Method for measuring the observability of a cat qubit

Abstract

The present invention relates to a method for measuring the observability of a cat qubit. The quantum system includes a command circuit and a nonlinear superconducting quantum circuit including a nonlinear element and a resonant portion. Each of the nonlinear superconducting quantum circuits has a resonant frequency f a and f b It has first and second modes. The method is a) by a command circuit, frequency f b The method includes (1) delivering microwave radiation to drive a second mode, thereby causing a nonlinear element to operate a jump operator [Equation 1] that stabilizes a two-dimensional manifold hosting a cat qubit (900), (2) mapping a measurement of the observable under measurement to a Pauli operator Z in a cat qubit manifold of a size value whose square coefficient is 2 or greater (910), and (3) measuring the value of the Pauli operator Z (920).

Description

[Technical Field]

[0001] The field of this invention relates to the measurement of observables of cat qubits. More specifically, the observable to be measured is a Pauli operator X, a Pauli operator Y, or a parity operator. [Background technology]

[0002] Generally, superconducting qubits can be implemented as two-level systems of superconducting electronic circuits. Such qubits are conserved in bosonic mode, and therefore can form a specific class of superconducting qubits known as bosonic qubits.

[0003] Recent technologies have shown that it is possible to stabilize cat qubits, which are bosonic qubits defined by a quantum manifold whose range is determined by the superposition of two coherent states, which are quantum states close to the classical state of the bosonic mode. For this purpose, a specific dissipative stabilization mechanism can be implemented, which involves manipulating a nonlinear transformation between two photons of a first mode, also known as the memory mode or cat qubit mode, which hosts the stabilized quantum manifold, and one photon of a strongly dissipative second mode, also known as the buffer mode.

[0004] 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, the cat qubit relies on a mechanism that dissipates photons in pairs. This process pins two coherent states to separate locations in phase space. By increasing this separation, an exponential decrease in the bit flip rate has been experimentally observed, while the phase flip rate only increases linearly. As a result, it has been proven that stabilized cat qubits benefit from a high noise bias, which means that the bit flip probability is exponentially smaller than the phase flip probability.

[0005] To determine the state of a cat qubit, a transmon is commonly used as a measuring device coupled to the cat qubit mode. For example, Lescanne et al. (2020) proposed using an asymmetric threaded superconducting quantum interference device, hereafter ATS, to manipulate a 2-1 photon conversion associated with a transmon for measuring the state of a cat qubit. However, as noted in the article, as a result of such an architecture, the bit flip time saturates in a few milliseconds. Our research has shown that this is because the stabilization rate of the quantum manifold is too low to resist the dispersion frequency shift induced by thermal excitation of the measuring device. In fact, the transmon has a pseudo-crossing Kerr term, thereby inducing an additional noise process, which is accompanied by a deviation rate from the quantum manifold resulting from a very large transmon-cat qubit dispersion shift.

[0006] Furthermore, the transmon problem has already been observed in the first implementations of the aforementioned stabilization method, namely in the article by Z. Leghtas et al. (2015), “Confining the state of light to a quantum manifold by engineered two-photon loss” (Science, Vol.347, No.6224), and in the article by S. Touzard et al. (2018), “Coherent Oscillations inside a Quantum Manifold Stabilized by Dissipation” (Physical Review X8, 023005), where the superconducting circuit element used as a four-wave mixed element was a transmon with a single Josephson junction.

[0007] Therefore, there is a need to measure the state of a cat qubit without using transmons. This need arises from a recent study, Berdou et al. (2022), “One hundred second bit-flip time in a two-photon dissipative oscillator” (arXiv:2204.09128, http: / / arxiv.org / pdf / 2204.09128.pdf), which achieved a bit-flip time on the order of 100 seconds by eliminating the measurement transmon and operating the ATS in a state assumed to be dynamically stable. On the other hand, the measurement device used in place of the transmon in Berdou et al. (2022) could not measure the observable of the cat qubit regarding the superposition of quanta of the coherent state of the cat qubit. Nevertheless, measuring such an observable is essential for any quantum application. [Prior art documents] [Non-patent literature]

[0008] [Non-Patent Document 1] “Exponential suppression of bit-flips in a qubit encoded in an oscillator”(Nature Physics,2020) [Non-Patent Document 2] “Confining the state of light to a quantum manifold by engineered two-photon loss”(Science,Vol.347,No.6224) [Non-Patent Document 3] “Coherent Oscillations inside a Quantum Manifold Stabilized by Dissipation”(Physical Review X8,023005) [Non-Patent Document 4] “One hundred second bit-flip time in a two-photon dissipative oscillator”(arXiv:2204.09128,http. / / arxiv.org / pdf / 2204.09128.pdf) [Overview of the Initiative] [Problems that the invention aims to solve]

[0009] This invention aims to improve this situation. [Means for solving the problem]

[0010] To this end, the applicant proposes a method for measuring the observables of cat qubits performed by a quantum system, wherein the observables to be measured include the parity operator, Pauli operator X, and Pauli operator Y.

[0011] The quantum system includes a command circuit for delivering microwave radiation and a nonlinear superconducting quantum circuit which includes a three- or four-wave mixed nonlinear element and at least one resonant portion to which the three- or four-wave mixed nonlinear element is connected, the nonlinear superconducting quantum circuit having a first mode at a first resonant frequency and a second mode at a second resonant frequency.

[0012] The method includes the following actions: a) A command circuit delivers microwave radiation at a frequency equal to the second resonance frequency to at least one resonance region to drive the second mode, thereby stabilizing the two-dimensional manifold hosting the cat qubit in a three- or four-wave mixed nonlinear element.

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[0013] According to one or more embodiments, the observable to be measured is the Pauli operator Y, and b3) is

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[0014] According to one or more embodiments, the observable to be measured is the Pauli operator X, and b) further precedes b0)b1).

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[0015] According to one or more embodiments, the observable to be measured is a parity operator, the parity operator is mapped to a) a Pauli operator Z in, and b) further, b0) b1)

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[0016] According to one or more embodiments, the command circuit includes one or more microwave sources for performing a),

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[0017] According to one or more embodiments, the observable to be measured is the Pauli operator X, and b3) is

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[0018] According to one or more embodiments, the observable to be measured is a parity operator, the parity operator is mapped to a) a Pauli operator X, and b) b)

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[0019] According to one or more embodiments, the cat qubit

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[0020] According to one or more embodiments, c) is the c1) displacement operator

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[0021] According to one or more embodiments, c) is the c1) displacement operator

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[0022] According to one or more embodiments, c) parametrically drives the three- or four-wave mixed nonlinear element by delivering microwave radiation equal to the absolute difference between the first and second resonance frequencies if the three- or four-wave mixed nonlinear element is a three-wave mixed nonlinear element, or equal to half the absolute difference between the first and second resonance frequencies if the three- or four-wave mixed nonlinear element is a four-wave mixed nonlinear element, thereby driving the three- or four-wave mixed nonlinear element.

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[0023] According to one or more embodiments, c) the command circuit c1) delivers microwave radiation at a frequency equal to the second resonant frequency to at least one resonant portion to drive the second mode, thereby to a three or four-wave mixed nonlinear element.

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[0024] According to one or more embodiments, the transmission line is weakly coupled to a first mode (a), and c) includes detecting the electromagnetic field in the transmission line by performing hetrodyne or homodyne detection.

[0025] According to one or more embodiments, the three- or four-wave mixed nonlinear element is a four-wave mixed nonlinear element formed by an asymmetric threaded superconducting quantum interference element.

[0026] According to one or more embodiments, the three- or four-wave mixed nonlinear element is a three-wave mixed nonlinear element formed by at least one loop including a first Josephson junction, a central inductance element, and a second Josephson junction, wherein the nonlinear superconducting quantum circuit is configured such that when a predetermined current of a certain intensity is applied, the second resonance frequency is approximately equal to twice the first resonance frequency.

[0027] Other features and advantages of the present invention will become apparent from the following description with respect to the drawings below, which are provided for direct and limited purposes. [Brief explanation of the drawing]

[0028] [Figure 1] A schematic diagram of a quantum system arranged to measure the observables of a cat qubit according to the present invention is shown. [Figure 2] Figure 1 shows one embodiment of the quantum system in which 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 resonance 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 resonance parts shown. [Figure 5] Figure 2 partially shows an exemplary implementation of the quantum system, and a portion of the external environment is also shown. [Figure 6] Figure 2 partially illustrates other exemplary implementations of the quantum system, showing a portion of the external environment. [Figure 7] Another embodiment of the quantum system in Figure 1 is partially shown, in which the cat qubit is stabilized using resonant dissipation stabilization. [Figure 8] Figure 7 partially shows an exemplary implementation of the quantum system. [Figure 9] The present invention provides a method for measuring the observability of a cat qubit. [Figure 10] The mapping operation of the method shown in Figure 9 is illustrated. [Figure 11] The projection of the mapping operation in Figure 10 onto the cat coding space is shown for the measurement of the parity operator according to the first embodiment. [Figure 12] Figure 11 shows the time evolution of different parameters involved in the mapping operation. [Figure 13] Figures 11 and 12 show Wigner tomography performed using experimental measurements of the implemented parity measurement. [Figure 14] Figure 13 shows the physical chip layout of the quantum system implementation shown in Figure 5, which is used to perform the Wigna Tomography. [Figure 15] The projection of the mapping operation in Figure 10 onto the cat coding space is shown for the measurement of the parity operator according to the second embodiment. [Figure 16] Figure 15 shows the time evolution of different parameters involved in the mapping operation. [Figure 17] Figure 10 shows the projection of the mapping operation for measuring the Pauli operator Y into the cat coding space. [Figure 18] This shows a vertical join process for reading the value of the Pauli operator Z mapped to the observable being measured. [Modes for carrying out the invention]

[0029] The drawings and the following description consist of clear, well-defined features for most parts. As a result, they are not only useful for understanding the invention, but can also be used to aid in its definition where necessary.

[0030] Figure 1 shows a schematic diagram of a quantum system.

[0031] Quantum system 1 is configured to perform both the stabilization of the cat qubit and the measurement of its observable.

[0032] To date, the applicant's research has generally concerned the stabilization of cat qubits. As mentioned above, R. Lescanne et al. (2020) demonstrated that such qubits can be stabilized by a nonlinear transformation between two photons in a first mode a, i.e., the memory mode or cat qubit mode, and one photon in a second mode b, i.e., the buffer mode.

[0033] A cat qubit is a so-called cat state, which is a superposition of two coherent states |α〉 and |-α〉.

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[0034] Cat state

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[0035] As a result, when α=0, the cat qubit manifold encompasses all superpositions of the |0〉 and |1〉 Fock states.

[0036] Stabilized cat qubits are known to benefit from a high noise bias, which means that the bit flip probability is exponentially smaller than the phase flip probability. More precisely, the effective error channel (i.e., bit error or "bit flip") is equal to the "size" of the Schrödinger's cat state of the cat qubit, i.e., the average number of photons.

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[0037] Current understanding suggests that this suppression should apply to a large class of physical noises that have a local effect on the phase space of a harmonic oscillator. This includes, but is not limited to, photon losses, thermal excitations, photon phase relaxations, and various nonlinearities induced by coupling with Josephson junctions.

[0038] Recent experiments on quantum superconducting circuits have observed this exponential suppression of bit flip errors at the average number of photons in the cat state.

[0039] Generally, since bit-flip errors are very rare, it is considered sufficient to use a single repetition code to correct the remaining errors. More specifically, a phase-flip error correction code is considered sufficient to correct the remaining phase flips. This can be, for example, a repetition code defined on a binary basis or any other state-of-the-art error correction code.

[0040] The cat qubit can be stabilized or confined in the following exemplary ways: a) Jump operator

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[0041] With respect to the present invention, quantum system 1 is configured to implement parametric dissipation stabilization by method a) or resonant dissipation stabilization by method f). In either case, the cat qubit is

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[0042] Parametric dissipation stabilization relies on generating the following terms: - a 2 This first term corresponds to the nonlinear transformation between the first mode a, i.e., the two photons in memory, and the second mode b, i.e., the one photon in the buffer. This term is at frequency |2f a -fb |(where f a is the resonance frequency of the first mode, and f b This requires a pump at the resonance frequency of the second mode. Such a pump is therefore a two-photon pump. - α 2 This second term is frequency f b This requires driving the second mode b. Such driving is therefore a two-photon drive.

[0043] Resonant dissipation stabilization is 2f a =f b This corresponds to a specific case, and therefore the 2-1 photon conversion is not activated by parametric pumping. As a result, the jump operator

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[0044] As shown in Figure 1, quantum system 1 includes a nonlinear superconducting quantum circuit 3 and a command circuit 5.

[0045] The nonlinear superconducting quantum circuit 3 is configured to enable three-wave or four-wave mixing between the first mode a and the second mode b. In the following, the first mode a is used as a memory for storing cat qubits, and the second mode b is used as a buffer between the cat qubits and the external environment.

[0046] The first mode a and the second mode b each correspond to the natural resonance frequencies of the nonlinear superconducting quantum circuit 3. As a result, the first mode a and the second mode b each have their own resonance frequencies. The first mode a is the resonance frequency

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[0047] To "have" a first mode and a second mode should be understood here as including components operating in a superconducting system that host these modes separately or simultaneously. In other words, the first mode a and the second mode b may be hosted in different subsets of the components of the superconducting circuit, or on the same subset of components.

[0048] The nonlinear superconducting quantum circuit 3 is intended to manipulate various nonlinear interactions between the first mode a and the second mode b by receiving microwave radiation delivered by the command circuit 5. The frequency of each microwave radiation is tuned to select a specific term within the rotational wave approximation.

[0049] The nonlinear superconducting quantum circuit 3 includes a nonlinear element 7 and at least one resonant portion 9.

[0050] The nonlinear element 7 is a jump operator for stabilizing the cat qubit.

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[0051] When parametric dissipation stabilization is implemented, the nonlinear element 7 is a four-wave mixed nonlinear element.

[0052] A four-wave mixed nonlinear element is, for example, an ATS. To those skilled in the art, an ATS is a 2-1 photon conversion, i.e., a jump operator.

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[0053] Unlike the initial implementations of this stabilization scheme proposed in Z. Leghtas et al. (2015) and S. Touzard et al. (2018), in which the superconducting circuit element used as the four-wave mixer is a transmon with 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 a transmon, and therefore exponential suppression of bit inversion can be observed.

[0054] Typically, an ATS has flux lines through which it can deliver radiation to modulate common-mode flux and / or differential-mode flux.

[0055] When resonant dissipation stabilization is implemented, the nonlinear element 7 is a three-wave mixed nonlinear element.

[0056] As mentioned above, in the case of resonance, 2f a =f b (f pThis may be particularly advantageous in certain situations where = 0), as is evident in European Patent Application No. 21306965.1 filed by the present applicant. In such resonance cases, the 2-1 photon conversion cannot be enabled by parametrically pumping a four-wave mixed nonlinear element, and instead a three-wave mixed element should be used.

[0057] However, as will be detailed later, measuring the observability of a cat qubit may involve manipulating a Hamiltonian that requires four-wave mixing. Typically, the observability of a cat qubit can be performed using a longitudinal coupling between the first mode a and the second mode b. Nevertheless, the longitudinal term a of the corresponding Hamiltonian † a † This still requires four-wave mixing operation. Therefore, in the case of resonance, both a three-wave mixing nonlinear element for stabilizing the cat qubit and a four-wave mixing nonlinear element for measuring the observability of the cat qubit may be required. Three-wave mixing can be achieved by operating the ATS at a flux operating point different from the 0-π flux operating point from which it normally operates, or by adding other nonlinear elements as described in the aforementioned European Patent Application No. 21306965.1.

[0058] The resonant portion 9 is connected to the nonlinear element 7 and arranged to provide a nonlinear superconducting quantum circuit 3, where the first mode a and the second mode b have their respective resonant frequencies f a and f b It has the following characteristics. More specifically, the first mode a and the second mode b "participate" in the nonlinear element 7, which means that some or all of the mode magnetic energy is stored in the nonlinear element 7. Such participation is φ for the first mode a. a For the second mode b, φ b This can be quantified by the zero-point fluctuation of the superconducting phase through the ATS, which is described as follows.

[0059] In the schematic diagram of quantum system 1 shown in Figure 1, the nonlinear superconducting quantum system 3 contains only one resonance region, i.e., resonance region 9. However, it should be understood that the nonlinear superconducting quantum system 3 contains at least one resonance region, typically two, and ultimately forms two electromagnetic modes.

[0060] Command circuit 5 is positioned to deliver microwave radiation.

[0061] With respect to quantum system 1, the command circuit 5 has at least the second resonance frequency f b The system is configured to drive the second mode b by delivering radiation of the same frequency to the resonant portion 9, because both parametric dissipative stabilization and resonant dissipative stabilization require the driving of such a second mode b.

[0062] Figure 2 shows a detailed architecture of quantum system 1 for implementing parametric dissipative stabilization.

[0063] As a result, the nonlinear element 7 is a four-wave mixed nonlinear element, or more precisely, an ATS to which a resonant portion 9, which is a linear microwave network b / a, is connected.

[0064] Those skilled in the art know that when biased at its flux operating point (0-π or vice versa), the Hamiltonian of ATS 7 has the following "sin-sin" form:

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[0065] Parametric pumping of the ATS 7 is typically performed by pumping the common-mode flux, because pumping the differential-mode flux would only displace the modes coupled to the ATS 7. For example, common-mode flux at frequency f p =|2f a -f b |,φ Σ (t) = ε 2ph cos(2πf p By pumping at t), the nonlinear resonance of the Hamiltonian is expressed in the rotating frame as follows:

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[0066] Referring again to Figure 2, when the linear microwave network b / a is coupled to the ATS 7, which acts as an inductance element, through the linear coupler 11, the nonlinear superconducting quantum circuit 3 participates in the ATS 7 at their respective resonance frequencies f a and f b It has a first mode a and a second mode b.

[0067] When an external DC magnetic field is set such that a 0 mod 2π flux links with one of the loops and a π mod 2π flux links with the other loop, it is ensured that the ATS Hamiltonian has its "sin-sin" form. For clarity, the mechanism for applying the external DC magnetic field is not shown in Figure 2, but it can be applied via the two lower mutual inductances of the ATS 7. A typical implementation consists of interleaving a bias tee connected to a DC current source to input a DC current into the system and pass microwave radiation through the other.

[0068] The microwave source 13 is configured to modulate the common-mode magnetic flux of the ATS 7.

[0069] For this purpose, the microwave network 15 is used to divide the radiation emitted by the microwave source 13 and supply it to each node of the ATS 7 in the correct phase. Alternatively, two different microwave sources can also be used, each simply coupled to one node of the ATS 7, with their relative phase and amplitude set to achieve the desired flux modulation.

[0070] Microwave source 13 has a frequency of |2f a -f b When set to |, 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 interacts with the linear coupler 21 and the frequency f b It is selectively coupled to the load 19 via a microwave filter 23 configured as a bandpass filter.

[0071] Alternatively, the microwave filter 23 uses frequency f a It can also be configured as a bandstop filter, installed between the external environment on the one hand and the first mode a and the second mode b on the other hand, to isolate the first mode a, thereby preventing the first mode a from incurring additional losses due to unwanted coupling with the load 19.

[0072] Instead, this is, f a >f b In the case of, can be configured as a low-pass (or, f b >f a In the case of, can be configured as a high-pass) filter. In other embodiments, the microwave filter 23 can be omitted if it can establish a coupling substantially only with the second mode and the load 19. Therefore, those skilled in the art will understand that the first mode a has a high quality factor and the second mode b has a low quality factor.

[0073] As already described, the second mode b is driven at its resonance frequency f b This two-photon drive is performed by a microwave source 17 set at the frequency f b

[0074] In the above, the load 19 can be regarded as part of the command circuit 5 in FIG. 2, and the linear coupler 21 and the microwave filter 23 can be regarded as part of the non-linear superconducting quantum circuit 3.

[0075] In addition to stabilizing the cat qubit, the command circuit 5 also enables the measurement of its observable. As will be described in detail later, particularly referring to FIGS. 9 and later, the measurement method, which is the subject of the present invention, includes the measurement as well as the mapping of the Pauli operator Z of the observable of the measurement target. For this purpose, as shown in FIG. 2, the command circuit 5 further includes a microwave source 25, a microwave source 27, and a microwave source 29.

[0076] The microwave source 25 is arranged to drive the first mode a by delivering microwave radiation of the frequency f a to the linear microwave network b / a. By such driving of the first mode a, the non-linear superconducting quantum circuit 3 is

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[0077] Such Hamiltonian H Z This can be included in the mapping of the observable being measured. When manipulated with respect to two-photon dissipation, the Hamiltonian H Z Therefore, a Pauli gate, more precisely a Pauli Y gate or a Pauli Z gate, can be applied to the cat qubit depending on the size of the cat qubit and the phase of the first mode a drive.

[0078] Furthermore, such Hamiltonian H Z This can itself be included in the measurement. When manipulated with respect to two-photon dissipation, the Hamiltonian H Z This allows the value of the Pauli operator Z to be measured through the execution of hetrodyne or homodyne detection in the second mode b.

[0079] The microwave source 27, in addition to the microwave source 13, is configured to modulate the common-mode flux within the ATS 7. As will be detailed later, such a microwave source 27 can also be included in the measurement itself. The microwave source 27 is positioned to deliver microwave radiation to cause the nonlinear superconducting quantum circuit 3 to manipulate the Hamiltonian. For example, such a Hamiltonian is obtained through a longitudinal coupling between a first mode a and a second mode b.

[0080] The microwave source 29 is configured to modulate the differential-mode flux within the ATS 7 via the microwave network 15. The microwave source 29 can be used to reduce the pseudo-Hamiltonian term induced by the microwave source 27. It should be noted that the microwave network 15 is used for convenience but can be omitted, and the microwave sources 29 and 27 can also be applied directly to both nodes of the ATS 7, with their relative phase and amplitude set to achieve the desired flux modulation.

[0081] For the sake of thoroughness, it should be noted that the microwave source 29 can also be used instead of the microwave source 17 to deliver microwave radiation of frequency f b to the linear microwave network b / a and drive the second mode b.

[0082] Figures 3 and 4 show the electrical equivalent diagrams of respective embodiments of the non-linear superconducting quantum circuit 3 in the form of a galvanic circuit. Such electrical equivalent diagrams are partial in that they both represent only the ATS 7 and the linear microwave network b / a, and thus do not represent the linear coupler 21 or the microwave filter 23. More specifically, the linear microwave network b / a includes a first resonance portion 31 and a second resonance portion 33.

[0083] The ATS 7 is realized as known in the art, for example, by R. Lescanne et al. (2020). The ATS 7 includes a first Josephson junction 35 and a second Josephson junction 37 in parallel, and an inductance element 39 in parallel therebetween. As a result, the ATS 7 has two connected loops, each loop including a Josephson junction in parallel with a shunt inductance element. The inductance element 39 can be realized either geometrically or in a junction chain. The ATS 7 has magnetic fluxes that are DC and AC biased in both of its loops. The DC bias sets the magnetic flux operating point of the ATS 7. This is a sweet spot in terms of frequency and can be operated near the so-called saddle point, which has a small cross Kerr term.

[0084] Both the first resonance portion 31 and the second resonance portion 33 are galvanically coupled to the ATS 7. The first resonance portion 31 imparts a first mode a having a resonance frequency f a to the non-linear superconducting quantum circuit 3, while the second resonance portion 33 imparts a second mode b having a resonance frequency f b to the non-linear superconducting quantum circuit 3.

[0085] "Galvanically coupled" should be understood here as meaning that there is a short conductive section connecting the first resonant portion 31 and the second resonant portion 33 to the ATS 7, i.e., a short conductive track or any other means to ensure a physically continuous conductive junction. The expression "short" means that the impedance of the conductive track is such that the resonance frequency f a and f b This means that the impedances of ATS 7, the first resonant section 31, and the second resonant section 33 are negligible. These short conductive sections correspond to the linear coupler 11 in Figure 2.

[0086] In the embodiment shown in Figure 3, the first resonant portion 31 includes a capacitance element 41 and an inductance element 43, which are connected in series. Similarly, the second resonant portion 33 includes a capacitance element 45 and an inductance element 47, which are connected in series.

[0087] In the embodiment shown in Figure 4, the first resonant portion 31 also includes a capacitance element 41 and an inductance element 43. However, in this embodiment, the capacitance element 41 and the inductance element 43 are connected in parallel. Similarly, the capacitance element 45 and the inductance element 47 of the second resonant portion 33 are connected in parallel.

[0088] In the respective embodiments of Figures 3 and 4, the microwave linear network b / a includes two resonant regions. However, as already explained, the microwave linear network b / a may include only one resonant region arranged to produce both the first mode a and the second mode b.

[0089] Figures 5 and 6 both show possible exemplary implementations of quantum system 1 in Figure 2. Unlike Figures 3 and 4, the external environment is shown. ATS 7 acts as the central reference point, to which the remaining components are connected. Filtering for the second mode b and the external environment are also present for this purpose. The microwave source is omitted for simplicity, but is arranged as in Figure 2 to operate the circuit.

[0090] In Figure 5, the linear microwave network b / a is formed by a capacitance element galvanically coupled to the ATS 7 to form a second mode b, and a parallel LC resonator capacitively coupled to the ATS 7 to form a first mode a. The coupling with the ATS 7 forms a linear coupler 11. The buffer is coupled to the external environment by the capacitance 19.

[0091] In Figure 6, the linear microwave network b / a consists of two parallel LC resonators, which are galvanically coupled to the ATS 7 to form the first mode a and the second mode b. The buffer is coupled to the external environment by capacitance 19. The diagrams of the ATS 7 and the linear microwave network b / a correspond to those shown in Figure 3, with the first mode corresponding to the first resonant portion 31 and the second mode b corresponding to the second resonant portion 33.

[0092] Figures 2-6 above illustrate an embodiment of quantum system 1 in which the nonlinear element 7 is an ATS and the cat qubit is stabilized using parametric dissipative stabilization.

[0093] However, as already explained, the method for measuring the observability of a cat qubit according to the present invention can also be implemented by using a three-wave mixed nonlinear element as the nonlinear element 7 and stabilizing the cat qubit using resonant dissipation stabilization. Figures 7 and 8 both relate to such embodiments.

[0094] Figure 7 partially shows quantum system 1. In this figure, a three-wave mixed nonlinear element 7 and at least one resonant portion 9 are combined to form a nonlinear superconducting quantum circuit 3. As already explained, such a nonlinear superconducting quantum circuit 3 is configured to perform a 2-1 photon conversion between the first mode a and the second mode b, which is here represented by a single arrow 49 and a double arrow 51 in the forward and backward directions.

[0095] The current source 53 is connected to the nonlinear superconducting quantum circuit 3 via wiring. Such a current source 53 can be considered as part of the command circuit.

[0096] The current source 53 is directly connected to the nonlinear superconducting quantum circuit 3, so that the current supplied by the current source 53 flows through at least some of the components of the nonlinear superconducting quantum circuit 3. The current source 53 facilitates the three-wave mixed interaction and the frequency matching state 2f a =f b It is configured to allow for both adjustments.

[0097] Similar to Figure 2, the second mode b is coupled to the load 19 via the linear coupler 21, and this coupling makes the second mode b dissipative. The microwave source 17 is f b Microwave radiation of the same frequency as is delivered to the resonant part 9 and is arranged to drive the second mode b. As mentioned earlier, two-photon driving is required for both parametric dissipative stabilization and resonant dissipative stabilization.

[0098] Here too, frequency f b There is a microwave filter 23 configured as a bandpass filter at frequency f. Alternatively, the microwave filter 23 is configured at frequency f. a It can also be configured as a bandstop filter, positioned between the external environment on the one hand and the first mode a and the second mode b on the other, to isolate the first mode a, and thus prevent the first mode a from incurring additional losses due to unwanted coupling with the load 19.

[0099] FIG. 8 shows an exemplary implementation of the quantum system 1 of FIG. 7, more specifically an exemplary implementation of a circuit that can implement a three-wave mixing non-linear element.

[0100] The three-wave mixing non-linear element is formed by at least one loop 55 including a first Josephson junction 57, a central inductance element 59, and a second Josephson junction 61.

[0101] When a predetermined current of a certain intensity is supplied by the current source 53, the resonance frequency f b becomes substantially equal to twice the resonance frequency f a . The circuit shown in FIG. 8 is particularly configured to symmetrically distinguish between the first mode a and the second mode b. Due to the high symmetry of such a circuit, the quality of 2-1 photon conversion can be improved.

[0102] The central inductance element 59 can be an inductance, one Josephson junction, or an array of Josephson junctions. The central inductance element 59 can therefore be arranged in series as a loop between the first Josephson junction 57 and the second Josephson junction 61. The series arrangement can include a first internal node connecting the pole of the first Josephson junction 57 to the pole of the central inductance element 59. The series arrangement can also include a second internal node connecting the pole of the second Josephson junction 61 to the other pole of the central inductance element 59. The series arrangement can also include a closed-loop node connecting the other pole of the first Josephson junction 57 to the other pole of the second Josephson junction 61.

[0103] At least one loop 55 can be connected to a common ground via a closed-loop node. The circuit can also include a first capacitor 63 and a second capacitor 65. The first capacitor 63 can be connected in a complementary series with the first Josephson junction 57 between the common ground and the first internal node of the loop. The second capacitor 65 can be connected in parallel with the second Josephson junction 61 between the common ground and the second internal node of the loop.

[0104] The first Josephson junction 57 and the second Josephson junction 61 are substantially identical, as are the capacitance elements 63 and 65. Here, the symmetry of the circuit suggests that the first mode a is a symmetrical superposition of the two resonators (indicated by the solid arrows), and the second mode b is an asymmetrical superposition of the two resonators (indicated by the dashed lines). Only the second mode b has a contribution through the central inductance element 59, which is advantageously used to preferentially couple the external environment to this second mode b and isolate the first mode a from the external environment.

[0105] Here, the method for measuring the observability of a cat qubit will be explained in detail with reference to Figure 9.

[0106] In quantum mechanics, an observable refers to a measurable or observable physical quantity. Mathematically, an observable is represented by a self-adjoint operator acting on the quantum state of a given entity. The conceivable result of an observable measurement is the eigenvalue of the corresponding operator.

[0107] Common examples of observables include the following: - Represents the position of a particle in space, and its eigenvalues ​​correspond to the position operators that the particle can take. - Represents the momentum of a particle, and its eigenvalues ​​correspond to the momentum operators that the particle can possess. - Represents the energy of a quantum entity, and its eigenvalues ​​correspond to the energy levels that the quantum entity can possess, - Represents the essential angular momentum of a particle, and has three components <S x >, <S y >, and <S z The spin operator takes the form of a vector operator consisting of , and the eigenvalues ​​of the spin operator correspond to the possible spin values ​​of the particle along different directions.

[0108] Regarding the present invention, the observable to be measured is the Pauli operator X of the cat qubit manifold, the Pauli operator Y of the cat qubit manifold, or the parity operator of the cat qubit manifold.

[0109] Two-photon dissipation stabilizes the coherent state |±α〉 and the cat state

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[0110] These states are exponentially close to the coherent state |±α〉 when α >> 1. With this definition, the Pauli operators of the cat qubit manifold are as follows:

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[0111] Finally, the parity operator takes the following form:

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[0112] It can be seen that the projection of the parity operator onto the cat qubit manifold is equal to the Pauli operator X. In the present invention, this property is utilized to enable the realization of Wigner tomography without using a transmon.

[0113] Referring to Figure 9, in operation 900, quantum system 1 stabilizes the two-dimensional manifold hosting the cat qubits. If the cat qubit modes were originally in a state belonging to the cat qubit manifold, they remain unchanged in operation 900. If the cat qubit modes were originally in a state outside the cat qubit manifold, this operation maps the parity operator to the Pauli operator X of the stabilized manifold.

[0114] For this purpose, command circuit 5 sends a jump operator to nonlinear superconducting quantum circuit 3.

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[0115] In the case of parametric dissipation stabilization, microwave source 13 is |2f a -f b By delivering microwave radiation of a frequency equal to |, the nonlinear element 7 is parametrically driven, and the microwave source 17 is f b The second mode b is driven by delivering microwave radiation of the same frequency as the first mode.

[0116] In the case of resonant dissipation stabilization, the microwave source 17 is f b By delivering microwave radiation of the same frequency as the first mode, the second mode b is driven to the nonlinear superconducting quantum circuit 3, relating to 2f. a =f b The intrinsic 2-1 photon conversion obtained from this is converted into a 2-photon dissipation.

[0117] In operation 910, the observable being measured is mapped to the Pauli operator Z.

[0118] "Mapping one observable to another" must be understood here as the quantum system 1 performing a process of transforming one observable into another.

[0119] From a mathematical standpoint, if O is the operator being measured, then the process represented by the unitary operator U is applied by quantum system 1, and UOU + = Z. As a result, the eigenvectors and eigenvalues ​​of the operator being measured are mapped to the eigenvectors and eigenvalues ​​of the Pauli operator Z in a bijective manner. The measurement of the Pauli operator Z therefore contains all knowledge about the operator O being measured.

[0120] Figure 10 illustrates in more detail the mapping operation 910 performed by quantum system 1 and describes the operation common to the various possible mappings described in the remainder of this specification.

[0121] Figure 10 also deals with an important aspect of the mapping performed in relation to the present invention, namely the variable value derived with respect to the size of the cat qubit manifold.

[0122] Jump operator

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[0123] In particular, mapping operation 910 maps the observable being measured to the initial size value α of the cat qubit being measured. i The final size value α may differ from the final size value α. T The aim is to improve the measurement fidelity by mapping the qubit manifold to the Pauli operator Z.

[0124] In fact, the advantage of the Pauli operator Z compared to the Pauli operators X and Y is that the eigenstates of the Pauli operator Z are stabilized by two-photon dissipation. As a result, their lifetimes become exponentially longer with the size of the cat qubit, thereby enabling the integration of very long-duration measurements and achieving a very good signal-to-noise ratio. If very long-duration measurements are not possible, for example, during a quantum error correction cycle, the stabilization of the Pauli operator Z by two-photon dissipation allows for alpha to be processed without loss of information. T This can be increased to any large value. In this case, the eigenstate of the observable being measured is a large amount of photons, 4|α T | 2 =|α T -(-α T )| 2 Only the coherent states that are different | α T > and |-α T These are mapped to >, which allows them to be efficiently distinguished even with short measurement times.

[0125] In operation 1100, quantum system 1 compresses the cat qubits, thus reducing the size α of the cat qubit manifold.

[0126] For this reason, quantum system 1 weakens or even turns off the driving of the second mode b.

[0127] Command circuit 5 adjusts the amplitude of microwave source 17 so that α reaches a value equal to less than 1, or in a favorable mode, equal to 0. Since the cat qubit manifold is defined up to α=0, operation 1100 sets the Pauli operator X to an initial size value α i From the cat qubit manifold of size |α| 2 Continuously up to a cat qubit manifold of <1, and with the Pauli operator Y having an initial size value α i From the cat qubit manifold of size |α| 2 Continuously up to a cat qubit manifold of <1, and the Pauli operator Z with an initial size value α i From the cat qubit manifold of size |α| 2We continuously map down to cat qubit manifolds with values ​​<1.

[0128] At the end of operation 1100, the cat qubit is essentially a superposition of only the |0〉 and |1〉 Fock states.

[0129] "Essentially" here should be understood as a superposition of several Fock states, which are primarily |0〉 and |1〉 Fock states. In fact, as already explained, the definition of a cat state

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[0130] In operation 1200, quantum system 1 is

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[0131] Similar to the Pauli operators mentioned earlier, each Pauli gate corresponds to a rotation around an axis of the Bloch sphere. The Pauli-X(θ) gate is an θ rotation around the x-axis, the Pauli-Y(θ) gate is an θ rotation around the y-axis, and the Pauli-Z(θ) gate is an θ rotation around the z-axis. These should be performed by quantum system 1.

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[0132] Finally, in operation 1300, quantum system 1 expands the cat qubit, and therefore the size α of the cat qubit manifold increases.

[0133] For this reason, quantum system 1 strengthens the driving of the second mode b, or turns it on if applicable.

[0134] Command circuit 5 is the final size value α such that α has a square coefficient of 2 or greater. T To reach |α T | 2 The amplitude of the microwave source 17 is adjusted so that ≥ 2. The jump operator is

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[0135] As explained below, |α T | 2 Final size value α that satisfies ≥ 2 T This represents a substantial improvement in measurement fidelity compared to the latest technology, which is |α T | 2 This is because increasing the amplitude of the measurement signal increases, and therefore the measurement fidelity improves.

[0136] In Figure 10, operations 1100, 1200, and 1300 are shown to be performed sequentially. However, in practice, the value of parameter α is in particular the control setpoint of the microwave source for driving the second mode b, here microwave source 17, thereby causing the cat qubit to reach a number of photons corresponding to the square coefficient of the setpoint value of α in the steady state. However, in the transition state, the actual average number of photons of the cat qubit † a> lags behind the squared coefficient of the setpoint value of α.

[0137] Therefore, operation 1200

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[0138] As a result, operation 1200 typically overlaps in time with both operations 1100 and 1300.

[0139] In operation 920, quantum system 1 measures the value of the Pauli operator Z. As already detailed, such a Pauli operator Z has size α T It is located within the cat qubit manifold.

[0140] The eigenstates of the Pauli operator Z, i.e., the coherent states |α|. T > and |-α T Because > is a quasi-classical state that can have a long lifetime, a large number of photons, or both, it can be read with high fidelity using various techniques, including those explicitly stated below: i) Displacement operator

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[0141] Mapping the observables in Pauli operators X, Y, and parity operators to Pauli operator Z has the advantage of forming a bijection. Therefore, the value of the observed observable can be obtained from the value of Pauli operator Z.

[0142] Figure 11 shows an exemplary implementation of the mapping operation 910 in an embodiment where the observable under observation is the parity operator. This figure shows the rotation of cat qubits in the Bloch sphere, as well as the change in the size of the cat qubit manifold, within the cat coding space.

[0143] As mentioned above, the jump operator was executed during operation 900.

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[0144] In the example shown in Figure 11, the stabilized cat qubit has an initial size of |α i | 2 = 2.

[0145] In operation 1000, quantum system 1 is

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[0146] For this purpose, the command circuit 5, and more specifically the microwave source 25, f a By delivering radiation of the same frequency as, drive the first mode a,

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[0147] In operation 1100, quantum system 1 turns off the driving of the second mode b, thereby the size of the cat qubit becomes equal to zero, i.e., |α| 2 = 0. As mentioned above, it is also possible to simply weaken the driving of the second mode b. In this case, the value of α will be a value whose squared coefficient is less than 1.

[0148] In operation 1200, quantum system 1 is

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[0149] The Pauli-X(θ) gate of a cat qubit manifold with α=0 is particularly easy to execute because it corresponds to a rotation of angle θ in phase space. The orientation of the cat qubit in phase space corresponds to the absolute phase φ of the oscillation of the quantum harmonic oscillator. a and the relative phase difference between the absolute phase of the microwave source

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[0150] At operation 1300, quantum system 1 turns on the second mode b drive, thereby increasing the size of the cat qubit to its final size |α T | 2 It reaches |α|. In the example shown in Figure 11, the final size is |α|. T | 2 = 2. Needless to say, it is also possible to simply strengthen the drive of the second mode b. In such a case, the value of α is the final size value α. T It is considered that the squared coefficient is 2 or greater.

[0151] In an ideal quantum system 1, the theoretical requirements for performing operations 1100, 1200, and 1300 with 100% fidelity are as follows: - Operation 1200

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[0152] However, in practice, true adiabatic operation is possible because it takes infinitely long. Therefore, the optimal times for operations 1100 and 1300 are obtained from a compromise between the requirement that they be performed fast enough so that single-photon loss does not degrade the fidelity of the operation compared to κ1, and the requirement that they be performed slow enough so that these operations can be considered adiabatic. It should be noted that this is the fate of the Fock states |0〉 and |1〉, which are problematic with respect to the adiabatic condition, as explained by Albert et al. (2016) in the article, “Holonomic Quantum Control with Continuous Variable Systems” (Physical Review Letters 116, 140502). In other words, when α>>1, the cat can be compressed or expanded quickly, but when α~1, it must be done slowly compared to κ2. Similarly, operation 1200 is not actually possible with a cat qubit in a manifold of strict size α=0 because it takes infinitely long to reach it during operation 1100. still,

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[0153] ​​​Figure 12 shows the pulse sequence over the period corresponding to the stabilization of the cat qubit and the mapping of the parity operator to the Pauli operator Z. More specifically, the cat qubit is initially of size α i It is stabilized within the manifold, while the Pauli operator Z has a magnitude of α T It is located within the cat qubit manifold. The 2-1 photon conversion represented by κ2(t) always occurs during stabilization and mapping. The squared coefficient |α(t)| 2 The curve indicates, firstly, stabilization and

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[0154] The applicant experimentally measured the parity operator of a cat qubit using the quantum circuit in Figure 5, according to the pulse sequence in Figure 11. The measured parity operator can be used to perform wigner tomography, thereby completely characterizing the quantum states of the quantum harmonic oscillator, such as the first mode a. More precisely, the wigner function Wρ(λ) of the quantum state ρ of the first mode a at complex amplitude λ is equal to the mean value of the parity operator after the quantum state ρ is replaced by λ: Wρ(λ) = <ΠD(λ)ρD(-λ)>. Initially, the state

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[0155] For this measurement, the applicant has specified size α T ​The aforementioned process (i) was used to read out the Pauli operator Z of the cat qubit. The pulse sequence used to perform it and the characteristics of the mode b displacement β, which depends on the number of photons in the first mode, are shown in Figure 18.

[0156] Figure 14 is an image of the physical layout of the chip that implements the circuit of Figure 5 used to perform the wigner tomography of Figure 13.

[0157] Figure 15 shows an exemplary implementation of the mapping operation 910 in another embodiment where the observable being measured is the parity operator. This figure shows the rotation of cat qubits in the Bloch sphere, as well as the change in the size of the cat qubit manifold, in the cat coding space.

[0158] Similar to Figure 11, the jump operator is executed throughout operation 900.

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[0159] In the example shown in Figure 15, the initial size of the stabilized cat qubit is |α| 2 = 2

[0160] In operation 1100, quantum system 1 turns off the drive of the second mode b. The cat qubit has size |α| 2 This is reached. As mentioned above, it is also possible to simply weaken the drive of the second mode b. In such a case, the value of α is defined as a value whose squared coefficient is less than 1.

[0161] In operation 1200, quantum system 1 is

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[0162] At operation 1300, quantum system 1 turns off the driving of the second mode b, and as a result the size of the cat qubit becomes the final size |α T | 2 It reaches |α|. In the example shown in Figure 15, the final size is |α|. T | 2 = 2. Needless to say, it is also possible to simply strengthen the driving of the second mode b. In such a case, the value of α is the final size value α whose squared coefficient is 2 or greater. T This is the result.

[0163] Figure 16 shows the pulse sequence over the period corresponding to the stabilization of the cat qubit and the mapping of the parity operator to the Pauli operator Z. More specifically, the cat qubit is initially of size α i It is stabilized within a manifold of size α, while the Pauli operator Z has size α T It is stabilized within the cat qubit manifold. The 2-1 photon conversion represented by κ2(t) is always on during stabilization and mapping. Square coefficient |α(t)| 2 The curve allows us to first observe stabilization, and then observe the compression and expansion of the cat qubit.

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[0164] Finally, Figure 17 shows an exemplary implementation of the mapping operation 910 in an embodiment where the observable being measured is the Pauli operator Y. This figure shows the rotation of cat qubits in the Bloch sphere, as well as the change in the size of the cat qubit manifold, within the cat coding space.

[0165] This mapping operation appears similar to that shown in Figure 11, but differs in that it does not necessarily map Pauli operator X to Pauli operator Y, because the observable being measured is the latter.

[0166] The method of the present invention does not simply consist of mapping the observable under measurement to the Pauli operator Z. In fact, the operation of expanding the cat qubit corresponds to strengthening or turning on the second mode b, and in practice, thereby allowing an arbitrarily large size, i.e., the squared coefficient |α|. T | 2 α is 2 or greater T It becomes possible to measure the Pauli operator Z in a cat qubit manifold of values. Such a size allows for a significant amplification of the signal-to-noise ratio, ultimately to the point where the measurement becomes a single-shot measurement, i.e., until the difference in the number of photons between the logical states corresponding to each possible value of the observable being measured is high enough to avoid any ambiguity. This is shown in Figure 18, in which the histogram of the measurement using the longitudinal combination of process (i) is |α T | 2 The histogram moves further and further away from the one corresponding to α=0 as it increases.

[0167] As overlap decreases, the signal-to-noise ratio increases, making it highly likely that high-fidelity single-shot measurements can be achieved.

[0168] This method differs from the parity measurement proposed in particular in U.S. Patent Application Publication No. 2022 / 0156622A1, which, in paragraphs

[0220] -

[0226] , cat state

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[0169] In addition, the measurement methods described in this invention include each mapping embodiment, in particular

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Claims

1. A method for measuring the observable of a cat qubit performed by a quantum system (1), wherein the observable to be measured includes a parity operator, a Pauli operator X, and a Pauli operator Y. The quantum system (1) includes a command circuit (5) for delivering microwave radiation, and a nonlinear superconducting quantum circuit (3) which includes a three- or four-wave mixed nonlinear element (7) and at least one resonant portion (9) to which the three- or four-wave mixed nonlinear element (7) is connected, wherein the nonlinear superconducting quantum circuit has a first mode (a) at a first resonant frequency and a second mode (b) at a second resonant frequency. The above method involves the following actions, namely, a) The command circuit (5) delivers microwave radiation at a frequency equal to the second resonance frequency to the at least one resonance portion (9) to drive the second mode (b), thereby causing the three or four-wave mixed nonlinear element (7) to stabilize the two-dimensional manifold hosting the cat qubit. [Math 1] (In the formula, κ ２ A jump operator (where is the two-photon dissipation rate, a is the annihilation operator for the first mode (a), and α is a complex number obtained from the delivered microwave radiation) [Math 2] An action to operate (900), b) an operation (910) to map the observable to be measured to the Pauli operator Z, wherein the mapping operation (910) depends on the observable to be measured and includes at least b1) turning off or weakening the drive of the second mode (b) (1100) so that α reaches a value whose square coefficient is less than 1, and then b2) turning on or strengthening the drive of the second mode (b) (1300) so that α reaches a value whose square coefficient is 2 or greater (α Ｔ ) including, and further, b3) [Math 3] Place the gate in b1) or b2) <a † A mapping operation (910) includes applying (1200) to the cat qubit at the moment when a> is less than 2, c) The operation (920) of measuring the value of the Pauli operator Z, A method that includes this.

2. The observable being measured is the Pauli operator Y, and b3) is [Math 4] The method according to claim 1, comprising applying a gate to a cat qubit (1200).

3. The observable to be measured is the Pauli operator X, and b) further precedes b0) and b1). [Math 5] (1000) includes applying the gate to the cat qubit, and (b) is [Math 6] The method according to claim 1, comprising applying the gate to the cat qubit (1200).

4. The observable being measured is the parity operator, the parity operator is mapped to the Pauli operator X in a), and b) further before b0) and b1). [Number 7] (1000) includes applying the gate to the cat qubit, and (b) is [Number 8] The method according to claim 1, comprising applying the gate to the cat qubit (1200).

5. The command circuit includes one or more microwave sources (13, 17) for performing a), [Number 9] The method according to any one of claims 2 to 4, wherein applying the gate (1200) to the cat qubit is done by shifting the phase of one or more microwave sources (13, 17) among the microwave sources (13, 17) by an angle of π.

6. The observable to be measured is the Pauli operator X, and b3) is [Number 10] The method according to claim 1, comprising applying the gate to the cat qubit (1200).

7. The observable being measured is the parity operator, the parity operator is mapped to the Pauli operator X in a), and b3) is 【Number 11】 The method according to claim 1, comprising applying the gate to the cat qubit (1200).

8. The cat qubit [Math 12] Applying a gate (1200) or [Number 13] Applying the gate (1000) means that the command circuit (5) delivers microwave radiation at a frequency equal to the first resonant frequency to the at least one resonant portion (9) to drive the first mode (a), thereby [Number 14] (In the formula, ε Ｚ The Hamiltonian H is expressed as (obtained from the amplitude and phase of the drive in the first mode (a)). Ｚ The method according to any one of claims 3 to 7, which is performed by operating [a specific device].

9. c) is the c1) displacement operator [Number 15] Applying this to the cat qubit, and then, using the command circuit (5), a resonance frequency f substantially equal to the second resonance frequency is set. ｂ By delivering microwave radiation of the same frequency as ', the four-wave mixed nonlinear element (7) of the quantum system (1) is parametrically driven, thereby to the four-wave mixed nonlinear element (7) [Number 16] (In the formula, g Ｌ The method according to any one of claims 1 to 8, comprising: c) operating a Hamiltonian H (where a is obtained from the amplitude of the microwave radiation and b is the annihilation operator of the second mode (b)), wherein the Hamiltonian H is obtained through a longitudinal coupling between the first mode (a) and the second mode (b); and c) performing heterodyne or homodyne detection in the second mode (b).

10. c) is the c1) displacement operator [Number 17] Applying this to the cat qubit, and then c2) [Number 18] The method according to any one of claims 1 to 8, comprising manipulating a Hamiltonian H represented as (wherein X is a frequency shift per photon and b is the annihilation operator of the second mode (b)), wherein the Hamiltonian H is obtained through a crossing Kerr between the first mode (a) and the second mode (b), and the crossing Kerr tends to modulate the second resonance frequency; and then c3) measuring the second resonance frequency.

11. c) The command circuit (5) in c1) delivers microwave radiation equal to the absolute difference between the first resonance frequency and the second resonance frequency if the three- or four-wave mixed nonlinear element (7) is a three-wave mixed nonlinear element, or equal to half the absolute difference between the first resonance frequency and the second resonance frequency if the three- or four-wave mixed nonlinear element is a four-wave mixed nonlinear element, thereby parametrically driving the three- or four-wave mixed nonlinear element (7), and thereby causing the three- or four-wave mixed nonlinear element (7) to [Number 19] (In the formula, g ｃ c) the method according to any one of claims 1 to 8, comprising: manipulating a Hamiltonian H represented as (a) obtained from the amplitude of the microwave radiation, where b) is the annihilation operator of the second mode (b), wherein the Hamiltonian H is obtained by a single-photon conversion of the first mode (a) and the second mode (b); and c) performing hetrodyne or homodyne detection in the second mode (b).

12. c) The command circuit (5) in c1) delivers microwave radiation at a frequency equal to the second resonance frequency to the at least one resonance portion (9) to drive the second mode (b), thereby causing the three or four-wave mixed nonlinear element (7) to [Number 20] Jump operator represented as [Math 21] c2) The command circuit (5) delivers microwave radiation at a frequency equal to the first resonance frequency to the at least one resonance portion (9) to drive the first mode (a), thereby to the three or four mixed nonlinear element (7) [Number 22] (where ε Ｚ is obtained from the amplitude and phase of the drive of the first mode (a)) and operating the Hamiltonian H represented as such, and c3) performing heterodyne or homodyne detection in the second mode (b). The method according to any one of claims 1 to 8, comprising:

13. The method according to any one of claims 1 to 8, wherein the transmission line is weakly coupled to the first mode (a), and c) performs heterodyne or homodyne detection to detect an electromagnetic field in the transmission line.

14. The method according to any one of claims 1 to 13, wherein the three or four-wave mixed nonlinear element (7) is a four-wave mixed nonlinear element formed by an asymmetric thread superconducting quantum interference element.

15. The method according to any one of claims 1 to 14, wherein the three or four-wave mixed nonlinear element (7) is a three-wave mixed nonlinear element formed by at least one loop (55) including a first Josephson junction (57), a central inductor element (59), and a second Josephson junction (61), and the nonlinear superconducting quantum circuit (3) is configured such that when a predetermined current of a certain intensity is applied, the second resonance frequency becomes approximately equal to twice the first resonance frequency.

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