DE102021203536B3 - Method for optimizing an arrangement of machine parts using quantum computing - Google Patents

Abstract

A method for optimizing an arrangement of machine parts (disk) using quantum computing, comprising the steps of: providing an optimization function (E0) from entries in the collection of lists as a function of a displacement vector (s→) that rearranges the measured values (Aik) of the respective columns in a describes each of the rows of the collection of lists; Encode the displacement vector(s→) into a quantum register(|s→〉) with the first number (Nobjects) of qudits(|sk〉k), where each of the qudits encodes a kth rearrangement in the displacement vector and the dimensionality of the qudits (|sk〉k) is less than, equal to or greater than the respective second number (NSegm) (V4); providing operators (A^ik) for each of which the quantum register (|s→〉) is an eigenvector; Combining the operators (A^ik) into a Hamiltonian (H0) whose expected value in the quantum register (|s→〉) is equal to the value of the optimization function (E0) over the displacement vector (s→) (V6).

Description

The invention relates to a method for optimizing an arrangement of machine parts using quantum computing.

In the production of transmissions, multi-plate clutches, such as multi-plate clutches, are assembled in one production step. The performance and longevity of the resulting clutch and/or transmission is critically dependent on the discrepancy between the thinnest and thickest longitudinal section of the stacked discs. The task here is to turn the clutch discs and stack them on top of each other in such a way that this discrepancy is minimal.

In the known prior art, this task is solved with classical principles for solving optimization problems, for example with branch-and-bound methods, which solve known knapsack problems, for example. However, these methods become impracticable with increasing combination possibilities due to computational effort and limited computing time.

the DE 10 2008 005 227 A1 discloses a multi-plate clutch. Primary-side and/or secondary-side slats are aligned with respect to each other with respect to their flatness deviations, ie arranged rotated relative to one another in the circumferential direction such that at least the flatness deviations of adjacent slats do not meet in the axial direction.

the DE 10 2009 048 620 A1 discloses a method for aligning the position of a plurality of friction plates relative to one another, the friction plates forming a friction plate set of a friction clutch, in particular a multi-disk friction clutch. According to the invention, a thickness of each friction plate is determined at several positions of the friction surfaces of the friction plates, with tooth positions of each tooth of the friction plate or tooth positions of friction plates to be paired being specified as positions and the respective friction plate or pairs of friction plates being aligned or . will.

the DE 10 2018 109 587 A1 is based on a method for creating at least one clutch disk pack, in particular for a double clutch, having at least one clutch disk unit, which is intended to include at least three clutch disks. It is disclosed that, in at least one method step, a mathematical arrangement and/or selection of all clutch plates provided for the clutch plate unit is carried out, taking into account at least one extreme condition of at least one clutch plate unit parameter of the clutch plate unit.

the U.S. 7,877,333 B2 discloses a method for solving optimization problems using quantum computing.

The invention was based on the object of how solutions to combinatorial arrangement problems based on classical computers can be transferred to a Hamilton operator that can be used for quantum computing.

The subject matter of claim 1 solves this problem.

• • Bereitstellen einer ersten Anzahl der Maschinenteile für ein Maschinenelement;
• • Unterteilen jedes der Maschinenteile jeweils in eine zweite, für das jeweilige Maschinenteil spezifische Anzahl von Abschnitten und Einlesen von jeweils einem Messwert in den jeweiligen Abschnitten unter Verwendung eines Sensors in eine Sammlung von Listen mit der ersten Anzahl an Reihen und den zweiten Anzahlen an Spalten;
• • Bereitstellen einer Optimierungsfunktion aus Einträgen der Sammlung von Listen in Abhängigkeit eines Verschiebungsvektors, der eine Umordnung der Messwerte der jeweiligen Spalten in einer jeden der Zeilen der Sammlung von Listen beschreibt, wobei ein zu berechnender Verschiebungsvektor, durch dessen Umordnung eine Diskrepanz der Messwerte in den jeweiligen Abschnitten der Maschinenteile minimiert wird, die Optimierungsfunktion minimiert;

• • Encodieren des Verschiebungsvektors in ein Quantenregister mit der ersten Anzahl an Qudits, wobei jedes der Qudits eine k-te Umordnung in dem Verschiebungsvektor codiert und die Dimensionalität der Qudits kleiner, gleich oder größer der jeweiligen zweiten Anzahl ist;
• • Bereitstellen von Operatoren, für die das Quantenregister jeweils ein Eigenvektor ist und ein Eigenwert eines der Operatoren jeweils ein von den Messwerten, umfassend die Messwerte mehrerer Abschnitte oder Maschinenteile, berechneter Wert ist;
• • Kombinieren der Operatoren zu einem Hamiltonoperator, dessen Erwartungswert im Quantenregister gleich dem Wert der Optimierungsfunktion über dem Verschiebungsvektor ist;
• • Berechnen eines Zustandes des Hamiltonoperators mittels eines Quantencomputing-Hardwaremoduls, eines von Quantencomputing inspirierten klassischen Hardwaremoduls oder eines von Quantencomputing inspirierten Hardwaremoduls mit besonderer Zweckbestimmung;
• • Decodieren zumindest von Teilen des berechneten Zustandes in die optimierte Anordnung der Maschinenteile und Auslesen der optimierten Anordnung, um die Maschinenteile entsprechend anzuordnen.

One aspect of the invention relates to a method for optimizing an arrangement of machine parts using quantum computing. The procedure includes the steps

• • providing a first number of machine parts for a machine element;
• • dividing each of the machine parts into a second number of sections specific to the respective machine part and reading a measurement value in each of the respective sections using a sensor into a collection of lists with the first number of rows and the second number of columns;
• • Providing an optimization function from entries in the collection of lists as a function of a displacement vector that describes a rearrangement of the measured values of the respective columns in each of the rows of the collection of lists, with a displacement vector to be calculated whose rearrangement causes a discrepancy in the measured values in the respective sections of the machine parts is minimized, the optimization function is minimized;

• • encoding the displacement vector into a quantum register with the first number of qudits, each of the qudits encoding a kth rearrangement in the displacement vector and the dimensionality of the qudits being less than, equal to or greater than the respective second number;
• • providing operators for each of which the quantum register is an eigenvector and an eigenvalue of each of the operators is a value calculated from the measured values comprising the measured values of a plurality of sections or machine parts;
• • Combining the operators into a Hamiltonian whose expectation value in the quantum register is equal to the value of the optimization function over the displacement vector;
• • Compute a state of the Hamiltonian using a quantum computing hardware module, a quantum computing inspired classical hardware module, or a quantum computing inspired special purpose hardware module;
• • Decoding at least parts of the calculated state into the optimized arrangement of the machine parts and reading out the optimized arrangement in order to arrange the machine parts accordingly.

Within the scope of the invention is a computer program for optimizing an arrangement of machine parts using quantum computing. The computer program comprises instructions that cause a quantum computing hardware module, a quantum computing inspired classical hardware module or a quantum computing inspired special purpose hardware module to carry out the steps of a method according to any one of the preceding claims when the computer program runs on one of these hardware modules.

Within the scope of the invention are a computer-readable data carrier on which the computer program is stored and a data carrier signal that represents the computer program. For example, the computer program is transferred to quantum computing hardware modules using the data carrier signal. According to one aspect of the invention, the data carrier signal transfers the computer program to a remote quantum computing hardware module.

Advantageous refinements of the invention result from the definitions, the dependent claims, the drawings and the description of preferred exemplary embodiments.

The machine parts are, for example, clutch discs. The machine element is a clutch, for example. The clutch discs are arranged in a stacked manner. This arrangement gives the clutch. For example, the first number is seven. This means that, for example, a clutch includes seven clutch discs. For example, the second number is 42 for each clutch plate. According to the invention, however, each clutch disc can have a different second number of segments. This means that, for example, each clutch disk is subdivided into 42 segments or into a second number of segments specific to the respective clutch disk. For example, a measured value is obtained in each of these segments for each clutch disc, for example a height/thickness of the respective segment. These readings are then entered into a seven-row, 42-column matrix, or a collection of seven-row, row-specific number-of-column lists.

A rearrangement, including a rotation or translation, of the readings of the respective columns in each of the rows of the matrix or collection of lists corresponds to a rotation of the machine parts.

The measured values are, for example, heights/thicknesses of the individual sections. For example, the measured values are measured with optical sensors. According to one aspect, the clutch disks are measured using a laser measuring machine.

The dimension of the qudit encoding the kth rearrangement in the displacement vector can, but need not, contain the same dimensionality as the number of segments N _Seg (k) of the kth machine part A ^k , k ∈ {1, ..., N _objects }.

According to one aspect of the invention, the dimension of the qudit is greater than the number N _Segm (k), which can be advantageous for hardware reasons.

According to a further aspect of the invention, the dimension of the qudit is smaller than the number N _segm (k) in order to optimally utilize the available qubits of a quantum computer and only search parts of the solution space.

According to another aspect of the invention, the dimension of the qudit is equal to the number N _segm (k), ie $${|s_k〉}_k,s_k\in \left\{\mathrm{0,}...,N_{\text{segment}}\left(k\right)-1\right\}.$$

The eigenvalues of the operators for which the quantum register is an eigenvector can be, for example, modified measured values including average measured values. A modified measurement is, for example, the modified segment height is B ^k , see below.

According to one aspect of the invention, the eigenvalue of one of the operators is exactly equal to one of the measured values in one of the sections of one of the machine parts.

Quantum computing hardware modules include parts of quantum computers based on the quantum circuit model with quantum gates, disposable quantum computers and adiabatic quantum computers.

Classical hardware modules inspired by quantum computing enable quantum computing on classical hardware, but are limited in terms of problems and size of the problems that can be solved with them due to memory and computing capacities, analogous to classical computer architectures.

Quantum-inspired computing with special-purpose hardware, for example, includes quantum-inspired digital annealer processors that are specifically designed to solve larger and more complex optimization problems. For example, special purpose hardware modules inspired by quantum computing are beneficial for Knapsack problems.

For example, quantum computers consisting of quantum computing hardware modules, e.g. based on the Noisy Intermediate Scale Quantum technology, are not yet ready for industrial applications, but quantum computing-inspired classical hardware modules and quantum computing-inspired special-purpose hardware modules are.

Quantum computing, circuit models with quantum lattices, adiabatic quantum computers and qubits are for example in U.S. 7,877,333 B2 , column 1, line 43, to column 3, line 3. These revelations of U.S. 7,877,333 B2 are incorporated by reference into the disclosure of this application for the meaning of the corresponding terms for this application.

By decoding at least parts of the calculated state, the solution space can already be restricted sufficiently by the information obtained from it to find the complete solution, for example using other methods.

According to one aspect of the invention, the calculated state is completely decoded and read out.

According to one aspect of the invention, for example in connection with adiabatic quantum computers, a ground state of the Hamiltonian is approximated by the calculated state.

$$+{\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{objects}}}{\displaystyle \sum _{i=1}^{N_{\text{Segm}}\left(k\right)}{\displaystyle \sum _{j=1}^{N_{\text{Segm}}\left(k\text{'}\right)}{c}_{i,j,k,k\text{'}}{A}_{w\left(i,k,s_k\right)}^k{A}_{w\left(i,k\text{'},s_{k\text{'}}\right)}^{k\text{'}}}}}$$

gegeben mit Verschichungsvektor $\overrightarrow{s},$

Maschinenteilen A^k, k ∈ {1, ... , N_objects} Messwerten $A_i^k\in ℝ,i\in \left\{\mathrm{1,}\dots ,N_{\text{Segm}}\left(k\right)\right\},w:{\mathbb{N}}_0^3\to N_0^3,\text{und}w\left(\cdot ,k,\cdot \right):{\mathbb{N}}_0^2\to \left\{\mathrm{1,}\dots ,N_{\text{Segm}}\left(k\right)\right\},k-\text{ter}$

Eintrag s_k ∈ {0,... , N_Segm(k) - 1} des Verschiebungsvektors $\overrightarrow{s}$

und Konstanten c₀, c_i,k, c_i,j,k,k' ∈ ℝ.In one aspect, the optimization function is of quadratic order and given by the formula $$E_0\left(\overrightarrow{s}\right)=c_0+{\displaystyle \sum _{k=1}^{N_{\text{objects}}}{\displaystyle \sum _{i=1}^{N_{\text{segment}}\left(k\right)}{c}_{i,k}{A}_{w\left(i,k,s_k\right)}^k}}$$

$$+{\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{objects}}}{\displaystyle \sum _{i=1}^{N_{\text{segment}}\left(k\right)}{\displaystyle \sum _{j=1}^{N_{\text{segment}}\left(k\text{'}\right)}{c}_{i,j,k,k\text{'}}{A}_{w\left(i,k,s_k\right)}^k{A}_{w\left(i,k\text{'},s_{k\text{'}}\right)}^{k\text{'}}}}}$$

given with shift vector $\overrightarrow{s},$

machine parts A ^k , k ∈ {1, ... , N _objects } measured values $A_i^k\in ℝ,i\in \left\{\mathrm{1,}...,N_{\text{segment}}\left(k\right)\right\},w:{\mathbb{N}}_0^3\to N_0^3,\text{and}w\left(\cdot ,k,\cdot \right):{\mathbb{N}}_0^2\to \left\{\mathrm{1,}...,N_{\text{segment}}\left(k\right)\right\},k-\text{ter}$

Entry s _k ∈ {0,... , N _segm (k) - 1} of the displacement vector $\overrightarrow{s}$

and constants c ₀ , c _i,k , c _i,j,k,k' ∈ ℝ.

gegeben mit Verschiebungsvektor $\overrightarrow{s},$

Maschinenteilen A^k, k ∈ {1, ... , N_objects}, Messwerten $A_i^k\in ℝ,i\in \left\{\mathrm{1,}\dots ,N_{\text{Segm}}\right\},w:{\mathbb{N}}_0^3\to N_0^3,\text{und}w\left(\cdot ,k,\cdot \right):{\mathbb{N}}_0^2\to \left\{\mathrm{1,}\dots ,N_{\text{Segm}}\right\},k-\text{ter}$

Eintrag s_k ∈ {0, ... ,N_Segm - 1} des Verschiebungsvektors $\overrightarrow{s}$

und Konstanten c₀, (c_i,k,c_i,j,k,k' ∈ ℝ.In the event that each machine part is divided into the same second number of sections, the optimization function is given by the formula $$E_0\left(\overrightarrow{s}\right)=c_0+{\displaystyle \sum _{k=1}^{N_{\text{objects}}}{\displaystyle \sum _{i=1}^{N_{\text{segment}}}{c}_{i,k}{A}_{w\left(i,k,s_k\right)}^k}}+{\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{objects}}}{\displaystyle \sum _{i,j=1}^{N_{\text{segment}}}{c}_{i,j,k,k\text{'}}{A}_{w\left(i,k,s_k\right)}^k{A}_{w\left(i,k\text{'},s_{k\text{'}}\right)}^{k\text{'}}}}$$

given with displacement vector $\overrightarrow{s},$

Machine parts A ^k , k ∈ {1, ... , N _objects }, measured values $A_i^k\in ℝ,i\in \left\{\mathrm{1,}...,N_{\text{segment}}\right\},w:{\mathbb{N}}_0^3\to N_0^3,\text{and}w\left(\cdot ,k,\cdot \right):{\mathbb{N}}_0^2\to \left\{\mathrm{1,}...,N_{\text{segment}}\right\},k-\text{ter}$

Entry s _k ∈ {0,...,N _Segm - 1} of the displacement vector $\overrightarrow{s}$

and constants c ₀ , (c _i,k ,c _i,j,k,k' ∈ ℝ.

The dimension of the displacement vector is equal to the first number, for example seven. The constants only depend on the indices. The task is to calculate the displacement vector that minimizes the optimization function. The task is not subject to any conditions/constraints because each displacement vector gives a valid configuration.

In another aspect, the quantum register is equal $$|\overrightarrow{s}〉:=\underset{k=1}{\overset{N_{\text{objects}}}{\otimes }}{|s_k〉}_k.$$

Qubits. Für den Fall, dass jedes Maschinenteil in dieselbe Anzahl von Abschnitten unterteilt ist und die Dimensionalität jedes Qubits gleich dieser Anzahl ist, werden N_objectsn_q Qubits benötigt. Nach einem weiteren Aspekt sind die Operatoren gegeben durch $${\widehat{A}}_i^k={\displaystyle \sum _{m=1}^{N_{\text{Segm}}\left(k\right)}{A}_{w\left(i,k,m\right)}^k}|m\rangle {\langle m|}_k\otimes 1_{\mathrm{1,}\dots ,k-\mathrm{1,}k+\mathrm{1,}\dots ,N_{\text{Segm}}\left(k\right)}.$$

The qudit |s _k 〉 _k is represented as a state in a first Hilbert space whose dimension is equal to the second number N _segm (k), say 42 for each of the machine parts. To encode one entry of the displacement vector, n _q (k) = [log ₂ (N _segm (k))] qubits are needed, where n _q (k) is rounded down. The quantum register is represented as a product state in a second Hilbert space, which is a product space of the first Hilbert spaces, with the number of subspaces equal to the first number. For example, the second Hilbert space is a product space of seven of the first Hilbert spaces. The product is a tensor product. So the quantum register requires ${\displaystyle {\sum }_{k=1}^{N_{\text{objects}}}{n}_q\left(k\right)}$

qubits. In case each machine part is divided into the same number of sections and the dimensionality of each qubit is equal to that number, N _objects n _q qubits are needed. According to a further aspect, the operators are given by $${\widehat{A}}_i^k={\displaystyle \sum _{m=1}^{N_{\text{segment}}\left(k\right)}{A}_{w\left(i,k,m\right)}^k}|m〉{〈m|}_k\otimes 1_{\mathrm{1,}...,k-\mathrm{1,}k+\mathrm{1,}...,N_{\text{segment}}\left(k\right)}.$$

gibt den Messwert des k-ten Maschinenteils im i-ten Abschnitt aus. Der Operator ${\widehat{A}}_i^k$

ist unabhängig vom Verschiebungsvektor.The operator ${\widehat{A}}_i^k$

outputs the measured value of the k-th machine part in the i-th section. The operator ${\widehat{A}}_i^k$

is independent of the displacement vector.

$$={\displaystyle \sum _{m=1}^{N_{\text{Segm}}\left(k\right)}{A}_{w\left(i,k,m\right)}^k}|m\rangle {\langle m|}_k\underset{n=1}{\overset{N_{\text{objects}}}{\otimes }}{|s_n\rangle }_n$$

$$={\displaystyle \sum _{m=1}^{N_{\text{Segm}}\left(k\right)}{A}_{w\left(i,k,m\right)}^k}|m\rangle {\langle m|}_k{|s_1\rangle }_1\otimes \cdots \otimes {|s_k\rangle }_k\otimes \cdots \otimes {|s_{N_{\text{objects}}}\rangle }_{N_{\text{objects}}}$$

$$={\displaystyle \sum _{m=1}^{N_{\text{Segm}}\left(k\right)}{A}_{w\left(i,k,m\right)}^k}{\delta }_{m,s_k}|m\rangle {\langle m|}_k{|s_1\rangle }_1\otimes \cdots \otimes {|s_k\rangle }_k\otimes \cdots \otimes {|s_{N_{\text{objects}}}\rangle }_{N_{\text{objects}}}$$

$$=A_{w\left(i,k,s_k\right)}^k\underset{n=1}{\overset{N_{\text{objects}}}{\otimes }}{|s_n\rangle }_n$$

$$=A_{w\left(i,k,s_k\right)}^k|\overrightarrow{s}\rangle$$

The following applies: $${\widehat{A}}_i^k|\overrightarrow{s}〉$$

$$={\displaystyle \sum _{m=1}^{N_{\text{segment}}\left(k\right)}{A}_{w\left(i,k,m\right)}^k}|m〉{〈m|}_k\underset{n=1}{\overset{N_{\text{objects}}}{\otimes }}{|s_n〉}_n$$

$$={\displaystyle \sum _{m=1}^{N_{\text{segment}}\left(k\right)}{A}_{w\left(i,k,m\right)}^k}|m〉{〈m|}_k{|s_1〉}_1\otimes \cdots \otimes {|s_k〉}_k\otimes \cdots \otimes {|s_{N_{\text{objects}}}〉}_{N_{\text{objects}}}$$

$$={\displaystyle \sum _{m=1}^{N_{\text{segment}}\left(k\right)}{A}_{w\left(i,k,m\right)}^k}{\delta }_{m,s_k}|m〉{〈m|}_k{|s_1〉}_1\otimes \cdots \otimes {|s_k〉}_k\otimes \cdots \otimes {|s_{N_{\text{objects}}}〉}_{N_{\text{objects}}}$$

$$=A_{w\left(i,k,s_k\right)}^k\underset{n=1}{\overset{N_{\text{objects}}}{\otimes }}{|s_n〉}_n$$

$$=A_{w\left(i,k,s_k\right)}^k|\overrightarrow{s}〉$$

ist herinitisch, denn $${\left({\widehat{A}}_i^k\right)}^{†}={\displaystyle \sum _{m=1}^{N_{\text{Segm}}\left(k\right)}{\left(A_{w\left(i,k,m\right)}^k\right)}^{\ast }}|m\rangle {\langle m|}_k\otimes 1_{\mathrm{1,}\dots ,k-\mathrm{1,}k+\mathrm{1,}\dots ,N_{\text{Segm}}\left(k\right)}$$

$$={\widehat{A}}_i^k$$

mit reellen Zahlen $A_{w\left(i,k,m\right)}^k.$

The operator ${\widehat{A}}_i^k$

is Herinitic, because $${\left({\widehat{A}}_i^k\right)}^{†}={\displaystyle \sum _{m=1}^{N_{\text{segment}}\left(k\right)}{\left(A_{w\left(i,k,m\right)}^k\right)}^{\ast }}|m〉{〈m|}_k\otimes 1_{\mathrm{1,}...,k-\mathrm{1,}k+\mathrm{1,}...,N_{\text{segment}}\left(k\right)}$$

$$={\widehat{A}}_i^k$$

with real numbers $A_{w\left(i,k,m\right)}^k.$

In addition, these operators commute: $$\left[{\widehat{A}}_i^k,{\widehat{A}}_j^{k\text{'}}\right]=0$$

According to a further aspect, the Hamilton operator is given by $$H_0=c_{0}1+{\displaystyle \sum _{k=1}^{N_{\text{objects}}}{\displaystyle \sum _{i=1}^{N_{\text{segment}}\left(k\right)}{c}_{i,k}{\widehat{A}}_i^k}}+{\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{objects}}}{\displaystyle \sum _{i=1}^{N_{\text{segment}}\left(k\right)}{\displaystyle \sum _{j=1}^{N_{\text{segment}}\left(k\text{'}\right)}{c}_{i,j,k,k\text{'}}{\widehat{A}}_i^k{\widehat{A}}_j^{k\text{'}}}}}$$

Since the individual operators in the Hamiltonian are Hermitian, the Hamiltonian is also Hermitian.

This Hamiltonian can be represented as a matrix with ${\left({\displaystyle {\prod }_{k=1}^{N_{\text{objects}}}{N}_{\text{segment}}\left(k\right)}\right)}^2$

nicht notwendigerweise eine Matrix sind. Es werden aber nur ${\displaystyle {\sum }_{k=1}^{N_{\text{objects}}}{n}_q\left(k\right)}$

n_q(k) Qubits mit n_q(k) = [log₂ (N_Segm(k))] benötigt, um eine derartige Dimensionalität für Quantencomputing zu encodieren. Für den Fall, dass die zweite Anzahl N_Segm für alle Maschinenteile gleich ist, ist der Hamiltonoperator gegeben durch $$H_0=c_{0}1+{\displaystyle \sum _{k=1}^{N_{\text{objects}}}{\displaystyle \sum _{i=1}^{N_{\text{Segm}}}{c}_{i,k}{\widehat{A}}_i^k}}+{\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{objects}}}{\displaystyle \sum _{i,j=1}^{N_{\text{Segm}}}{c}_{i,j,k,k\text{'}}{\widehat{A}}_i^k{\widehat{A}}_j^{k\text{'}}}}$$

Entries, where the operators ${\widehat{A}}_i^k$

are not necessarily a matrix. But only ${\displaystyle {\sum }_{k=1}^{N_{\text{objects}}}{n}_q\left(k\right)}$

n _q (k) qubits with n _q (k) = [log ₂ (N _Segm (k))] are needed to encode such a dimensionality for quantum computing. In the event that the second number N _segm is the same for all machine parts, the Hamiltonian is given by $$H_0=c_{0}1+{\displaystyle \sum _{k=1}^{N_{\text{objects}}}{\displaystyle \sum _{i=1}^{N_{\text{segment}}}{c}_{i,k}{\widehat{A}}_i^k}}+{\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{objects}}}{\displaystyle \sum _{i,j=1}^{N_{\text{segment}}}{c}_{i,j,k,k\text{'}}{\widehat{A}}_i^k{\widehat{A}}_j^{k\text{'}}}}$$

Einträgen. Es werden aber nur N_objectsn_q Qubits benötigt, um eine derartige Dimensionalität für Quantencomputing zu encodieren.This Hamilton operator can be represented as a matrix with ${\left(N_{\text{segment}}^{N_{\text{objects}}}\right)}^2$

entries. However, only N _objects n _q qubits are required to encode such a dimensionality for quantum computing.

$$=\langle \overrightarrow{s}|H_0|\overrightarrow{s}\rangle$$

$$=\langle \overrightarrow{s}|\left(c_{0}1+{\displaystyle \sum _{k=1}^{N_{\text{objects}}}{\displaystyle \sum _{i=1}^{N_{\text{Segm}}\left(k\right)}{c}_{i,k}{\widehat{A}}_i^k}}+{\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{objects}}}{\displaystyle \sum _{i=1}^{N_{\text{Segm}}\left(k\right)}{\displaystyle \sum _{j=1}^{N_{\text{Segm}}\left(k\text{'}\right)}{c}_{i,j,k,k\text{'}}{\widehat{A}}_i^k{\widehat{A}}_j^{k\text{'}}}}}\right)|\overrightarrow{s}\rangle$$

$$=c_0+{\displaystyle \sum _{k=1}^{N_{\text{objects}}}{\displaystyle \sum _{i=1}^{N_{\text{Segm}}\left(k\right)}{c}_{i,k}{A}_{w\left(i,k,s_k\right)}^k}}$$

$$+{\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{objects}}}{\displaystyle \sum _{i=1}^{N_{\text{Segm}}\left(k\right)}{\displaystyle \sum _{j=1}^{N_{\text{Segm}}\left(k\text{'}\right)}{c}_{i,j,k,k\text{'}}{A}_{w\left(i,k,s_k\right)}^k{A}_{w\left(i,k\text{'},s_{k\text{'}}\right)}^{k\text{'}}}}}$$

$$=E_0\left(\overrightarrow{s}\right)$$

The following then applies to the expectation value of the Hamilton operator in the quantum register: $$E\left(|\overrightarrow{s}〉\right)$$

$$=〈\overrightarrow{s}|H_0|\overrightarrow{s}〉$$

$$=〈\overrightarrow{s}|\left(c_{0}1+{\displaystyle \sum _{k=1}^{N_{\text{objects}}}{\displaystyle \sum _{i=1}^{N_{\text{segment}}\left(k\right)}{c}_{i,k}{\widehat{A}}_i^k}}+{\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{objects}}}{\displaystyle \sum _{i=1}^{N_{\text{segment}}\left(k\right)}{\displaystyle \sum _{j=1}^{N_{\text{segment}}\left(k\text{'}\right)}{c}_{i,j,k,k\text{'}}{\widehat{A}}_i^k{\widehat{A}}_j^{k\text{'}}}}}\right)|\overrightarrow{s}〉$$

$$=c_0+{\displaystyle \sum _{k=1}^{N_{\text{objects}}}{\displaystyle \sum _{i=1}^{N_{\text{segment}}\left(k\right)}{c}_{i,k}{A}_{w\left(i,k,s_k\right)}^k}}$$

$$+{\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{objects}}}{\displaystyle \sum _{i=1}^{N_{\text{segment}}\left(k\right)}{\displaystyle \sum _{j=1}^{N_{\text{segment}}\left(k\text{'}\right)}{c}_{i,j,k,k\text{'}}{A}_{w\left(i,k,s_k\right)}^k{A}_{w\left(i,k\text{'},s_{k\text{'}}\right)}^{k\text{'}}}}}$$

$$=E_0\left(\overrightarrow{s}\right)$$

This means that the calculation of the energy of the system in the quantum register is identical to the classical calculation of the optimization function. For quantum computing this means that the calculation of a state is identical to the classical minimization of the optimization function for the corresponding displacement vector.

beschrieben werden mit Normierungsfaktor M, sodass Trρ = 1.In general, all quantum mechanical states can be represented by the density operator $$\rho =\frac{1}{M}{\displaystyle \sum _{\overrightarrow{s}}{a}_{\overrightarrow{s}}}|\overrightarrow{s}〉〈\overrightarrow{s}|$$

can be described with a normalization factor M, so that Trρ = 1.

According to a further aspect, the state of the Hamilton operator is calculated using an adiabatic quantum computer. A quantum mechanical system that is in the ground state, i.e. in the state of minimum energy, of a time-independent system remains in the ground state even if the system changes if the change occurs sufficiently slowly, i.e. adiabatically. The idea of the adiabatic quantum computer is to construct a system that has a ground state that is still unknown at the time, which corresponds to the solution of a certain problem, and another one whose ground state can easily be prepared experimentally. Then, the system, which is easy to prepare, is transferred to the system whose ground state one is interested in, and whose state is then measured. If the transition is slow enough, this is the solution to the problem, see also Edward Farhi, Jeffrey Goldstone, Sam Gutmann, Michael Sipser: Quantum Computation by Adiabatic Evolution; arXiv.org > quant-ph > arXiv:quant-ph/0001106.

According to a further aspect, the state of the Hamiltonian is calculated on a quantum gate-based quantum computer.

• • Darstellen eines initialen Quantenzustands auf einem Quantencomputer;
• • Generieren eines Ansatz-Quantenzustandes durch Anwenden eines Quantengatters auf den initialen Quantenzustand, wobei das Quantengatter durch klassische Parameter parametrisiert ist;
• • Bestimmen des Erwartungswertes des Hamiltonoperators in dem Ansatz-Quantenzustand auf einem Quantencomputer;
• • Aktualisieren der Parameter des Quantengatters, um den Erwartungswert kleiner zu machen, mittels klassischen Optimierungsalgorithmen;
• • Wiederholen der vorangehenden Schritte bis der Erwartungswert konvergiert.

For example, the calculated state is calculated using a variational quantum eigensolver algorithm, VQE algorithm for short. The VQE algorithm is based on quantum mechanical variation methods and divides the calculation of the ground state into first processes that can be calculated using classical computers and second processes that can be calculated using quantum computing. For example, the VQE algorithm is described in Ä variational eigenvalue solver on a photonic quantum processor”, A. Peruzzo et al., Nat. Comm., 5, 4213 (2014) and includes the steps:

• • Representation of an initial quantum state on a quantum computer;
• • generating an initial quantum state by applying a quantum gate to the initial quantum state, the quantum gate being parameterized by classical parameters;
• • determining the expectation value of the Hamiltonian in the approach quantum state on a quantum computer;
• • Updating the parameters of the quantum gate to make the expectation value smaller using classical optimization algorithms;
• • Repeat the previous steps until the expected value converges.

• • die Maschinenteile N_Disks-Scheilben sind, die gestapelt und zu einer Kupplung als Maschinenelement angeordnet werden;
• • die Abschnitte jeder der Scheiben Segmente sind, wobei alle Scheiben die gleiche Anzahl an Segmenten haben;
• • als Messwerte Höhen der Segmente gemessen werden;
• • eine durch den Verschiehungsvektor bewirkte Umordnung Drehungen der Scheiben entspricht;
• • der Hamiltonoperator durch $$H_0={\displaystyle \sum _{i=1}^{N_{\text{Segm}}}{\left({\displaystyle \sum _{k=1}^{N_{\text{Disks}}}\left({\widehat{A}}_{\text{avg}}-{\widehat{A}}_i^k\right)}\right)}^2}$$
  
  gegeben ist mit $${\widehat{A}}_{\text{avg}}:=\frac{1}{N_{\text{Segm}}}{\displaystyle \sum _{j=1}^{N_{\text{Segm}}}\frac{1}{N_{\text{Disks}}}}{\displaystyle \sum _{i=1}^{N_{\text{Disks}}}{\widehat{A}}_i^j};$$
  
• • die optimierte Anordnung gegeben ist durch einen Stapel, für den Abweichungen der über die Scheiben summierten Höhen in jedem Segment minimal sind.

Another aspect includes the characteristics that

• • the machine parts are N _Disks that are stacked and arranged into a clutch as a machine element;
• • the portions of each of the disks are segments, with all disks having the same number of segments;
• • heights of the segments are measured as measured values;
• • a rearrangement caused by the displacement vector corresponds to rotations of the disks;
• • the Hamilton operator through $$H_0={\displaystyle \sum _{i=1}^{N_{\text{segment}}}{\left({\displaystyle \sum _{k=1}^{N_{\text{disks}}}\left({\widehat{A}}_{\text{avg}}-{\widehat{A}}_i^k\right)}\right)}^2}$$
  
  is given with $${\widehat{A}}_{\text{avg}}:=\frac{1}{N_{\text{segment}}}{\displaystyle \sum _{j=1}^{N_{\text{segment}}}\frac{1}{N_{\text{disks}}}}{\displaystyle \sum _{i=1}^{N_{\text{disks}}}{\widehat{A}}_i^j};$$
  
• • the optimized arrangement is given by a stack for which deviations of the heights summed over the slices are minimal in each segment.

For example, N _Disks = N _objects .

The  _avg operator returns the average height across all machine parts slices.

In the manufacture of clutches, the performance and longevity of the resulting clutch is critically dependent on the discrepancy between the thinnest and thickest longitudinal section of the stacked clutch plates. The classic task of turning the clutch disks and stacking them on top of each other in such a way that this discrepancy is minimal is thus solved according to the invention by means of quantum computing.

wobei $A_i^k$

der Messwert des k-ten Maschinenteils im i-ten Abschnitt ist, beispielsweise die Höhe/Dicke dieses Abschnittes, und $$A_{\text{avg}}=\frac{1}{N_{\text{Segm}}}{\displaystyle \sum _{i=1}^{N_{\text{Segm}}}\frac{1}{N_{\text{Disks}}}}{\displaystyle \sum _{k=1}^{N_{\text{Disks}}}{A}_i^k}.$$

In the classic case, this task corresponds to a minimization of the function. $$E_0={\displaystyle \sum _{i=1}^{N_{\text{segment}}}{\left({\displaystyle \sum _{k=1}^{N_{\text{disks}}}\left(A_{\text{avg}}-A_{{\pi }_k\left(i\right)}^k\right)}\right)}^2},$$

whereby $A_i^k$

is the measured value of the k-th machine part in the i-th section, for example the height/thickness of this section, and $$A_{\text{avg}}=\frac{1}{N_{\text{segment}}}{\displaystyle \sum _{i=1}^{N_{\text{segment}}}\frac{1}{N_{\text{disks}}}}{\displaystyle \sum _{k=1}^{N_{\text{disks}}}{A}_i^k}.$$

$$=\frac{1}{N_{\text{Segm}}}{\displaystyle \sum _{i=1}^{N_{\text{Segm}}}\frac{1}{N_{\text{Disks}}}}{\displaystyle \sum _{k=1}^{N_{\text{Disks}}}{A}_{w\left(i,k,s_k\right)}^k}|\overrightarrow{s}\rangle$$

$$=A_{\text{avg}}1|\overrightarrow{s}\rangle$$

The following applies: $${\widehat{A}}_{\text{avg}}|\overrightarrow{s}〉=\frac{1}{N_{\text{segment}}}{\displaystyle \sum _{i=1}^{N_{\text{segment}}}\frac{1}{N_{\text{disks}}}}{\displaystyle \sum _{k=1}^{N_{\text{disks}}}{\widehat{A}}_i^k}|\overrightarrow{s}〉$$

$$=\frac{1}{N_{\text{segment}}}{\displaystyle \sum _{i=1}^{N_{\text{segment}}}\frac{1}{N_{\text{disks}}}}{\displaystyle \sum _{k=1}^{N_{\text{disks}}}{A}_{w\left(i,k,s_k\right)}^k}|\overrightarrow{s}〉$$

$$=A_{\text{<figure-callout id="1" label="avg" filenames="DE102021203536B3\_0144.png,DE102021203536B3\_0146.png" state="{{state}}">avg</figure-callout>}}1|\overrightarrow{s}〉$$

This means that in the quantum register the value  _{avg is} an eigenvalue of the operator  _avg .

= (s₀,...,sN_Disks-1)^T wie folgt zusammen: $${\pi }_k\left(i\right)=\left(i+s_k\text{mod}{N}_{\text{Segm}}\right)+1$$

mit Inversem $${\pi }_k^{-1}\left(i\right)=\left(i-s_k\text{mod}{N}_{\text{Segm}}\right)+1.$$

The mapping π _k (i) is related to the displacement vector $\overrightarrow{s}={\left(s_0,...,s_{N_{\text{disks}}-1}\right)}^T$

= (s ₀ ,...,sN _Disks -1) ^T as follows: $${\pi }_k\left(i\right)=\left(i+s_k\text{model}{N}_{\text{segment}}\right)+1$$

with inverse $${\pi }_k^{-1}\left(i\right)=\left(i-s_k\text{model}{N}_{\text{segment}}\right)+1.$$

$$={\displaystyle \sum _{i=1}^{N_{\text{Segm}}}{\left(N_{\text{Disks}}{A}_{\text{avg}}-{\displaystyle \sum _{k=1}^{N_{\text{Disks}}}{A}_{{\pi }_k\left(i\right)}^k}\right)}^2}$$

$$={\displaystyle \sum _{i=1}^{N_{\text{Segm}}}{\left(\frac{1}{N_{\text{Segm}}}{\displaystyle \sum _{j=1}^{N_{\text{Segm}}}{\displaystyle \sum _{k=1}^{N_{\text{Disks}}}{A}_j^k}}-{\displaystyle \sum _{k=1}^{N_{\text{Disks}}}{A}_{{\pi }_k\left(i\right)}^k}\right)}^2}$$

$$={\displaystyle \sum _{i=1}^{N_{\text{Segm}}}{\left({\displaystyle \sum _{k=1}^{N_{\text{Disks}}}\left(\frac{1}{N_{\text{Segm}}}{\displaystyle \sum _{j=1}^{N_{\text{Segm}}}{A}_j^k-A_{{\pi }_k\left(i\right)}^k}\right)}\right)}^2}$$

$$={\displaystyle \sum _{j=1}^{N_{\text{Segm}}}{\displaystyle \sum _{k=1}^{N_{\text{Disks}}}\left(\frac{1}{N_{\text{Segm}}}{\displaystyle \sum _{j=1}^{N_{\text{Segm}}}{A}_j^k-A_{{\pi }_k\left(i\right)}^k}\right)}}\left(\frac{1}{N_{\text{Segm}}}{\displaystyle \sum _{j=1}^{N_{\text{Segm}}}{A}_j^{k\text{'}}-A_{{\pi }_{k\text{'}}\left(i\right)}^{k\text{'}}}\right)$$

$$={\displaystyle \sum _{i=1}^{N_{\text{Segm}}}{\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{Disks}}}(\frac{1}{N_{\text{Segm}}^2}{\displaystyle \sum _{j,j\text{'}=0}^{N_{\text{Segm}}-1}{A}_j^k{A}_{j\text{'}}^{k\text{'}}-A_{{\pi }_{k\text{'}}\left(i\right)}^{k\text{'}}}\frac{1}{N_{\text{Segm}}}{\displaystyle \sum _{j=1}^{N_{\text{Segm}}}{A}_j^k}}}$$

$$-A_{{\pi }_k\left(i\right)}^k\frac{1}{N_{\text{Segm}}}{\displaystyle \sum _{j=1}^{N_{\text{Segm}}-1}{A}_j^{k\text{'}}+A_{{\pi }_k\left(i\right)}^k{A}_{{\pi }_{k\text{'}}\left(i\right)}^{k\text{'}}})$$

$$=\frac{1}{N_{\text{Segm}}}{\displaystyle \sum _{i,j=1}^{N_{\text{Segm}}}{\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{Disks}}}\left(A_j^k{A}_i^{k\text{'}}-A_{{\pi }_{k\text{'}}\left(i\right)}^{k\text{'}}{A}_j^k-A_{{\pi }_k\left(i\right)}^k{A}_j^{k\text{'}}+A_{{\pi }_k\left(i\right)}^k{A}_{{\pi }_{k\text{'}}\left(i\right)}^{k\text{'}}\right)}}$$

$$=\frac{1}{N_{\text{Segm}}}{\displaystyle \sum _{i,j=1}^{N_{\text{Segm}}}{\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{Disks}}}\left(A_j^k{A}_i^{k\text{'}}-A_j^k\underset{i\to {\pi }_{k\text{'}}^{-1}\left(i\right)}{\underbrace{A_i^{k\text{'}}}}-A_{{\pi }_k\left(i\right)}^k{A}_j^{k\text{'}}+A_{{\pi }_k\left(i\right)}^k{A}_{{\pi }_{k\text{'}}\left(i\right)}^{k\text{'}}\right)}}$$

$$=\frac{1}{N_{\text{Segm}}}{\displaystyle \sum _{i,j=1}^{N_{\text{Segm}}}{\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{Disks}}}\left(A_{{\pi }_k\left(i\right)}^k{A}_{{\pi }_{k\text{'}}\left(i\right)}^{k\text{'}}-A_{{\pi }_k\left(i\right)}^k{A}_j^{k\text{'}}\right)}}$$

$$=\frac{1}{N_{\text{Segm}}}{\displaystyle \sum _{i,j=1}^{N_{\text{Segm}}}{\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{Disks}}}\left(N_{\text{Segm}}{A}_{{\pi }_k\left(i\right)}^k{A}_{{\pi }_{k\text{'}}\left(j\right)}^{k\text{'}}{\delta }_{i,j}-A_{{\pi }_k\left(i\right)}^k{A}_j^{k\text{'}}\right)}}$$

$$={\displaystyle \sum _{i=1}^{N_{\text{Segm}}}{\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{Disks}}}\left(A_{{\pi }_k\left(i\right)}^k{\displaystyle \sum _{j=1}^{N_{\text{Segm}}}{A}_{{\pi }_{k\text{'}}\left(j\right)}^{k\text{'}}{\delta }_{i,j}}-\frac{1}{N_{\text{Segm}}}{A}_{{\pi }_k\left(i\right)}^k{\displaystyle \sum _{j=1}^{N_{\text{Segm}}}\underset{j\to {\pi }_{k\text{'}}\left(j\right)}{\underbrace{A_{{\pi }_{k\text{'}}\left(j\right)}^{k\text{'}}}}}\right)}}$$

$$={\displaystyle \sum _{i,j=1}^{N_{\text{Segm}}}{\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{Disks}}}{A}_{{\pi }_k\left(i\right)}^k{A}_{{\pi }_{k\text{'}}\left(i\right)}^{k\text{'}}\left({\delta }_{i,j}-\frac{1}{N_{\text{Segm}}}\right)}}$$

The following applies: $$E_0={\displaystyle \sum _{i=1}^{N_{\text{segment}}}{\left({\displaystyle \sum _{k=1}^{N_{\text{disks}}}\left(A_{\text{avg}}-A_{{\pi }_k\left(i\right)}^k\right)}\right)}^2}$$

$$={\displaystyle \sum _{i=1}^{N_{\text{segment}}}{\left(N_{\text{disks}}{A}_{\text{avg}}-{\displaystyle \sum _{k=1}^{N_{\text{disks}}}{A}_{{\pi }_k\left(i\right)}^k}\right)}^2}$$

$$={\displaystyle \sum _{i=1}^{N_{\text{segment}}}{\left(\frac{1}{N_{\text{segment}}}{\displaystyle \sum _{j=1}^{N_{\text{segment}}}{\displaystyle \sum _{k=1}^{N_{\text{disks}}}{A}_j^k}}-{\displaystyle \sum _{k=1}^{N_{\text{disks}}}{A}_{{\pi }_k\left(i\right)}^k}\right)}^2}$$

$$={\displaystyle \sum _{i=1}^{N_{\text{segment}}}{\left({\displaystyle \sum _{k=1}^{N_{\text{disks}}}\left(\frac{1}{N_{\text{segment}}}{\displaystyle \sum _{j=1}^{N_{\text{segment}}}{A}_j^k-A_{{\pi }_k\left(i\right)}^k}\right)}\right)}^2}$$

$$={\displaystyle \sum _{j=1}^{N_{\text{segment}}}{\displaystyle \sum _{k=1}^{N_{\text{disks}}}\left(\frac{1}{N_{\text{segment}}}{\displaystyle \sum _{j=1}^{N_{\text{segment}}}{A}_j^k-A_{{\pi }_k\left(i\right)}^k}\right)}}\left(\frac{1}{N_{\text{segment}}}{\displaystyle \sum _{j=1}^{N_{\text{segment}}}{A}_j^{k\text{'}}-A_{{\pi }_{k\text{'}}\left(i\right)}^{k\text{'}}}\right)$$

$$={\displaystyle \sum _{i=1}^{N_{\text{segment}}}{\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{disks}}}(\frac{1}{{\mathrm{<figure-callout\; id="2"\; label="N\; segment"\; filenames="DE102021203536B3\_0144.png,DE102021203536B3\_0146.png"\; state="\{\{state\}\}">N</figure-callout>}}_{\text{segment}}^{}}{\displaystyle \sum _{j,j\text{'}=0}^{N_{\text{segment}}-1}{A}_j^k{A}_{j\text{'}}^{k\text{'}}-A_{{\pi }_{k\text{'}}\left(i\right)}^{k\text{'}}}\frac{1}{N_{\text{segment}}}{\displaystyle \sum _{j=1}^{N_{\text{segment}}}{A}_j^k}}}$$

$$-A_{{\pi }_k\left(i\right)}^k\frac{1}{N_{\text{segment}}}{\displaystyle \sum _{j=1}^{N_{\text{segment}}-1}{A}_j^{k\text{'}}+A_{{\pi }_k\left(i\right)}^k{A}_{{\pi }_{k\text{'}}\left(i\right)}^{k\text{'}}})$$

$$=\frac{1}{N_{\text{segment}}}{\displaystyle \sum _{i,j=1}^{N_{\text{segment}}}{\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{disks}}}\left(A_j^k{A}_i^{k\text{'}}-A_{{\pi }_{k\text{'}}\left(i\right)}^{k\text{'}}{A}_j^k-A_{{\pi }_k\left(i\right)}^k{A}_j^{k\text{'}}+A_{{\pi }_k\left(i\right)}^k{A}_{{\pi }_{k\text{'}}\left(i\right)}^{k\text{'}}\right)}}$$

$$=\frac{1}{N_{\text{segment}}}{\displaystyle \sum _{i,j=1}^{N_{\text{segment}}}{\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{disks}}}\left(A_j^k{A}_i^{k\text{'}}-A_j^k\underset{i\to {\pi }_{k\text{'}}^{-1}\left(i\right)}{\underbrace{A_i^{k\text{'}}}}-A_{{\pi }_k\left(i\right)}^k{A}_j^{k\text{'}}+A_{{\pi }_k\left(i\right)}^k{A}_{{\pi }_{k\text{'}}\left(i\right)}^{k\text{'}}\right)}}$$

$$=\frac{1}{N_{\text{segment}}}{\displaystyle \sum _{i,j=1}^{N_{\text{segment}}}{\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{disks}}}\left(A_{{\pi }_k\left(i\right)}^k{A}_{{\pi }_{k\text{'}}\left(i\right)}^{k\text{'}}-A_{{\pi }_k\left(i\right)}^k{A}_j^{k\text{'}}\right)}}$$

$$=\frac{1}{N_{\text{segment}}}{\displaystyle \sum _{i,j=1}^{N_{\text{segment}}}{\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{disks}}}\left(N_{\text{segment}}{A}_{{\pi }_k\left(i\right)}^k{A}_{{\pi }_{k\text{'}}\left(j\right)}^{k\text{'}}{\delta }_{i,j}-A_{{\pi }_k\left(i\right)}^k{A}_j^{k\text{'}}\right)}}$$

$$={\displaystyle \sum _{i=1}^{N_{\text{segment}}}{\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{disks}}}\left(A_{{\pi }_k\left(i\right)}^k{\displaystyle \sum _{j=1}^{N_{\text{segment}}}{A}_{{\pi }_{k\text{'}}\left(j\right)}^{k\text{'}}{\delta }_{i,j}}-\frac{1}{N_{\text{segment}}}{A}_{{\pi }_k\left(i\right)}^k{\displaystyle \sum _{j=1}^{N_{\text{segment}}}\underset{j\to {\pi }_{k\text{'}}\left(j\right)}{\underbrace{A_{{\pi }_{k\text{'}}\left(j\right)}^{k\text{'}}}}}\right)}}$$

$$={\displaystyle \sum _{i,j=1}^{N_{\text{segment}}}{\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{disks}}}{A}_{{\pi }_k\left(i\right)}^k{A}_{{\pi }_{k\text{'}}\left(i\right)}^{k\text{'}}\left({\delta }_{i,j}-\frac{1}{N_{\text{segment}}}\right)}}$$

$$c_{i,j,k,k\text{'}}={\delta }_{i,j}-\frac{1}{N_{\text{Segm}}}$$

$$w\left(i,k,s_k\right)=\left(s_k+i\text{mod}{N}_{\text{Segm}}\right)+1$$

$$E_0={\displaystyle \sum _{i=1}^{N_{\text{Segm}}}{\left({\displaystyle \sum _{k=1}^{N_{\text{Disks}}}\left(A_{\text{avg}}-A_{{\pi }_k\left(i\right)}^k\right)}\right)}^2}$$

ein Spezialfall von $$E_0\left(\overrightarrow{s}\right)=c_0+{\displaystyle \sum _{k=1}^{N_{\text{objects}}}{\displaystyle \sum _{i=1}^{N_{\text{Segm}}\left(k\right)}{c}_{i,k}{A}_{w\left(i,k,s_k\right)}^k}}$$

$$+{\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{objects}}}{\displaystyle \sum _{i=1}^{N_{\text{Segm}}\left(k\right)}{\displaystyle \sum _{j=1}^{N_{\text{Segm}}\left(k\text{'}\right)}{c}_{i,j,k,k\text{'}}{A}_{w\left(i,k,s_k\right)}^k{A}_{w\left(i,k\text{'},s_{k\text{'}}\right)}^{k\text{'}}}}}$$

ist.The identifications $$c_{i,k}=c_0=0$$

$$c_{i,j,k,k\text{'}}={\delta }_{i,j}-\frac{1}{N_{\text{segment}}}$$

$$w\left(i,k,s_k\right)=\left(s_k+i\text{model}{N}_{\text{segment}}\right)+1$$

$$E_0={\displaystyle \sum _{i=1}^{N_{\text{segment}}}{\left({\displaystyle \sum _{k=1}^{N_{\text{disks}}}\left(A_{\text{avg}}-A_{{\pi }_k\left(i\right)}^k\right)}\right)}^2}$$

a special case of $$E_0\left(\overrightarrow{s}\right)=c_0+{\displaystyle \sum _{k=1}^{N_{\text{objects}}}{\displaystyle \sum _{i=1}^{N_{\text{segment}}\left(k\right)}{c}_{i,k}{A}_{w\left(i,k,s_k\right)}^k}}$$

$$+{\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{objects}}}{\displaystyle \sum _{i=1}^{N_{\text{segment}}\left(k\right)}{\displaystyle \sum _{j=1}^{N_{\text{segment}}\left(k\text{'}\right)}{c}_{i,j,k,k\text{'}}{A}_{w\left(i,k,s_k\right)}^k{A}_{w\left(i,k\text{'},s_{k\text{'}}\right)}^{k\text{'}}}}}$$

is.

$$={\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{Disks}}}{\displaystyle \sum _{i=1}^{N_{\text{Segm}}}{\widehat{B}}_i^k{\widehat{B}}_i^{k\text{'}}}}$$

gegeben mit $${\widehat{B}}_i^k={\displaystyle \sum _{m=1}^{N_{\text{Segm}}}{B}_{w\left(i,k,m\right)}^k}|m\rangle {\langle m|}_k\otimes 1_{\mathrm{1,}\dots ,k-\mathrm{1,}k+\mathrm{1,}\dots ,N_{\text{Segm}}}$$

According to a further aspect of the invention, the Hamiltonian is through $$H_0={\displaystyle \sum _{i=1}^{N_{\text{segment}}}{\left({\displaystyle \sum _{k=1}^{N_{\text{disks}}}{\widehat{B}}_i^k}\right)}^2}$$

$$={\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{disks}}}{\displaystyle \sum _{i=1}^{N_{\text{segment}}}{\widehat{B}}_i^k{\widehat{B}}_i^{k\text{'}}}}$$

given with $${\widehat{B}}_i^k={\displaystyle \sum _{m=1}^{N_{\text{segment}}}{B}_{w\left(i,k,m\right)}^k}|m〉{〈m|}_k\otimes 1_{\mathrm{1,}...,k-\mathrm{1,}k+\mathrm{1,}...,N_{\text{segment}}}$$

encodiert. Die modifizierte Segmenthöhe ist $B_i^k=A_{\text{avg}}-A_i^k.$

Die Zeitentwicklung ist gegeben durch den k ↔ k' Kopplungsterm ${\displaystyle {\sum }_{i=1}^{N_{\text{Segm}}}{\widehat{B}}_i^k{\widehat{B}}_i^{k\text{'}}}$

zwischen den Qudits.According to this aspect, instead of calculating the difference between average height and respective segment height, only the differences in the operator ${\widehat{B}}_i^k$

encoded. The modified segment height is $B_i^k=A_{\text{avg}}-A_i^k.$

The time evolution is given by the k ↔ k' coupling term ${\displaystyle {\sum }_{i=1}^{N_{\text{segment}}}{\widehat{B}}_i^k{\widehat{B}}_i^{k\text{'}}}$

between the qudits.

• • eine dritte Anzahl N_E an Maschinenelementen bereitgestellt wird;
• • eine Gesamtmenge D an Maschinenteilen die Mächtigkeit ${\displaystyle {\sum }_{i=1}^{N_E}{N}_{\text{object}}}\left(i\right)$
  
  wobei N_objects(i) die Anzahl der Maschinenteile für das Maschinenelement i ist für alle i ∈ {1,...,N_E};
• • eine gemeinsame Partition P : D → {1,...,N_E} bestimmt wird, wobei die Partition P eine Zuordnung der Maschinenteile auf die Maschinenelemente ist und wobei die Mächtigkeit |P^-1(i)| gleich der Anzahl N_objects(i) an Maschinenteilen für das Maschinenelement i ist für alle i ∈ {1, ... , N_E}.

Another aspect includes the characteristics that

• • a third number N _E of machine elements is provided;
• • a total amount D of machine parts the thickness ${\displaystyle {\sum }_{i=1}^{N_E}{N}_{\text{objects}}}\left(i\right)$
  
  where N _objects (i) is the number of machine parts for machine element i for all i ∈ {1,...,N _E };
• • a common partition P : D → {1,...,N _E } is determined, where the partition P is an assignment of the machine parts to the machine elements and where the size |P ^-1 (i)| equal to the number N _objects (i) of machine parts for the machine element i for all i ∈ {1, ... , N _E }.

This aspect is particularly advantageous in the event that the machine elements are clutches and the machine parts are clutch discs. If the clutch discs of a clutch cannot be turned in such a way that the discrepancy described is minimized, this clutch is usually discarded for quality reasons. By taking into account an exchange of clutch disks between the clutches by means of the partition, ideally no more clutches have to be discarded. In the aspects described above, the partition was fixed, the arrangement of the clutch discs was optimized separately, i.e. locally, for each clutch. According to this aspect, the arrangement of the clutch disks is optimized globally, ie across all clutches. The global optimization allows to maximize the number of clutches each having an optimized clutch disc arrangement.

Due to the common partition, the number of machine parts per machine element is variable. If, for example, 70 clutch discs are provided and the third number N _E is three, For example, a first clutch with 30 clutch discs, a second clutch with 25 clutch discs and a third clutch with 15 clutch discs are produced, with the first, second and third clutch each having an optimized arrangement of the clutch discs.

In an embodiment of this aspect, the number of machine parts does not depend on the machine elements. Each machine element has the same number of machine parts. For example, 70 clutch discs are provided and 10 clutches are to be produced, each with 7 clutch discs. In the global optimization, the 70 clutch disks are then distributed across all clutches in such a way that the 7 resulting clutches each have an optimized arrangement of 10 clutch disks.

According to one aspect, the number of clutches that fall below a quality threshold with regard to the arrangement of their clutch disks is determined, for example, by considering a clutch i that falls below the quality threshold. For a random partition P, a randomly chosen clutch disc from P ^-1 (i) is swapped with a randomly chosen clutch disc from P ^-1 (j) for any i ≠ j. This step is repeated until all clutches that fall below the quality threshold are resolved, or until a given time limit is reached.

According to a further aspect, the number of clutches that falls below a quality threshold with regard to the arrangement of their clutch disks is determined by considering a clutch i that falls below the quality threshold. Then a clutch j is randomly considered. The clutch discs of clutches i and j are randomly swapped and/or rearranged. This step is repeated until all clutches that fall below the quality threshold are resolved, or until a given time limit is reached.

As an alternative to random exchange, in one aspect the most advantageous exchange is calculated and applied. Other aspects in this context concern simulated annealing and evolutionary algorithms.

The invention can be used, for example, in the assembly of drive train systems, for example electrified drive trains, and their components.

• 1 eine Darstellung einer Diskrepanz im Sinne der Erfindung,
• 2 eine Darstellung der Diskrepanz aus 1 nach Optimierung,
• 3 ein Ausführungsbeispiel eines Maschinenteils,
• 4 eine Darstellung für eine Verschiebung mittels eines erfindungsgemäßen Verschiebungsvektors und
• 5 ein Flussdiagramm eines Ausführungsbeispiels des erfindungsgemäßen Verfahrens.

The invention is illustrated in the following exemplary embodiments. Show it:

• 1 a representation of a discrepancy within the meaning of the invention,
• 2 a representation of the discrepancy 1 after optimization,
• 3 an embodiment of a machine part,
• 4 a representation for a displacement by means of a displacement vector according to the invention and
• 5 a flowchart of an embodiment of the method according to the invention.

In the figures, the same reference symbols denote the same or functionally similar reference parts. For the sake of clarity, only the relevant reference parts are highlighted in the individual figures.

The example of 1 could be based on four machine parts Disk in the form of discs stacked together to form a clutch. Each of the slices Disk is divided into 3 Seg sections.

According to the invention, however, the four discs could also be subdivided into different numbers of sections Seg. For example, the first disk could be divided into 2, the second disk into 3, the third disk into 4, and the fourth disk into 5 sections Seg.

In each of these sections Seg, the height/thickness of the section is measured as a measured value M using a sensor L, for example using a laser measuring machine.

3 shows one of the machine parts Disk, for example a clutch disk.

The individual measured values M are converted into the in 1 shown matrix entered. A row of this matrix corresponds to a slice Disk. A column of this matrix corresponds to a section Seg of the respective disk.

In the above example, where the first disk is divided into 2, the second disk into 3, the third disk into 4, and the fourth disk into 5 sections Seg, the individual readings are entered into a collection of lists. A first line has 2, a second line 3, a third line 4 and a fourth line 5 entries.

The sum of the measured values M in 1 over the first sections Seg of all slices Disk gives 14. The sum of the measured values M over the second sections Seg of all slices Disk gives 14. The sum of the measured values M over the third sections Seg of all slices Disk gives 26. The discrepancy of these sums is 12.

In 2 the disk Disk 4 was rotated cyclically: Seg3→Seg2, Seg2→Seg1, Seg1→Seg3. Notation: Seg_(before rotation)→Seg_(after rotation). Disk 3 was rotated cyclically: Seg3→Segl, Seg2→Seg3, Seg1→Seg2. Disk 2 was not rotated. Disk 1 was rotated in the same way as Disk 3. These rotations correspond to displacements of the columns of the matrix by a displacement vector.

After the rotation, the sum of the measured values M over the first sections Seg of all disks is 20. The sum of the measured values M over the second sections Seg of all disks is 14. The sum of the measured values M over the third sections Seg of all disks is 20 The discrepancy of these sums is now optimized to 6.

4 also shows a matrix representation. The individual entries A7 are the height measurement values M. The first number N _objects is equal to seven, for example. For example, the second number N _Segm is equal to 42 for each slice. In 4 the displacement vector was applied to the second disk Disk2. A total of all entries in the second column is shown.

This in 5 The method illustrated is one embodiment of the general method of the present invention. In the exemplary embodiment, all machine parts are divided into an equal number N _segm of sections and the dimensionality of each of the qudits (|s _k 〉 _k ) is equal to the second number N _segm .

In a first in 5 In the method step V1 shown, a first number N _objects , the machine parts disk for a machine element is provided. For example, seven clutch discs are provided for one clutch.

In a method step V2, each of the machine parts Disk is subdivided into at least a second number N _Segm of sections Seg. A measured value M is read into a matrix using a sensor L in the respective sections Seg. The matrix has the first number N _objects of rows and the second number N _segments of columns.

bereitgestellt. Der Verschiebungsvektors $\overrightarrow{s}$

beschreibt eine Umordnung der Messwerte M der jeweiligen Spalten in einer jeden der Zeilen der Matrix. Die Optimierungsfunktion E₀ kann folgende Form aufweisen: $$E_0\left(\overrightarrow{s}\right)=c_0+{\displaystyle \sum _{k=1}^{N_{\text{objects}}}{\displaystyle \sum _{i=1}^{N_{\text{Segm}}}{c}_{i,k}{A}_{w\left(i,k,s_k\right)}^k}}+{\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{objects}}}{\displaystyle \sum _{i,j=1}^{N_{\text{Segm}}}{c}_{i,j,k,k\text{'}}{A}_{w\left(i,k,s_k\right)}^k{A}_{w\left(i,k\text{'},s_{k\text{'}}\right)}^{k\text{'}}}}$$

In a third method step V3, an optimization function E ₀ from entries in the matrix as a function of a displacement vector $\overrightarrow{s}$

provided. The displacement vector $\overrightarrow{s}$

describes a rearrangement of the measured values M of the respective columns in each of the rows of the matrix. The optimization function E ₀ can have the following form: $$E_0\left(\overrightarrow{s}\right)=c_0+{\displaystyle \sum _{k=1}^{N_{\text{objects}}}{\displaystyle \sum _{i=1}^{N_{\text{segment}}}{c}_{i,k}{A}_{w\left(i,k,s_k\right)}^k}}+{\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{objects}}}{\displaystyle \sum _{i,j=1}^{N_{\text{segment}}}{c}_{i,j,k,k\text{'}}{A}_{w\left(i,k,s_k\right)}^k{A}_{w\left(i,k\text{'},s_{k\text{'}}\right)}^{k\text{'}}}}$$

durch dessen Verschiebungen eine Diskrepanz der Messwerte M in den jeweiligen Abschnitten Seg der Maschinenteile Disk minimiert wird, minimiert die Optimierungsfunktion E₀.A displacement vector to calculate $\overrightarrow{s},$

whose displacements minimize a discrepancy in the measured values M in the respective sections Seg of the machine parts Disk, which minimizes the optimization function E ₀ .

in ein Quantenregister $|\overrightarrow{s}\rangle$

mit der ersten Anzahl N_objccts an Qudits encodiert. Die Dimensionalität jedes der Qudits (|s_k)_k) ist gleich der zweiten Anzahl N_Segm. Das Quantenregister $|\overrightarrow{s}\rangle$

kann folgende Form aufweisen: $$|\overrightarrow{s}\rangle :=\underset{k=1}{\overset{N_{\text{objects}}}{\otimes }}{|s_k\rangle }_k$$

In a fourth method step V4, the displacement vector $\overrightarrow{s}$

into a quantum register $|\overrightarrow{s}〉$

encoded with the first number N _objccts of qudits. The dimensionality of each of the qudits (|s _k ) _k ) is equal to the second number N _Segm . The Quantum Register $|\overrightarrow{s}〉$

can have the following form: $$|\overrightarrow{s}〉:=\underset{k=1}{\overset{N_{\text{objects}}}{\otimes }}{|s_k〉}_k$$

bereitgestellt, für die das Quantenregister $|\overrightarrow{s}\rangle$

jeweils ein Eigenvektor ist und ein Eigenwert eines der Operatoren ${\widehat{A}}_i^k$

jeweils einer der Messwerte M in einem der Abschnitte Seg jeweils eines der Maschinenteile Disk ist. Die Operatoren können folgende Form aufweisen: $${\widehat{A}}_i^k={\displaystyle \sum _{m=1}^{N_{\text{Segm}}\left(k\right)}{A}_{w\left(i,k,m\right)}^k}|m\rangle {\langle m|}_k\otimes 1_{\mathrm{1,}\dots ,k-\mathrm{1,}k+\mathrm{1,}\dots ,N_{\text{Segm}}}$$

In a fifth method step V5, operators ${\widehat{A}}_i^k$

provided for the the quantum register $|\overrightarrow{s}〉$

each is an eigenvector and an eigenvalue of one of the operators ${\widehat{A}}_i^k$

one of the measured values M in one of the sections Seg is one of the machine parts Disk in each case. The operators can have the following form: $${\widehat{A}}_i^k={\displaystyle \sum _{m=1}^{N_{\text{segment}}\left(k\right)}{A}_{w\left(i,k,m\right)}^k}|m〉{〈m|}_k\otimes 1_{\mathrm{1,}...,k-\mathrm{1,}k+\mathrm{1,}...,N_{\text{segment}}}$$

zu einem Hamiltonoperator H₀ kombiniert, dessen Erwartungswerts im Quantenregister ${\widehat{A}}_i^k$

gleich dem Wert der Optimierungsfunktion E₀ über dem Verschiebungsvektor $\overrightarrow{s}$

ist. Der Hamiltonoperator kann folgende Form aufweisen: $$H_0=c_{0}1+{\displaystyle \sum _{k=1}^{N_{\text{objects}}}{\displaystyle \sum _{i=1}^{N_{\text{Segm}}}{c}_{i,k}{\widehat{A}}_i^k}}+{\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{objects}}}{\displaystyle \sum _{i,j=1}^{N_{\text{Segm}}}{c}_{i,j,k,k\text{'}}{\widehat{A}}_i^k{\widehat{A}}_j^{k\text{'}}}}$$

In a sixth method step V6, the operators ${\widehat{A}}_i^k$

combined into a Hamiltonian H ₀ whose expectation value is in the quantum register ${\widehat{A}}_i^k$

equal to the value of the optimization function E ₀ over the displacement vector $\overrightarrow{s}$

is. The Hamilton operator can have the following form: $$H_0=c_{0}1+{\displaystyle \sum _{k=1}^{N_{\text{objects}}}{\displaystyle \sum _{i=1}^{N_{\text{segment}}}{c}_{i,k}{\widehat{A}}_i^k}}+{\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{objects}}}{\displaystyle \sum _{i,j=1}^{N_{\text{segment}}}{c}_{i,j,k,k\text{'}}{\widehat{A}}_i^k{\widehat{A}}_j^{k\text{'}}}}$$

In a seventh method step V7, a basic state of the Hamilton operator H ₀ is calculated, for example using an adiabatic quantum computer.

In an eighth method step V8, the basic state is decoded into the optimized arrangement of the machine parts disk and the optimized arrangement is read out in order to arrange the machine parts disk accordingly.

Reference List

V1-V8

process steps

disk

machine part

seg

section

M

height, reading

L

sensor

readings

No objects

first number

NSegm

second number

Sum

total

E0

optimization function

quantum register

operator

H0

Hamilton operator

expected value

Claims (10)

Method for optimizing an arrangement of machine parts (A ^k , k; ∈ {1,..., N _objects }, Disk) by means of quantum computing, comprising the steps • providing a first number (N _objects ) of the machine parts (A ^k , k ∈ { 1, ... , N _objects }; Disk) for a machine element (V1); • Subdividing each of the machine parts (A ^k ; Disk) into a second number (N _Segm (k)) of sections (Seg) specific to the respective machine part (A ^k ; Disk) and reading in one measured value in each case $\left(A_i^k\in ℝ,i\in \left\{\mathrm{1,}...,N_{\text{segment}}\left(k\right)\right\}\right)$

in the respective sections (Seg) into a collection of lists with the first number (N _objects ) of rows and the second numbers (N _segm ) of columns (V2) using a sensor (L); • Providing an optimization function (E ₀ ) from entries in the collection of lists depending on a shift vector $\left(\overrightarrow{s}\right),$

which is a rearrangement of the readings $\left(A_i^k\right)$

of the respective columns in each of the rows of the collection of lists describing a displacement vector to be calculated $\left(\overrightarrow{s}\right),$

due to its rearrangement, there is a discrepancy in the measured values $\left(A_i^k\right)$

in the respective sections (Seg) of the machine parts (A ^k , k ∈ {1, ... , N _objects }; Disk) is minimized, the optimization function (E ₀ ) is minimized (V3); • Encoding the displacement vector $\left(\overrightarrow{s}\right)$

into a quantum register $\left(|\overrightarrow{s}〉\right)$

with the first number (N _objects ) of qudits $\left({|s_k〉}_k\right),$

where each of the qudits encodes a kth rearrangement in the displacement vector and the dimensionality of the qudits $\left({|s_k〉}_k\right)$

is less than, equal to or greater than the respective second number (N _Segm ) (V4); • Providing operators $\left({\widehat{A}}_i^k\right),$

for the the quantum register $\left(|\overrightarrow{s}〉\right)$

each is an eigenvector and an eigenvalue of one of the operators $\left({\widehat{A}}_i^k\right)$

one of the measured values $\left(A_i^k\in ℝ,i\in \left\{\mathrm{1,}...,N_{\text{segment}}\left(k\right)\right\}\right),$

including the measurements $\left(A_i^k\right)$

multiple sections (Seg) or machine parts (A ^k , k ∈ {1, ... , N _objects }; Disk), calculated value is (V5); • Combining operators $\left({\widehat{A}}_i^k\right)$

to a Hamilton operator (H ₀ ), whose expectation value is in the quantum register $\left(|\overrightarrow{s}〉\right)$

equal to the value of the optimization function (E ₀ ) over the displacement vector $\left(\overrightarrow{s}\right)$

is (V6); • computing a state of the Hamiltonian (H ₀ ) by means of a quantum computing hardware module, a quantum computing inspired classical hardware module or a quantum computing inspired special purpose hardware module (V7); • Decoding at least parts of the calculated state into the optimized arrangement of the machine parts (disk) and reading out the optimized arrangement in order to arrange the machine parts (disk) accordingly (V8). procedure after claim 1 , where the optimization function (E ₀ ) is of quadratic order and given by the formula $$E_0\left(\overrightarrow{s}\right)=c_0+{\displaystyle \sum _{k=1}^{N_{\text{objects}}}{\displaystyle \sum _{i=1}^{N_{\text{segment}}\left(k\right)}{c}_{i,k}{A}_{w\left(i,k,s_k\right)}^k}}$$

$$+{\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{objects}}}{\displaystyle \sum _{i=1}^{N_{\text{segment}}\left(k\right)}{\displaystyle \sum _{j=1}^{N_{\text{segment}}\left(k\text{'}\right)}{c}_{i,j,k,k\text{'}}{A}_{w\left(i,k,s_k\right)}^k{A}_{w\left(i,k\text{'},s_{k\text{'}}\right)}^{k\text{'}}}}}$$

is given with displacement vector $\overrightarrow{s},$

Machine parts A ^k , k ∈ {1,..., N _objects }, measured values $A_i^k\in ℝ,i\in \left\{\mathrm{1,}...,N_{\text{segment}}\left(k\right)\right\},w\left(\cdot ,k,\cdot \right):{\mathbb{N}}_0^2\to \left\{\mathrm{1,}...,N_{\text{segment}}\left(k\right)\right\},k-\text{ter}$

Entry s _k ∈ {0,..., N _segm (k) - 1} of the displacement vector $\overrightarrow{s}$

and constants c ₀ , c _i,k , c _i,j,k,k' ∈ ℝ. A method according to any one of the preceding claims, wherein the quantum register $\left(|\overrightarrow{s}〉\right)$

same $$|\overrightarrow{s}〉:=\underset{k=1}{\overset{N_{\text{objects}}}{\otimes }}{|s_k〉}_k$$

is. Method according to one of the preceding claims, wherein the operators $\left({\widehat{A}}_i^k\right)$

through $${\widehat{A}}_i^k={\displaystyle \sum _{m=1}^{N_{\text{segment}}\left(k\right)}{A}_{w\left(i,k,m\right)}^k}|m〉{〈m|}_k\otimes 1_{\mathrm{1,}...,k-\mathrm{1,}k+\mathrm{1,}...,N_{\text{segment}}\left(k\right)}$$

(63) are given. Method according to one of the preceding claims, wherein the Hamiltonian (H ₀ ) by $$H_0=c_{0}1+{\displaystyle \sum _{k=1}^{N_{\text{objects}}}{\displaystyle \sum _{i=1}^{N_{\text{segment}}\left(k\right)}{c}_{i,k}{\widehat{A}}_i^k}}+{\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{objects}}}{\displaystyle \sum _{i=1}^{N_{\text{segment}}\left(k\right)}{\displaystyle \sum _{j=1}^{N_{\text{segment}}\left(k\text{'}\right)}{c}_{i,j,k,k\text{'}}{\widehat{A}}_i^k{\widehat{A}}_j^{k\text{'}}}}}$$

given is. Method according to one of the preceding claims, wherein the state of the Hamilton operator (H ₀ ) is calculated by means of an adiabatic quantum computer. Method according to one of the preceding claims, wherein the state of the Hamiltonian (H ₀ ) is calculated on a quantum gate based quantum computer. Method according to one of the preceding claims, wherein • the machine parts (Disk) are N _Disks which are stacked and arranged to form a clutch as a machine element; • the sections (seg) of each of the slices are segments, with all slices having the same number of segments; • as measured values $\left(A_i^k\right)$

heights of the segments are measured; • one through the displacement vector $\left(\overrightarrow{s}\right)$

rearrangement effected corresponds to rotations of the discs; • the Hamiltonian (H ₀ ) through $$H_0={\displaystyle \sum _{i=1}^{N_{\text{segment}}}{\left({\displaystyle \sum _{k=1}^{N_{\text{disks}}}\left({\widehat{A}}_{\text{avg}}-{\widehat{A}}_i^k\right)}\right)}^2}$$

is given with ${\widehat{A}}_{\text{avg}}:=\frac{1}{N_{\text{segment}}}{\displaystyle {\sum }_{j=1}^{N_{\text{segment}}}\frac{1}{N_{\text{disks}}}}{\displaystyle {\sum }_{i=1}^{N_{\text{disks}}}{\widehat{A}}_i^j};$

• the optimized arrangement is given by a stack for which deviations of the heights summed over the slices are minimal in each segment. procedure after claim 8 , where the Hamiltonian (H ₀ ) is given by $$H_0={\displaystyle \sum _{i=1}^{N_{\text{segment}}}{\left({\displaystyle \sum _{k=1}^{N_{\text{disks}}}{\widehat{B}}_i^k}\right)}^2}$$

$$={\displaystyle \sum _{k,k\text{'}=1}^{N_{\text{disks}}}{\displaystyle \sum _{i=1}^{N_{\text{segment}}}{\widehat{B}}_i^k{\widehat{B}}_i^{k\text{'}}}}$$

is given with $${\widehat{B}}_i^k={\displaystyle \sum _{m=1}^{N_{\text{segment}}}{B}_{w\left(i,k,m\right)}^k}|m〉{〈m|}_k\otimes 1_{\mathrm{1,}...,k-\mathrm{1,}k+\mathrm{1,}...,N_{\text{segment}}}$$

and $B_i^k=A_{\text{avg}}-A_i^k.$

Method according to one of the preceding claims, wherein • a third number N _E of machine elements is provided; • a total amount D of machine parts the thickness ${\displaystyle {\sum }_{i=1}^{N_E}{N}_{\text{objects}}}\left(i\right)$

where N _objects (i) is the number of machine parts for machine element i for all i ∈ {1,..., _NE }; • a common partition P : D → {1, ... , N _E } is determined, where the partition is an assignment of the machine parts to the machine elements and where the size |P ^-1 (i)| is equal to the number N _objects (i) of machine parts for the machine element i for all i ∈ {1,..., N _E }.

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