How to calibrate a quantum system to stabilize a cat qubit

Abstract

A method for calibrating a quantum system for stabilizing a cat qubit, wherein the quantum system (1) has a first resonant frequency (ω) for hosting the cat qubit. a A first mode (a) having ) and a second resonant frequency (ω b The quantum system (1) includes a nonlinear superconducting quantum circuit (3) having a second mode (b) having ω, and the quantum system (1) includes a nonlinear element (7) arranged to induce a 2:1 photon exchange between the first mode (a) and the second mode (b) either via a parametric pump and a 4-wave mixing coupling the first mode (a) and the second mode (b), or via a DC bias and a 3-wave mixing coupling the first mode (a) and the second mode (b), and an instruction circuit (5) for delivering microwave radiation or a current bias to the nonlinear superconducting quantum circuit (3) to provide the parametric pump or the DC bias and the first mode drive, wherein the method is: an operation to acquire a set of instruction circuit operation parameters, each instruction circuit operation parameter being such that if the nonlinear element (7) performs 4-wave mixing, the parametric pump frequency (ω) p ) and the first mode drive frequency (ω z ) and a couple that associates, or if the nonlinear element (7) performs 3-wave mixing, the DC bias value (I DC ) and the first mode drive frequency (ω z A method comprising: an operation to acquire a set of instruction circuit operation parameters that constitutes one of a couple relating to ); an operation to apply the set of instruction circuit operation parameters to the instruction circuit (5) for a period of more than 10 ns; an operation to perform at least one measurement in a group including a first mode photon number measurement and a second mode electromagnetic field measurement; an operation to repeat operations a) to c) until calibration conditions are satisfied; and an operation to derive a calibrated set of instruction circuit operation parameters based on the measurement results acquired in operation c).

Description

[Technical Field]

[0001] This invention relates to a method for calibrating a quantum system to stabilize cat qubits. [Background technology]

[0002] State-of-the-art technology has shown that it is possible to stabilize cat qubits (i.e., boson qubits defined by a quantum manifold spanned by the superposition of two coherent states), which are quasi-classical states of boson modes. To achieve this objective, a specific dissipative stabilization mechanism is required, which essentially involves cleverly designing a nonlinear conversion between two photons of a first mode (also known as the memory mode or cat qubit mode) that hosts the stabilized quantum manifold and one photon of a strongly dissipative second mode (also known as the buffer mode). This process is activated by bringing the system close to a 2:1 photon conversion resonance thanks to a parametric pump or DC bias.

[0003] The possibility of realizing such a qubit was demonstrated by R. Lescanne et al. in “Exponential suppression of bit-flips in a qubit encoded in an oscillator” (Nature Physics, 2020). As mentioned above, CAT qubit stabilization relies on a mechanism that dissipates photons in pairs. This process binds two coherent states to separate locations in phase space. By increasing this separation, an exponential decrease in the bit flip rate was experimentally observed while the phase flip rate simply increased linearly. As a result, it was demonstrated that stabilized CAT qubits benefit from high-noise bias: this means that the bit flip probability is exponentially smaller than the phase flip probability.

[0004] Transmon qubits (or transmons) are commonly used as measuring devices in which a transmon qubit (or transmon) is coupled to a cat qubit mode for the purpose of determining the state of a cat qubit. For example, in 2020, Lescanne et al. proposed parametrically pumping an asymmetrically threaded superconducting quantum interference device (ATS) to cleverly design a 2:1 photon conversion related to transmons for measuring cat qubit states.

[0005] Therefore, initially, we might think that "when designing a cat qubit circuit, the frequencies required to drive the buffer mode and the parametric pump or DC bias need to be known." However, due to the finite precision of circuit simulation and fabrication, these parameters need to be calibrated in situ when the circuit is in its final low-temperature environment.

[0006] Traditionally, this calibration is performed through a complex process involving the use of transmons to read out memory modes. More precisely, this process involves implementing various operating points for the cat qubit (i.e., driving buffers and parametric pumps or DC biases) and using transmons to perform Wigner tomography of the memory mode state to detect whether it is a cat qubit state and whether it is correct.

[0007] This process, which is cumbersome as a Wigner tomography technique, has not only taken a very long time, but it has also been demonstrated that transmons have a detrimental effect on cat qubits. In fact, as pointed out by Lescanne et al. in 2020, the presence of transmons results in bit flip times that saturate in milliseconds. The patent applicant's research has revealed that "this is due to the quantum manifold stabilization rate being too small to resist the dispersive frequency shift induced by the thermal excitation of the transmon." Indeed, the transmon has a spria-crossed Kerr term that induces an additional noise process with an escape rate from the quantum manifold given by a large transmon cat qubit dispersive shift.

[0008] Furthermore, these challenges with transmons were already observed in the very first implementations of the stabilization scheme described above (Z. Legtas et al. (2015) in the article “Confining the state of light to a quantum manifold by engineered two-photon loss” (Science, Vol.347, No.6224) and S. Touzard et al. (2018) in the article “Coherent Oscillations inside a Quantum Manifold Stabilized by Dissipation” (Physical Review X 8, 023005)), where the superconducting circuit element used as the four-wave mixer was a transmon with a single Josephson junction, and no exponential bit flip suppression was observed. [Prior art documents] [Non-patent literature]

[0009] [Non-Patent Document 1] R.Lescanne et al. “Exponential suppression of bit-flips in a qubit encoded in an oscillator”(Nature Physics,2020) [Non-Patent Document 2] Z.Leghtas et al.(2015)in the article “Confining the state of light to a quantum manifold by engineered two-photon loss”(Science,Vol.347,No.6224) [Non-Patent Document 3] S.Touzard et al.(2018)in the article “Coherent Oscillations inside a Quantum Manifold Stabilized by Dissipation”(Physical Review X 8,023005) [Overview of the Initiative] [Problems that the invention aims to solve]

[0010] Therefore, there is a need to provide a method for calibrating a system to stabilize cat qubits that do not require the use of transmons. Furthermore, there is a need for a faster calibration method as part of the startup sequence of a superconducting processor containing hundreds of cat qubit systems. [Means for solving the problem]

[0011] The present invention aims to improve this situation. To achieve this objective, the applicant proposes a method for calibrating a quantum system for stabilizing a cat qubit. The quantum system includes a nonlinear superconducting quantum circuit having a first mode having a first resonant frequency and a second mode having a second resonant frequency for hosting the cat qubit, and a nonlinear element arranged to induce a 2:1 photon exchange between the first mode and the second mode either via a parametric pump and a four-wave mix coupling the first mode and the second mode, or via a DC bias and a three-wave mix coupling the first mode and the second mode, and an instruction circuit for delivering microwave radiation or a current bias to the nonlinear superconducting quantum circuit to provide the parametric pump or the DC bias and the first mode drive, the method being: a) An operation to acquire a set of instruction circuit operating parameters, each set consisting of either a couple relating the parametric pump frequency to the first mode drive frequency if the nonlinear element performs four-wave mixing, or a couple relating the DC bias value to the first mode drive frequency if the nonlinear element performs three-wave mixing. b) An operation to apply the set of instruction circuit operation parameters to the instruction circuit for a period of more than 10 ns. c) An operation to perform at least one measurement within a group, including a first mode photon number measurement and a second mode electromagnetic field measurement. d) Repeat operations a) to c) until the calibration conditions are satisfied. e) This includes deriving the instruction circuit operation parameters of the calibrated set based on the measurement results obtained in operation c).

[0012] This method is advantageous because it can be performed without involving the use of transmons. Furthermore, depending on the measurement being performed, significant time savings can be achieved through the aforementioned Wigner-based measurement.

[0013] In various embodiments, this method may present one or more of the following features: -The repetition of operation a) within operation d) includes performing a two-dimensional sweep of the parametric pump frequency and the first mode drive frequency or the DC bias value and the first mode drive frequency. - The calibration condition for operation d) is satisfied when a two-dimensional sweep is performed. - Operation e) includes determining the saddle point in the measurement of operation c), - Operation a) within operation d) includes performing a saddle point search algorithm on the parametric pump frequency and the first mode drive frequency or on the DC current value and the first mode drive frequency based on the measurement in operation c), - The calibration condition for operation d) is satisfied when the saddle point search algorithm converges to exceed the selection threshold. - Operation e) involves returning a set of instruction circuit operation parameters for the final iteration of step a), and - Operations b) and c) overlap.

[0014] Other features and advantages of the present invention will readily appear in the following description of the accompanying drawings illustrating exemplary embodiments of the invention. [Brief explanation of the drawing]

[0015] [Figure 1] A schematic diagram of the quantum system arranged to measure the observable variables of the cat qubit according to the present invention is shown. [Figure 2] Figure 1 shows one embodiment of the quantum system, where the cat qubit is stabilized using parametric dissipative stabilization. [Figure 3] Figure 2 shows a partial electrical equivalent diagram of an exemplary implementation of a nonlinear superconducting quantum circuit of the quantum system, with only the ATS and two resonant regions shown. [Figure 4] Figure 2 shows a partial electrical equivalent diagram of another exemplary implementation of the nonlinear superconducting quantum circuit of the quantum system, with only the ATS and two resonant regions shown. [Figure 5] Figure 2 partially shows an exemplary implementation of the quantum system, and a portion of the external environment is shown. [Figure 6] Another embodiment of the quantum system shown in Figure 1 is partially illustrated, where the cat qubit is stabilized by using resonant dissipative stabilization. [Figure 7] Figure 6 partially shows an exemplary implementation of the quantum system. [Figure 8] This shows the memory and buffer state evolution in a standard calibration using buffer-driven methods. [Figure 9] This specification demonstrates the evolution of memory and buffer states using a new calibration method proposed herein that utilizes memory drive. [Figure 10] A general diagram of a method for calibrating a quantum system to stabilize a cat qubit is shown. [Figure 11] This is a 2D color map representing simulation measurement values ​​(average number of photons in memory and magnitude of the buffer electromagnetic field) in a steady state to indicate the point of application conditions. [Figure 12] This is a 2D color map similar to Figure 11, corresponding to experimental results illustrating methods for calibrating quantum systems or stabilizing chat qubits. [Figure 13] Figure 12 shows the Wigner measurement results of the cat qubit calibrated by the saddle point determined on the figure. [Modes for carrying out the invention]

[0016] The accompanying drawings and the following description include most of the clearly and well-defined features. As a result, these may be useful not only in understanding the invention but also in contributing to its definition if necessary.

[0017] A. Cat Qubit Quantum system 1 is deployed to stabilize the cat qubit.

[0018] To date, the applicant's research has generally concerned the stabilization of cat qubits. As previously explained, R. Lescanne et al. (2020) demonstrated that such cat qubits can be stabilized by a nonlinear transformation between two photons in a first mode a (memory mode or cat qubit mode) and one photon in a second mode b (buffer mode).

[0019] A cat qubit is a so-called cat state, which is a superposition of two coherent states |α> and ├α>.

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[0020] Stabilized CAT qubits are known to benefit from high-noise bias: this means that the bit flip probability is exponentially smaller than the phase flip probability. More precisely, the effective error channel (e.g., bit error or "bit flip") is the "size" of the Schroedinger CAT state of the CAT qubit (i.e., the average number of photons).

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[0021] According to current knowledge, this suppression should be applied to a large class of physical noise processes that have local effects on the phase space of a harmonic oscillator. This includes, but is not limited to, photon loss induced by coupling to a Josephson junction, thermal excitation, photon phase dissipation, and various nonlinearities.

[0022] Recent experiments in the context of quantum superconducting circuits have observed this exponential suppression of bit flipping error due to the average number of photons in the cat state.

[0023] It is generally accepted that "the use of a single repeating code is sufficient to correct residual errors because bit inversion errors are sufficiently rare; that is, more specifically, a phase inversion error correction code is sufficient to correct residual phase inversion." This could be a repeating code defined in, for example, a dual-base error correction code or any other state-of-the-art error correction code.

[0024] B. Stabilization Scheme Cat qubits can be stabilized or confined by the following exemplary scheme: a) Jump operator

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[0025] In the context of the present invention, quantum system 1 is configured to implement parametric dissipative stabilization according to scheme a) or resonant dissipative stabilization according to scheme f). In either case, the cat qubit is

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[0026] Parametric dissipation stabilization depends on generating the following terms: -a 2 This first term corresponds to a nonlinear transformation between two photons in the first mode a (memory) and one photon in the second mode b (buffer). This term has a frequency of |2f a -f b | requires a pump: where f a is the resonant frequency of the first mode a, and f b is the resonant frequency of the second mode b. Therefore, such a pump is a two-photon pump. -α 2 This second term is frequency f b This requires the driving of the second mode b. Therefore, such driving is a two-photon drive.

[0027] Resonant dissipation stabilization is specific to certain cases (2f a =f bThis corresponds to ). Therefore, here, the 2:1 photon conversion is activated not by parametric pumping, but by a DC bias that tunes the mode frequency to the frequency matching condition. As a result, in addition to the precise tuning made possible by the DC bias, the jump operator

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[0028] Although this calibration method may not be sufficient to fully tune scheme e), it can be the first step in its calibration set.

[0029] C. Quantum system for stabilizing CAT qubits As shown in Figure 1, quantum system 1 includes a nonlinear superconducting quantum circuit 3 and an instruction circuit 5.

[0030] The nonlinear superconducting quantum circuit 3 is arranged to enable three-wave or four-wave mixing between the first mode and the second mode b. Hereinafter, the first mode a is used as memory to host the cat qubit, while the second mode b is used as a buffer between the cat qubit and the external environment.

[0031] Each of the first mode a and the second mode b corresponds to the natural resonant frequency of the nonlinear superconducting quantum circuit 3. As a result, each of the first mode a and the second mode b has its own resonant frequency. The first mode a is the resonant frequency

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[0032] By "having" the first and second modes, it should be understood here that the nonlinear superconducting quantum circuit 3 includes components operating within a superconducting region that host the modes independently or simultaneously. In other words, the first mode a and the second mode b can be hosted in various subsets of the components of the superconducting circuit or on the same subset of components.

[0033] The nonlinear superconducting quantum circuit 3 is intended to be exposed to microwave radiation delivered by the instruction circuit 5 in order to cleverly design various nonlinear interactions between the first mode a and the second mode b. The frequency of each microwave radiation is tuned to select a specific term within the rotational wave approximation.

[0034] The nonlinear superconducting quantum circuit 3 includes a nonlinear element 7 and at least one resonant section 9.

[0035] The nonlinear element 7 is a jump operator for stabilizing the cat qubit.

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[0036] In cases where parametric dissipation stabilization is implemented, the nonlinear element 7 is a four-wave mixed nonlinear element.

[0037] A four-wave mixed nonlinear element is, for example, an ATS. As successfully demonstrated by R. Lescanne et al. (2020), the ATS performs parametric dissipation stabilization by using a 2:1 photon conversion (i.e., a jump operator).

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[0038] In contrast to the first implementation of this stabilization scheme proposed by Z. Leghtas et al. (2015) and S. Touzard et al. (2018), in which the superconducting circuit element used as a four-wave mixer is a transmon having a single Josephson junction, the solution developed by R. Lescanne et al. (2020) utilizes an ATS design that has a much lower cross-Kerr term than the transmon and thus allows for the observation of exponential suppression of bit inversion.

[0039] Typically, an ATS has flux lines that can be delivered to modulate the common flux and / or differential flux.

[0040] In cases where resonant dissipation stabilization is implemented, the nonlinear element 7 is a three-wave mixed nonlinear element.

[0041] As mentioned above, the resonant case is 2f a =f b (f p This corresponds to a specific situation of =0) and may be particularly advantageous (as developed in European Patent Application No. 21306965.1 filed by the present patent applicant). In such resonant cases, the 2:1 photon conversion is not activated by parametric pumping of a 4-wave mixed nonlinear element; rather, a 3-wave mixed nonlinear element should be used instead, along with a DC bias to bring the system to 2:1 photon resonance.

[0042] However, as detailed below, measuring the observable variables of a cat qubit can involve cleverly designing a Hamiltonian that requires four-wave mixing. Typically, a longitudinal connection between the first mode a and the second mode b can be used to measure the observable variables of a cat qubit. Moreover, the longitudinal term a of the corresponding Hamiltonian... † a † This still requires that the four-wave mixing be skillfully designed. Thus, the resonant condition may require both a three-wave mixing nonlinear element to stabilize the cat qubit and a four-wave mixing nonlinear element to measure the observable variables of the cat qubit. Three-wave mixing can be achieved by operating the ATS at a flux application point different from the normally operated 0-π flux application point by using or adding another nonlinear element, as described in the aforementioned European Patent Application No. 21306965.1.

[0043] The resonant section 9 has each resonant frequency f a and f b The nonlinear superconducting quantum circuit 3 has a first mode a and a second mode b, and is arranged to be connected to a nonlinear element 7. Specifically, the first mode a and the second mode b "participate" in the nonlinear element 7: this means that some or all of the mode magnetic energy is stored within the nonlinear element 7. Such participation results in zero-point fluctuations of the superconducting phase across the entire ATS (with respect to the first mode a, φ). a And regarding the second mode b φ b It can be quantified by (represented by ).

[0044] In the schematic diagram of quantum system 1 shown in Figure 1, the nonlinear superconducting quantum system 3 contains only one resonant region (i.e., resonant region 9). However, it should be understood here that the nonlinear superconducting quantum system 3 contains at least one resonant region (usually two or more resonant regions to form two electromagnetic modes (cat qubit mode and buffer mode)).

[0045] The instruction circuit 5 is positioned to deliver microwave radiation.

[0046] In the context of quantum system 1, instruction circuit 5 requires such driving of the second mode b as both parametric dissipative stabilization and resonant dissipative stabilization, and therefore the second resonant frequency f b It is arranged to drive at least the second mode b by delivering radiation to the resonant section 9 at a frequency equal to .

[0047] C.1. Parametric dissipative stabilization quantum systems Figure 2 shows the detailed architecture of quantum system 1 for implementing parametric dissipative stabilization.

[0048] As a result, the nonlinear element 7 is a four-wave mixed nonlinear element, and more precisely, an ATS to which a resonant section 9, which is a linear microwave network b / a, is connected.

[0049] It is known to those skilled in the art that when biased at its point of flux application (0-π, or vice versa), the Hamiltonian of ATS7 has the following "sin-sin" form:

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[0050] Parametric pumping of the ATS7 is usually performed by pumping the common flux, since pumping the differential flux only moves the modes coupled to the ATS7. For example, frequency f p =|2f a -f b |, φ Σ (t) = ε 2ph cos(2πf p By pumping the common flux at t), Hamilton's nonlinear resonant region can be written in a rotating coordinate system as follows:

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[0051] Referring again to Figure 2, when the linear microwave network b / a is coupled to the ATS7, which acts as an inductive element, via the linear coupler 11, the nonlinear superconducting quantum circuit 3 participates in the ATS7 at their respective resonant frequencies f a and f b This is a network that has a first mode a and a second mode b.

[0052] When an external DC magnetic field is set up such that a flux of 0 mod 2π passes through one of the loops and a flux of π mod 2π passes through the other loop, it is guaranteed that the ATS Hamiltonian has its "sin-sin" form. For clarification, the setup for applying an external DC magnetic field is not shown in Figure 2, but the external DC magnetic field can be applied through the two lower mutual inductances of the ATS7. A typical implementation is essentially interleaving a bias tee connected to a DC current source to input a DC current into the system while allowing microwave radiation to pass through.

[0053] The microwave source 13 is a setup that modulates the common magnetic flux within the ATS7.

[0054] For this purpose, the microwave network 15 is used to split the radiation emitted by the microwave source 13 and to apply this radiation to each node of the ATS7 with the correct phase. Alternatively, two different microwave sources may be used, each simply coupled to a single node of the ATS7, and their relative phases and amplitudes set to achieve the desired flux modulation.

[0055] Microwave source 13 has a frequency of |2f a -f b When set at |, the nonlinear superconducting quantum circuit 3 performs a 2:1 photon conversion between the first mode a and the second mode b. In order to convert this 2:1 photon conversion into a 2-photon dissipation, the second mode b is coupled with the linear coupler 21 at frequency f b It is selectively coupled to the load 19 via a microwave filter 23 configured as a band-pass filter having [specific characteristics].

[0056] Alternatively, the microwave filter 23 uses frequency f a It can be configured as a band-stop filter in and placed between the external environment on the one hand and between the first mode a and the second mode b on the other hand in order to isolate the first mode a and thus prevent the first mode a from suffering additional losses resulting from undesirable coupling to the load 19.

[0057] Alternatively, the microwave filter 23 is f a >f b In that case, as a low-pass filter, f b >f a In that case, they can each be configured as high-pass filters. In other embodiments, the microwave filter 23 may be omitted when coupled with the load 19, and substantially only the second mode b may be established. Thus, those skilled in the art will understand that the first mode a has a high-quality coefficient while the second mode b has a low-quality coefficient.

[0058] As explained earlier, the second mode b has a resonant frequency f bThis is driven at frequency f. b This is performed by a microwave source 17 set up in [location].

[0059] In the above, the load 19 can be considered as part of the instruction circuit 5 in Figure 2, while the linear coupler 21 and microwave filter 23 can be considered as part of the nonlinear superconducting quantum circuit 3.

[0060] In addition to stabilizing the cat qubit, instruction circuit 5 also enables the measurement of the cat qubit's observable variables. As detailed below, and particularly with reference to Figure 9 onward, the measurement method that is the subject of the present invention includes not only the mapping of the observable variables to be measured to the Pauli operator Z, but also the measurement of the observable variables. To achieve this objective, and as shown in Figure 2, instruction circuit 5 further includes microwave sources 25, 27, and 29.

[0061] Microwave source 25 has a frequency f a The microwave radiation in is directed to the linear microwave network b / a to drive the first mode a. This driving of the first mode a is directed to the nonlinear superconducting quantum circuit 3.

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[0062] In addition to microwave source 13, microwave source 27 is set up to modulate the common magnetic flux within ATS7. As detailed below, such microwave source 27 may be involved in such measurements. Microwave source 27 is positioned to deliver microwave radiation to the nonlinear superconducting quantum circuit 3 to cleverly design a Hamiltonian. For example, such a Hamiltonian results in a longitudinal coupling between a first mode a and a second mode b.

[0063] Finally, microwave source 29 is set up to modulate the differential flux within the ATS7 via the microwave network 15. Microwave source 29 may be used to reduce the spurious Hamilton term induced by microwave source 27. It should be noted that the microwave network 15 is used for convenience but may be omitted, and that microwave sources 29 and 27 may be applied directly to both nodes of the ATS7, with their relative phase and amplitude set to achieve the desired flux modulation.

[0064] For completeness, the microwave source 29 drives the second mode b at frequency f b It should also be noted that it can be used in place of microwave source 17 to deliver microwave radiation to the linear microwave network b / a.

[0065] C.1.1. Galvanic Cat Qubit Figures 3 and 4 show the electrical equivalent diagrams of each embodiment of the nonlinear superconducting quantum circuit 3, which takes the form of a galvanic circuit. These electrical equivalent diagrams are partial, as both figures represent only the ATS 7 and the linear microwave network b / a, and therefore do not represent the linear coupler 21 or the microwave filter 23. Specifically, the linear microwave network b / a includes a first resonant section 31 and a second resonant section 33.

[0066] The ATS7 is implemented as known in the art (e.g., in R. Lescanne et al. (2020)). The ATS7 includes a first Josephson junction 35 and a second Josephson junction 37 in parallel, and an inductive element 39 in parallel between them. As a result, the ATS7 has two connecting loops, each containing a Josephson junction in parallel with a shunt inductive element. The inductive element 39 can be implemented either geometrically or by a series of junctions. The ATS7 has both DC and AC biased loop fluxes. The DC bias sets the flux point of the ATS7. The DC bias can be operated near a so-called saddle point, which is a sweep spot in frequency and has a small crossing Kerr term.

[0067] Both the first resonant section 31 and the second resonant section 33 are galvanically coupled to the ATS7. The first resonant section 31 has a resonant frequency f a The first mode a having is given to the nonlinear superconducting quantum circuit 3, while the second resonant section 33 has a resonant frequency f b A second mode b having the following characteristics is provided to the nonlinear superconducting quantum circuit 3.

[0068] It should be understood here that "galvanic coupling" means that there is a short conductive section (i.e., a short conductive track, or any other means that ensures a physically continuous conductive junction) connecting the first resonant section 31 and the second resonant section 33 to the ATS7. The expression "short" means that the conductive track is at the resonant frequency f a and f b This means that the ATS7 has an impedance that is negligible compared to the impedances of the first resonant section 31 and the second resonant section 33. These short conductive sections correspond to the linear coupler 11 in Figure 2.

[0069] In the embodiment shown in Figure 3, the first resonant section 31 includes a capacitive element 41 and an inductive element 43 connected in series. Similarly, the second resonant section 33 includes a capacitive element 45 and an inductive element 47 connected in series.

[0070] In the embodiment shown in Figure 4, the first resonant section 31 also includes a capacitive element 41 and an inductive element 43. However, in this embodiment, the capacitive element 41 and the inductive element 43 are connected in parallel. Similarly, the capacitive element 45 and the inductive element 47 of the second resonant section 33 are connected in parallel.

[0071] In the respective embodiments of Figures 3 and 4, the microwave linear network b / a includes two resonators. However, as previously described, the microwave linear network b / a may include only one resonator arranged to generate both the first mode a and the second mode b.

[0072] C.1.2. Capacitive Cat Qubit Figure 5 shows a possible exemplary implementation of quantum system 1 in Figure 2. In contrast to Figures 3 and 4, the external environment is represented. ATS7 acts as a central reference point to which the rest of the components are connected. The filtering of the second mode b and the external environment are also present to achieve this purpose. The microwave source is omitted for simplicity, but is arranged in the same way as in Figure 2 to execute the circuit.

[0073] In Figure 5, the linear microwave network b / a is formed by a capacitor galvanically coupled to ATS7 to form the second mode b and a parallel LC resonator strongly capacitively coupled to ATS7 to form the first mode a. The coupling to ATS7 forms a linear coupler 11. The buffer is coupled to the external environment by capacitance 19.

[0074] Figures 2-5 discussed above illustrate one embodiment of quantum system 1 in which the nonlinear element 7 is an ATS and the cat qubit is stabilized by using parametric dissipative stabilization.

[0075] C.2. Resonant dissipative stabilization quantum systems However, as previously described, the method for measuring the observable variables of a cat qubit according to the present invention can also be implemented by stabilizing the cat qubit with a three-wave mixed nonlinear element as a nonlinear element 7 and having resonant dissipative stabilization. Both Figures 6 and 7 relate to such an embodiment.

[0076] Figure 6 partially shows quantum system 1. In this figure, the three-wave mixed nonlinear element 7 and at least one resonant section 9 are put together to form a nonlinear superconducting quantum circuit 3. As previously described, such a nonlinear superconducting quantum circuit 3 is arranged to substantially perform a two-to-one photon conversion between the first mode a and the second mode b (symbolized here by a round-trip single arrow 49 and a double arrow 51).

[0077] The current source 53 is connected to the non-linear superconducting quantum circuit 3 via a wire. Such a current source 53 can be considered as part of the command circuit 5.

[0078] The current source 53 is directly connected to the non-linear superconducting quantum circuit 3 such that the current applied by the current source 53 flows through at least a subset of the components of the non-linear superconducting quantum circuit 3. The current source 53 is configured to enable a three-wave mixing interaction and to tune the frequency matching condition 2f a =f b to be satisfied.

[0079] Similar to FIG. 2, the second mode b is coupled to the load 19 via the linear coupler 21, and such coupling dissipates the second mode b. The microwave source 17 is arranged to deliver microwave radiation at a frequency equal to f b to the resonator 9 in order to drive the second mode b. Note that two-photon driving is required for both parametric dissipative stabilization and resonant dissipative stabilization.

[0080] Here again, there is a microwave filter 23 configured as a band-pass filter having a frequency f b . Alternatively, the microwave filter 23 can be configured as a band-stop filter at the frequency f a and can be placed between the external environment on the one hand and the first mode a and the second mode b on the other hand in order to separate the first mode a and thus prevent the first mode a from being troubled by additional losses coming from unwanted coupling to the load 19.

[0081] FIG. 7 shows an exemplary implementation of the quantum system 1 of FIG. 6, and more specifically a circuit capable of realizing a three-wave mixing non-linear element.

[0082] The three-wave mixing non-linear element is formed by at least one loop 55 including a first Josephson junction 57, a central inductive element 59 and a second Josephson junction 61.

[0083] When a predetermined current of a constant intensity is applied by the current source 53, the resonant frequency f b The resonant frequency is f a This is substantially equivalent to twice the original value. The circuit shown in Figure 8 is configured to symmetrically distinguish between the first mode a and the second mode b. The high symmetry of such a circuit results in improved quality of 2:1 photon conversion.

[0084] The central inductive element 59 may be an inductance, a single Josephson junction, or an array Josephson junction. Thus, the central inductive element 59 may be arranged as a series loop between the first Josephson junction 57 and the second Josephson junction 61. The series arrangement may include a first inner node connecting the poles of the first Josephson junction 57 to the poles of the central inductive element 59. The series arrangement may also include a second inner node connecting the poles of the second Josephson junction 61 to another pole of the central inductive element 59. The series arrangement may also include a closed-loop node connecting another pole of the first Josephson junction 57 to another pole of the second Josephson junction 61.

[0085] At least one loop 55 may be connected to a common ground via a closed-loop node. The circuit may also include a first capacitor 63 and a second capacitor 65. The first capacitor 63 may be connected in parallel with a first Josephson junction 57 between the common ground and a first inner node of the loop. The second capacitor 65 may be connected in parallel with a second Josephson junction 61 between the common ground and a second inner node of the loop.

[0086] The first Josephson junction 57 and the second Josephson junction 61 are substantially identical, and the capacitive elements 63 and 65 are also substantially identical. Thus, the symmetry of the circuit means that the first mode a is a symmetric superposition of the two resonators (as indicated by the thick arrow) and the second mode b is an inversely symmetric superposition of the two resonators (as indicated by the dashed arrow). Note that "only the second mode b contributes to the entire central inductive element 59, which is advantageously used to preferentially couple the external environment to this second mode b while isolating the first mode a from the external environment."

[0087] D. Calibration Method A method for calibrating a quantum system to stabilize a cat qubit will now be described.

[0088] D.1. Necessity of Cat Qubit Stabilization and Calibration Stabilizing a cat qubit requires designing a two-to-one photon exchange Hamiltonian with an intensity g2. This Hamiltonian is activated by a DC bias or a parametric pump. In the resonant stabilization case, the mode frequencies ω a and ω b are tuned due to the DC bias to approach the frequency matching condition 2ω a ≈ ω b . In the parametric dissipative stabilization case, the pump frequency ω p needs to be tuned to approach the frequency matching condition 2ω a ≈ ω b + ω p . The two frequency matching conditions are identical considering ω p = 0 in the DC bias case. In the second case, the intensity of the two-to-one photon exchange ratio g2 is tunable and thus can be selected by modifying the pump amplitude ε p . Stabilizing a cat qubit also requires dissipating the buffer mode and driving it with a microwave tone at a frequency ω b close to the resonant frequency ω d of the buffer mode.

[0089] As the frequency matching condition is approached, in both the parametric and resonant cases, the quantum system dynamics can be written as follows. [Equation] Here, a and b are the annihilation operators of the memory and buffer modes respectively, and Δ p = 2ω a - ω b - ω p is the distance from the frequency matching condition, and Δb =ω b -ω d g2 is the detuning between buffer mode and buffer drive, g2 is the 2:1 photon exchange ratio between memory mode and buffer, and ε d L is the buffer drive amplitude, b is a buffer mode jump operator, and κ b This is the buffer loss rate.

[0090] In addition to these dynamics, typical spurious terms exist as follows:

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[0091] Calibrating the CAT qubit system's point of action involves the operation parameter (ω), which is the parameter of the microwave tone transmitted over the quantum system to stabilize the CAT qubit in memory mode. p ,ε p ,ω d ,ε d ) or (I DC ,ω d ,ε d Its essence lies in judging ). Usually, ω d This is due to the finite precision of the design and simulation phase (which essentially involves finding precise circuit parameters and geometry), the imperfections in the fabrication of the device, and finally, some of the spurious terms mentioned above, which affect the design target ω b Unlike and ωp is 2ω a -ω b This is different. For example, due to the AC Stark shift, the resonant frequency of a mode will shift depending on whether the parametric pump is on or off.

[0092] D.2. Conventional Calibration As explained above, conventional cat qubit calibration involves the parameter of the microwave tone transmitted on the chip to stabilize the cat qubit in memory, ω. p , ε p , ω d , ε d Its essence lies in determining such operational parameters.

[0093] According to an article by Lescanne et al. (2020), the procedure was as follows: Buffer mode is well coupled to the environment, so the frequency ω b It can be easily found by spectroscopy on the coupling port. The memory mode is well isolated from its environment, so its frequency ω a This cannot be determined by spectroscopy. Instead, the pump frequency ω p This is determined. When performing buffer mode spectroscopy while the pump tone is on, the characteristic is that the pump frequency is 2ω a -ω b As it approaches this value, it appears in the spectroscopic view. Therefore, the first mode frequency ω a =(ω b +ω p ) / 2 can be inferred by performing buffer spectroscopy while sweeping the pump tone frequency. These two measurement results are used, for example, for Z gates and the buffer drive frequency ω used as the frame rotation frequency for readout. d (≒ω b ), pump frequency ω p and memory drive frequency ω z =(ω d +ω p This gives a rough calibration of ) / 2.

[0094] Equivalently, in the resonant case, spectroscopy is first used to determine the buffer frequency. Next, this spectroscopy is performed for various DC biases, and the evolution of the buffer frequency with respect to the DC bias is observed. When the buffer frequency resonates at twice the memory frequency, a feature for setting the DC bias appears in the buffer spectroscopy: this is the frequency matching condition 2ω a =ω b This means that it is close. From this, the buffer drive frequency ω under frequency matching conditions d ≒ω b and memory drive frequency ω z =ω d / 2 is inferred.

[0095] In either case, this first rough calibration is not sufficient to calibrate the cat qubit system with sufficient precision. Therefore, a second step is performed, which, if the frequency is well selected, involves applying buffer drive and pump (or resonant case buffer drive and DC bias) for a sufficiently long period of time for the memory to converge to the cat state, and then measuring the observable variables of the memory state.

[0096] This measurement is repeated for various couples of buffer drive frequency and pump amplitude (or buffer drive frequency and DC bias value for the resonant case) until a couple that provides satisfactory convergence to the cat state is determined.

[0097] To better illustrate the calibration method according to the present invention, and to highlight the difference from the conventional method described in this paragraph, the time evolution of the system under the conventional calibration sequence is then depicted in the simulation in Figure 8.

[0098] In this figure, starting from a vacuum, two curves plot the average number of photons in memory and the size of the buffer field over time. The thick curve is plotted for couples where convergence to the cat state is reached, while the dashed curve is plotted for couples where convergence is not reached. The clear difference between the thick and dashed curves demonstrates why this method of calibration is experimentally feasible.

[0099] On each curve, a vertical line was drawn corresponding to the time when the Wigner representation shown below was taken. Thus, the two columns of Winger representations (where the orthogonal coordinate axes are the orthogonal components of the square root of the number of photons) each correspond to snapshots taken at five different times indicated by the vertical lines in the curves described above.

[0100] The first row corresponds to the memory state, and the second row to the buffer state. The Wigner row clearly shows how the memory state converges to the CAT state.

[0101] In Lescanne et al. (2020), the memory observable variable was parity, and was measured via neighbor transmons. This observable variable could also be the vacuum probability. Various ω d and ω p By measuring these observable variables, its center is determined by definition (Δ p ,Δ d A diamond-shaped feature can be found where ) = (0,0). Since the observable variables of the memory state were directly evaluated, the system is well calibrated.

[0102] However, this method requires an ancilla transmon and lacks a high dynamic range: that is, the diamond shape may not be precise enough to accurately determine the center, and within the central region, the observable variables measured hardly fluctuate, making it difficult to pinpoint the center. This is critical because the best point of application may be offset from the precise geometric center of the diamond due to the spurious effects mentioned earlier.

[0103] Another measurement method that does not use transmons was used by Berdou et al. 2023 (https: / / journals.aps.org / prxquantum / abstract / 10.1103 / PRXQuantum.4.020350). This method involved directly evaluating the memory field by using a separate port coupled to the memory. While this method provides a clearer center of the diamond, it is not practical for real-world applications because the coupled memory port reduces the quality factor of the memory and leads to high phase inversion or decoherence rates.

[0104] D.3. A New Calibration Paradigm The patent applicant had a fundamentally different idea for fine-tuning parameters after the first "rough" calibration step.

[0105] Instead of proceeding step-by-step toward stabilizing the cat qubit and tweaking the parameters, the applicant has chosen to perform a measurement method that does not lead to stabilizing the qubit and thus the cat, but provides meaningful information about the quantum system, thereby enabling calibration.

[0106] Most importantly, this method does not require the use of transmons and is therefore compatible with measurements performed solely on a buffer.

[0107] This is the memory mode frequency mode ω a A frequency close to ω z and amplitude ε z Frequency ω z This is achieved by replacing buffer driving with driving in memory mode. This alters the dynamics of the system as shown below:

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[0108] "Term a † and b † It is important to note that the prefactor of b is the same as the previous Hamiltonian and has simply been rewritten to better fit the new driving scheme. "Gating and reading out the cat qubit is done by ω z (and ω p ) is known to be ω d It is also known that (and vice versa)

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[0109] Furthermore, this method does not stabilize the cat qubit as shown in Figure 9, but it provides information that enables calibration. In fact, ω p and ω z Δ p and Δ a Optimizing ω d =2ω z -ω p This will provide the service.

[0110] As mentioned earlier, in the resonant case, correcting the DC bias directly affects the memory's resonant frequency and buffer mode. Therefore, Δ p =2ω a -ω b This includes not only the DC bias which is brought close to 0, but also Δ a =ω a -ω z Thanks to this, it can be corrected and tuned. ω z When this becomes known, ω d It is simply 2ω z That is the case.

[0111] Figure 9 is identical to Figure 8 except for the parameters used to generate it. In fact, the only difference is that "Figure 9 is obtained while using a memory-driven and parametric pump (or memory-driven and DC-biased resonant case) instead of the buffer-driven and parametric pump (or buffer-driven and DC-biased resonant case) of Figure 8 - as described above."

[0112] The first Wigner row in Figure 9 indicates that "even when the memory drive and parametric pump (or memory drive and DC bias in the resonant case) are tuned in a way that using these parameters would induce a cat state in the setup in Figure 8, the setup in Figure 9 causes their convergence to a state different from the cat state in memory." Again, the clear difference between the thick curve and the dashed curve in Figure 9 is the difference that makes calibration experimentally feasible.

[0113] Figure 10 provides a general diagram of the calibration method according to the present invention.

[0114] In the first operation 1000, the new set of instruction circuit operation parameters (also called "CCOP") is determined. As discussed above, there can only be two types of CCOP: -Memory drive frequency ω z and parametric pump frequency ω p A combination that associates or -Memory drive frequency ω z A coupling that associates the DC bias value with the DC bias value.

[0115] The DC bias value should be understood as any value that allows defining the DC bias, either as current or voltage.

[0116] After the CCOP is determined, it is applied to the instruction circuit 5 in operation 1010, where a constant memory drive is applied in conjunction with either a parametric pump or a DC bias for a given period of time. It should be noted that this does not mean that the DC bias is systematically turned off between different iterations: i.e., depending on the situation, the DC bias may always be on and can only be modified. It is important that "operation 1010 lasts long enough for the quantum system to show a measurable change." The applicant has determined that a period greater than 10 ns and less than 100 μs provides the best results. The lower limit (10 ns) arises from the time it takes for the dynamics to begin and is currently constrained with respect to g2 by state-of-the-art technology (currently up to a maximum of a few MHz according to current state-of-the-art technology). If g2 is increased, the lower limit may be further reduced. The upper limit (100 μs) arises from the time it takes to converge to a steady state or to acquire sufficient data and obtain a proper SNR in the case of electromagnetic buffer field measurements. The upper limit may be unlimited, for example, in the case of continuous measurements of a buffer electromagnetic field.

[0117] Simultaneously with or immediately after operation 1010, a measurement is performed in operation 1020. This measurement aims to provide data on the quantum circuit that will enable calibration.

[0118] Even if this drive sequence does not stabilize the cat qubit in memory mode, this drive sequence still provides useful information. The reason for this is Δ a When optimized, the maximum number of photons exists in memory, while Δ p When optimized, the two-photon loss rate is maximized, and the number of photons in memory is reduced. This usually leads to a saddle point structure in the measurement, and this behavior similarly affects the buffer field. As a result, various couples of ω p and ω z By analyzing the memory photon count or buffer field measurement results obtained in relation to this, as shown in Figure 11, we can determine which couple has the best Δ p and Δ aIt is possible to define what kind of compromise is offered.

[0119] The applicant has found that one of either a memory-mode photon count measurement or a buffer-mode electromagnetic field measurement is useful for calibration. The memory-mode photon count measurement corresponds to the average photon count measurement and usually occurs after operation 1010. If the device does not contain a transmon, this is advantageously performed via longitudinal coupling between the buffer and the memory mode by parametric pumping a four-wave mixing element at the buffer frequency, as described by the applicant in European Patent Application No. 23180476. The buffer-mode electromagnetic field measurement may correspond to acquiring the signal coming out of the buffer or arriving at the load 19 via the coupler 21. This usually occurs when the signal is continuously generated in a pulse sequence converted from the memory drive. Thus, in this case, operations 1020 and 1010 occur simultaneously. Such a situation is advantageous because the measurement can be performed in a single experiment, and the lengths of operations 1020 and 1010 are increased so that the signal-to-noise ratio is appropriate. Conversely, if measurement 1020 is performed following operation 1010, the experiment must be repeated several times to ensure a suitable signal-to-noise ratio, and all results must be averaged.

[0120] As a result, it should be noted that "since measurements can be taken as soon as CCOP is applied, operations 1010 and 1020 may partially or completely overlap, but naturally, no significant measurement results should be expected approximately 10 nanoseconds before the application of CCOP."

[0121] Next, in operation 1030, a test may be performed to determine whether sufficient measurement results are available for calibration. Operation 1030 may be very simple or very complex according to this method.

[0122] In the first embodiment, the method proposes performing a two-dimensional sweep of CCOP. In this case, operation 1030 checks only whether the two-dimensional sweep has been completed, and operation 1000 checks whether ω in the two-dimensional sweep p (or DC bias I DC ) and ω z The essence of it lies in selecting the next couple of values.

[0123] In a second embodiment, instead of a two-dimensional sweep of parameters, the method proposes using an optimalizer that attempts to converge to an optimal couple given previous measurement results in order to speed up calibration. Several well-known saddle point search algorithms exist for efficiently converging to a saddle point.

[0124] One example involves inferring (from the first rough calibration) two points that straddle a saddle point (for example, the white dots shown in Figure 11). From there, several measurements can be taken along a linear path between the two points (dashed line in Figure 11). The measurements along the path are analyzed, and new points are explored near its maximum area. If these points have lower values, the interpolated path is updated. This process is repeated until the interpolated path reaches a path that passes through a saddle point where the maximum value of the path can no longer be minimized. At this point, the maximum value of this final path is the saddle point.

[0125] Therefore, the optimalizer consists of minimizing the maximum value of the path, and the new path is updated via gradient descent. Given the presence of noise in the measurement record, a threshold is set with respect to a gradient descent equivalent to the measurement noise. This threshold is effectively the saddle point search algorithm convergence threshold. This noise may be reduced by averaging the same measurement multiple times. In this case, operation 1030 checks whether the gradient descent is still progressing. Furthermore, in this case, the CCOP decision of operation 1000 is the result of the gradient descent.

[0126] It will readily become clear that many other types of optimizers can be used to determine the saddle point without departing from the scope of the present invention.

[0127] If operation 1030 is negative, operations 1000-1020 are repeated. Otherwise, the calibration method has reached its end, and the calibrated CCOP can be determined in operation 340.

[0128] In the example of the first embodiment described above, once the number of photons in the memory is measured, ω p and ω z Which couple of values ​​is Δ p and Δ a It is necessary to determine within the measurement 2D map whether it corresponds to an optimized saddle point. The same applies if the measurement is performed to find the electromagnetic field amplitude in buffer mode.

[0129] In the example of the second embodiment described above, due to the nature of gradient descent, operation 1040 essentially consists of simply finding the final CCOP of operation 1000.

[0130] The above embodiment was described in relation to parametric dissipation stabilization, but this situation is "ω p It is similar to the case of resonance stabilization, except that "the DC bias value is tested instead."

[0131] Finally, Figure 12 represents data from recent experiments. The quantum device includes a parametric pump ATS, and the measurement performed was a memory mean photon count measurement via longitudinal coupling to a buffer mode. The saddle point was recovered by the method described above using a two-dimensional sweep, leading to the Wigner shown on Figure 13, which provides evidence that this calibration method provides cat state stabilization within the memory mode.

Claims

1. A method for calibrating a quantum system to stabilize a cat qubit, The quantum system (1) has a first resonant frequency (ω) for hosting the cat qubit. ａ A first mode (a) having a second resonant frequency (ω ｂ The quantum system (1) includes a nonlinear superconducting quantum circuit (3) having a second mode (b) having ), and the quantum system (1) includes a nonlinear element (7) arranged to induce a 2:1 photon exchange between the first mode (a) and the second mode (b) via either a parametric pump and a four-wave mixer coupling the first mode (a) and the second mode (b), or a DC bias and a three-wave mixer coupling the first mode (a) and the second mode (b), and an instruction circuit (5) for delivering microwave radiation or current bias to the nonlinear superconducting quantum circuit (3) to provide the parametric pump or the DC bias and the first mode drive. The aforementioned method is: a) An operation to acquire a set of instruction circuit operation parameters, wherein each set of instruction circuit operation parameters is such that if the nonlinear element (7) performs four-wave mixing, the parametric pump frequency (ω ｐ ) and the first mode drive frequency (ω ｚ ) and a couple that associates, or if the nonlinear element (7) performs three-wave mixing, the DC bias value (I ＤＣ ) and the first mode drive frequency (ω ｚ An operation to obtain parameters that constitute one of the couples that associate with ) b) An operation to apply the set of instruction circuit operation parameters to the instruction circuit (5) for a period of more than 10 ns, c) An operation to perform at least one measurement within a group, including a first mode photon number measurement and a second mode electromagnetic field measurement. d) Repeat operations a) to c) until the calibration conditions are satisfied. e) A method comprising the operation of deriving a calibrated set of instruction circuit operation parameters based on the measurement results obtained in operation c).

2. The repetition of operation a) in operation d) is the operation value of the parametric pump frequency (ω ｐ ), and the first mode drive frequency (ω ｚ ), or the operation value of the DC bias value (I ＤＣ ), and the first mode drive frequency (ω ｚ ), and the method according to claim 1, comprising performing a two-dimensional sweep of the operation values.

3. The method according to claim 2, wherein the calibration conditions for operation d) are satisfied when the two-dimensional sweep is performed.

4. The method according to any one of claims 1 to 3, wherein operation e) includes determining a saddle point in the measurement result of operation c).

5. The repetition of operation a) in operation d) is the parametric pump frequency (ω ｐ ) and the first mode drive frequency (ω ｚ ) or the DC current value (I) based on the measurement in operation c) ＤＣ ) and the first mode drive frequency (ω ｚ The method according to claim 1, comprising performing a saddle point search algorithm on )

6. The method according to claim 5, wherein the calibration condition for operation d) is satisfied when the saddle point search algorithm converges to a selection threshold.

7. The method according to claim 5 or 6, wherein operation e) includes returning the set of instruction circuit operation parameters for the final iteration of step a).

8. The method according to any one of claims 1 to 7, wherein operations b) and c) overlap.

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