Resonant cat qubit circuit device and quantum measurement method

Abstract

The present disclosure relates to, among other things, a nonlinear superconducting quantum circuit having a first mode and a second mode. The first mode and the second mode each have a corresponding resonance frequency. The circuit is configured such that the resonance frequency of the second mode is substantially 2N times the resonance frequency of the first mode when a predetermined current of constant strength is applied to the circuit. The circuit thus inherently performs a 2N to 1 photon exchange between the first mode and the second mode, respectively, when resonating. This forms an improved nonlinear superconducting quantum circuit. When the circuit receives the predetermined current and the second mode is suitably driven, the circuit can stabilize a cat qubit. In a symmetric version of the circuit, a flux line is developed to allow radio frequency coupling to substantially only couple with the second mode and not the first mode for quantum measurement. This allows the stabilization function to be turned off and quantum measurement to be performed.

Description

Technical Field

[0001] This disclosure relates to the field of quantum technology, and more specifically, to a method for measuring qubits in superconducting quantum circuits. Background Technology

[0002] In recent years, there has been increasing interest in developing quantum technologies for various applications such as quantum computing and communication. Superconducting quantum circuits are a promising platform for realizing quantum computers, with cat qubits stored in superconducting resonators being a particularly attractive candidate.

[0003] A cat qubit is defined as a specific, selected two-dimensional submanifold of a mainstream consisting of multiple coherent states. For example, the mainstream of a two-component cat qubit is the span of two coherent states with equal amplitudes and opposite phases; if the mainstream is two-dimensional, then the submanifold is equal to the mainstream. As another example, the mainstream of a four-component cat qubit is the span of four coherent states with equal amplitudes and a phase difference of 90°. The two-dimensional submanifold is chosen as an even-numbered photon-odd-even submanifold. In summary, the mainstream of a 2N-component cat qubit is the span of 2N coherent states with equal amplitudes and a phase difference of π / N. The two-dimensional submanifold is chosen as a 0-mode, N-photon-number submanifold.

[0004] In this context, superconducting quantum circuits can be designed to exhibit specific quantum dynamics, such as quantum manifolds with stable coherent states. The following article focuses on quantum manifolds that stabilize two coherent states through dissipation:

[0005] - "Exponential suppression of coded qubit flipping in oscillators," Lescanne R. et al., published in Nature Physics, 2020;

[0006] - "Through coherent oscillations within dissipatively stable quantum manifolds," Touzard S. et al., Physical Review X, 2018; and

[0007] - "Confining optical states within quantum manifolds by designing two-photon losses." Leghtas Z. et al., Science, 2015.

[0008] The stabilization of 2N coherent states requires a nonlinear transition between 2N photons carrying a stable quantum manifold in a first mode (also known as a cat qubit mode) and a single photon in a second mode (called a buffer mode), and vice versa. To achieve this nonlinear transition, existing solutions involve applying an external time-varying excitation to a superconducting quantum circuit, in the form of one or more micro-wave tunes at a specific frequency (called parametric pumps), to bridge the energy gap between the 2N photons of the first mode and a single photon of the second mode. These solutions are also known as parametric pumping techniques.

[0009] The applicant disclosed a series of superconducting quantum circuits in patent application EP21306965.1, in which the resonant frequency of the second mode is approximately 2N times the resonant frequency of the first mode when a predetermined current of constant strength is applied to the circuit. This provides a technique referred to below as a “resonant cat qubit circuit” and a “resonant cat qubit”, because the 2N to 1 photon conversion does not require a parametric pump to bridge the bandgap in the process. Once this nonlinear conversion process is enabled, the coherent manifold is stabilized by coupling the second mode (or “buffered mode”) to an environment containing a microwave source (driven resonantly) and a load (dissipating the second mode). Through the nonlinear conversion process, this single-photon drive is converted to 2N photon drive in the first mode or cat qubit mode. Similarly, single-photon loss is converted to 2N photon loss in cat qubit mode.

[0010] While abandoning the use of parametric pump technology has significant advantages, the realization of resonant cat qubit circuits presents new challenges in the measurement of cat qubits.

[0011] In previous implementations, the state of the cat qubit mode was determined by measuring the Wigner function. This measurement is typically based on the dispersive interaction between the cat qubit mode and a two-level system. In a two-level system, the dispersive interaction embedded with a Ramsey sequence can measure the parity of the field. In their paper "Tracing Photon Transitions by Repeated Quantum Nondestructive Parity Measurements" published in *Nature*, Vol. 511, pp. 444-448 (2014) (https: / / doi.org / 10.1038 / nature13436), Sun, L., Petrenko, A., Leghtas, Z., et al., described a method for measuring the conventional Wigner function in the context of superconducting circuits.

[0012] Coherent state stabilization suppresses dispersion coupling. Therefore, the Wigner function cannot be measured when the mainstream mode is stable. In previous stabilized cat qubit implementations involving parametric pumping techniques, the Wigner function was measured after the stabilization process was neutralized, achieved by shutting down the parametric pump that enables 2N-to-1 photon switching between the cat qubit mode and the buffer mode. Due to the large bandwidth of the microwave circuit, this can be accomplished almost instantaneously (tens of nanoseconds).

[0013] However, this is impossible in the context of resonant cat qubits. In fact, since stability is self-sustaining, there is no need to turn off the parametric pump.

[0014] One possible solution is to use a magnetic field to detune the resonant frequency of the cat qubit circuit, thereby disabling the 2N-to-1 photon conversion between the cat qubit mode and the buffer mode. However, given the limitations of resonant cat qubit circuits, there is currently no sufficiently fast way to apply such a magnetic field. Summary of the Invention

[0015] The present invention aims to improve this situation. To this end, the applicant provides a quantum device comprising:

[0016] - A nonlinear superconducting quantum circuit having a first mode, a second mode, and a central hybrid component. The first mode and the second mode each have corresponding resonant frequencies. The central hybrid component includes:

[0017] At least one loop comprising a first Josephson junction, a central inductor, and a second Josephson junction arranged in a series topology generally symmetrical about an axis, the axis mapping the first Josephson junction to the second Josephson junction, the loop including: a first inner node connecting one pole of the first Josephson junction to one pole of the central inductor; a second inner node connecting one pole of the second Josephson junction to the other pole of the inductor; and a closed node connecting the other pole of the first Josephson junction to the other pole of the second Josephson junction;

[0018] A first circuit portion connected between a common ground terminal and a first inner node of the loop; a second circuit portion connected between a common ground terminal and a second inner node of the loop, such that the first and second circuit portions are substantially symmetrical about the axis; and a third circuit portion connected between the closed node and the common ground terminal and substantially symmetrical about the axis.

[0019] The nonlinear superconducting quantum circuit is configured such that when a predetermined current of constant intensity is applied, the resonant frequency of the second mode is approximately 2N times the resonant frequency of the first mode, generating a phase difference between one or more Josephson junctions, thereby enabling the nonlinear superconducting quantum circuit to be expanded into at least one form. The Hamiltonian of the sum of the dominant term and a series of auxiliary terms, where g 2N is a scalar corresponding to the intrinsic coupling strength, a is the annihilation operator for the first mode, and b is the annihilation operator for the second mode. It is the reduced Planck constant, thus inherently allowing for a resonant 2N-to-1 photon exchange between the first and second modes, where N is a positive integer.

[0020] The nonlinear superconducting quantum circuit is located on a dielectric substrate and is separated from the common ground plane by an exposed portion of the dielectric substrate.

[0021] - A magnetic flux line located opposite a portion of the at least one loop including the central inductor element, the magnetic flux line including a first slot and a CPW (coplanar waveguide) portion.

[0022] The CPW portion is arranged to be coupled to a current source, a microwave source (configured to apply microwave radiation), and a load, and is defined by end lead bonding.

[0023] The first slot is patterned in the ground plane to expose the dielectric substrate and extends generally parallel to the axis. One end of the first slot is connected to the slot of the CPW portion at the height of the end lead bonding, and the other end of the first slot is connected to the exposed dielectric substrate portion, which defines a portion of the at least one loop including the central inductor element.

[0024] The magnetic flux lines are configured to allow the current source to couple to induce the predetermined current in the at least one loop, to allow the microwave radiation source to couple to induce the second mode drive at a frequency substantially equal to the resonant frequency of the second mode or 2N times the resonant frequency of the first mode, and to allow the load to be coupled substantially only to the second mode.

[0025] According to various embodiments, the quantum device may include one or more of the following features:

[0026] - The magnetic flux line further includes a second slot patterned in the ground plane to expose the dielectric substrate and extending generally parallel to the axis. One end of the second slot is connected to another slot of the CPW portion at the height of the end wire bond, and the other end of the second slot terminates in the ground plane at a distance greater than 1 μm from the portion of the at least one loop including the central inductor element.

[0027] - The second slot includes a return portion that is closest to the portion of the at least one loop containing the central inductor element, the return portion extending substantially perpendicular to the axis in a turning direction away from the first slot;

[0028] - The length of the return section is greater than 1 μm, and can reach up to the width of the nonlinear superconducting circuit;

[0029] -The first slot is generally symmetrical about the axis;

[0030] - Both the first slot and the second slot are offset in a direction perpendicular to the axis and opposite to the direction of rotation;

[0031] - The offset is greater than 1 μm and can reach up to the width of the nonlinear superconducting circuit;

[0032] - Select the length of the first slot or the location of the wire bond based on the expected coupling strength between the second mode and the load;

[0033] - The coupling strength between the load and the second mode is more than 100 times higher than the coupling strength between the load and the first mode;

[0034] - The device also includes a current source, a microwave source, and a load coupled to the coplanar waveguide portion (CPW) to stabilize the coherent manifold in the device;

[0035] The present invention also relates to a quantum measurement method for the device described above, comprising:

[0036] a) Modify the current output of the current source so that the predetermined current in the nonlinear superconducting circuit induces a resonant frequency of the second mode that is significantly different from the resonant frequency of the first mode by 2N times.

[0037] b) Perform quantum measurements on the first mode of the nonlinear superconducting circuit.

[0038] According to various embodiments, the method may include one or more of the following features:

[0039] The modification of operation a) is performed by applying a square wave pulse to the current;

[0040] - Operation b) is a quantum non-destructive measurement, and the method further includes the following operation: c) after performing operation b), restoring the current output of the current source to obtain a predetermined current;

[0041] The present invention also relates to a quantum computing system comprising at least one device according to the present invention. Attached Figure Description

[0042] A non-limiting example will now be described with reference to the accompanying drawings, in which:

[0043] Figure 1 This demonstrates how to incorporate a resonant cat qubit circuit into a device to stabilize quantum information;

[0044] Figure 2 Examples of various circuits for isolating the first mode from the environment are shown;

[0045] Figure 3 An example of how to stabilize quantum information using a resonant cat qubit circuit with an incorporated device is shown;

[0046] Figure 4a and 4b This illustrates how the current is tuned to achieve frequency matching conditions;

[0047] Figure 5 Several examples of circuit symbols used interchangeably throughout the specification are shown;

[0048] Figure 6 It shows Figure 1 Examples of the device;

[0049] Figure 7 A general circuit notation representation of the resonant cat qubit circuit used in the device according to the present invention is shown;

[0050] Figure 8a A top view of a device according to the present invention is shown, which includes a resonant cat qubit circuit without magnetic flux lines; Figure 8b The circuit symbol representation of the differential mode of the resonant cat qubit circuit is shown.

[0051] Figure 9a This is a top view of a first embodiment of the device of the present invention; Figure 9b for Figure 9a The circuit symbol representation of the differential mode of the resonant cat qubit circuit shown in the figure during device operation;

[0052] Figure 10 This is a top view of a second embodiment of the device of the present invention;

[0053] Figure 11 This is a top view of a third embodiment of the device of the present invention;

[0054] Figure 12 This is a top view of a fourth embodiment of the device of the present invention;

[0055] Figure 13 Images of the manufacturing device corresponding to the first embodiment; and

[0056] Figure 14 This is an image of the manufacturing device corresponding to the fourth embodiment. Detailed Implementation

[0057] Before describing the measurement method of this invention, the applicant will first describe the resonant cat qubit and its basic principles. This will help to better understand the properties of the resonant cat qubit circuit, why its stability is unique, and the unique challenges that arise from it.

[0058] First, the background of the resonant cat qubit will be explained.

[0059] To realize a resonant cat qubit, a nonlinear superconducting quantum circuit with a first mode and a second mode is provided. The first and second modes each have corresponding resonant frequencies. The circuit is configured such that, when a predetermined current of constant strength is applied to the circuit, the resonant frequency of the second mode is approximately 2N times the resonant frequency of the first mode. Therefore, the circuit inherently performs a resonant 2N-to-1 photon exchange between the first and second modes, respectively. N is a positive integer (i.e., N is any positive integer greater than or equal to 1), and therefore 2N is an even number, such as 2, 4, 6, or greater. Thus, the expression "2N photons" represents a discrete and even number of photons defined by the integer 2N.

[0060] This superconducting quantum circuit improves the resonant 2N-to-1 photon exchange between the first and second modes. In fact, in the superconducting quantum circuit, when a predetermined current of constant strength is applied to the circuit, the resonant frequency of the second mode is approximately 2N times that of the first mode. This contrasts with known dissipative-based cat qubit implementations, where external time-varying excitation (e.g., excitation performed via parametric pumping) is used to bridge the gap between the two mode frequencies and achieve resonant 2N-to-1 photon exchange. While the external time-varying excitation used in the prior art relaxes the restrictions on the mode frequencies, it introduces some undesirable effects, such as heating the modes. This heating effect shortens the coherence time of the circuit and leads to kinetic instability. As a result, any nonlinear mixing processes, including resonant 2N-to-1 photon exchange, are eventually disrupted. In this respect, due to the absence of external time-varying excitation, the rate of resonant 2N-to-1 photon exchange can be increased by one to two orders of magnitude compared to the prior art.

[0061] The first and second modes of this superconducting quantum circuit can each correspond to the circuit's intrinsic resonant frequencies. For example, the first and second modes can be either electromagnetic or mechanical modes. Each of the first and second modes can have a corresponding intrinsic resonant frequency. For example, the first mode (and correspondingly the second mode) can be a mechanical mode, while the second mode can be an electromagnetic mode (and correspondingly the first mode is an electromagnetic mode). The first mechanical mode and the second electromagnetic mode can be coupled in the circuit via the piezoelectric effect. Each of the first and second modes can have a corresponding resonant frequency; for example, the first mode can have... The type of resonant frequency, the second mode can have The resonant frequency of type, where ω a and ω bThis can be the angular frequency of each corresponding mode. "Having" a first mode and a second mode means that the superconducting quantum circuit can include components that operate in the superconducting state, these components either independently of each other or simultaneously carrying these modes. In other words, the first and second modes can be carried in different subsets of components in the superconducting circuit, or (optionally) in the same subset of components.

[0062] Superconducting quantum circuits can operate at temperatures close to absolute zero (e.g., 100 mK or lower, typically 10 mK) and be as isolated from the environment as possible to avoid energy loss and decoherence, except for certain specific couplings. For example, only the second mode can couple to the dissipative environment, while the first mode can remain isolated from the environment.

[0063] Superconducting quantum circuits can be fabricated as one or more patterned layers of superconducting material (e.g., aluminum, tantalum, niobium, etc., known in the art) deposited on a dielectric substrate (e.g., silicon, sapphire, etc.). Each of the one or more patterned layers can define a lumped element resonator. Capacitive elements can be formed from two adjacent superconducting material plates (on corresponding layers of the one or more patterned layers). Inductive elements can be formed from superconducting wires. Alternatively, at least one of the one or more patterned layers can define portions of a transmission line, the resonant frequency of each portion of the transmission line depending on its length. The transmission line can be, for example, a coplanar waveguide or a microstrip line. Alternatively, the circuit can be embedded in a three-dimensional architecture containing high-quality three-dimensional modes fabricated into a bulk superconductor by machining or micromachining, which can be used as either mode.

[0064] The configuration of the nonlinear circuit and the predetermined current of constant intensity ensures that when the predetermined current is applied to the circuit, the resonant frequency of the second mode of the circuit is approximately 2N times the resonant frequency of the first mode (also known as the "frequency matching condition", which has the form 2Nf). a =f b f a f is the resonant frequency of the first mode. b This is the resonant frequency of the second mode, or equivalent to the form 2Nω. a =ω bIn other words, the predetermined current is an external current that induces an internal DC bias in the circuit. The internal current is adjusted so that the components / hardware constituting the circuit are in a specific state: that is, the corresponding resonant frequency of the second mode of the circuit (also called the second resonant frequency) is approximately 2N times the corresponding resonant frequency of the first mode (also called the first resonant frequency). Therefore, the nonlinearity of the circuit inherently performs a resonant 2N-to-1 photon exchange, which destroys 2N photons in the first mode at the first resonant frequency and generates one photon in the second mode at the second resonant frequency; conversely, it destroys one photon in the second mode at the second resonant frequency and generates 2N photons in the first mode at the first resonant frequency. In practice, the resonant 2N-to-1 photon exchange can occur at a given rate, hereinafter referred to as g. 2N Photon exchange between the two modes can be achieved through components in the circuit (e.g., nonlinear components). A predetermined current is applied such that this type 2Nω a =ω b The frequency matching condition occurs, enabling the circuit to achieve resonant 2N-to-1 photon exchange dynamics. The predetermined current at which the frequency matching condition is met is also called the bias point of the circuit. If the circuit parameters are chosen such that spurious dynamics disappear under the frequency matching condition, this bias point can also be called the optimal bias point, as shown in the following example.

[0065] A predetermined current can be applied directly (i.e., in an electrical manner) to the circuit, causing it to flow through at least one subset of the circuit's elements and be distributed into the various possible branches. The current flowing in the circuit branches is called the internal current, which is determined according to Kirchhoff's current law. In various examples, the predetermined current can be applied directly by a current source connected to the circuit. In various examples, the circuit can have a planar geometry, so that the path of the predetermined current merges with a portion of a superconducting loop in the same planar geometry (e.g., an on-chip current path).

[0066] Alternatively, a predetermined current can be applied indirectly, i.e., by inducing an internal current in the circuit through mutual inductance (e.g., using a coil). In other words, an external inductor couples with the circuit inductance via shared mutual inductance, inducing a current in the circuit that flows through at least one subset of the circuit's components. Thus, the internal current is the current induced in the circuit through mutual inductance. Since there is no need for a direct (current) connection to the circuit, the predetermined current path can be on the same horizontal plane as the circuit, or it can be formed by an external coil located above or below the circuit, with its axis perpendicular to the superconducting circuit plane. In various examples, when mutual inductance is shared with a coil, the coil can be made of a multi-turn material that allows current to flow, thereby generating a magnetic field. The coil can consist of any number of coils as desired, which increases the mutual inductance. The coil can be made of any material that allows current to flow to generate a magnetic field and thus induce an internal current. For example, the coil can be made of a superconducting or non-superconducting material. Alternatively, a permanent magnet that directly generates a constant magnetic field can be used instead of the coil. However, this makes tuning the magnetic field impractical.

[0067] Applying a predetermined current generates an internal current flowing through a subset of inductive elements, producing a superconducting phase drop on each inductive element. The phase drop can be calculated based on the circuit geometry and parameters and is a function of the predetermined current. The following example illustrates how a predetermined current (or equivalent superconducting phase drop) can be experimentally tuned to a bias point to achieve resonant 2N-to-1 photon exchange dynamics. Therefore, the application of the current should be adjusted according to the circuit configuration to induce intrinsic resonant 2N-to-1 photon exchange.

[0068] The resonant cat qubit circuit inherently performs resonant 2N-to-1 photon exchange. In other words, the circuit is configured to autonomously / natively perform resonant 2N-to-1 photon exchange, meaning that no external time-varying excitation is required to bridge the gap between the first mode frequency (2N times the first mode frequency) and the second mode frequency. In other words, the predetermined current value is set in a way that allows the components carrying the first and second modes to perform resonant 2N-to-1 photon exchange. It does not depend on any external microwave devices, such as parametric pumps. The resonant 2N-to-1 photon exchange between the first and second modes is a quantum dynamic that destroys 2N photons of the first mode at the corresponding resonant frequency of the first mode and generates one photon of the second mode at the corresponding resonant frequency of the second mode, and vice versa. This resonant 2N-to-1 photon exchange is achieved by applying a predetermined current of constant strength to the circuit when the resonant frequency of the second mode is approximately 2N times the resonant frequency of the first mode. Here, "approximately" means that the predetermined current value makes the frequency of the second mode equal to 2N times the frequency of the first mode (also known as the frequency matching condition in some applications), until the resonant 2N-to-1 photon exchange rate g is reached. 2N A predetermined threshold of the same order of magnitude.

[0069] Since the superconducting quantum circuit reliably and intrinsically executes the quantum dynamics, the use of parametric pumps is unnecessary, thereby improving the quality of resonant 2N-to-1 photon exchange. In fact, removing the parametric pumps eliminates the occurrence of harmful parasitic interactions that affect the quality of resonant 2N-to-1 photon exchange.

[0070] The circuit can be integrated into a device, which may also include a current source configured to apply a predetermined current of constant intensity to the circuit, such that the frequency of the second mode is approximately 2N times the frequency of the first mode. The current source can be directly connected or inductively coupled to the circuit, such that an induced internal current flows through at least some or all of the circuit's components. In other words, the applied current can induce an internal current flowing through the circuit surface in specific components of the circuit. The current source is a room-temperature device and therefore does not contain superconducting elements. The current source can initially apply a predetermined current to the circuit through a wire at room temperature, which, as the temperature decreases, is then connected to a superconducting wire that applies the current to the superconducting circuit. A low-pass filter can also be used to filter the current along the current path to reduce the effects of low-frequency noise.

[0071] In various examples, the device may also include a load, a microwave source, and a coupler. The coupler can be configured to connect a second mode of the superconducting quantum circuit to the load. The load is a dissipative element, for example, an element with a given resistance (as opposed to a superconducting element), located outside the superconducting circuit. The load dissipates photons exchanged from the first mode to the second mode through resonant 2N-to-1 photon exchange. In other words, photons destroyed from the first mode are evacuated to the environment via the second mode through the load. The microwave source can be configured to apply microwave radiation at a frequency approximately equal to the second mode frequency or approximately 2N times the first mode frequency. In other words, the microwave source can be configured to control the amplitude and phase of the microwave radiation. Thus, the microwave source drives photons in the form of microwave radiation to the second mode, which in turn drives 2N photons in the first mode through resonant 2N-to-1 photon exchange. The coupler is a component that can be connected by current, capacitance, or inductance to the circuit element carrying the second mode and mediates the interaction between the second mode, the load, and the microwave source.

[0072] Alternatively, the coupler can also be configured to couple circuit elements carrying a second mode of superconducting quantum circuit to a load and a microwave source.

[0073] The load can be a resistor, a matched transmission line, or a matched waveguide. "Matching" should be understood as the transmission line or waveguide terminating at a resistor at an end different from the end connected to the element carrying the second mode, the value of which should be chosen such that most of the power flowing to the load is absorbed. The load can be contained within a microwave source.

[0074] In various examples, the microwave source can be placed at room temperature and connected to the circuit via a coaxial cable. In various examples, an attenuator can be placed between the microwave source and the circuit (i.e., along the path of the microwave radiation applied by the microwave source) to thermalize the microwave radiation using a cryogenic environment. This allows microwave radiation to be applied without increasing noise. This microwave radiation serves a different purpose than a parametric pump and does not require obtaining resonant 2N-to-1 photon exchange dynamics. Here, the dissipation of the load, the microwave radiation, and the resonant 2N-to-1 photon exchange dynamics inherently performed by the circuit work together to stabilize 2N coherent states in the first mode. In fact, without microwave radiation, the stable manifold is a manifold composed of Fock states {|0>,|1>}. Due to some residual single-photon dissipation in the first mode, state |1> decays towards state 0>, so in the first mode, only the vacuum is stable in the long run. In other words, the device allows for a quantum manifold of stable coherent states outside of a vacuum.

[0075] Optionally, the device may include a bandpass filter or a bandstop filter connected to the circuit for both the first and second modes. The bandpass or bandstop filter may be configured to allow only the second mode to couple to the load. Optionally, the device may also include a microwave filter to prevent the first mode from dissipating in the load. This microwave filter may be staggered between the load and the coupler. From a circuit perspective, the filter is designed to prevent microwave photons at the first resonant frequency from escaping the circuit. This can be achieved by implementing a bandstop filter at the first resonant frequency or a bandpass filter at the second resonant frequency, since the photons of the second mode are the only photons that need to be dissipated in the environment. For some circuits where the two modes have different symmetries, the filter may not be necessary, and appropriate symmetry of the coupler may be sufficient to prevent the dissipation of the first mode.

[0076] Therefore, this device can stabilize 2N coherent states in the first mode (i.e., the quantum manifold of coherent states). For example, applying microwave radiation to a microwave source in the second mode via a microwave filter, once the photon exchange conversion is achieved through resonance (2N to 1), can be considered as driving 2N photons in the first mode; while a load that only dissipates photons in the second mode, once the photon exchange conversion is achieved through resonance (2N to 1), can be considered as dissipating 2N photons in the first mode. The 2N photon driving and 2N photon dissipation enable the stabilization of 2N coherent states in the first mode.

[0077] The second mode of single-photon drive can be achieved using Hamiltonians. Formal description, where ∈ b This represents the single-photon drive intensity of the microwave source in the second mode. Single-photon dissipation in the second mode can be represented using the Lindblad operator. Formal description, where κ bThis indicates the single-photon dissipation generated by coupling with the load in the second mode.

[0078] 2N photon drive can be achieved using Hamiltonian Formal description, where ∈ 2N =2∈ b g 2N / κ b It is an effective 2N photon drive, g 2N This refers to the 2N-to-1 nonlinear conversion rate between the first and second modes. The 2N photon loss can be represented using the Lindblad operator. Formal description, in which It is the dissipation rate of 2N photons. The amplitude α of the stable coherent state is ultimately determined by... Provided.

[0079] A method for realizing a resonant cat qubit circuit is also provided. The method includes providing a superconducting quantum circuit as described above and applying a predetermined current of constant strength to the circuit or a device containing the circuit, such that the resonant frequency of the second mode is approximately 2N times the resonant frequency of the first mode, where N is a positive integer. This effectively achieves resonant 2N to 1 photon exchange between the two modes respectively.

[0080] The method also includes using a device incorporating this circuit to stabilize a quantum manifold consisting of 2N coherent states, each with the same amplitude and a π / N phase difference between them. This is achieved 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 (e.g., a cat qubit).

[0081] A quantum computing system is also provided. This quantum computing system may include at least one of a superconducting quantum circuit and / or a device containing such a circuit. The quantum computing system can be configured to use the superconducting quantum circuit and / or a device for executing high-quality quantum computing protocols. In other words, the quantum system can use the superconducting quantum circuit and / or a device containing such a circuit to perform fault-tolerant quantum computing. This is due to the inherent resonant 2N-to-1 photon exchange dynamics combined with external dissipation and microwave radiation, which stabilizes a quantum manifold consisting of 2N first-mode coherent states (also known in some applications as "cat qubit states"). In this quantum manifold, the 2N coherent states have the same amplitude, and there is a π / N phase difference between each coherent state. The quantum computing system can use these coherent states to define logical qubits, which are naturally protected from errors due to the stability of the quantum manifold, especially bit-flipping errors. Therefore, the quantum system can define operations (e.g., CNOT, Hadamard, and / or Tofoli gates) that perform computations on the logical qubits. This opens up a complete paradigm for implementing quantum algorithms in a fault-tolerant manner.

[0082] The resonant cat qubit circuit will now be discussed in more detail.

[0083] When a predetermined current is applied to a circuit, the circuit may possess a specific Hamiltonian. As is known in the art, a Hamiltonian is an operator corresponding to the total energy of a superconducting circuit (e.g., including kinetic and potential energy). The Hamiltonian can be used to calculate the time evolution of the circuit. The Hamiltonian can be designed to exhibit the desired quantum dynamics, particularly the resonant 2N-to-1 photon exchange between the first and second modes. The Hamiltonian of the circuit is here a function of the specific set of parameters of the superconducting quantum circuit and the parameters of the predetermined current. The term "parameter" refers to any and all types of physical parameters of the circuit and / or the predetermined current, such as capacitance, inductance, resistance, frequency, phase difference, energy level, zero-point fluctuation of the phase, Josephson energy or critical current, and other parameters such as the voltage and / or current level of the applied predetermined current (e.g., DC bias). The specific meaning of this set of parameters is that it consists of the circuit parameters and the predetermined current parameters. In other words, the Hamiltonian does not depend on any other parameters besides these specific parameters. In other words, the Hamiltonian depends on and only depends on the circuit and the predetermined current. For ease of explanation, the parameter set can be represented as set P1 U P2, where U represents the union operator, parameter set P1 consists only of circuit parameters, and parameter set P2 consists only of predetermined current (e.g., induced DC bias) parameters.

[0084] Therefore, the total energy of a superconducting quantum circuit does not depend on the parameters or predetermined current of any external device. In fact, the Hamiltonian depends only on the parameter set and / or predetermined current of the superconducting quantum circuit, which can be predetermined according to quantum engineering specifications. Thus, the Hamiltonian is independent of time, and especially independent of any time-varying excitation (e.g., from a parametric pump).

[0085] In various examples, the Hamiltonian can contain linear terms. Linear terms describe the presence of modes carried by a circuit element. In other words, each linear term describes the presence of the corresponding mode among the first and second modes carried in the circuit.

[0086] The Hamiltonian can also contain nonlinear terms. These nonlinear terms describe the interaction between the first and second modes. Nonlinear terms can also be called "mixing terms," ​​similar to the mixing that occurs in classical nonlinear microwave circuits. Nonlinear terms can contain a constant that acts as a pre-factor. This constant describes the strength of the interaction between the first and second modes. The pre-factor of the nonlinear term can be smaller than or much smaller than the system frequency. Nonlinear terms can also be called resonant or non-resonant terms, depending on their compatibility with energy conservation.

[0087] The Hamiltonian can be expanded as a sum of terms. The statement "expandable as a sum of terms" should be understood as allowing a Taylor series approximation of the sum of terms; it's important to note that this represents the energy of a superconducting quantum circuit. The total number of terms may be finite or infinite, but according to the Taylor series approximation, the sum of several terms always remains finite. This sum may contain a dominant term and a series of auxiliary terms. The statement "dominant term" should be understood as a term that has a significant impact on the system's dynamics. For a term to be dominant, it must satisfy the following two conditions: First, the magnitude of the term (whether in absolute value or measured using an appropriate norm and compared with other terms in the Taylor series approximation) should significantly contribute to the magnitude of the nonlinear part of the Hamiltonian. Second, the term should have resonant properties, i.e., it should be compatible with energy conservation. A series of auxiliary terms refers to terms whose sum magnitude is lower than a predetermined magnitude. These can themselves be derived from the Taylor series approximation, therefore, auxiliary terms will be omitted below. The Hamiltonian can be expanded as at least one of the following forms: The sum of the dominant term and a series of auxiliary terms, which are omitted here and below.

[0088] In the dominant item In this context, 'a' is the annihilation operator for the first mode, and 'b' is the annihilation operator for the second mode. Conversely, It is the first mode's generation operator. It is the generator operator for the second mode. Dominant term. yes The polynomial in the expression describes the resonant 2N-to-1 photon exchange. Therefore, it contains... The dominant term describes the annihilation of the first-mode 2N photon and the production of the second-mode 1 photon. Since the Hamiltonian is a Hermitian operator, the dominant term also includes a reciprocal term. (Also known as Hermitian conjugation, abbreviated as hc), in which the first mode 2N photons are produced and the second mode 1 photons are annihilated.

[0089] scalar g 22 It is a function of a parameter set, which consists of circuit parameters and / or predetermined current parameters. Due to the scalar g... 2N It is the prefactor of the dominant term, representing the strength of the interaction between the first and second modes, also known as the "intrinsic coupling strength," i.e., the term... (Correspondingly, its Hermitian conjugate). In other words, g 2N The rate of resonant 2N-to-1 photon exchange is described.

[0090] When a predetermined current is applied to induce resonant 2N-to-1 photon exchange (i.e., tuning the predetermined current at the bias point), the rate of the resonant 2N-to-1 exchange term, i.e., the constant g, is... 2N The constant g is non-zero, and the frequency matching condition ensures that this term is in a resonant state. Furthermore, in the sense of the Taylor series approximation, this constant g... 2N It is relatively large compared to other nonlinear terms, therefore the other nonlinear terms are auxiliary terms and will not be discussed further.

[0091] Therefore, when a predetermined current is applied to induce a 2N-to-1 photon exchange (i.e., satisfying the frequency matching condition 2Nω), a =ω b When ), the dominant term is a resonant term, meaning it is compatible with energy conservation. For any N, the dominant term describes nonlinear interactions; it is an odd power of the annihilation operator and the generating operator.

[0092] Hamiltonians can contain other dominant terms, such as Kerr terms. Or cross Kerr terms These nonlinear terms can have significant amplitudes and are essentially resonant regardless of frequency matching conditions (this only occurs under even powers of annihilation and generation operators). These Kerr and cross-Kerr terms can be considered detrimental to the desired engineering dynamics. However, the applicant found that the corresponding constants of other nonlinear terms (i.e., the constant K) a K b , χ ab (e.g., ...) can be reduced by selecting circuit parameters, as shown in the example below. In particular, the applicant found that they disappear near the optimal bias point. Therefore, these items will not be considered further in the following examples, but are described only for completeness.

[0093] This circuit can be configured to perform inherent 2-to-1 resonant photon exchange between the first and second modes, respectively. In other words, N equals 1. In this case, the dominant term of the Hamiltonian can be the 2-to-1 interacting Hamiltonian. Alternatively, N can be greater than 1, for example, N = 2. The coherent state of the first mode is the eigenstate of the annihilation operator acting on the first mode; for example, for a given coherent state |α>, we get a|α> = α|α>, where α is the complex amplitude. In the cat qubit paradigm, the inherent 2-to-1 photon exchange stabilizes the quantum manifold of the first-mode coherent state. In the cat qubit paradigm, the cat qubit state can be defined from the coherent state; for example, the logical qubit state |0> can be defined as |α>, and the logical qubit state |1> can be defined as |-α>. Both of these logical states belong to the coherent state quantum manifold stabilized by the 2-to-1 photon exchange. Due to the stability of the quantum manifold achieved through the intrinsic 2-to-1 photon exchange, the cat qubit state is naturally protected from errors such as bit flips. In fact, bit flip errors can be autonomously suppressed exponentially. This allows for fault-tolerant computation of quantum gates (e.g., CNOT gates, Tofoli gates, and / or Hadamard gates) acting on the logical qubit state.

[0094] In various examples, a superconducting circuit may have a symbolic representation, such as consisting of a set of interconnected dipoles. The term "symbolic representation" should be understood as specifying the symbol and arrangement of the lines for the set of interconnected dipoles. This set of interconnected dipoles (also called components) constitutes a circuit structure (or topology) that is functionally equivalent to a nonlinear superconducting circuit.

[0095] In other words, as is the classic practice in the field of superconducting circuits, the configuration of a nonlinear superconducting circuit is designed to achieve the function defined by its symbolic representation, namely, the function of the theoretical set of interconnected dipoles shown in the symbolic representation. To put it another way, while the circuit can be constructed using patterned layers of superconducting material, it should be understood that the circuit allows for symbolic representation using dipoles (e.g., capacitors, inductors, and / or Josephson junctions). Although the example dipoles describe discrete elements, it will be readily understood by those skilled in the art that these elements correspond to equivalent circuits of distributed elements within a specific frequency range (e.g., low frequencies), as is known in the art.

[0096] The interconnection of symbolically represented groups of interconnected dipoles can be described by a network topology. In this symbolically represented 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 that starts from a given node and returns to the starting node without passing through any node more than once. Each branch can include, for example, two or more components connected in parallel or series. For example, a pair of components including an inductor and a capacitor (i.e., an LC resonator) can be implemented using distributed elements in a patterned layer of superconducting material, such as:

[0097] - Two adjacent plates form a capacitor, which is connected in parallel with an inductor formed by superconducting wires;

[0098] A superconducting transmission line with two different boundary conditions at its ends (one end grounded and short-circuited, the other end open-circuited) forms a so-called λ / 4 resonator. This transmission line can be, for example, a coplanar waveguide or a microstrip type.

[0099] A superconducting transmission line with two identical boundary conditions at its ends (open-open or short-short circuit) forms a so-called λ / 2 resonator. This transmission line can be, for example, a coplanar waveguide or microstrip type; or...

[0100] - A three-dimensional cavity sculpted in a block of superconducting material, which resonates at a given frequency (which depends on its size).

[0101] As is known in the art, such distributed elements can have high-frequency modes that are irrelevant to and unimportant to the dynamic characteristics described herein. Therefore, these distributed elements can be represented symbolically. This symbolic representation can be refined by adding elements (e.g., adding a series inductor at each wire connection or a parallel capacitor between any two nodes in the circuit) or by adding nodes and branches to account for other modes of the distributed element. Thus, the symbolic representation can better describe the distributed element without altering the circuit's operating principle. Therefore, as is known in the art, those skilled in the art consider the physical circuit (i.e., the circuit actually manufactured) and its symbolic representation to be equivalent. In fact, compared to the basic model, the refined dipole of the symbolic representation only adjusts the zero-point ripples of the resonant frequency or phase. When designing the circuit, the final geometry can be fully and accurately simulated using a finite element solver that can easily provide the frequencies of the modes, the dissipation from the load, and the zero-point ripples of the phase at both ends of the Josephson junction—the only unknowns in calculating the resonant 2N-to-1 photon exchange rate under any configuration.

[0102] The symbolic representation of a circuit, such as a collection of connected dipoles or components, may contain at least one superconducting loop, i.e., a series of connected components forming a loop (described by dipoles in the symbolic representation). This at least one loop may contain one or more Josephson junctions. Each Josephson junction may consist of a thin insulating layer that separates two superconducting leads, allowing Cooper pairs to tunnel through the insulating layer. Each Josephson junction may have the following potential energy: Where E J It is a node The Josephson energy is given by Φ, where Φ is the integral of the voltage across the node and Φ0 is the flux quantum. In various examples, the Josephson energy can be tuned during fabrication by selecting the surface area and thickness (i.e., room temperature resistance) of the insulating barrier.

[0103] Josephson junctions can be called nonlinear inductors. In fact, by using inductive elements... Potential energy (where E) L The inductor energy is compared with the potential energy of the node. In the first order, the node behaves as an inductor. When a predetermined current is applied to the circuit, an internal current flows through the node. This current comes either from the direct connection or from the mutual inductance with the superconducting loop (in which the node is embedded), which causes a phase drop at the node. Therefore, applying a predetermined current will change the potential energy of the Josephson junction, as shown below:

[0104]

[0105] The first term on the right-hand side of the above equation is a cosine nonlinearity, describing the nodal points. The effective Josephson energy change. Since the inductance of the Josephson junction is inversely proportional to the Josephson energy, the DC offset... The inductance of a node can be tuned, thereby tuning the frequencies of the modes participating in that node. In the example, the DC offset can be tuned to achieve the frequency matching condition 2Nω. a =ω b The second term describes the nonlinearity corresponding to sinusoidal nonlinearity. The Taylor series expansion of sinusoidal nonlinearity includes the function that provides odd-wave mixing. Odd-power terms, therefore including terms describing the resonant 2N-to-1 photon exchange. Specifically, when the phase drop reaches... At this critical point, the inductance of the Josephson junction approaches infinity, at which point the junction behaves as an open circuit. Therefore, the cosine nonlinearity of the potential energy disappears (i.e., the potential energy drops to zero), while the sinusoidal nonlinearity reaches its peak.

[0106] When DC current flows through the loop or a DC phase drop occurs at both ends of the loop, another linear inductor in the loop can maintain the same inductance.

[0107] In the example, the embedded inductor energy E L Energy E in at least one loop J Josephson knots can be used with ratios To describe. When biased by a predetermined current, it is equivalent to the effective magnetic flux passing through the loop. To describe it, an internal DC current is generated in the superconducting loop, and at the nodes, it depends only on... Phase drop of β In the example, the phase drop can be numerically calculated using the solution of the following equation: in

[0108] The solution to the above equation can be called For example, when inductance E L The loop contains two nodes, each with an energy of E. J When, the phase drop at each node is expressed as

[0109] The Hamiltonian of a circuit can be determined in any way. For example, it can be determined by first determining the equivalent inductance of each junction when a predetermined overcurrent is applied to the circuit. This equivalent inductance (which depends on the DC phase drop across the junction) is determined by... Given, among which This is the reduced magnetic flux Φ0 / 2π. The frequency of the mode can be calculated algebraically or numerically by replacing the nodes in the circuit with their equivalent inductances; for more complex circuit layouts, it can also be calculated using microwave simulations employing the finite element method. These frequencies directly give the linear part of the Hamiltonian. The mode frequency can be calculated along with the mode geometry, which describes the oscillating phase difference that exists between any two points in the circuit when the mode is excited.

[0110] The magnitude of the oscillating phase difference applied to the Josephson junction is crucial for calculating the nonlinearity of the system. In the quantum state, the phase difference between the nodes can be written as... in This is the previously calculated DC phase offset. and The phase difference is the zero-point oscillation of the phase at the node, which is proportional to the geometry of the mode until normalized. The phase difference may include other terms in “…”, such as terms related to other modes. Since these terms are irrelevant to this paper, they will be omitted below.

[0111] The nonlinear term of the Hamiltonian can be expressed by considering the potential energy of the nodes. The Taylor approximation is performed, and the contribution of each node in the circuit is summed to calculate the total. A complete description of the circuit only requires finding the DC phase drop at the nodes, the frequency of each mode, and the zero-point oscillation of the phase at the nodes associated with each mode of the system.

[0112] In various examples, since the circuits are respectively at frequency ω a / 2π and ω b The first and second modes are carried at / 2π, and the linear part of the Hamiltonian can be written as... in It is the reduced Planck constant.

[0113] Given that at least one of the interactions provided by the Josephson knot is of the form of in It is the DC phase drop caused by the predetermined current at the node. and These are the zero-point oscillations of the phase at the nodes associated with the first and second modes, respectively. Therefore, the Hamiltonian can be expanded using a Taylor series, for example, in any desired form relative to the Taylor series expansion, to obtain the resonant 2N-to-1 photon-exchange Hamiltonian. in At resonance 2Nω a =ω b The resonant 2N to 1 photon exchange rate under certain conditions. Therefore, the resonant 2N to 1 photon exchange depends only on the circuit parameters and the predetermined current, such as the parameters mentioned above.

[0114] In contrast, as mentioned earlier, existing cat qubit circuits rely on using parametric pumps to perform 2-to-1 photon swapping. In these prior art implementations, the 2-to-1 photon interaction Hamiltonian is of the form: In these cases, the coupling term g2(t) is modulated by a parametric pump, where the pump injection frequency is ω. p =2ω a -ω b The external time-varying parameter. Parametric pumps are used in the prior art to induce resonance in nonlinear interactions, but have adverse effects.

[0115] Examples and descriptions of circuits and devices will now be discussed with reference to the accompanying drawings. In the following text, the terms "resonant cat qubit circuit," "circuit," and "superconducting circuit" are used interchangeably and all refer to a circuit that performs 2N-to-1 photon swapping to allow the cat qubit to remain stable when the second mode is properly driven and exhibits dissipative characteristics.

[0116] Figure 1 An example of a circuit device containing a resonant cat qubit is shown.

[0117] For ease of explanation, N will be equal to 1 in the following text, but the same applies when N is any other integer value. The device includes a nonlinear superconducting circuit 100 that inherently performs a 2-to-1 photon exchange between a first mode a101 and a second mode b102 (represented by a round-trip single arrow 1200 and a double arrow 1100). A current source 103 is connected to the nonlinear superconducting circuit 100 via a wire. In other words, the current source 103 is directly connected to the circuit such that a predetermined current flows through at least one subset of elements of the nonlinear superconducting circuit 100. The current source 103 is configured to apply the predetermined current to the circuit 100. The current source 103 is capable of both achieving three-wave mixing interaction and adjusting the frequency matching condition 2f. a =f b The circuit components carrying the second mode 102 are coupled to the load 105 via coupler 104. This coupling makes the second mode dissipative. The device also includes an operating frequency f. b A microwave source 106 is used to drive the circuit 100. This device integrates a microwave filter 107, configured for a frequency of f. b A bandpass filter. Alternatively, filter 107 can be configured with a frequency of f. a A band-stop filter can be placed between the environment and the two modes to isolate the first mode, thereby preventing the first mode from suffering additional losses due to unnecessary coupling with the load 105.

[0118] Figure 2 An example of the coupling of a second mode of a superconducting circuit with a load is shown, as well as a filter that can be integrated into a device to allow only the second mode to couple with the load.

[0119] Figure 2 Schematic diagram a) shows the coupler 200 connected to the device via coupler 104 and filter 107, in order to... Figure 1 The second mode 102 of the circuit 100 is coupled to the load and ensures that the operating frequency is f. b The microwave source 106 drives the circuit 100, while only allowing the second mode (modeled as resistor 105) at the second resonant frequency f. b The loss at the first resonant frequency f does not allow the first mode to be at the first resonant frequency f. a Lower loss. Figure 2 At least two possibilities for coupling 200 are provided: frequency f b The bandpass filter (also shown in) Figure 2 In the schematic diagrams b) and d), the frequency is f. a Band-stop filters (also shown in) Figure 2(See schematic diagrams c) and e). Alternatively, when the circuit is fabricated using a 3D architecture, another possible solution is to use a waveguide high-pass filter to couple the second mode to the load, thanks to its high frequency selectivity.

[0120] Figure 2 The schematic diagrams b) to e) in the diagram illustrate different possibilities for coupling 200.

[0121] Schematic diagram b) shows the bandpass filter 210 at frequency f. b A capacitive coupling (201) is applied to the input port. The bandpass filter (LC oscillator) 210 is configured to operate at a frequency... At resonance, its impedance Adjust the bandpass width.

[0122] Schematic diagram c) shows the band-stop filter 220 at frequency f. a Capacitive coupling 201 is applied to the input port. The band-stop filter 220 (using an LC stub) is configured to operate at a frequency... At resonance, its impedance Adjust the band stop width.

[0123] Schematic diagram d) shows the bandpass filter 210 at frequency f b Inductive coupling 202 is applied to the input port. The bandpass filter 210 (using an LC oscillator) is configured to operate at a frequency... At resonance, its impedance Adjust the bandpass width. This configuration is very convenient because the device's microwave radiation (e.g., RF) input port can also be used for input current biasing.

[0124] Schematic diagram e) shows the band-stop filter 220 at frequency f. a An inductive coupling (202) is applied to the input port. This band-stop filter (using an LC stub) is configured to operate at a frequency... At resonance, its impedance Adjustable band width.

[0125] Figure 3 A stable example of a first-mode coherent state quantum manifold realized by 2N-to-1 photonic resonant exchange is shown.

[0126] Figure 3Schematic diagram a) illustrates a linear scheme involving single-photon drive (microwave tuning of the mode frequency α), characterized by an amplitude of ε1 and a single-photon dissipation rate of κ1. After a time interval 1 / k1, the mode state converges to a single coherent state 301 with an amplitude of α = 2ε1 / κ1. This is the dynamically stable steady state. However, this steady state is unique and cannot be encoded with information. Due to the uncertainty principle of quantum mechanics, this state is represented as an ambiguity point in the mode orthogonal space.

[0127] Figure 3 Schematic diagram b) illustrates a first mode that undergoes two-photon drive of intensity ε2 and two-photon dissipation of rate κ2, the mode having an amplitude of Furthermore, there are two stable steady states (302, 303) with opposite phases. Since there are two possible states, information can be encoded: state |0>302 is looped out with a solid circle, and state |1>303 is looped out with a dashed circle. Due to the dynamic stability of converging to the two states, this encoding is highly robust to bit-flip errors (bit-flip errors cause the system to flip between states |0> and |1>). This encoding does not correct for another error channel, namely phase-flip errors. However, additional error correction schemes can be added to handle this error separately. This stability is achieved by coupling to an additional mode and designing a 2-to-1 photon exchange between the first and second modes.

[0128] Figure 3 Schematic diagram c) illustrates the case where N=2, stabilizing four states in phase space using 4-photon drive and 4-photon dissipation. In this 4D manifold, state |0> can be encoded as a superposition of two coherent states 304 (shown by solid lines), and state |1> as another superposition of two coherent states 305 (shown by dashed lines). The remaining two degrees of freedom in the 4D manifold can be used as an error manifold for performing first-order quantum error correction. This stabilization is achieved by coupling to an additional mode and designing a 4-to-1 photon exchange between the first and second modes.

[0129] Figure 3 Schematic diagram d) illustrates the increase in the dimension of the stable manifold. The increase in dimension allows for higher-order error correction compared to lower N values. A 6-dimensional coherent manifold can be stabilized and second-order quantum error correction performed via 6-photon drive and 6-photon dissipation (i.e., N=3). This stabilization is achieved by coupling to an additional mode and designing a 6:1 photon exchange between the first and second modes.

[0130] Figure 4a and Figure 4b An example of how to determine the current intensity applied to a circuit to achieve 2N-to-1 resonant photon exchange is shown.

[0131] Figure 4a The tuning of a predetermined current applied to the superconducting circuit is illustrated. For a given value of N, the bias point is where the resonant frequencies of the first and second modes satisfy the matching condition 2Nω. a =ω b The point. In other words, the equation can have a constant g. 2N The error range is on the order of magnitude, and this constant describes the rate of resonant 2N to 1 photon exchange. The bias point, corresponding to the experimentally determined 401 position, can be achieved by changing the circuit current. Virtual 2Nω a Frequency line 402 and ω b The anti-crossing point between the spectral frequency lines 403. The parameters of the circuit components can be selected within a range where the respective frequencies of the two modes are close to the frequency matching condition.

[0132] For some circuits, there exists an optimal choice of circuit element parameters. Therefore, the bias point is the optimal bias point. At this optimal bias point, as mentioned earlier, stray even-order terms (such as Kerr terms and cross-Kerr terms) will also be canceled out.

[0133] Figure 4b The optimal bias point 404 is shown to coincide with node 405, where the DC phase at the node drops to π / 2. To achieve this optimal bias point, circuit parameters can be adjusted during manufacturing. Alternatively, a separate tuning knob can be added. This additional tuning knob, for example, can be implemented by adding a SQUID (two parallel nodes, whose frequencies are independently controlled by external current bias) in a second mode, and can serve as an additional degree of freedom to achieve frequency matching and Kerr vanishing conditions. Those skilled in the art will recognize that this is merely a matter of implementation.

[0134] Figure 5 Several examples of circuit symbols used interchangeably in this specification are shown.

[0135] Figure 5 Schematic diagram a) illustrates two example implementations of current bias: direct bias via electroplated connection and mutual inductance bias represented by transformer 503 (which generates a magnetic field to induce an internal current in the circuit). The first circuit (schematic diagram a) on the left) shows a current source 501 electrically coupled to a superconducting loop. As an example, at least one terminal of the current source is directly connected to the superconducting loop 510, which is isolated from the rest of the circuit for convenience. However, this is only a matter of implementation. As shown below, the current source can be connected in any way to apply a predetermined current. When a predetermined current I is applied… ext At this time, a phase drop will occur at node 502. This is because of part of the internal current I DCA current will flow through junction 502. The second circuit (located to the right of the first circuit) shows a current source that is inductively coupled to the superconducting loop via mutual inductance 503. This is shown as transformer 503 in the diagram. The current source generates a magnetic field, which in turn induces a current in the superconducting loop. This current is related to the phase drop at the junction.

[0136] Therefore, the current source can be implemented in any way to produce a phase drop at the node. Figure 5 Schematic diagram a) is illustrated using circuit 530 as an example, which equivalently summarizes the two circuits 510 and 520 in schematic diagram a). When a predetermined current is applied (through either a standard current source or a transformer), the magnetic flux... This refers to the effective magnetic field of the bias loop caused by a predetermined current applied to the circuit. In the following text, when the magnetic flux is integrated onto the loop surface, it can also be referred to as external magnetic flux. Inductance 505 represents the total self-inductance of the superconducting loop embedded at node 502. This notation is also used when the circuit is placed in a global magnetic field, which can be generated by a magnetic coil located outside the circuit plane with its axis perpendicular to the circuit plane. The external magnetic flux can be expressed as an angle. Where Φ0 is the magnetic flux quantum. For this application, it can be assumed that the system is in... The middle is periodic, with a period of 2π, and about Symmetric. Therefore, the analysis can be restricted to the interval. Inside.

[0137] Figure 5 Schematic diagram b) illustrates another description of a transformer as a current source. The transformer (labeled 550 here) can be two unconnected (i.e., not in direct contact) circuit branches that share mutual inductance due to their proximity. Alternatively, transformer 550 can be part of a circuit where two loops share a common conductor in their electrical connection. Transformer 550 can be effectively implemented in practice.

[0138] Figure 5 Schematic diagram c) shows an array of series nodes instead of an inductor. Unless otherwise stated, the array may contain only a single node.

[0139] Figure 5 The schematic diagram d) illustrates the Y-Δ transformation of the inductor, which can be used to change the topology of a circuit without altering its characteristics, thereby producing an alternative circuit that simplifies analysis while following the same principles as other embodiments.

[0140] This circuit can be configured to perform resonant 2N-to-1 photon exchange when a predetermined current is applied to induce a phase difference between one or more Josephson junctions. The following example illustrates how to design the energy of a Josephson junction (more broadly, a nonlinear inductive device) to perform resonant 2N-to-1 photon exchange. The presence of at least one loop containing one or more Josephson junctions contributes to the inherent resonance of the circuit. In practice, the energy of the Josephson junctions can depend on the parameters of the components carrying the first and second modes, as well as the predetermined current that induces the internal current in at least one loop. In practice, the one or more Josephson junctions included in the loop provide the mixing capability to achieve resonant 2N-to-1 photon exchange. That is, the energy of one or more Josephson junctions describes the interaction of the first and second modes generated by resonant 2N-to-1 photon exchange when a predetermined current is applied through at least two nodes of the circuit.

[0141] The following examples assume the circuit operates at the optimal bias point according to the principles described above. This helps improve the readability of the mathematical expressions provided below. However, it should be noted that the following examples also apply to cases with non-optimal bias points.

[0142] Figure 6 Another embodiment is shown, wherein the circuit's symbolic representation includes at least one loop comprising one or more Josephson junctions. The circuit 600 according to this example is also configured to perform resonant 2N-to-1 photon switching when a predetermined current is applied to generate a phase difference between the one or more Josephson junctions. The circuit 600 according to this example is specifically configured to symmetrically distinguish between a first mode and a second mode. The high symmetry of the circuit achieves improved quality of the resonant 2N-to-1 photon switching.

[0143] refer to Figure 6 The schematic diagram a) is now being discussed, along with an example of a circuit 600 having at least one loop 610. In circuit 600, at least one loop 610 may include a first Josephson junction 601, a central inductor element 603, and a second Josephson junction 602 arranged in series. The central inductor element 603 may be an inductor, a single Josephson junction, or an array of Josephson junctions. Thus, the central inductor element 603 may be arranged between the first and second Josephson junctions to form a series loop. The series arrangement may include a first inner node connecting one terminal of the first Josephson junction 601 to one terminal of the inductor element 603. The series arrangement may also include a second inner node connecting one terminal of the second Josephson junction 602 to the other terminal of the inductor element 603. The series arrangement may also include a closed node connecting the other terminal of the first Josephson junction 601 to the other terminal of the second Josephson junction 602.

[0144] The at least one loop 610 can be connected to a common ground terminal via a closed-loop connection. The circuit may also include a first capacitor 604 and a second capacitor 605. The first capacitor 604 may be connected in parallel with a first Josephson junction 601 between the common ground terminal and the first inner node of the loop. The second capacitor 605 may be connected in parallel with a second Josephson junction 603 between the common ground terminal and the second inner node of the loop.

[0145] Therefore, the superconducting quantum circuit 600 is configured to perform resonant 2N-to-1 photon exchange between the first and second modes, respectively, when a predetermined current is applied. Josephson junctions 601 and 602 are substantially identical, as are capacitors 604 and 605. Thus, the symmetry of the circuit implies that the actual modes of the system are a symmetrical superposition of two resonators 606 (as indicated by the solid arrow) and an antisymmetric superposition of two resonators 607 (as indicated by the dashed arrow). The first mode is a symmetrical superposition 606, and the second mode is an antisymmetric superposition 607. It can be noted that only the second mode contributes to the central inductor 603, which can be advantageously used to preferentially couple the environment to the second mode while isolating the first mode from the environment.

[0146] There is no optimal bias point in this circuit. In fact, because a non-zero phase drop is required across the inductor, the nodes cannot be biased simultaneously. and The maximum budget. However, this is not a problem because the optimal bias point is not ideal. In fact, at such a point, the Josephson junction acts as an open circuit, thus leaving only the parallel Josephson capacitor, which contradicts the fact that in this particular embodiment, the junction is the primary inductor element in the symmetry mode. The optimal bias point can be achieved by adding a loop. An exemplary implementation is Figure 6 The schematic diagrams a) and b) show symmetrical versions of the circuit, where the two resonators are made identical and nonlinear by replacing the inductor 605 with a loop 610. Figure 6 As shown in schematic diagrams a) and b), the coupling between two identical nonlinear resonators can be capacitive or inductive.

[0147] Here, circuit 600 consists of two identical resonators strongly coupled through a central inductor. This means that the bare detuning between the two resonators (before adding coupling) is zero, and the perturbation description of circuit 600 no longer holds. Therefore, the analysis here differs from that of circuit 600. Assuming the system is perfectly symmetric (i.e., the two nodes and two capacitors are identical), the system can be decomposed into symmetric modes (i.e., mode 1a) and antisymmetric modes (i.e., mode 2b). Based on these characteristic modes, the contribution of each node to the system's Hamiltonian can be calculated:

[0148]

[0149] The above formula can be decomposed into:

[0150]

[0151] The quadratic part of the first term gives the system The effective inductive energy at the node at the operating point. Combining the charging energy of the capacitor and the inductive energy of the center inductor, the frequency can be defined. and and phase and The zero-point fluctuation.

[0152] By expanding the second term to the desired order, we can obtain the resonant 2N-to-1 photon-exchange Hamiltonian. in

[0153] Due to the system's symmetry and the frequency matching condition that the system's bias point must satisfy, the expression can be further simplified. In fact, due to 2Nω a =ω b It can be proven therefore Thus obtain

[0154] Figure 6 Schematic diagram b) illustrates an example device for stabilizing a coherent manifold, employing circuit 600. Viewed from bottom to top, the nonlinear superconducting circuit 600 is located at the bottom, with an inductor sharing mutual inductance 608 with another inductor terminating at the top in an environment 620. Environment 620 consists of a load, a microwave source, and a DC current source. In the same manner as described above, the same input port of the device can be conveniently used to introduce DC current for external magnetic flux and microwave radiation. In practice, as previously stated, a transmission line is used to connect the circuit to the environment. To maintain the symmetry of circuit 600, this transmission line segment closest to circuit 609 is typically a differential transmission line.

[0155] Figure 6Schematic diagram c) illustrates an example of a planar superconducting pattern 650 and its equivalent circuit. The superconducting circuit shown is fabricated using a coplanar waveguide (CPW) structure, where the background (gray area) represents the superconducting metal remaining on the dielectric substrate. Each black cross represents a Josephson junction. Compared to the device in schematic diagram b), the inductor element is replaced by a single node. In this embodiment, the central superconducting loop 610 is diluted using three CPW transmission lines (referred to as stubs, i.e., 611, 612, 613) to control the degree of nonlinearity of the system. The open-circuit stubs 611 and 612 on the left and right sides provide the required capacitance and, in series, some stray inductance. At the bottom, the loop is not directly grounded but is interleaved with short-circuit stubs 613 to provide ground inductance. The circuit modes and their operating principles remain unchanged, but the increased transmission line length from the stubs can limit the degree of nonlinearity of the circuit. Transmission slot lines 609 are used to couple the mode inductance to the environment to preferentially couple to the second mode and maintain the symmetry of the circuit. The figure shows the transition structure between the CPW and the slotted line 614 to couple to an environment (and current source) with a common geometry 615 (e.g., CPW line or coaxial cable).

[0156] The design of the magnetic flux lines according to the present invention will now be discussed. Figure 6 Compared to the embodiment shown in schematic c), this design offers better properties for the stability of coherent manifolds. It should be noted that in order to stabilize coherent manifolds using a resonant cat circuit, the following is required:

[0157] - Induction of bias current in the center loop to activate nonlinear conversion;

[0158] - Coupled to a second mode or buffer mode to enable driving and dissipation; and

[0159] - Importantly, it is not coupled to the first mode or the cat qubit mode.

[0160] Compared to the embodiment proposed in application EP21306965.1, this invention aims to provide a solution that fully satisfies these requirements using a single flux line. In particular, the circuit design 650 is improved to better achieve coupling between DC and the center loop, as well as coupling between RF and buffer modes.

[0161] Figure 7 A general circuit diagram of a resonant cat qubit circuit is shown to remind... Figure 6The circuit presented here has key characteristics. Specifically, the circuit is constructed around a central loop 710, which includes two substantially equal Josephson junctions 711 and 712 and a central inductor 713. The arrangement of this loop such that the two substantially equal Josephson junctions define an axis of symmetry A. Details of the circuit portions 721 and 722 to the left and right of the central loop, and the circuit portion 723 below it, are not important to the invention, which is locally focused around the central loop, except that they are symmetric about the axis of symmetry A. These circuit portions will be omitted in the following figure. Due to the symmetry of the circuit, all modes it supports are either symmetric or antisymmetric with respect to the axis of symmetry. Symmetric modes are also called common modes, and antisymmetric modes are also called differential modes. The following section focuses on two modes of the circuit: the first mode has a general geometry and stores cat qubits; the second mode has a differential geometry and is used as a buffer mode in the context of cat qubits. The mode geometry is emphasized by showing the voltage configuration around the central loop. In the following text, common mode is represented by solid lines, differential mode by dashed lines; current is represented by thick arrows, and potential difference by thin arrows.

[0162] Figure 8a An embodiment of a quantum device is shown, which includes a symmetric resonant cat qubit circuit that has no magnetic flux lines and is therefore not connected to the environment.

[0163] like Figure 8a As shown, device 800 includes a ground plane 820 and a symmetrical resonant cat qubit circuit 840 for performing 2N-to-1 photon swapping. The gray area represents superconducting material on a dielectric substrate, patterned to form the circuit, and thus defined by the exposed substrate portion (white area). Most of the superconducting material resides within the ground plane 820 surrounding the circuit. To explain the coupling strategy, it is useful to represent the mode currents in the ground plane immediately adjacent to the central loop (circled by the corresponding reference numeral 844). Importantly, it should be noted that the common-mode current is zero at the circuit's axis of symmetry A, while the differential-mode current is not zero at the same location.

[0164] Figure 8b express Figure 8a The described configuration includes a general equivalent circuit 850 for differential mode, which helps characterize the coupling. This equivalent circuit comprises a horizontally arranged inductor 851 and capacitor 852, such that the + / - direction of the circuit polarity is symmetrical with respect to the polarity of the capacitor. Figure 8a Polarity direction matching of the intermediate differential mode.

[0165] Figure 9a A first embodiment of a quantum device according to the present invention is shown, the device comprising a symmetric resonant cat qubit circuit and a magnetic flux line.

[0166] like Figure 9aAs shown, device 900 includes an environment 910, a ground plane 920, a magnetic flux line 930, and a resonant cat qubit circuit 940 for performing 2N-to-1 photon exchange.

[0167] Environment 910 includes current sources ( Figure 9a (Not shown in the diagram) microwave source 914 and load 916. In this embodiment, the current source is not shown because the focus of this example is the radio frequency characteristics of the first mode (common mode or cat qubit mode) and the second mode (differential mode or buffer mode).

[0168] As shown by the triangle at the connection point, the ground plane 920 is connected to the common ground terminal. By exposing the dielectric substrate portion where the common ground terminal 920 is located, a magnetic flux line 930 is formed on the common ground terminal 920.

[0169] The flux line 930 includes a coplanar waveguide (CPW) portion 932, a first slot 934, and a second slot 936. More precisely, the CPW portion 932 includes slots 9320 and 9322 that define a centerline, which is further connected to a current source, a microwave source 914, and a load 916. The CPW portion 932 contains multiple wire bonds to ensure that the ground plane has the same potential on both sides of the CPW portion 932. For simplicity, in Figure 9a as well as Figures 10 to 12 In the diagram, the only wire bond represented is wire bond 938, which is the wire bond closest to the resonant cat qubit circuit 940 and separates the CPW portion 932 of the flux line 930, the first slot 934, and the second slot 936.

[0170] In the example shown here, the resonant cat qubit circuit 940 and Figure 6 The circuit shown in schematic a) is similar, comprising two Josephson junctions 941 and 942 and a central inductor element 943 (which is an inductor). In other embodiments, the central inductor element 943 may be a Josephson junction or an array of Josephson junctions.

[0171] As mentioned above, the main focus of this figure is to illustrate the common mode and differential mode arising in this embodiment. To better explain these modes, the following conventions are used:

[0172] - Solid arrows indicate common mode.

[0173] - The dashed arrow indicates the differential mode.

[0174] - The thick arrow indicates current circulation.

[0175] - The thin arrow indicates the potential difference.

[0176] therefore, Figure 9aThe arrows (whether solid or dashed, thick or thin) indicate the paths of common-mode and differential-mode currents, respectively. For example, Figure 9a The differential-mode current flows from the environment to the common-mode waveguide (CPW) section 932, and the return current flows along the first slot 934.

[0177] like Figure 9a As shown, the second slot 936 is much shorter than the first slot 934, which connects the gap between the resonant cat qubit circuit 940 and the rest of the ground plane 920. In other words, the first slot 934 divides the ground plane 920 into two unconnected parts 9210 and 9220.

[0178] In this embodiment, the second slot 936 can be omitted. The two slots extend substantially parallel to the A-axis. The A-axis forms the axis of symmetry between the resonant cat qubit circuit 940 and the first slot 934. The symmetry of the resonant cat qubit circuit 940 is significant because the common-mode current flows in opposite directions in portions 9210 and 9220 of the ground plane, resulting in zero common-mode current at the A-axis, while the differential-mode current is not zero, inducing coupling with the differential mode. This causes the microwave radiation emitted by the environment 910 to couple almost exclusively with the differential mode and not with the common mode, which is crucial for the stability of the cat qubit. "Almost exclusively coupled" or "substantially coupled" should be understood as the differential-mode (and correspondingly common-mode) coupling being much greater than the common-mode (and correspondingly differential-mode) coupling. Preferably, this ratio is greater than 100. Notably, slot 934 extends to the center loop of the circuit and completely cuts the ground plane in half, thereby enabling strong coupling with the differential mode. This is consistent with... Figure 6 The implementation in schematic c) presents a stark contrast (where the slot does not completely cut through the ground plane). In the prior art, the cut is not fully extended, creating a residual current path between the two halves of the ground plane, which facilitates DC current coupling but at least partially shunts differential-mode RF coupling. Figure 2 As shown, the coupling rate can be further improved by using an interleaved microwave filter (which allows DC and microwave radiation at differential-mode frequencies to pass through).

[0179] Figure 9b The diagram shows the equivalent circuit of the differential mode of the resonant cat qubit circuit 940, where the device operates to ensure 2N to 1 photon exchange, and the microwave source (and corresponding load) and the differential mode are substantially coupled only to this differential mode. In this case, the presence of magnetic flux lines will affect... Figure 8b The equivalent circuit. Specifically, the cut caused by slot 934 is represented by capacitor 952, and the remaining inductance path from the left ground plane 9210 to the right ground plane 9220 (which extends from the left ground plane to the right ground plane via wire bonding 938) is represented by inductor 950. Figure 8b In the circuit, inductor 851 is now split in two, 954 and 956, while main capacitor 958 remains largely unchanged. This equivalent circuit demonstrates that the coupling with the environment can be adjusted by choosing the location of wire bond 938. In fact, by placing the wire bond away from the resonant cat qubit circuit, the impedance of the reactive path (composed of inductor 950 and capacitor 952) increases, thus the differential-mode current tends to flow more through the resistor, i.e., the coupling with the environment increases.

[0180] The applicant’s tests show that this embodiment is very effective in coupling the microwave source and the load to differential mode rather than common mode, but at the cost of less than ideal coupling of the current source’s current (hereinafter referred to as DC coupling) to provide the intended current.

[0181] Intuitively, the applicant believes that while placing the first slot 934 at the center of the A-axis allows for largely exclusive coupling of the microwave source 914 and load 916 with the differential mode, the second slot 936 must be very short (if not nonexistent) to respect the symmetry of the device, which is why the DC coupling is less than satisfactory.

[0182] Therefore, the applicant prepared Figure 10 The second embodiment shown, wherein with Figure 9a Similar components share the same reference number, except for the first two digits. Figure 9a Wire bonding 938 in Figure 10 The reference number is 1038. This also applies to... Figure 11 and Figure 12 .

[0183] For simplicity, only descriptions of... Figure 9a The difference. Figure 11 and Figure 12 The same applies. Only current source 1012 is shown in the diagram because the focus of the diagram is to show DC coupling.

[0184] In this embodiment, the only difference is that the second slot 1036 is almost as long as the first slot 1034, except that the second slot 1036 does not extend all the way to the exposed dielectric substrate portion, which corresponds to the boundary of the portion of the resonant cat qubit circuit 1040 (which includes the central inductor element 1043). For example, the distance between the end of the second slot 1036 and the aforementioned exposed dielectric substrate portion can be as low as 1 μm or as high as tens of μm. In this embodiment, as shown by the thick dashed line, DC current flows from the coplanar waveguide portion 1032 along the first slot 1034 and the second slot 1036, and spreads out in the ground plane portion 10220, spreading out in a direction generally perpendicular to the A-axis and away from the first slot 1034 (hereinafter referred to as "direction").

[0185] The applicant's performance analysis of this embodiment shows that DC coupling is significantly improved, but at the cost of decreased differential-mode coupling performance and increased common-mode coupling, which is undesirable. The applicant found that this is because the second slot 1036 becomes correlated with the resonant cat qubit circuit 1040, thereby breaking the device symmetry. Incidentally, it should be noted that the second slot 1036 can be relative to... Figure 10 A-axis mirroring implementation.

[0186] To overcome the newly introduced defects, the applicant proposed shifting the magnetic flux lines along a direction perpendicular to the A-axis to re-establish common-mode decoupling.

[0187] This effort ultimately yielded results. Figure 11 The illustrated embodiment shows that, according to this embodiment, the first slot 1134 and the second slot 1136 are offset in a direction opposite to the turning direction. This offset can be as small as 1 μm and as large as the width of the device, meaning that the first slot 1134 and the second slot 1136 will be located at the end of the device.

[0188] The applicant's analysis of the performance of this embodiment shows that common-mode coupling exhibits an optimal point where coupling with the environment disappears while differential-mode coupling is retained, but differential-mode coupling can be improved.

[0189] To overcome the newly introduced defects, the applicant proposed adding a return section generally along the turning direction on the second slot 1236.

[0190] This effort ultimately led to Figure 12 The illustrated embodiment. According to this embodiment, the first slot 1234 and the second slot 1236 are still axially offset, but the second slot 1236 supplements the return portion 12360. This return portion can be as small as 1 μm or as large as the device width, i.e., extending to the end of the device.

[0191] The applicant's performance analysis of this embodiment shows that it is the best performing embodiment, with high DC coupling and differential-mode coupling, while the common-mode coupling is very low.

[0192] The applicant also found that the return section can be used with the second and third embodiments, but some improvements are needed.

[0193] Figure 13 and Figure 14 Figure 9 and 10 respectively show the results. Figure 12 Images of the manufacturing device corresponding to the embodiments are shown. In each image, the enlarged portion better illustrates how the first slot divides the ground plane, and how the first and second slots are positioned relative to the central hybrid component.

[0194] The tests conducted by the applicant showed that Figure 13The circuit shown can achieve differential κ up to 100MHz. b / 2π coupling target. However, DC coupling with the central loop can be improved because, depending on the precise wire bonding configuration, the coupling amount is approximately 0.1 flux quantum (Φ0) per mA current. Figure 14 The circuit shown achieves this, and it also achieves differential-mode targeting coupling at frequencies up to 50 MHz k. b / 2π, and the DC coupling with the central loop is approximately 1 flux quantum per mA of current, which is a typical target parameter in this field.

[0195] The measurement method according to the present invention will now be discussed.

[0196] Cat qubits belong to the family of boson qubits encoded in harmonic oscillators, also known as cat qubit modes. Unlike two-level systems, harmonic oscillators have an infinite number of energy levels, which can be used to encode information.

[0197] Tomography is an operation that provides a comprehensive understanding of the state of a quantum system. This operation requires measuring various observables. For a two-level system, measuring the observables X, Y, and Z is sufficient to fully characterize the system's state. For a harmonic oscillator with an infinite number of energy levels, assumptions must be made about the system to perform a tomographic scan with a finite number of measurements. Typically, it is assumed that the system resides in a low-energy Fock state subspace. Using a harmonic oscillator in a superconducting circuit, various forms of measurement can be experimentally performed, such as the Husimi-Q function or the Wigner function and their corresponding characteristic functions. These functions are defined on the phase space of the oscillator, and their orthogonal functions are called I and Q. These functions can be directly measured by sampling a finite portion of the phase space. Based on this finite sampling and the assumption that the system has a low-energy subspace, the system state can be reconstructed using a maximum likelihood algorithm, thus completing the tomographic scan.

[0198] More specific assumptions can be made about the subspace in which the system resides. For example, in the cat qubit paradigm, the stability of the coherent state manifold restricts the possible states to a range of 2N coherent states. Therefore, under this assumption, fewer measurements are required to perform a system tomography scan. For instance, for a two-component cat qubit defined over two coherent states {|α>,|-α>}, the state can be measured using the full Wigner function, or the effective X, Y, or Z can be measured as in a two-level physical system. The former is primarily used for tuning and characterizing the operation of superconducting circuits, while the latter is used for computation. In certain cases, the measurement of X is also the parity of the photon number in that state, while Z is a measure of whether the particle swarm is in one coherent state or another. Another example is in the so-called four-component cat qubit paradigm, where qubits are defined in an even-numbered manifold with four coherent state spans {|α>,|iα>,|-α>,|-iα>}}. Clearly, measuring the full Wigner function of the system is more comprehensive than measuring some observables while making assumptions about the possible states. Therefore, the average of these observables can usually be reconstructed using the full Wigner function.

[0199] In the context of cat qubits, the Wigner function or its characteristic function is generally easier to use. In fact, while the Husimi-Q function also contains all information in principle, it is more sensitive to noise in the context of cat qubits. The Wigner function at a point β in phase space can be determined by measuring its parity after shifting the field by a certain amount -β (as described in the article "Tracing Photon Transitions by Repeated Quantum Nondestructive Parity Measurements" by Sun, L., Petrenko, A., Leghtas, Z. et al., published in Nature, Vol. 511, pp. 444–448 (2014, https: / / doi.org / 10.1038 / nature13436)). By measuring the average value of the shift operator D(-β), the characteristic function at a point in phase space can be determined (as described in the article "Quantum Error Correction of Oscillator-Grid-Encoded Qubits" by Campagne-Ibarcq, P., Eickbusch, A., and Touzard, S., et al., published in Nature, Vol. 584, pp. 368–372 (2020, https: / / doi.org / 10.1038 / s41586-020-2603-3). The Wigner function will be used below because it contains more information about the cat qubit. For example, in a two-component cat qubit, the Wigner function is the same for measurement X as for measurement 0. However, it should be noted that the characteristic function of either the Husimi-Q function or the Wigner function can also be used.

[0200] To measure the Wigner function, the system's state must first be displaced, and then the parity must be measured. The displaced state is equivalent to transmitting a finite-duration pulse with a frequency close to the mode frequency, placing the mode frequency within the pulse's spectrum. This pulse is generated by a microwave source connected to the mode via a transmission line coupled to the mode. This coupling can be capacitive, inductive, or current-dependent. The amplitude and phase of this pulse determine the amplitude and phase of the displacement. Parity can be measured indirectly by directly coupling the mode to the two-stage system and mapping the parity of the fields in the mode to the states of the two-stage system. This mapping can be accomplished by implementing a Hamiltonian that maps the mode... The photon number operator is coupled to the Z or X operator of the two-level system. or The former can be achieved through the so-called dispersive interaction (Sun, L., Petrenko, A., Leghtas, Z. et al., “Tracking photon transitions by repeated quantum non-removing parity measurements”, Nature, Vol. 511, pp. 444–448 (2014, https: / / doi.org / 10.1038 / nature13436), while the latter can be achieved through the so-called longitudinal interaction (see, for example, S. Touzard, A. Kou, N.E. Frattini et al., “Gated displacement readout of superconducting qubits”, Physical Review Letters, Vol. 122, No. 080502), which can be activated by a parametric pump at the frequency of a two-stage system. The state of the two-stage system will rotate about the Z-axis (correspondingly the X-axis) at a speed depending on the number of photons in the mode. By adjusting the duration of the interaction to... This ensures that for an even number of photons in the oscillator, the two-level system accumulates an integer number of rotations, while for an odd number of photons, it accumulates a half-integer number of rotations. In the case of dispersive interaction, if the two-level system starts in state |+>, it will eventually be in state |+> when there are an even number of photons in the mode, and in state |-> when there are an odd number of photons. By measuring X, it can be determined whether the qubit is in state |+> or |->, thus determining the photon number parity. In the case of longitudinal interaction, if the two-level system starts in state |0>, it will eventually be in state |0> when there are an even number of photons in the mode, and in state |1> when there are an odd number of photons. By measuring Z, it can be determined whether the qubit is in state |0> or |1>, thus determining the photon number parity.

[0201] Unfortunately, the fundamental interaction required to perform Wigner tomography, namely the interaction with the photon number operator... The coupling is incompatible with the previously described coherent state stability. It depends on... or The same applies to the characteristic function of the coupling. This is no coincidence: it is actually the expected effect of the stabilization mechanism, which aims to suppress spurious coupling. Therefore, the Wigner function cannot be measured when the cat qubit is stably turned on.

[0202] There are indeed some measurements that are compatible with stability. For example, the system can be determined to be in a stable coherent state by coupling the mode to a heterodyne detector (or to a homodyne detector if there are only two coherent states).

[0203] For a two-component cat qubit, this corresponds to a measurement of Z. However, although it is theoretically possible to measure X and Y of a two-component cat state in a steady state, measuring X and Y requires designing a nonlocal Hamiltonian, which is currently only theoretical. As mentioned earlier, the experimental measurement of X reduces to a parity measurement, which depends on the interaction protected by the steady state, much like in Wigner tomography.

[0204] For a four-component cat qubit, measuring which stable coherent state the system is in does not measure the observables of the qubit, nor does it project the system outside the code space.

[0205] Therefore, only a small number of measurements can be performed in a steady state, which is insufficient for calibration or quantum computing.

[0206] In existing technologies, for stable two-component cat qubits that rely on parametric pumps, this problem can be solved by simply turning off the parametric pumps, which enable 2-to-1 photon exchange, a key factor for stability. However, this solution is unusable for the aforementioned resonant cat qubit circuit due to the inherent resonant characteristics of the stabilization mechanism, as it does not require parametric pumps. Since the frequency of the buffer mode is tuned to twice that of the cat qubit mode, energy conservation is built-in. Therefore, the 2-to-1 photon exchange dynamics are always on, a principle that also holds true for the more general 2N-to-1 photon conversion case.

[0207] Compared to previously proposed implementations, the resonant cat qubit device contains a magnetic flux line that is well coupled to the buffer (or differential mode or second mode) under both DC and RF conditions, which provides new possibilities for readout. Specifically, this magnetic flux line can be rapidly (within tens of nanoseconds) adjusted with a predetermined current. By suddenly adjusting the predetermined current value, the resonant frequencies of both the first and second modes change, causing the system to deviate from the resonant frequency matching condition, which is a factor in achieving 2N to 1 photon conversion. If ω′ a / 2π and ω′ b / 2π represents the new frequency of the mode, expressed in Δ. a =ω′ a -ω aand Δ b =ω′ b -ω b This indicates that under the condition of resonant frequency matching, the frequency ω a / 2π and ω b A detuning of / 2π will shut down the 2N-to-1 photon conversion, provided that |2Δ a -Δ b |>g 2N , where g 2N This refers to the intensity of the nonlinear conversion. This is typically achieved by adjusting the magnetic flux within the central loop of the device by a small fraction of the magnetic flux quanta (typical units of magnetic flux known in the art). Due to the DC coupling of the flux lines, this can be achieved by adjusting the output current of the current source to less than 1 mA. Hereinafter, we refer to this new predetermined current value as the tomographic bias point, rather than the stable bias point effective for 2N to 1 photon conversion. The current can be adjusted in any suitable manner, such as by applying a square wave pulse, a ramp, or any other signal correction shape, to address signal distortion between the current source and the device.

[0208] Another approach involves exploiting the fact that the buffered modes in some embodiments exhibit a degree of Kerr nonlinearity to detune the mode's frequency from the resonant frequency matching condition. These embodiments are preferably those where the central inductor is a single Josephson junction. In this case, a sudden, parametric pump is applied to the buffered mode via a flux line or another RF coupling line, at a frequency different from any nonlinear transition in the device. This causes an AC-Stark shift in the buffer frequency, detuning the effective resonant frequency of the buffered mode (and to a lesser extent, the cat qubit mode) from the frequency that satisfies the frequency matching condition. This solution does not require rapid adjustment of the DC current but has been shown to have some adverse effects on the device, such as reducing the coherence time of modes used for tomography and two-level systems, or modes that typically heat the device.

[0209] Once the system becomes detuned from the resonant frequency matching condition, any quantum measurement can be performed on the cat qubit mode because coherent state stability is no longer valid, and the dynamic characteristics of the cat qubit mode are in a free state. Extra care must be taken when performing measurements because the cat qubit mode frequency at the tomographic bias point is now different from the frequency at the stable bias point. However, the accumulated phase of the cat qubit state due to the frequency difference between the two points can be calculated and compensated for when manipulating the cat qubit.

[0210] In a preferred embodiment, in addition to abruptly adjusting the predetermined current to shut down the 2N-to-1 photon conversion, the drive of the buffer mode is also turned off because it no longer serves any purpose.

[0211] If the quantum measurement is non-destructive (QND), stability can be advantageously restored by restoring the output of the current source to a stable DC bias point. It should also be restored if the buffer mode drive is turned off. Similarly, the accumulated phase due to the cat qubit mode detuning between the two bias points should be calculated and compensated.

Claims

1. A quantum device, comprising: - A nonlinear superconducting quantum circuit, the nonlinear superconducting quantum circuit having a first mode and a second mode and a central hybrid component (840, 940, 1040, 1140, 1240), the first mode and the second mode each having a corresponding resonant frequency, the central hybrid component (840, 940, 1040, 1140, 1240) comprising: At least one loop (710) comprising a first Josephson junction (841, 941, 1041, 1141, 1241), a central inductor (843, 943, 1043, 1143, 1243), and a second Josephson junction (842, 942, 1042, 1142, 1242) arranged in a series topology generally symmetrical about an axis (A), the axis (A) mapping the first Josephson junction (841, 941, 1041, 1141, 1241) to the second Josephson junction (842, 942, 1042, 1142, 1242), the at least one loop (710) comprising connecting ... A first inner node connecting one pole of a Josephson junction (841, 941, 1041, 1141, 1241) to one pole of the central inductor (843, 943, 1043, 1143, 1243), a second inner node connecting one pole of a second Josephson junction (842, 942, 1042, 1142, 1242) to the other pole of the central inductor (843, 943, 1043, 1143, 1243), and a closed node connecting the other pole of the first Josephson junction (841, 941, 1041, 1141, 1241) to the other pole of the second Josephson junction (842, 942, 1042, 1142, 1242). A first circuit portion (721) is connected between the common ground terminal and the first inner node of the loop (710), a second circuit portion (722) is connected between the common ground terminal and the second inner node of the loop (710), such that the first circuit portion (721) and the second circuit portion (722) are generally symmetrical about the axis (A), and a third circuit portion (723) is connected between the closed node and the common ground terminal and is generally symmetrical about the axis (A). The nonlinear superconducting quantum circuit is configured such that when a predetermined current of constant intensity is applied, the resonant frequency of the second mode is approximately 2N times the resonant frequency of the first mode, generating a phase difference between one or more Josephson junctions (841, 941, 1041, 1141, 1241; 842, 942, 1042, 1142, 1242), thereby enabling the nonlinear superconducting quantum circuit to be expanded into at least one form of The Hamiltonian of the sum of the dominant term and a series of auxiliary terms, where It is a scalar corresponding to the intrinsic coupling strength. It is the annihilation operator of mode one. It is the annihilation operator of mode two. It is the reduced Planck constant, thereby inherently performing a resonant 2N-to-1 photon exchange between the first mode and the second mode, where N is a positive integer. The nonlinear superconducting quantum circuit is located on a dielectric substrate and is separated from the common ground plane (820, 920, 1020, 1120, 1220) by the exposed portion of the dielectric substrate. - Magnetic flux lines (930, 1030, 1130, 1230), said magnetic flux lines being positioned relative to a portion of said at least one loop (710) containing said central inductor element (843, 943, 1043, 1143, 1243), said magnetic flux lines including a first slot (934, 1034, 1134, 1234) and a coplanar waveguide portion (932, 1032, 1132, 1232). The coplanar waveguide portions (932, 1032, 1132, 1232) are arranged to be coupled to a current source (1012), a microwave source (914, 1114, 1214) configured to apply microwave radiation, and a load (916, 1116, 1216), and are defined by end lead bonding (938, 1038, 1138, 1238). The first slot (934, 1034, 1134, 1234) is patterned in the ground plane to expose the dielectric substrate, extending generally parallel to the axis (A). One end of the first slot (934, 1034, 1134, 1234) is connected at the height of the end wire bond (938, 1038, 1138, 1238) to the slot (9320, 10320, 11320, 12320) of the coplanar waveguide portion (932, 1032, 1132, 1232). The other end of the first slot (934, 1034, 1134, 1234) is connected to the exposed dielectric substrate portion, which defines the at least one loop (710) containing the central inductor element (843, 943, 1043, 1143, 1243). The magnetic flux lines (930, 1030, 1130, 1230) are configured to allow the current source (1012) to couple to induce the predetermined current in at least one loop (710), allow the microwave source (914, 1114, 1214) to couple to induce the drive of the second mode at a frequency substantially equal to the resonant frequency of the second mode or 2N times the resonant frequency of the first mode, and allow the load (916, 1116, 1216) to be coupled substantially only to the second mode.

2. The quantum device according to claim 1, wherein, The magnetic flux lines (930, 1030, 1130, 1230) further include second slots (936, 1036, 1136, 1236), which are patterned in the ground plane to expose the dielectric substrate and extend generally parallel to the axis (A). One end of the second slot (936, 1036, 1136, 1236) is connected to the end lead bonding (938, 1038, 1138, 1238) at the height of the end lead bonding. Another slot (9322, 10322, 11322, 12322) of the coplanar waveguide portion (932, 1032, 1132, 12322), and the other end of the second slot (936, 1036, 1136, 1236) terminates within the ground plane (820, 920, 1020, 1120, 1220) at a distance greater than 1 µm from the portion of the at least one loop (710) containing the central inductor element (843, 943, 1043, 1143, 1243).

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

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

5. The quantum device according to any one of claims 1 to 4, wherein, The first slots (934, 1034, 1134, 1234) are generally symmetrical about the axis (A).

6. The quantum device according to claim 3 or 4, wherein, The first slot (934, 1034, 1134, 1234) and the second slot (936, 1036, 1136, 1236) are both offset in a direction perpendicular to the axis (A) and opposite to the direction of rotation.

7. The quantum device according to claim 6, wherein, The offset is greater than 1 μm and can reach up to the width of a nonlinear superconducting quantum circuit.

8. The quantum device according to any one of claims 1 to 4, wherein, The length of the first slot (934, 1034, 1134, 1234) or the position of the end wire bond (938, 1038, 1138, 1238) is selected according to the coupling strength between the second mode and the load (916, 1116, 1216).

9. The quantum device according to any one of claims 1 to 4, wherein, The coupling strength between the load and the second mode is more than 100 times higher than the coupling strength between the load and the first mode.

10. The quantum device according to any one of claims 1 to 4, further comprising a current source (1012), a microwave source (914, 1114, 1214), and a load (916, 1116, 1216) coupled to the coplanar waveguide portions (932, 1032, 1132, 1232), configured to stabilize the coherent manifold in the quantum device.

11. A quantum measurement method for the quantum device according to claim 10, comprising: a) Modify the current output of the current source so that the predetermined current in the nonlinear superconducting quantum circuit induces a resonant frequency of the second mode that is significantly different from the resonant frequency of the first mode by 2N times. b) Perform quantum measurement on the first mode of the nonlinear superconducting quantum circuit.

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

13. The quantum measurement method according to claim 11 or 12, wherein, Operation b) is a quantum non-destructive measurement, and the quantum measurement method further includes the following operations: c) After performing operation b), restore the current output of the current source to obtain the predetermined current.

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

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