Quantum systems for stabilizing bosonic particle bits

Abstract

本発明は、ボソニック量子ビットを安定化させるための量子システム(1)に関し、これは、3つのマイクロ波源(11、13、15)を含むコマンド回路(5)と、四波混合非線形素子(7)と共鳴部分(9)を含む非線形超伝導量子回路(3)と、を含み、量子回路(3)は、それぞれ第一及び第二の共鳴周波数fa、fbの第一及び第二のモードを有する。量子回路(3)は、TIFF2026517463000070.tif6170として表されるハミルトニアンHを操作するように配置され、ハミルトニアンHにより、第二のモードの散逸を通じて、ボソニック量子ビットを安定化させる有効な散逸系D[α2+λα†α-α2]が得られる。このために、第一のマイクロ波源(11)は、|2fa-fb|と等しい周波数で放射を送達し、第二のマイクロ波源(13)は、fbと等しい周波数で放射を送達し、第三のマイクロ波源(15)は、fbと等しい周波数で放射を送達して、第二のモードを駆動する。

Description

[Technical Field]

[0001] The field of this invention relates to the stabilization of bosonic qubits. [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] Cat qubits offer the advantage of having an exponentially decreasing bit flip rate, while the phase flip rate increases only linearly. As a result, stabilized cat qubits can benefit from a high noise bias, which means the bit flip probability is much smaller than the phase flip probability.

[0005] Due to their noise structure, cat qubits can be coupled with repeating codes rather than the conventional surface codes used for typical superconducting qubits. However, the noise bias of cat qubits may not be sufficient for some large-scale algorithms to use simple repeating codes, and therefore, there is a need to discover bosonic qubits with even higher noise biases.

[0006] For this purpose, recent research has focused on a specific class of cat qubits, namely squeezed cat qubits. For example, Q.Xu et al. (2022), in their article, “Autonomous quantum error correction and fault-tolerant quantum computation with squeezed cat qubits” (arXiv:2210.13406), propose implementing an autonomous quantum error correction (AQEC) scheme that uses squeezed cat codes for the primary error source of continuous variable systems, namely excitation loss. This article demonstrates how encoding a cat state with an average of only 4 photons can reduce the number of photons required by approximately 10 -15 The noise bias has been published. As another example, T. Hillmann et al. (2022) proposed and analyzed the error correction performance of dissipatively stabilized squeezed cat qubits in their article, “Quantum error correction with dissipatively stabilized squeezed cat qubits” (arXiv:2210.13359,Phys.Rev.A 107,032423).

[0007] However, there are actually several obstacles to the possibility of stabilizing a squeezed cat qubit, and therefore, at present, this remains purely theoretical. In particular, such stabilization requires manipulating many Hamiltonian terms to obtain the desired dissipative system, and therefore requires using too many pumps, to the point of threatening the system's stability. In addition, one of these terms, namely

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[0008] [Non-Patent Document 1] Q.Xu et al.(2022), “Autonomous quantum error correction and fault-tolerant quantum computation with squeezed cat qubits”(arXiv:2210.13406) [Non-Patent Document 2] T. Hillmann et al. (2022), “Quantum error correction with dissipatively stabilized squeezed cat qubits” (arXiv:2210.13359,Phys.Rev.A 107,032423) [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] For this purpose, the applicant has provided a quantum system for stabilizing bosonic qubits, - A command circuit including a first microwave source, a second microwave source, and a third microwave source, each arranged to deliver microwave radiation, - An asymmetric threaded superconducting quantum interference element, and a nonlinear superconducting quantum circuit including at least one resonance portion to which the asymmetric threaded superconducting quantum interference element is connected, We propose a quantum system comprising the asymmetric threaded superconducting quantum interference element having flux lines through which radiation can be delivered to modulate common-mode and differential-mode fluxes, wherein the nonlinear superconducting quantum circuit has a first mode at a first resonance frequency and a second mode at a second resonance frequency, the second resonance frequency being twice the first resonance frequency and the second mode being dissipative.

[0011] Asymmetric threaded superconducting quantum interference devices are The first microwave source delivers radiation through the magnetic flux lines to modulate the common-mode magnetic flux at a frequency equal to the absolute value of the difference between twice the first resonance frequency and the second resonance frequency. The second microwave source delivers radiation through the flux lines to modulate the common-mode flux at a frequency equal to the second resonance frequency. • When a third microwave source delivers radiation at a frequency equal to the second resonance frequency to at least one resonance portion, thereby driving the second mode, Nonlinear superconducting quantum circuits

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[0012] According to one or more embodiments, the first microwave source and the second microwave source are configured such that the phases of the first microwave source and the second microwave source are substantially equal to each other.

[0013] According to one or more embodiments, the second microwave source and the third microwave source are configured such that the phases of the second microwave source and the third microwave source are substantially equal to each other.

[0014] According to one or more embodiments, the first microwave source, the second microwave source, and the third microwave source are configured such that the phases of the first microwave source, the second microwave source, and the third microwave source are substantially equal to each other.

[0015] According to one or more embodiments, the third microwave source is arranged to drive the second mode by delivering radiation through the flux lines to modulate the differential mode flux at a frequency equal to the second resonant frequency.

[0016] According to one or more embodiments, the second microwave source is arranged to deliver radiation having an amplitude such that the amplitude of λ is 1 or less.

[0017] According to one or more embodiments, the second microwave source is arranged to deliver radiation having an amplitude such that the amplitude of λ is substantially equal to 1.

[0018] According to one or more embodiments, the respective phases of the first mode and the second mode through the asymmetric threaded superconducting quantum interference element are zero-point fluctuations φ a , φ b It has, compared

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[0019] Furthermore, the applicant also proposes a method for stabilizing bosonic particle bits, which is executed by the above-mentioned quantum system and includes the following operations, namely, - Operating of delivering, by a first microwave source, radiation for modulating a common-mode magnetic flux through a magnetic flux line at a frequency equal to the absolute value of the difference between twice a first resonance frequency and a second resonance frequency; - Operating of delivering, by a second microwave source, radiation for modulating a common-mode magnetic flux through a magnetic flux line at a frequency equal to the second resonance frequency; - Operating of delivering, by a third microwave source, radiation at a frequency equal to the second resonance frequency to at least one resonance part to drive a second mode. The method also includes the above operations.

[0020] By executing this method, a non-linear superconducting quantum circuit will

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[0021] Other features and advantages of the present invention will become apparent from the following description of the drawings provided for illustrative and non-limiting purposes.

Brief Description of the Drawings

[0022] [Figure 1] A schematic representation of a quantum system including a nonlinear superconducting quantum circuit and a command circuit according to the present invention is shown. [Figure 2] The wigner functions for a standard cat qubit and a bosonic qubit stabilized by the quantum system shown in Figure 1 are shown. [Figure 3] This is an explicit diagram of the quantum system in Figure 1. [Figure 4] Figure 3 shows a partial electrical equivalent diagram of the first galvanic implementation example of the quantum system. [Figure 5] Figure 3 shows a partial electrical equivalent diagram of a second galvanic implementation example of the quantum system. [Figure 6] Figure 3 shows a partial electrical equivalent diagram of a capacitive implementation example of a quantum system. [Figure 7] Figure 1 shows the bit inversion rate of a bosonic qubit stabilized by the quantum system, depending on the number of photons. [Figure 8] The non-adiabatic error in the quantum system shown in Figure 1 is a result of the implementation of the Z gate. [Figure 9] Figure 1 shows a comparison of non-adiabatic errors between a bosonic qubit stabilized by the quantum system and a theoretical squeezed cat qubit. [Figure 10] Figure 1 shows the bit inversion rate of a bosonic qubit stabilized by the quantum system, corresponding to the ratio of the zero-point fluctuations of the superconducting phases of the bosonic qubit mode and the buffer mode. [Modes for carrying out the invention]

[0023] 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.

[0024] A. Cat Qubit To date, the applicant's research has generally focused on the stabilization of cat qubits. The possibility of realizing such qubits was demonstrated by R. Lescanne et al. in “Exponential suppression of bit-flips in a qubit encoded in an oscillator” (Nature Physics, 2020), which demonstrated that such cat qubits can be stabilized by a nonlinear transformation between a first mode a, i.e., two photons in the memory mode or cat qubit mode, and a second mode b, i.e., one photon in the buffer mode.

[0025] A cat qubit is a so-called cat state, which is a superposition of two coherent states |α> and |-α>.

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[0026] 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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[0027] 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.

[0028] 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.

[0029] A.1 Stabilization method Cat qubits can be stabilized or confined by the following exemplary methods: a) Jump operator

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[0030] The following sections will focus on a specific stabilization method, namely dissipative squeezing stabilization.

[0031] A.2 Squeezed Cat Qubit Cat qubits have the characteristic of being able to implement quantum gates such as Z or CNOT gates while maintaining noise bias. The characteristics used to evaluate the quality of a quantum gate are its execution time, i.e., the time the quantum gate operates and the associated error probability. In the case of cat qubits, the phase inversion error probability at the optimal time of the gate is ratio

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[0032] In general, bit flip errors are very rare, so the use of a single iterative code is sufficient to correct any remaining errors; more specifically, a phase flip error correction code is considered sufficient to correct any remaining phase flips. This can be, for example, a binary-based iterative code or any other modern error correction code.

[0033] However, even though cat qubits possess a high noise bias, a particularly advantageous feature when considering the realization of quantum error correction and, therefore, the design of reliable quantum computers, such a noise bias may be insufficient for large-scale quantum algorithms with simple iterative codes. In this regard, it has recently attracted particular attention that certain classifications of cat qubits have a much higher noise bias, namely the squeezed cat qubit.

[0034] In the case of squeezed cat qubits, the non-adiabatic error during the quantum gate, such as a Z gate or CNOT gate, is also lower. Overall, by reducing the non-adiabatic error, the ratio

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[0035] The squeezed state |α,ζ> is, |α,ζ>=D(α)S(ζ)|0> Defined as such, D(α) is the displacement operator,

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[0036] A squeezed cat qubit is a coherent superposition of two squeezed states with opposite displacement amplitudes and the same squeezing:

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[0037] Note that in order to have a proper squeezed cat state, the complex phases of α and ζ must be as follows: α = |α|e iθ ζ=re 2iθ

[0038] Dissipative squeezing stabilization is based on the following jump operator:

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[0039] This jump operator has a dissipation rate κ. b Loss buffer mode b and four-wave mixed devices, typically Josephson junctions or ATS, coupled to cat quantum modes and Hamiltonian

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[0040] The dissipation of a cat qubit is D[a 2 -α 2 Although it takes the form of ], the theoretical squeezed cat qubit is a more complex one, as follows:

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[0041] As a result, the theoretical implementation of stabilizing the squeezed cat qubit relies on generating four specific terms: i)a 2 b † +hc: 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 has a frequency of |2f a -f b |Requires a pump, in the formula, f a is the resonance frequency of the first mode a, and f b This is the resonance frequency of the second mode b. ii)a † a † +hc: This second term corresponds to the longitudinal coupling between the first mode a and the second mode b. This term corresponds to frequency f b It can be obtained using a pump. iii)

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[0042] The so-called "standard cat qubit" is the first term a 2 b † +hc and the fourth term α2 b † It should be noted that stabilization is achieved only by generating +hc.

[0043] However, a major obstacle to the use of squeezed cat qubits is the difficulty in manipulating such dissipative squeezing stabilization. This obstacle stems, in particular, from the need to manipulate numerous terms using pumps, which compromises system stability, increases thermal population, or shortens coherence time. In addition, generating a third term requires frequency f a and f b Assuming that each of these is typically 5 gigahertz (GHz), then the extremely high frequency 2f is approximately 15 gigahertz (GHz). a +f b It must be noted that this is necessary. The amount of energy that must be injected into the system to reach such frequencies will have a detrimental effect on the system's coherence. Furthermore, the third item

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[0044] Therefore, although squeezed cat qubits theoretically offer a higher noise bias than cat qubits, they currently appear unstabilized due to several problems in implementing the necessary dissipation. On the other hand, cat qubits can be stabilized, but require extreme usage conditions to achieve sufficient noise bias, and thus are used to run large-scale quantum algorithms. This was the subject of a recent publication by E. Gouzien et al. (2023), “Computing 256-bit Elliptic Curve Logarithm in 9 Hours with 126133 Cat Qubits” (arXiv:2502.06639).

[0045] B. Moon Cat Qubit Figure 1 shows a schematic diagram of quantum system 1, which is configured to stabilize a specific type of bosonic qubit, namely a moon cat qubit.

[0046] Quantum system 1 includes a nonlinear superconducting quantum circuit 3 and a command circuit 5.

[0047] The nonlinear superconducting quantum circuit 3 is configured to enable four-wave mixing of the first mode a and the second mode b. In this configuration, the first mode a hosts the moon cat qubit, while the second mode b is dissipative and used as a buffer between the moon cat qubit and the external environment.

[0048] In the following explanation, the first mode a is called memory mode, and the second mode b is called buffer mode.

[0049] Memory mode a and buffer mode b correspond to the natural resonance frequencies of the nonlinear superconducting quantum circuit 3. As a result, memory mode a and buffer mode b each have their own resonance frequencies. Memory mode a is the resonance frequency

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[0050] To "have" memory mode and buffer mode should be understood here as including components operating in a superconducting system that host these modes separately or simultaneously. In other words, memory mode a and buffer mode b may be hosted in different subsets of the components of the superconducting circuit, or on the same subset of components.

[0051] Memory mode a has a high quality factor Q a Buffer mode b has a low quality factor Q b It has. As can be understood, the quality factor (Q factor) can be determined in various ways, for example, by (a) spectroscopic linewidth measurement, where the quality factor is given by Q = f / Δf, where f is the resonant frequency and Δf is the spectroscopic linewidth, or by (b) time-domain measurement, where a tone is transmitted and the return signal is measured after a predetermined time has elapsed, and Q = f * τ, where τ is the characteristic decay time. Of course, those skilled in the art will know of various other methods for determining the quality factor of a particular mode.

[0052] The nonlinear superconducting quantum circuit 3 is intended to manipulate various nonlinear interactions between memory mode a and buffer 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.

[0053] As detailed below, the nonlinear superconducting quantum circuit 3 is intended to interact with the command circuit 5 to manipulate a specific Hamiltonian H to obtain an effective dissipative system that stabilizes the two-dimensional manifold hosting the moon cat qubit through the dissipation of buffer mode b.

[0054] Such a Hamiltonian H can be expressed in the following form:

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[0055] The nonlinear superconducting quantum circuit 3 includes a four-wave mixed nonlinear element 7 and at least one resonant portion 9.

[0056] The four-wave mixed nonlinear element 7 is parametrically driven by the command circuit 5 and positioned to cause the nonlinear superconducting quantum circuit 3 to manipulate the Hamiltonian H necessary for stabilizing the moon cat qubit.

[0057] More specifically, the four-wave mixed nonlinear element 7 is an asymmetric threaded superconducting quantum interference element, or hereafter referred to as ATS. To those skilled in the art, ATS is a 2-1 photon conversion, that is, the first term a of the Hamiltonian H mentioned above. 2 b † It is known that +hc can be manipulated and used to perform dissipative stabilization, as successfully demonstrated by R. Lescanne et al. (2020).

[0058] Unlike the initial implementations of this stabilization scheme proposed by Z. Leghtas et al. in the article, “Confining the state of light to a quantum manifold by engineered two-photon loss” (Science, Vol.347, No.6224, 2015), and by S. Touzard et al. in the article, “Coherent Oscillations inside a Quantum Manifold Stabilized by Dissipation” (Physical Review X8, 023005, 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 flipping can be observed.

[0059] An ATS has flux lines through which it can deliver radiation to modulate common-mode flux and / or differential-mode flux.

[0060] Those skilled in the art will see that when biased at the flux operating point (0-π or vice versa), the Hamiltonian H of ATS 7 ATS It is known to have the following "sin-sin" form:

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[0061] Common mode flux modulation φ Σ (t) can be performed by delivering microwave radiation in different phases through the flux lines of ATS 7, differential mode flux modulation φ Δ (t) can be performed by delivering microwave radiation in phase through the flux lines of ATS 7.

[0062] 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.

[0063] For example, common-mode magnetic flux at frequency f p =|2f a -f b |,φ Σ (t) = ε p 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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[0064] To manipulate the longitudinal Hamiltonian between memory mode and buffer mode, the common-mode flux is at frequency f b You have to pump it: φ Σ (t) = ε l cos(2πf b t)

[0065] In the rotating frame, the parametric part of the Hamiltonian is expressed as follows: [Number]

[0066] This Hamiltonian can be expressed as follows to highlight the desired dynamics: [Number] However, [Number] is.

[0067] The first term corresponds to the desired longitudinal coupling.

[0068] The second term is the effective drive of the buffer mode b at the resonance frequency f b . This term can be used to manipulate the term α 2 b † of the Hamiltonian H. However, since this term needs to be adjusted independently of λ, a buffer drive 15 or a direct drive is required.

[0069] The third term is spurious and corresponds to a potentially harmful non-linearity. Depending on the noise level that the moon cat qubit stabilization has to cope with, this may lead to the ratio φ b / φ a being constrained so that this last term is not too harmful.

[0070] The 2-1 photon conversion between the memory mode a and the buffer mode b is at the frequency f p = |2f a - fb It is obtained by parametrically pumping with |. An advantage is that the pump frequency is f p The parametric pumping works as well as possible by satisfying g2, where g2 is the two-photon coupling rate.

[0071] 2f a =f b (f p A specific situation where (=0) is called the resonance case and can be advantageous, as described in European Patent Application No. EP21306965.1 filed by the present applicant. However, in such resonance cases, the 2-1 photon conversion cannot be operated by parametrically pumping a four-wave mixed nonlinear element such as an ATS. Instead, a three-wave mixed nonlinear element should be used. As will be discussed later, the longitudinal term of the Hamiltonian H still requires the four-wave mixing operation. Thus, the resonance situation, while compatible with the proposed Moon Cat qubit stabilization, requires another three-wave mixed nonlinear element. In other words, in the resonance case, the first term a 2 b † +hc and the second term a † a † To generate +hc, both three-wave and four-wave mixing are necessary. Three-wave mixing can be achieved by operating the ATS at a flux point different from the 0-π flux point on which it normally operates, or by adding other nonlinear elements as described in the aforementioned European Patent Application EP21306965.1.

[0072] In the present invention, the applicant uses ATS to stabilize the moon cat qubit, and therefore, |2f a -f b We propose verifying g2, or proposing the advantages of being outside the resonance range for 2-1 photon conversion.

[0073] The resonant portion 9 is coupled to or connected to the ATS 7, and the resonant frequencies f of the nonlinear superconducting quantum circuit 3 are respectively a and f bThe system is arranged to provide a memory mode a and a buffer mode b having φ. More specifically, memory mode a and buffer mode b "engage" in the ATS 7, which means that some or all of the mode magnetic energy is stored in the ATS 7. Such involvement can be quantified by the zero-point fluctuation of the superconducting phase through the ATS, and for memory mode a, φ a , for buffer mode b φ b It is written as follows.

[0074] In the schematic diagram of quantum system 1 shown in Figure 1, the nonlinear superconducting quantum system 3 contains only one resonance region, namely resonance region 9. This single resonance region can be configured to generate both memory mode a and buffer mode b. However, the nonlinear superconducting quantum system 3 typically contains two resonance regions, each forming memory mode a and buffer mode b.

[0075] Command circuit 5 is positioned to deliver microwave radiation.

[0076] Command circuit 5 is configured to parametrically drive ATS 7 to perform a nonlinear conversion between two photons in memory mode a and one photon in buffer mode b, and to parametrically drive ATS 7 to perform a longitudinal coupling between memory mode a and buffer mode b, thereby driving buffer mode b.

[0077] For this purpose, as shown in Figure 1, the command circuit 5 includes a first microwave source 11, a second microwave source 13, and a third microwave source 15.

[0078] The first microwave source 11 is positioned to parametrically drive the ATS 7. More specifically, the first microwave source 11 operates at a frequency of |2f a -f b It is used to implement a parametric pump that delivers radiation to ATS 7.

[0079] In the following explanation, the first microwave source 11 will be referred to as a two-photon pump.

[0080] The second microwave source 13 is also positioned to parametrically drive the ATS 7. More specifically, the second microwave source 13 operates at frequency f b It is used to implement a parametric pump that delivers radiation to ATS 7.

[0081] In the following explanation, the second microwave source 13 will be referred to as the vertical pump.

[0082] Finally, the third microwave source 15 has a frequency f b The system is configured to drive buffer mode b by delivering its radiation to the resonant portion 9.

[0083] In the following explanation, the third microwave source 15 will be referred to as the buffer drive.

[0084] As already explained, the combined effect of the two-photon pump 11 and the buffer drive 15 allows the nonlinear superconducting quantum circuit 3, and more precisely the ATS 7, to perform the nonlinear conversion between two photons in memory mode a and one photon in buffer mode b, as well as drive buffer mode b.

[0085] Considering the stabilization of the 2-dimensional manifold hosting the Moon Cat qubit, which is the objective of quantum system 1, such combinatorial effects will be manipulated by the nonlinear superconducting quantum circuit 3 in the Hamiltonian H term a 2 b † +hc and α 2 b † It contributes to the generation of +hc.

[0086] Furthermore, a parametric pump implemented using a vertical pump 13 enables vertical coupling between memory mode a and buffer mode b.

[0087] As a result of this vertical combination, term a † a †+hc is generated, which contributes to the stabilization of the Hamiltonian H and, therefore, the 2-dimensional manifold hosting the Moon Cat qubit.

[0088] In response to radiation delivered by command circuit 5, nonlinear superconducting quantum circuit 3 manipulates the Hamiltonian H, which is expressed as follows: H / h = g2(a 2 +λa † a-α 2 )b † +hc In the formula, -g2 is the two-photon coupling ratio or interaction strength, which is proportional to the amplitude of the two-photon pump 11. -a is the extinction operator for memory mode a, -λ is a complex number of phase and amplitude obtained from the amplitude of radiation induced by the vertical pump 13. -α is a complex number obtained from buffer drive 15, -b is the extinction operator for buffer mode b, - Specify the Hermitian conjugate.

[0089] It should be noted that in the representation of the Hamiltonian H, both parameters λ and α are defined with respect to the two-photon pump 11. In particular, the phase and amplitude from which the complex number λ is derived are defined with respect to the phase and amplitude of the two-photon pump 11, respectively.

[0090] In the following explanation, the complex number λ will be referred to as the Moon Cat parameter.

[0091] Such Hamiltonian H is a term

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[0092] The Hamiltonian H stabilizes the 2-dimensional manifold hosting the moon cat qubit through the dissipation of buffer mode b, providing an effective dissipation system D[a]. 2 +λa † a-α 2 ] can be obtained.

[0093] By adjusting the relative phase of the microwave source, a desired number of photons can be obtained in the memory.

[0094] More specifically, an advantage is that the phase difference between the two-photon pump 11 and the longitudinal pump 13 is substantially zero. In other words, the two-photon pump 11 and the longitudinal pump 13 are configured such that their respective phases are substantially equal to each other.

[0095] An advantage is that the phase difference between the vertical pump 13 and the buffer drive 15 is virtually zero. In other words, the vertical pump 13 and the buffer drive 15 are configured such that their respective phases are substantially equal to each other.

[0096] Preferably, the two-photon pump 11, the longitudinal pump 13, and the buffer drive 15 are configured such that their respective phases are substantially equal to each other.

[0097] The phrase "substantially equal" implies that, ideally, the phases of the microwave sources involved are exactly equal to each other. However, achieving such equivalence in practice is difficult. Typically, the phase of one microwave source is 5% different from the phase of another.

[0098] Figure 2 shows the wigner functions, or wigner tomography, of a standard cat qubit and a moon cat qubit stabilized by quantum system 1.

[0099] A standard cat qubit and a moon cat qubit stabilized by quantum system 1 produce the same number of photons.

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[0100] It can be seen that each of the two blobs has a crescent shape. This is why bosonic qubits stabilized by quantum system 1 are called "moon cat qubits."

[0101] B.1 Exemplary Implementation of Quantum Systems Figure 3 provides a more explicit schematic representation of Figure 1.

[0102] The resonant portion 9 takes the form of a linear microwave network connected to the ATS 7, thereby coupling to the ATS 7 which acts as an inductance element via a linear coupler 17, and the nonlinear superconducting quantum circuit 3 interacts with the respective resonant frequencies f of the ATS 7. a and f b It has memory mode a and buffer mode b.

[0103] The two-photon pump 11 and the longitudinal pump 13 are configured to modulate the common-mode magnetic flux within the ATS 7.

[0104] In the example shown in Figure 3, the command circuit 5 further includes a microwave source 19 configured to modulate the differential mode flux within the ATS 7.

[0105] Such a microwave source 19 has a required frequency f bThe radiation can be used to drive buffer mode b by delivering it to the resonant portion 9. In the sense of the present invention, the microwave source 19 can therefore also be considered a “third microwave source” or “buffer drive” rather than the buffer drive 15.

[0106] It should be noted that the microwave source 19 can also be used to compensate for parasitic drive in buffer mode b caused by the longitudinal pump 13. The amplitude and phase of such compensatory drive can be calculated analytically, but this is fine-tuned experimentally.

[0107] To clearly distinguish the role of microwave sources in the Hamiltonian, a microwave network 21 with the correct phase offset is shown in Figure 3. Alternatively, each microwave source can be simply coupled to a single node of the ATS 7, and their relative phases and amplitudes can be set to achieve the desired flux modulation. In this case, to modulate common-mode flux, the microwave sources must deal with phase-shifted circuits, and to modulate differential-mode flux, the microwave sources must deal with in-phase circuits.

[0108] As mentioned above, the two-photon pump 11 is |2f a -f b By delivering microwave radiation to ATS 7 at a frequency substantially equal to |, the nonlinear superconducting quantum circuit 3 is made to perform a 2-1 photon conversion between memory mode a and buffer mode b, and thus the term a of the Hamiltonian H 2 b † +hc is positioned to operate. To convert this 2-1 photon conversion to 2-photon dissipation, buffer mode b is set to linear coupler 25 and frequency f b It is selectively coupled to the load 23 via a microwave filter 27 configured as a bandpass filter.

[0109] Alternatively, the microwave filter 27 uses frequency f aIt can be configured as a bandstop filter, placed between the external environment on the one hand and memory mode a and buffer mode b on the other, to isolate memory mode a, and thus prevent additional losses from unwanted coupling with load 23 in memory mode a.

[0110] Alternatively, this is f a >f b (or f b >f a In the case of ), it can be configured as a low-pass (or high-pass) filter. In other embodiments, the microwave filter 27 can be omitted if a coupling can be established between the load 23 and substantially only buffer mode b. As mentioned above, memory mode a has a high quality factor Q a It has a low quality factor Q in buffer mode b. b It holds.

[0111] As mentioned above, the buffer drive 15 has a second resonant frequency f b By delivering microwave radiation to the resonant part 9 at a frequency substantially equal to that of the Hamiltonian H term α 2 b † It is configured to drive buffer mode b by manipulating +hc.

[0112] Alternatively, buffer mode b is frequency f b It is driven by a microwave source 19 configured as follows.

[0113] In the above, the load 23 can be seen as part of the command circuit 5 in Figure 1, while the linear coupler 25 and microwave filter 27 can be seen as part of the nonlinear superconducting quantum circuit 3.

[0114] As mentioned above, the vertical pump 13 has a second resonant frequency f b By delivering microwave radiation to the ATS 7 at a frequency substantially equal to that of the Hamiltonian H, and performing a longitudinal coupling between memory mode a and buffer mode b, the term a of the Hamiltonian H is determined. † a † It is positioned to operate +hc.

[0115] B.1.1 Galvanic Implementation Example Figures 4 and 5 show the electrical equivalent diagrams of each embodiment of the nonlinear superconducting quantum circuit 3, which takes the form of a galvanic circuit.

[0116] Such electrical equivalent diagrams are partial because they both represent only the ATS 7 and the resonant section 9, and therefore do not represent the linear coupler 25 or the microwave filter 27. More specifically, the resonant section 9 includes the first resonant section 29 and the second resonant section 31.

[0117] The ATS 7 is implemented as known in the art, for example, in R. Lescanne et al. (2020). The ATS 7 includes a parallel first Josephson junction 33 and a second Josephson junction 35, and an inductance element 37 in parallel between them. As a result, the ATS 7 has two connected loops, each loop including a shunt inductance element 37 and Josephson junctions 33, 35 in parallel. The inductance element 37 can be implemented geometrically or as a junction chain. The ATS 7 has DC and AC biased flux in both of its loops. The DC bias sets the operating point of the ATS 7. This is the sweet spot in frequency and can be operated near the so-called saddle point, which has a small crossing Kerr term.

[0118] Both the first resonance section 29 and the second resonance section 31 are galvanically coupled to the ATS 7. The first resonance section 29 has a resonance frequency f a A memory mode a having is applied to the nonlinear superconducting quantum circuit 3, while the second resonance portion 31 has a resonance frequency f b A buffer mode b having the following characteristics is applied to the nonlinear superconducting quantum circuit 3.

[0119] "Galvanically coupled" should be understood here as meaning that there is a short conductive section connecting the first resonant section 29 and the second resonant section 31 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 29, and the second resonant section 31 are negligible. These short conductive sections correspond to the linear coupler 17.

[0120] In the embodiment shown in Figure 4, the first resonant portion 29 includes a capacitance element 39 and an inductance element 41, which are connected in series. Similarly, the second resonant portion 31 includes a capacitance element 43 and an inductance element 45, which are connected in series.

[0121] In the embodiment shown in Figure 5, the first resonant portion 29 also includes a capacitance element 39 and an inductance element 41. However, in this embodiment, the capacitance element 39 and the inductance element 41 are connected in parallel. Similarly, the capacitance element 43 and the inductance element 45 of the second resonant portion 31 are connected in parallel.

[0122] In the respective embodiments shown in Figures 4 and 5, the nonlinear superconducting quantum circuit 3 includes two resonant regions. However, as already described, the nonlinear superconducting quantum circuit 3 may include only one resonant region arranged to generate memory mode a and buffer mode b.

[0123] B.1.2 Capacitive Implementation Example Figure 6 shows the electrical equivalent diagram of one embodiment of the nonlinear superconducting quantum circuit 3, which takes the form of a capacitance circuit.

[0124] Unlike Figures 4 and 5, the external environment is shown. ATS 7 acts as a central reference point to which the remaining components are connected. The filtering in buffer mode b and the external environment are also present for this purpose. The microwave source is omitted for simplicity, but it is positioned as in Figure 3 to operate the circuit.

[0125] In Figure 6, the linear microwave network is formed by a capacitance element galvanically coupled to the ATS 7 to form buffer mode b, and a parallel LC resonator capacitively coupled to the ATS 7 to form memory mode a. The coupling with the ATS 7 corresponds to the linear coupler 17. Both memory mode a and buffer mode b are strongly coupled to the ATS 7, which is indicated by the zero-point phase variation of the two modes at the central inductance of the ATS 7. The buffer is coupled to the external environment by a capacitance element corresponding to the linear coupler 25.

[0126] B.2 Characteristics of the Moon Cat Qubit In the present invention, the nonlinear superconducting quantum circuit 3 is

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[0127] When the phases of the second microwave 13 and the buffer drive 15 (or alternatively 19) are properly adjusted, λ and α 2 λ and α have the same complex topology and, therefore, can be considered real numbers without loss of generality. Thus, unless otherwise specified, λ and α are assumed to be real numbers.

[0128] Even and odd cat states

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[0129] The performance index of the implementation of quantum system 1 is given by term a 2 b † +hc and a † a † This is the relative amplitude of +hc. In this regard, if we can establish a comparison between the Moon Cat Qubit of the present invention and the theoretical Squeezed Cat Qubit, then Δ can be expressed as follows using the complex squeezing parameter ζ=re iθ It can be expressed as a function of: λ = 2tanh(γ)

[0130] Similar to the case of the squeezed cat, the bit flip probability exponentially decreases by increasing the value of λ, or equivalently γ for the squeezed cat.

[0131] B.3 Performance of the moon cat qubit FIG. 7 shows the bit flip probability for different values of the moon cat parameter λ as a function of the photon number. More specifically, for each value of the moon cat parameter λ, there is a solid line corresponding to the bit flip rate of the moon cat qubit stabilized by the quantum system 1 and a dotted line corresponding to the bit flip rate of an equivalent squeezed cat qubit. By "equivalent" it should be understood that for a given value of the moon cat parameter λ, the parameter obtained by the above formula

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[0132] For λ = 1 and the photon number being

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[0133] FIG. 7 shows that when the value of the moon cat parameter λ is less than or equal to 1, the moon cat qubit (solid line) shows the same or even a slightly greater improvement in terms of bit flip compared to the squeezed cat qubit (dotted line).

[0134] As already explained, both the phase and amplitude of the Moon Cat parameter λ are derived from the amplitude of the radiation induced by the longitudinal pump 13. Advantageously, the amplitude of the radiation induced by the longitudinal pump 13 is tuned to equal the Moon Cat parameter λ to 1. Beyond this value, the Moon Cat qubit stabilized by quantum system 1 is affected by saturation, and therefore its performance is lower than the theoretical performance of a squeezed Cat qubit.

[0135] Quantum System 1 also exhibits fewer non-adiabatic errors during gate implementation. While the following explanation focuses on the Z gate, it can be directly generalized to the CNOT gate as well.

[0136] Known cat qubits, i.e., term a 2 b † +hc and term α 2 b † Two types of errors can occur when implementing a Z-gate using a cat quantum stabilized by generating only +hc: -

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[0137] A compromise must be found between the single-photon loss error and the non-adiabatic error, because the former is proportional to the gate duration, while the latter is proportional to the reciprocal of the duration.

[0138] The quantum system 1 has fewer non-adiabatic errors than known cat qubits. The non-adiabatic errors in the Z gate of the non-quantum system 1 are given by the following equation:

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[0139] Figure 8 shows the non-adiabatic errors of the quantum system 1 in the Z gate as a function of the gate time in units of 1 / κ2 for a constant number of photons

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[0140] Figure 9 shows a comparison of the non-adiabatic errors between the moon cat qubit (solid line) stabilized by the quantum system 1 and the squeezed cat qubit (dotted line), indicating that the quantum system 1 has performance equivalent to that of the squeezed cat qubit.

[0141] The way the moon cat qubit is stabilized by the quantum system 1 suggests that the pseudo-term of form b † b(b † +b) is activated by the longitudinal pump 13, where b is the photon annihilation operator of the dissipative buffer mode b. However, this pseudo-term does not affect the quantum system 1 if its amplitude is small enough. As mentioned above, its amplitude is

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[0142] Finally, Figure 10 shows that in memory mode a, the number of photons is approximately

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[0143]

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[0144] In Figure 10, the curves correspond to the following experimentally relevant parameters: - ratio

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Claims

1. A quantum system (1) for stabilizing a bosonic qubit, - A command circuit (5) including a first microwave source (11), a second microwave source (13), and a third microwave source (15, 19), each arranged to deliver microwave radiation, - A nonlinear threaded superconducting quantum interference element (7) and a nonlinear superconducting quantum circuit (3) including at least one resonance portion (9) to which the asymmetric threaded superconducting quantum interference element (7) is connected, wherein the asymmetric threaded superconducting quantum interference element (7) has flux lines through which radiation can be delivered to modulate common-mode and differential-mode fluxes, the nonlinear superconducting quantum circuit (3) has a first mode (a) at a first resonance frequency and a second mode (b) at a second resonance frequency, the second resonance frequency being different from twice the first resonance frequency, and the second mode (b) being dissipative. Includes, The aforementioned asymmetric threaded superconducting quantum interference element (7) is - The first microwave source (11) delivers radiation through the magnetic flux lines to modulate the common-mode magnetic flux at a frequency equal to the absolute value of the difference between twice the first resonance frequency and the second resonance frequency. - The second microwave source (13) delivers radiation through the magnetic flux line to modulate the common-mode magnetic flux at a frequency equal to the second resonance frequency, - When the third microwave source (15, 19) delivers radiation to the at least one resonant portion (9) at a frequency equal to the second resonant frequency, driving the second mode (b), The nonlinear superconducting quantum circuit (3) [Math 1] The Hamiltonian H, represented by the formula, is arranged to manipulate the Hamiltonian H, in which g ２ A is proportional to the amplitude of the first microwave source (11), α is the annihilation operator of the first mode (a), λ is obtained from the amplitude of the second microwave source (13), each having a phase and amplitude defined relative to the phase and amplitude of the first microwave source (11), α is obtained from the third microwave source (15, 19), and is a complex number defined relative to the first microwave source (11), b is the annihilation operator of the second mode (b), h and c are Hermitian conjugates, and the Hamiltonian H stabilizes the two-dimensional manifold hosting the bosonic qubit in the first mode (a) through the dissipation of the second mode (b), with respect to the effective dissipation system D[α ２ +λα † α-α ２ A quantum system (1) is obtained from this.

2. The quantum system (1) according to claim 1, wherein the first microwave source (11) and the second microwave source (13) are configured such that the phases of the first microwave source (11) and the second microwave source (13) are substantially equal to each other.

3. The quantum system (1) according to claim 1 or 2, wherein the second microwave source (13) and the third microwave sources (15, 19) are configured such that the phases of the second microwave source (13) and the third microwave sources (15, 19) are substantially equal to each other.

4. The quantum system (1) according to any one of claims 1 to 3, wherein the first microwave source (11), the second microwave source (13), and the third microwave source (15, 19) are configured such that the phases of the first microwave source (11), the second microwave source (13), and the third microwave source (15, 19) are substantially equal to each other.

5. The quantum system (1) according to any one of claims 1 to 4, wherein the third microwave source (19) is arranged to drive the second mode (b) by delivering radiation through the flux lines to modulate the differential mode flux at a frequency equal to the second resonance frequency.

6. The quantum system (1) according to any one of claims 1 to 5, wherein the second microwave source (13) is arranged to deliver radiation having an amplitude such that the amplitude of λ is 1 or less.

7. The quantum system (1) according to claim 6, wherein the second microwave source (13) is arranged to deliver radiation having an amplitude such that the amplitude of λ is substantially equal to 1.

8. The phases of the first mode (a) and the second mode (b) through the asymmetric threaded superconducting quantum interference element (7) are, respectively, zero-point fluctuations φ ａ , φ ｂ It has, compared [Math 2] The quantum system (1) according to any one of claims 1 to 7, wherein is less than 3.

9. A method for stabilizing a bosonic particle bit, performed by a quantum system (1) according to any one of claims 1 to 8, - The first microwave source (11) delivers radiation through the magnetic flux lines to modulate the common-mode magnetic flux at a frequency equal to the absolute value of the difference between twice the first resonance frequency and the second resonance frequency, - The operation of the second microwave source (13) to deliver radiation through the magnetic flux line to modulate the common-mode magnetic flux at a frequency equal to the second resonance frequency, - The operation of driving the second mode (b) by delivering radiation at a frequency equal to the second resonance frequency to the at least one resonance portion (9) using the third microwave source (15, 19), Includes, By performing the above method, the nonlinear superconducting quantum circuit (3) is [Math 3] Operate the Hamiltonian \(H\) represented by the formula, where \(g\) ２ is proportional to the amplitude of the first microwave source (11), \(\alpha\) is the annihilation operator of the first mode (a), \(\lambda\) is obtained from the amplitude of the second microwave source (13), and each is a complex number having a phase and an amplitude defined relative to the phase and amplitude of the first microwave source (11) respectively, \(\alpha\) is obtained from the third microwave source (15) and is a complex number defined relative to the first microwave source (11), \(b\) is the annihilation operator of the second mode (b), \(h\), \(c\) are Hermitian conjugates, and the Hamiltonian \(H\) stabilizes a two-dimensional manifold hosting the bosonic qubit in the first mode (a) through the dissipation of the second mode (b). The effective dissipation system \(D[\alpha\) ２ +\lambda\alpha † \alpha-\alpha ２ is obtained.

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