave radiation at a frequency substantially equal to the resonant frequency of the second mode or 2N times the resonant frequency of the first mode, and coupled to the nonlinear superconducting quantum circuit to drive the second mode; and a load substantially coupled to the second mode only. c) Applying the method of any one of claims 7 to 9 to determine the amplitude, duration, and phase of the electromagnetic pulse in step 0) using the associated displacement amplitude and displacement phase for a given complex displacement in the set of complex displacements, d) Repeating actions b) and c) using different complex displacements, A quantum tomography method, including [the specified method].

11. A method for adjusting the operation of a quantum device, a) A step of obtaining multiple sets of parameters for the operation of a quantum device, b) For each set of parameters of operation a), the step of performing the method according to claim 10, c) A step of deriving an adjusted parameter set based on the result of operation b), A method for adjusting the operation of a quantum device, including the above.

12. A quantum computing device comprising a nonlinear superconducting quantum circuit (100, 600, 700, 800) having a symbolic representation comprising a first mode and a second mode, each having its own resonant frequency, and at least one loop (610, 710, 810) including one or more Josephson junctions (601, 701, 801-802), 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 (601, 701, 801-802), 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 device further comprises: a current source configured to provide the predetermined current of constant strength; a microwave source configured to apply microwave radiation at a frequency substantially equal to the resonant frequency of the second mode or 2N times the resonant frequency of the first mode, and coupled to the nonlinear superconducting quantum circuit to drive the second mode; and a load substantially coupled to the second mode only. The device operates using the adjusted parameter set determined by the method of claim 11. Quantum computing devices.

13. A quantum computing device comprising a nonlinear superconducting quantum circuit (100, 600, 700, 800) having a symbolic representation comprising a first mode and a second mode, each having its own resonant frequency, and at least one loop (610, 710, 810) including one or more Josephson junctions (601, 701, 801-802), 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 (601, 701, 801-802), 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 device further comprises: a current source configured to provide the predetermined current of constant strength; a microwave source configured to apply microwave radiation at a frequency substantially equal to the resonant frequency of the second mode or 2N times the resonant frequency of the first mode, and coupled to the nonlinear superconducting quantum circuit to drive the second mode; and a load substantially coupled to the second mode only. The device performs the method according to any one of claims 1 to 6. Quantum computing devices.

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