Methods and arrangements for optimally designing complex resonator networks

Abstract

To fabricate a quantum microwave circuit, a network model is provided. Each resonator element of the network model is characterized by one or more respective physical quantities that define the contribution of the respective resonator element to one or more resonator modes of the microwave circuit. The values ​​of the respective parameters (x) constitute a vector (1), whose initial form is (2). The characteristic of the microwave circuit in the i-th resonator mode as a quantity s i The real part is defined relative to the target damping constant and the imaginary part relative to the target resonant frequency of the respective resonator mode. Starting from the initial form (2) above, a numerical optimization method finds the vector (1) that gives an extremum of the objective function that depends on the quantities above. A physical instance of the microwave circuit is fabricated such that each physical quantity has the found value of the parameter (x).

Description

[Technical Field]

[0001] This disclosure relates to quantum computing. More specifically, this disclosure relates to methods and arrangements for optimally designing complex resonator networks for use in quantum computing systems. Furthermore, this disclosure relates to quantum computing systems. [Background technology]

[0002] A quantum computing system comprises one or more quantum processing units (QPUs), which are placed in a cryostat to operate at temperatures very close to absolute zero. A QPU typically includes multiple qubits and auxiliary circuits, such as multiple microwave resonator elements, built on one or more substrate materials, including sapphire, fused silica, or crystalline quartz. Often, the microwave resonator elements form a complex resonator network that must be carefully designed to exhibit the required frequency characteristics. Parameters of concern include the target frequency and lifetime, the latter of which is sometimes said to represent the linewidth in the frequency spectrum.

[0003] Figure 1 shows a network model of a simple microwave circuit. A transmission line 101 runs between the input port 102 and the output port 103. Coupled to the transmission line 101 are three resonator elements 104, 105, and 106. Each of the resonator elements 104, 105, and 106 is characterized by one or more respective physical quantities that define the contribution of each resonator element to one or more resonator modes of the microwave circuit. Examples of such physical quantities include, but are not limited to, the physical dimensions of the resonator element components, the length of the transmission line portion from the coupling point of the resonator elements to the input port 102 and the output port 103, and the distance between adjacent resonator element components.

[0004] The values ​​of the aforementioned physical quantities may be considered degrees of freedom or free parameters in the overall problem of ensuring that the microwave circuit performs its intended task as effectively as possible. Because the effects of various free parameter values ​​are intricately intertwined, finding the optimal combination of parameter values ​​is not straightforward. [Overview of the Initiative] [Problems that the invention aims to solve]

[0005] This summary is provided to introduce in a simplified form the set of concepts described further below in a mode for carrying out the invention. This summary is not intended to identify any important or essential features of the claims, nor is it intended to be used to limit the scope of the claims.

[0006] The objective is to provide methods and configurations for optimally designing complex resonator networks for efficient use in quantum computing systems. These methods and configurations should enable the accurate and computationally efficient discovery of the optimal combination of free parameter values ​​in the design. [Means for solving the problem]

[0007] These and further advantageous objectives are achieved by solving a multidimensional optimization problem in which the impedance of a microwave circuit is evaluated in the Laplace domain using complex-valued frequencies, the real part of which is the target decay constant of a suitable resonator mode and the imaginary part of which is the corresponding target resonant frequency. The objective function, from which the extrema will be found by numerical optimization, depends on both the complex-valued frequencies and the free parameter values.

[0008] According to a first aspect, a method for manufacturing a microwave circuit for use in a quantum computing system is provided. The method includes providing a network model of the microwave circuit. The network model includes at least a plurality of resonator elements, each of the resonator elements being characterized by one or more respective physical quantities that define the contribution of each resonator element to one or more resonator modes of the microwave circuit. The method includes representing the values of the physical quantities by respective parameters (x) that together constitute a vector

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[0009] ​ According to an embodiment, the quantity is the impedance of the microwave circuit in the i-th resonator mode

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[0010] According to an embodiment, the impedance of the microwave circuit in the i-th resonator mode

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[0011] According to an embodiment, using the numerical optimization method

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[0012] According to an embodiment, the microwave circuit includes k ports, where k is a positive integer. And the impedance between the n-th port and the m-th port of the microwave circuit is the matrix element Z of a k×k square matrix such that n ∈ [1, k] and m ∈ [1, k] nm and may be described as. The method may be performed on the diagonal elements (Z nn ) of the matrix.

[0013] According to an embodiment, the quantity is the admittance of the microwave circuit in the i-th resonator mode

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[0014] According to an embodiment, the microwave circuit includes k ports, where k is a positive integer. The admittance between the n-th port and the m-th port of the microwave circuit is the matrix element Y of a k×k square matrix such that n ∈ [1, k] and m ∈ [1, k] n,m and may be described as. And the method may be performed on the diagonal elements (Y n,n ) of the matrix.

[0015] According to an embodiment, using the numerical optimization method may

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[0016] According to one embodiment, the microwave circuit includes k ports, where k is a positive integer. The quantity may be the scattering characteristic S of the microwave circuit, where 1 ≤ n ≤ k and 1 ≤ m ≤ k is the scattering parameter between the nth port and the mth port in the ith resonator mode.

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[0017] The accompanying drawings, included to provide a further understanding of the present invention and constituting part of this specification, illustrate embodiments of the present invention and, together with the description, aid in explaining the principles of the present invention. [Brief explanation of the drawing]

[0018] [Figure 1] This figure shows a network model of a simple microwave circuit. [Figure 2] This diagram shows the method. [Modes for carrying out the invention]

[0019] In the following description, the accompanying drawings are referenced, which form part of the disclosure and illustrate certain embodiments in which the disclosure may be incorporated. It is understood that other embodiments may be used, and structural or logical modifications may be made, without departing from the scope of the disclosure. Therefore, since the scope of the disclosure is defined by the accompanying claims, the following detailed description should not be understood as restrictive.

[0020] For example, disclosures relating to a described method may also apply to a corresponding device or system configured to perform the method, and vice versa. For instance, if a step of a particular method is described, a corresponding device may include a unit for performing the step of the described method, even if such a unit is not explicitly described or illustrated in the figures. Conversely, if a particular apparatus is described based on a functional unit, a corresponding method may include a step for performing the described function, even if such a step is not explicitly described or illustrated in the figures. Furthermore, it is understood that features of the various exemplary embodiments described herein may be combined with each other unless otherwise specified.

[0021] A microwave circuit generally has k ports, as shown in the network model in Figure 1, for example, where k is a positive integer. In Figure 1, k=2. Of interest is the impedance of any part of the microwave circuit, i.e., the impedance of any of the ports, or the impedance between any pair of ports. Generally, the impedance between the nth port and the mth port of a microwave circuit is given by the matrix element Z of a k×k square matrix such that n∈[1, k] and m∈[1, k]. nm It is described as follows: If admittance Y or scattering characteristic S is considered instead of impedance Z, the matrix element Y of the admittance matrix n,m or the matrix element S of the scattering matrix n,mHowever, it may be used to describe a typical k-port microwave network.

[0022] Frequency itself is a real-valued quantity. Therefore, the frequency characteristics of a microwave circuit are often described using graphs that show parameter values, such as the S21 scattering parameter or other S-parameters, or the impedance Z, as a function of frequency. Such frequency characteristics of a microwave circuit depend on selected values ​​for several physical quantities within the circuit elements, as described above. The values ​​of these physical quantities are represented by their respective parameters x, and they are grouped together to form a vector.

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[0023] However, such methods have been found to be quite irregular and, therefore, can sometimes lead to the use of objective functions in numerical optimization that are not well suited to standard numerical optimization methods.

[0024] Therefore, it is proposed to evaluate the impedance of a microwave circuit in the Laplace domain using a complex-valued frequency s. The real part of s relates to the damping constant of the resonator mode of the microwave circuit, and the imaginary part of s relates to the real-valued resonant frequency.

[0025] Figure 2 schematically illustrates a method for fabricating a microwave circuit for use in a quantum computing system. Step 201 of the method includes selecting a target value for the complex-valued frequency s. Given that the microwave circuit to be designed may have multiple resonator modes of interest, each of the multiple target values ​​s i It is advantageous to define such that the index i takes values ​​from 1 to the total number of resonator modes of interest.

[0026] Step 202 of the method includes providing a network model of a microwave circuit. Very often, experienced designers of microwave circuits can predict with great accuracy the approximate effects of various circuit elements on the frequency characteristics, and therefore it is not unreasonable to assume that, at least qualitatively, the network model provided in step 202 represents a relatively feasible practical implementation. The network model provided in step 202 includes at least several resonant elements. Furthermore, the network model may include other circuit elements such as transmission lines, and / or even optical or (micro)mechanical elements, and / or any combination of any of the elements enumerated above.

[0027] Each of the resonator elements is characterized by one or more respective physical quantities that define the contribution of each resonator element to one or more resonator modes of the microwave circuit. It may be possible to model the influence of other types of circuit elements on resonator modes, not just pure resonator elements, through the respective physical quantities of those circuit elements. Step 203 of the method includes identifying at least the physical quantities related to the resonator elements and expressing the values ​​of these physical quantities in terms of their respective parameters. These parameters are represented above as parameter x, and together they form a vector

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[0028] Step 205 of the method is vector

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[0029] Step 206 of the method includes performing a numerical optimization portion. The impedance Z of the microwave circuit in the i-th resonator mode is:

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[0030] The use of an objective function is typical of numerical optimization methods. The objective function represents a quantity that takes an extremum when the best possible combination of free parameter values ​​is found, due to a chosen form of the objective function. In one embodiment, the use of numerical optimization in step 206 is:

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[0031] Step 207 of the method is the output step of the numerical optimization process, and the target value s is determined according to the numerical optimization. i Provides the best match with (vector)

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[0032] If the numerical optimization in step 206 fails to give a final parameter value x that sufficiently converges to the optimization goal, it is possible to return to step 202, redesign the network model, and then repeat steps 203, 204, 205, and 206. This loop may be repeated until sufficient convergence is achieved.

[0033] The target damping constant for each resonator mode is s i Let the real part be s, and the target resonant frequency of each resonator mode be s i Instead of using it directly as the imaginary part, describe the impedance Z of the microwave circuit in the i-th resonator mode.

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[0034] Returning to the impedance matrix method described earlier, the microwave circuit to be designed may contain k ports, where k is a positive integer. As previously explained, the impedance between the nth port and the mth port of such a microwave circuit is given by the matrix element Z of a k×k square matrix such that n∈[1, k] and m∈[1, k]. nm It may be described as follows. The experiment uses the method described above for the diagonal elements (Z nn Performing the operation on ) is approximately 2 times faster than performing it on the other elements of the matrix. k We showed that the computational cost could potentially be reduced by an order of magnitude.

[0035] As already noted above, instead of impedance, the admittance or scattering characteristics of the microwave network may be considered. In such cases, the use of numerical optimization methods is:

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[0036] It will be apparent to those skilled in the art that, with advances in technology, the basic concept of the present invention may be implemented in various ways. Therefore, the present invention and its embodiments are not limited to the examples described above, but rather may be modified within the scope of the claims. [Explanation of Symbols]

[0037] 101 Transmission line 102 Input Ports 103 Output Ports 104 Resonator element 105 Resonator element 106 Resonator elements

Claims

1. A method for manufacturing microwave circuits for use in quantum computing systems, A step of providing a network model of the microwave circuit, wherein the network model includes at least a plurality of resonator elements, and each of the resonator elements is characterized by one or more respective physical quantities that define the contribution of each resonator element to one or more resonator modes of the microwave circuit. Vector [Math 1] A step in which the value of the physical quantity is expressed by each parameter (x) that together constitutes the physical quantity, The aforementioned vector [Math 2] Initial form [Math 3] The steps include selecting an initial value for the parameter (x) in order to form, The characteristics of the microwave circuit in the i-th resonator mode are expressed as a complex number s having a real part and an imaginary part. i A step in which s is described as a quantity that depends on i The real part of is defined in relation to the target damping constant of each resonator mode, s i The imaginary part of the above is defined in relation to the target resonant frequency of each resonator mode, a step and The aforementioned initial form [Math 4] Starting from, a numerical optimization method is used to obtain the extrema of the objective function that depends on the quantity, the vector [Math 5] The steps include finding the value of the parameter (x) that constitutes the and The steps include: manufacturing a physical instance of the microwave circuit having the found value of the parameter (x) for each of the aforementioned physical quantities; Methods that include...

2. The aforementioned quantity is the impedance of the microwave circuit in the i-th resonator mode. [Math 6] and s i The real part of is the target damping constant for each of the resonator modes, and s i The method according to claim 1, wherein the imaginary part of is the target resonant frequency of each of the resonator modes.

3. The impedance of the microwave circuit in the i-th resonator mode [Number 7] However, the sum over j points around the target i-th resonator mode [Number 8] And each s i,j The real part of is the damping constant of each resonator mode at each j-th point, and each s i,j The method according to claim 2, wherein the imaginary part of is the resonant frequency of each resonator mode at each j-th point.

4. Using the aforementioned numerical optimization method, [Number 9] The vector that minimizes [Number 10] This includes finding the value of the parameter (x) that constitutes the parameter, The method according to any one of claims 1 to 3, wherein the index i varies across multiple resonator modes in the above equation.

5. The microwave circuit includes k ports, where k is a positive integer. The impedance between the nth port and the mth port of the microwave circuit is such that n∈[1, k] and m∈[1, k], and the matrix element Z of the k×k square matrix nm It is described as, The method is performed on the diagonal elements (Z nn ) of the matrix, the method according to any one of claims 1 to 4.

6. The above quantity is the admittance of the microwave circuit in the i-th resonator mode. [Math 11] and s i The real part of is the target damping constant for each of the resonator modes, and s i The method according to claim 1, wherein the imaginary part of is the target resonant frequency of each of the resonator modes.

7. The microwave circuit includes k ports, where k is a positive integer. The admittance between the nth port and the mth port of the microwave circuit is such that n∈[1, k] and m∈[1, k], and the matrix element Y of the k×k square matrix n,m It is described as, The above method involves the diagonal elements (Y n,n The method according to claim 6, which is performed on )

8. Using the aforementioned numerical optimization method, [Math 12] The vector that minimizes [Number 13] This includes finding the value of the parameter (x) that constitutes the parameter, The method according to claim 7, wherein the index i varies across multiple resonator modes in the above equation.

9. The microwave circuit includes k ports, where k is a positive integer. The aforementioned quantity is the scattering characteristic S of the microwave circuit, where 1 ≤ n ≤ k and 1 ≤ m ≤ k, and is the scattering parameter between the nth port and the mth port in the i-th resonator mode. [Number 14] It is represented by a matrix, s i The real part of the above is the target damping constant for each of the resonator modes, s i The imaginary part of the above is the target resonant frequency of each of the resonator modes, Using the aforementioned numerical optimization method, [Number 15] The vector that minimizes [Number 16] This includes finding the value of the parameter (x) that constitutes the parameter, The method according to claim 1, wherein the index i varies across multiple resonator modes in the above equation.

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