Compact resonators with shifted higher-order modes for quantum buses

Abstract

The resonator is based on a coplanar waveguide (CPW) structure that includes a first end having a first width and configured to be coupled to a first qubit, a middle portion having a second width narrower than the first width, and a second end having a third width wider than the second width and configured to be coupled to a second qubit.

Description

[Technical field]

[0001] The present invention relates generally to superconducting devices, and more particularly to coupling qubits together. [Background technology]

[0002] Superconducting quantum computing is an implementation of a quantum computer using superconducting electronic circuits. Quantum computing utilizes quantum phenomena for information processing and communication. Various models of quantum computing and quantum simulation exist. The basic building block of a gate-based quantum computer is the quantum bit (qubit). A qubit is a general concept of a bit that has two possible states, but due to its quantum nature, can be in a superposition of both states. A quantum gate is a general concept of a logic gate, but a quantum gate represents the change that occurs to one or more qubits after a gate is applied to those qubits, given an initial state of those qubits.

[0003] There are many different implementations of qubits using superconducting circuits. A common element is typically a Josephson junction. A Josephson junction is a weak coupling between two superconducting electrodes that allows dissipationless tunneling of Cooper pairs between the electrodes. Due to the relationship between the voltage across the junction and the tunneling current, a Josephson junction can in some cases be considered as a nonlinear lossless inductor. One of the most successful implementations of qubits is the so-called transmon qubit. A transmon qubit contains a Josephson junction in parallel with a shunt capacitor, thus forming an anharmonic oscillator. In this case, the two states of the qubit are considered as the two lowest energy levels of the anharmonic oscillator (called the ground state and the first excited state). The energy difference between the two states corresponds to a frequency in the several GHz regime (often around 5 GHz). To read out the qubit state, a microwave signal is applied to a microwave readout cavity that is coupled to the qubit (for example via a coupling capacitance). The transmitted (or reflected) microwave signal passes through multiple thermal isolation stages and low-noise amplifiers used to block or reduce noise and improve the signal-to-noise ratio. From the amplitude and / or phase of the returned / output microwave signal, information about the qubit state can be inferred. The microwave signal that conveys quantum information about the qubit state is typically weak (e.g., on the order of a few microwave photons). To measure this weak signal, low-noise quantum-limited amplifiers (QLAs) such as Josephson amplifiers and traveling-wave parametric amplifiers (TWPAs) can be used as preamplifiers (i.e., first amplification stage) at the output of the quantum system to boost the quantum signal with the minimum amount of added noise required by quantum mechanics to improve the signal-to-noise ratio of the output chain. In addition to Josephson amplifiers, certain Josephson microwave components using Josephson amplifiers or Josephson mixers such as Josephson circulators, Josephson isolators, and Josephson mixers can be used in scalable quantum processors.

[0004] The ability to incorporate more qubits is critical to the realization of the potential of quantum computers. Applicants have recognized that improvements can be made in two major aspects to improve the computational power and reliability of quantum computers. The first is the number of qubits themselves. The more qubits in a quantum processor, the more states can in principle be manipulated and stored. The second is a low error rate suitable for precisely manipulating the qubit states and performing sequential operations that provide consistent results and do not simply provide unreliable data. Therefore, in order to improve the fault tolerance of quantum computers, a large number of physical qubits must be used to store the logical qubits. In this way, the localized information is delocalized such that the quantum computer is less susceptible to localization errors and the performance of measurements in the eigenbasis of the qubits, similar to the parity checks of classical computers, thereby resulting in more fault-tolerant qubits. Summary of the Invention

[0005] In one embodiment, a resonator includes a coplanar waveguide (CPW) structure having a first width and including a first end configured to be coupled to a first qubit, a middle portion having a second width narrower than the first width, and a second end having a third width wider than the second width and configured to be coupled to a second qubit.

[0006] In one embodiment, the first width is substantially equal to the third width.

[0007] In one embodiment, the middle portion of the CPW structure is folded back.

[0008] In one embodiment, at least one of the first end or the second end has an S-shaped structure. The S-shaped structure can surround one or more bonding structures. At least one of the one or more bonding structures can be an under bump metallurgy (UBM).

[0009] In one embodiment, the first width and the third width are based on widths that provide capacitance and inductance at the first end and second end, respectively, that increase the frequency of modes above the fundamental frequency of the CPW structure.

[0010] In one embodiment, the coupling between the first qubit and the first end is capacitive.

[0011] In one embodiment, the length of the first end and the length of the second end are each less than the length of the middle portion.

[0012] In one embodiment, the inductance of the intermediate portion is higher than the inductance of each of the first end and the second end, and the capacitance of the intermediate portion may be lower than the capacitance of each of the first end and the second end.

[0013] In one embodiment, the first mode of the resonator is above twice the fundamental frequency of the resonator.

[0014] According to one embodiment, a quantum bus system includes a first qubit and a second qubit, a CPW structure including a first end having a first width coupled to the first qubit, a middle portion having a second width narrower than the first width, and a second end having a third width wider than the second width coupled to the second qubit.

[0015] In one embodiment, the middle portion of the CPW structure is folded back and at least one of the first end or the second end has an S-shaped structure. The S-shaped structure may surround one or more bonding structures.

[0016] In one embodiment, the first width and the third width are based on widths that result in capacitance and inductance at the first end and second end, respectively, that increase the frequency of modes above the fundamental frequency of the CPW structure.

[0017] In one embodiment, the coupling between the first qubit and the first end is capacitive.

[0018] In one embodiment, the length of the first end and the length of the second end are each less than the length of the middle portion.

[0019] In one embodiment, the inductance of the intermediate portion is higher than the inductance of each of the first end and the second end, and the capacitance of the intermediate portion is lower than the capacitance of each of the first end and the second end.

[0020] In one embodiment, the first mode of the resonator is above twice the fundamental frequency of the resonator.

[0021] According to one embodiment, a method of coupling a qubit includes coupling a first end of a CPW structure to a first qubit, a second end of the CPW structure to a second qubit, and a middle portion of the CPW structure having a width narrower than a width of the first end and a width of the second end.

[0022] In one embodiment, the frequency of a mode above the fundamental frequency of the CPW structure is increased by adjusting the geometry of the first and second ends of the CPW structure.

[0023] In one embodiment, the coupling between the first qubit and the first end is capacitive.

[0024] In one embodiment, the first mode of the CPW structure is configured to be at least twice the fundamental frequency of the resonator.

[0025] These and other features will become apparent from the following detailed description of illustrative embodiments, which is to be read in conjunction with the accompanying drawings.

[0026] The drawings are diagrams of example embodiments. The drawings do not depict all embodiments. Other embodiments may be used in addition or instead of these. Obvious or unnecessary details may be omitted to save space or for a more effective illustration. Some embodiments may be practiced with additional components or steps, or without all components or steps shown, or both. The same numbers in different drawings refer to the same or similar components or steps. [Brief description of the drawings]

[0027] [Figure 1A] FIG. 2 illustrates an example architecture of a coplanar waveguide transmission line resonator according to an example embodiment. [Figure 1B] FIG. 1 illustrates a symbolic representation of a coplanar waveguide transmission line coupled between two load impedances. [Diagram 2] FIG. 1 illustrates an example resonator for transmitting half and full wavelengths. [Figure 3A] FIG. 1 illustrates the resonance of a conventional coplanar waveguide transmission line resonator. [Figure 3B] FIG. 1 illustrates resonance of a coplanar waveguide transmission line resonator using a paddle structure in accordance with an example embodiment. [Figure 4A] 1 is a graph showing selected fundamental frequencies of a resonator. [Figure 4B] 4B is a graph showing the first harmonic of the resonator of FIG. 4A. [Diagram 5] FIG. 2 illustrates an example resonator structure having a middle portion and two ends having a width greater than the width of the middle portion, according to one example embodiment. [Figure 6] FIG. 1 illustrates an example coplanar waveguide transmission line resonator having wraparound ends for under-bump metallization that can be used for flip-chip packaging, according to an example embodiment. [Figure 7]FIG. 1 illustrates an example floor plan of a multi-qubit computing system, according to an example embodiment. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0028] In the following detailed description, by way of example, numerous specific details are set forth in order to provide a thorough understanding of the relevant teachings. However, it will be apparent that the present teachings may be practiced without such details. In other instances, well-known methods, procedures, components, or circuits, or combinations thereof, have been described in a relatively general manner without detailed description so as not to unnecessarily obscure aspects of the present teachings.

[0029] The present disclosure relates generally to superconducting devices, and more particularly to efficient readout of interconnected quantum bits in a quantum processor. The connections between quantum bits, sometimes referred to herein as qubits, are typically mediated by bus resonators. Presented herein are resonators that have relatively low loss and reduce the effects of undesired extraneous modes, thereby improving the quality of qubit readout. Additionally, the bus resonators occupy a relatively small area, such that the total area of ​​the entire quantum processor is reduced. Such size minimization allows the quantum processor to operate at higher frequencies.

[0030] Example Architecture FIG. 1A shows an example architecture 100 of a coplanar waveguide (CPW) transmission line resonator according to an example embodiment. FIG. 1B shows a symbolic representation of a CPW transmission line coupled between two load impedances ZL, respectively. The load impedances ZL represent a combined load of a coupling capacitor and a wide portion (L1) of the transmission line. The resonator 100 can be used to connect qubits to each other. The connection is necessary to create entanglement between the qubits. For example, there may be a grid of qubits connected to each other using a CPW transmission line resonator between each pair of qubits. The resonator 100 allows these qubits to interact with each other by providing the necessary coupling for entanglement of the qubits.

[0031] The architecture 100 illustrates a coplanar waveguide structure including an intermediate portion 104, sometimes referred to herein as a main portion, between a first end 102 and a second end 106. The first end 102 may be coupled (e.g., capacitively or inductively) to a first qubit, while the second end 106 may be coupled (e.g., capacitively or inductively) to a second qubit. As shown, the main portion 104 is of a suitable length L 2 While the first and second ends 102 and 106 may each have a different length, in one embodiment, the first end 102 and the second end 106 have a similar length L 1 In one embodiment, the length L 1 Length L for 2 The ratio of the width W of the first end 102 is at least 0.2. 1 and the width W of the second end 106 3 are the width W of the intermediate portion 104, 2 In one embodiment, the widths of the first end 102 and the second end are equal (i.e., W 1 =W 3 ).

[0032] In one embodiment, the actual dimensions of the midsection and end sections are based on the midsection 104 having an inductance that is a predetermined multiple (e.g., twice) that of the end sections 102, 106. The dimensions of the midsection and end sections are further based on the midsection having a capacitance that is a predetermined fraction (e.g., half) that of the end sections 102, 106. Thus, the inductance and capacitance of the resonator are different in different parts of the resonator. The resonator 100 transmits standing microwave modes. The effective (i.e., lumped) circuit parameters of the resonator 100 are obtained by multiplying the waveform by the inductance L or capacitance C per unit length. By optimizing the inductance and capacitance per unit length along the length of the transmission line 170, a bus resonator with less interference with higher order modes is obtained. The following equations can be used to obtain the equations for the resonance conditions:

[0033]

number

[0034] Where:

number

[0035] The resonance condition for the fundamental mode can be written as follows:

[0036]

number

[0037] The resonance condition for the first harmonic can be written as follows:

[0038] Z 2 tan(β 1 l 1 )=Z 1 tan(β 2 l 2 / 2) (Formula 5)

[0039] In the above equations, we use the fact that the imaginary part of the impedance seen from the center of the resonator structure is zero for the fundamental mode, and the imaginary part of the admittance is zero for the first excitation mode. Equations 4 and 5 can be easily solved numerically.

[0040] To better understand the features of this novel resonator 100, it will be useful to describe the general operation of a resonator. The resonant frequencies of a resonator, also referred to herein as normal modes or simply modes, are typically equally spaced multiples (harmonics) of a lowest frequency, referred to as the fundamental frequency. For example, for a typical λ / 2 resonator, there are multiple modes on the transmission line implementing the resonator, including the fundamental frequency mode and additional frequencies based on multiples of two from the fundamental frequency. Unlike conventional resonators whose modes are simple multiples of the fundamental frequency, the use of the "paddle" structure of the resonator 100 shifts the higher order modes further (e.g., by a factor of 2.3 or more), thereby substantially reducing the effect of the higher order modes on qubit coupling. For example, for a half-wave resonator, the frequency of the second mode is pushed even higher than twice the fundamental frequency. Moving the frequency of the higher order modes further away (i.e., higher) from the fundamental frequency reduces the unwanted Purcell loss experienced by the qubits and reduces the spurious coupling between qubits introduced by the higher resonant frequency.

[0041] Reference is now made to FIG. 2, which shows an example resonator transmitting half wavelength 220 and full wavelength 230. The standing waves on the resonator, which acts as a transmission line, are only multiples of half wavelengths. As shown in FIG. 2, the width of the resonator is narrower at the middle section 204 than at the ends 202 and 206, thereby forming a "paddle" structure for the resonator. In this way, the speed of the traveling wave passing between both ends of this transmission line changes. The change in boundary conditions depends on the appearance of the standing wave and affects the fundamental mode from the second excited mode.

[0042] 3A and 3B show the resonance of a conventional CPW transmission line resonator 300A and a CPW transmission line resonator 300B using a paddle structure, respectively. For purposes of illustration, chart 300A is based on a capacitance of 165e-12 F / m and an inductance of 2*pi*1e-7 H / m. Chart 300A shows a fundamental frequency (e.g., fundamental mode) that is approximately 7 GHz and a first harmonic that is twice the frequency, i.e., 1.5 GHz. For comparison, FIG. 3B shows a chart based on a resonator with a paddle structure, where the first and second ends of the resonator, respectively, have higher capacitance but lower inductance than the middle portion. For purposes of illustration and not limitation, C 1 =165e-12 / m, L 1 =2*pi*1e-7 / m, C 2 = 105e-12 / m, and L 2 =4*pi*1e-7 / m. This chart shows that there is a significant increase in the separation between the fundamental frequency and the first harmonic (i.e., 2.4 times the frequency in the example of Figure 3B). Note that while the fundamental frequency is shown shifted to a lower frequency, the fundamental frequency can be set (i.e., controlled) by appropriate selection of the capacitance and inductance of each portion of the resonator. For example, the fundamental frequency can be maintained at 7 GHz, lowered below 7 GHz, or increased based on design choice.

[0043] A notable feature of the present architecture is that the higher order modes are shifted in frequency above a factor of 2 (2.4 in the example shown in Figure 3B). The further apart the modes are in frequency, the lower the stray coupling between the fundamental frequency and the corresponding harmonics, thereby providing substantially better readout of the qubit of interest.

[0044] 4A-B show an example transmission line used in the simulator to select the fundamental frequency and the first harmonic, respectively. In the example of FIG. 4A-B, the total length of the resonator, including the wide and narrow portions, is 7 mm. FIG. 4A shows that the fundamental frequency of 6.4 GHz was selected. In contrast to a conventional resonator with the first harmonic at 12.8 GHz, the high frequency finite element simulation software (HFSS) shows that by using the paddle structure described herein, the first harmonic is significantly shifted to 17.5 GHz, thereby providing an additional nearly 5 GHz separation from the fundamental frequency.

[0045] Reference is now made to FIG. 5, which illustrates an exemplary resonator structure having a middle portion 510 and two ends 502 and 530 with a width greater than the width of the middle portion 510, according to one exemplary embodiment. In the embodiment of FIG. 5, the resonator 500 is folded back to provide a compact design. More specifically, the middle portion 510 is folded back, such as with a serpentine structure, thereby reducing the footprint. In one embodiment, the structure of the narrow portion of the central strip is selected to be significantly smaller than 10 μm (e.g., 2 μm) and the wide portion of the central strip is selected to be significantly wider than 10 μm (e.g., 30 μm). The narrow portion of the resonator is close to half of the total length, leaving both ends wider, each close to a quarter of the total length. The width of the narrow portion is in the range that can be reliably manufactured using optical lithography. Even narrower centerlines (sub-μm) can also be used using other manufacturing techniques, such as electron beam lithography. Leaving open areas at the ends provides better modal suppression of the slotline modes with fewer ground points. In one embodiment, the folded intermediate portion, which corresponds to the guide portion, can be tightly meandered with a pitch of up to 60 μm to minimize the occupied area.

[0046] In some embodiments, the first end 502 and / or the second end 530 may have an "S" structure. For example, the S structure may be used to surround the connection points 540(A)-540(D). For example, each of the connection points 540(A)-540(D) ​​may receive a wire bond via one or more pads, bumps, wire bonds, etc. It is noted that the resonator 500 may also be used in bump-bonding packaging. In this regard, FIG. 6 illustrates an example CPW transmission line resonator having ends that wrap around an under-bump metallization (UBM) that may be used in flip-chip packaging, according to an example embodiment. In this manner, the large open areas at the ends of the resonator 600 may receive the relatively large bump bonds 602(A)-602(D), respectively. The UBM may be deposited on each area of ​​the resonator structure 600 and lifted off as appropriate.

[0047] 5, in one embodiment, intermediate portion 510 has an inductance that is at least twice the inductance of ends 502 and 530. Alternatively or additionally, intermediate portion 510 has a capacitance that is at least half the capacitance of each of ends 502 and 530.

[0048] FIG. 7 illustrates an example floor plan of a multi-qubit computing system 700 according to one example embodiment. The floor plan 700 includes a qubit at each corner, with each pair of qubits coupled to each other by a corresponding resonator. For example, each qubit (704(A)-704(D)) may be a transmon qubit, which is a superconducting charge qubit with reduced susceptibility to charge noise. In some embodiments, the two superconductors are capacitively shunted to reduce susceptibility to charge noise. Transmons achieve reduced sensitivity to charge noise by increasing the ratio of Josephson energy to charge energy, which can be achieved by using large shunt capacitors.

[0049] As shown in Figure 7, each pair of qubits are coupled to each other via a CPW transmission line resonator having a structure described herein. For example, qubit 704(A) and qubit 704(B) are coupled to each other by resonator 706, qubit 704(B) and qubit 704(D) are coupled to each other by resonator 708, qubit 704(D) and qubit 704(C) are coupled to each other by resonator 710, and qubit 704(A) and qubit 704(C) are coupled to each other by resonator 712. In various embodiments, each pair of qubits is capacitively or inductively coupled to an adjacent resonator. Use of the architecture of Figure 7 provides a highly area-efficient, interference-free multi-qubit computing system 700.

[0050] conclusion The description of various embodiments of the present teachings has been presented for illustrative purposes, but is not intended to be exhaustive or limited to the disclosed embodiments. Many modifications and variations will be apparent to those skilled in the art without departing from the scope and spirit of the disclosed embodiments. The terms used in this specification have been selected to best explain the principles of the embodiments, practical applications, or technical improvements to the art found in the market, or to enable those skilled in the art to understand the embodiments disclosed herein.

[0051] While what is described above are considered to be the best mode and / or other embodiments, it should be understood that various modifications may be made thereto, that the subject matter disclosed herein can be implemented in various forms and embodiments, and that the teachings are applicable to many applications, only some of which are described herein. It is intended by the following claims to claim all applications, modifications and variations that fall within the true scope of the teachings presented.

[0052] The components, steps, features, objects, advantages and benefits described herein are merely illustrative. None of them, nor any description thereof, are intended to limit the scope of patent protection. Although various advantages have been described herein, it should be understood that not all embodiments necessarily include all advantages. Unless otherwise specified, all measurements, values, ratings, positions, dimensions, sizes and other specifications described herein, including the following claims, are approximate and not precise. They are intended to have a reasonable range consistent with the functions to which they relate and what is customary in the technical field to which they pertain.

[0053] Many other embodiments are contemplated, including those that have fewer, additional, or different, or combinations of, components, steps, features, objects, advantages, and benefits, including those that differ in the arrangement and / or ordering of components and / or steps.

[0054] Although the above has been described in conjunction with exemplary embodiments, it should be understood that the term "exemplary" means only an example, not best or optimal. Unless directly stated above, nothing described or illustrated is intended to, and should not be construed as, providing to the public any component, step, feature, object, advantage, benefit, or equivalent, whether claimed or not.

[0055] The terms and expressions used herein shall be understood to have the ordinary meaning given to such terms and expressions with respect to their respective fields of study and research, unless a specific meaning is otherwise stated herein. Relative terms such as first and second may be used only to distinguish one entity or operation from another, without necessarily requiring or implying any actual such relationship or order between the entities or operations. The terms "comprises," "comprises," or other variations thereof are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that includes a list of elements may include not only those elements, but also other elements not expressly listed or inherent to such process, method, article, or apparatus. An element preceded by "a" or "an" does not, without further constraints, preclude the presence of additional identical elements in the process, method, article, or apparatus that includes the element.

[0056] The Abstract is provided to enable the reader to quickly grasp the nature of the technical disclosure. It is submitted with the understanding that it will not be used to interpret or limit the scope or meaning of the claims. Moreover, in the above Detailed Description, it can be seen that various features have been grouped together in various embodiments to streamline the disclosure. This method of disclosure should not be interpreted as reflecting an intention that the claimed embodiments have more features than are expressly recited in each claim. Instead, as the following claims reflect, inventive subject matter lies in some but not all features of a single disclosed embodiment. Thus, the following claims are hereby incorporated into the Detailed Description, with each claim standing on its own as separately claimed subject matter.

Claims

1. 1. A resonator including a coplanar waveguide (CPW) structure, The CPW structure is a first end having a first width and configured to be coupled to a first qubit; an intermediate portion having a second width narrower than the first width; a second end having a third width greater than the second width and configured to be coupled to a second qubit; wherein the intermediate portion is between the first end and the second end.

2. The resonator of claim 1 , wherein the first width is substantially equal to the third width.

3. The resonator of claim 1 or 2, wherein the intermediate portion of the CPW structure is folded back.

4. 4. The resonator of claim 1, wherein at least one of the first end or the second end has an S-shaped configuration.

5. The resonator of claim 4 , wherein the S-shaped structure surrounds one or more coupling structures.

6. The resonator of claim 5 , wherein at least one of the one or more coupling structures is an under bump metallurgy (UBM).

7. 7. The resonator of claim 1, wherein the first width and the third width are based on widths that result in capacitance and inductance at the first end and the second end, respectively, that increase the frequency of harmonic modes exhibited by the CPW structure.

8. 8. The resonator of claim 1, wherein the coupling between the first qubit and the first end is capacitive.

9. 9. The resonator according to claim 1, wherein the length of said first end portion and the length of said second end portion are each shorter than the length of said intermediate portion.

10. 10. The resonator of claim 1, wherein an inductance of the intermediate portion is higher than an inductance of each of the first end and the second end.

11. The resonator of claim 10 , wherein the capacitance of the middle portion is lower than the capacitance of each of the first end and the second end.

12. 12. A resonator according to any preceding claim, wherein the frequency of a first harmonic mode exhibited by the CPW structure is above twice the fundamental frequency of a fundamental mode exhibited by the CPW structure.

13. A first qubit; and a second qubit; and Coplanar Waveguide (CPW) Structure 1. A quantum bus system comprising: The CPW structure is a first end having a first width and coupled to the first qubit; an intermediate portion having a second width narrower than the first width; a second end coupled to the second qubit, the second end having a third width greater than the second width; wherein the intermediate portion is between the first end and the second end.

14. the intermediate portion of the CPW structure is folded back; The system of claim 13 , wherein at least one of the first end or the second end has an S-shaped configuration.

15. The system of claim 14 , wherein the S-shaped structure surrounds one or more bonding structures.

16. 16. The system of claim 13, wherein the first width and the third width are based on widths that result in capacitance and inductance at the first end and the second end, respectively, that increase the frequency of harmonic modes exhibited by the CPW structure.

17. 17. The system of claim 13, wherein the coupling between the first qubit and the first end is capacitive coupling.

18. 18. The system of claim 13, wherein the length of the first end and the length of the second end are each less than the length of the intermediate portion.

19. an inductance of the intermediate portion is higher than an inductance of each of the first end portion and the second end portion; 19. The system of claim 13, wherein the capacitance of the intermediate portion is lower than the capacitance of each of the first end and the second end.

20. 20. A system according to any one of claims 13 to 19, wherein the frequency of a first harmonic mode exhibited by the CPW structure is more than twice the fundamental frequency of a fundamental mode exhibited by the CPW structure.

21. 1. A method of coupling qubits, comprising the steps of: Coupling a first end of a coplanar waveguide (CPW) structure to a first qubit; coupling a second end of the CPW structure to a second qubit; and providing an intermediate portion of the CPW structure between the first end and the second end such that the intermediate portion has a width narrower than a width of the first end and a width of the second end.

22. 22. The method of claim 21, further comprising increasing a frequency of a harmonic mode exhibited by the CPW structure by adjusting a geometry of the first end and the second end of the CPW structure.

23. 23. The method of claim 21 or 22, wherein the coupling between the first qubit and the first end is capacitive coupling.

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