Method for operating a quantum register

Abstract

According to different embodiments, a method for operating a quantum register (102) can have the steps of: carrying out a first adaptation (300) of a first parameter of a first quantum circuit (VQCp) in order to optimize a target function by actuating (103) the quantum register (102) according to the first quantum circuit (VQCp); carrying out a first determination of a second quantum circuit (VQCp+l) that has the first quantum circuit (VQCp) with an adapted first parameter, resulting from the first adaptation (300), and a quantum gate; carrying out a second adaptation (300) of a second parameter of the quantum gate in order to optimize the target function by actuating (103) the quantum register (102) according to the second quantum circuit (VQCp+l); and carrying out a second determination in order to determine whether the second quantum circuit (VQCp+l) with an adapted second parameter, resulting from the adaptation (300), satisfies a stored convergence criterion.

Description

[0001]R.401124 - 1 - Description Title Method for operating a quantum register State of the art The present disclosure relates to methods for operating a quantum register. Quantum computers and calculations performed therewith (also referred to as quantum computing) are one of the key technologies for the future, as they enable calculations that are inaccessible or difficult to access with classical computers. Quantum computing has promising applications, e.g. in mathematics (number theory, linear algebra, differential equations) and in physics (determining the electronic structure of molecules and materials, real-time simulation of quantum dynamic effects such as protein folding and optical properties). In this regard, quantum computers are often used as coprocessors (similar to graphics chips) that complement the classical computer rather than replace it.Further hopes are also placed on the application of quantum computers for time-critical optimization problems, such as those faced by navigation and traffic control systems, as well as artificial intelligence (AI), materials research, and medicine. One aspect of materials research lies in the quantitative exploration of a quantum mechanical system (hereinafter referred to as QS), such as a molecule. The simulation of a molecule (e.g., a protein) is often based on quantum mechanics, whose mathematical description is very complex, which complicates its implementation on classical computers and is often compensated for by significant reductions in accuracy. So-called exact diagonalization is one of the most important methods in quantum mechanics, especially in solid-state physics, for calculating physical properties, especially for strongly correlated electron systems, such as high-temperature superconductors or manganese oxides.The so-called exact diagonalization (ED) offers an exact mathematical solution to the QS. It uses matrix diagonalization to decompose the complete Hilbert space of all states into eigenstates that can be combined to form any thermalized state. However, the resource requirements for the ED increase so rapidly with the size of the QS (e.g., the number of atoms in the QS) that even modern supercomputers are rarely sufficient to obtain a solution of sufficient accuracy within a reasonable computing time. To reduce computing time, approximations are often used, including the so-called finite temperature Lanczos method (FTLM), in which the ground state and the first excited states are calculated very precisely, but the higher excited states are calculated only imprecisely. Nevertheless, even the approximations R.401124 - 2 - the same exponential increase in computational effort, so that in practice only relatively small systems have been considered so far (e.g. on the order of 40 spins for the Heisenberg model). In contrast, a quantum computer uses quantum mechanical effects, such as superposition, for calculations, which simplifies simulation and requires less compromise in terms of accuracy. When using a quantum computer, so-called hybrid, i.e. quantum-classical, algorithms are generally used, which outsource only part of the computation to the quantum computer. Quantum-classical algorithms make it possible to exploit the advantages of both computer types and reduce the demands on the quantum computer, for example with regard to the number of qubits, the circuit depth, and the tolerance to errors in gate operations (the so-called noise).Popular representatives of the hybrid algorithms are the so-called "Variational Quantum Eigensolver" (VQE). The VQE uses a variational quantum circuit to prepare a so-called initial state on the quantum computer. The quantum computer then measures the energy of this initial state. A classical (i.e., non-quantum mechanical) computer optimizes the quantum circuit to converge the initial state to the ground state of the QS (i.e., at a temperature T = 0 Kelvin). One aim is to calculate the state of the QS above 0 Kelvin (also referred to as thermalized or excited), which is, however, inaccessible to a VQE. Disclosure of the Invention According to various embodiments, a method is provided for operating a quantum register (QR) to determine whether a determined second quantum circuit (VQC) p+l) satisfies a stored convergence criterion, the method comprising: first adjusting a first parameter of a first quantum circuit (e.g. VQC p , where p≥1) for optimizing an objective function by controlling the quantum register according to the first quantum circuit; first determining the second quantum circuit (e.g. VQC p+lwhere l≥1), which has the first quantum circuit with an adapted first parameter resulting from the first adaptation, and (e.g. downstream of this) a quantum gate; second adaptation of a second parameter of the quantum gate for (e.g. by) optimizing the objective function by controlling the quantum register according to the second quantum circuit; second determination of whether the second quantum circuit with an adapted second parameter resulting from the second adaptation satisfies the stored convergence criterion. This method provided herein simplifies the reproducible initialization, simulation, and calculation of a quantum mechanical state, for example if this is to represent a temperature of the QS above 0 Kelvin. Intuitively, the method makes it easier to create a model of a QS that can be calculated using a QR and whose size scales with the number of qubits.The method described herein can, for example, be implemented and carried out in a resource-saving and cost-efficient manner R.401124 - 3 - since it does not necessarily scale exponentially with the size of the quantum system and requires only minimal resources of the quantum computer. Compared to the popular VQE, the method provided here does not require an approximation (e.g., as a ansatz state), does not require the calculation of the coefficients of a wavefunction, reduces the complexity of preparing the quantum state (e.g., no preparation is necessary), and / or optimizes the parameters of the quantum circuit (instead of the coefficients of the wavefunction). Compared to VQE, the method provided here is accessible to a wider range of applications and allows the calculation of physical properties including their temperature dependence. Likewise, the method provided here facilitates the calculation of excited states.Various embodiments are given below. Embodiment 1 is a method for operating a QR as given above. Embodiment 2 is the method according to embodiment 1, further comprising: determining an eigenvector and / or an eigenvalue (e.g. related to the eigenvector) of a mapping (e.g. a matrix and / or a Hamiltonian), wherein the objective function is a function of the eigenvalue of the mapping related to the eigenvector of the mapping, wherein the mapping preferably has or consists of a (e.g. physical) state model (e.g. a state of a physical system); and / or wherein the mapping preferably maps the eigenvector to the eigenvalue. This facilitates more complex calculations in quantum physics.The method preferably comprises a step of determining at least one physical property of a technical, in particular physical, system based on the determined eigenvector and / or the determined eigenvalue, wherein the mapping represents in particular a Hamiltonian function of the technical system. The physical property can be, for example, a magnetic susceptibility of a solid. Embodiment 3 is the method according to embodiment 1 or 2, wherein the first adaptation is based on first data (e.g.Measurement data or a frequency distribution) representing a first state of the quantum register that directly results from, or is at least based on, the control of the quantum register according to the first quantum circuit; and / or wherein the second adaptation is based on second data representing a second state of the quantum register that directly results from, or is at least based on, the control of the quantum register according to the second quantum circuit. This makes it easier to perform more complex calculations, for example, when a physical system is to be calculated.Embodiment 4 is the method according to one of embodiments 1 to 3, wherein the objective function: during the first adaptation is a function of the first parameter (and / or is independent of the second parameter) and / or depends at least on the first data; and / or during the second adaptation is a function of the second parameter (and optionally of the first parameter) and / or R.401124 - 4 - depends at least on the second data. This makes it easier to perform more complex calculations, for example when a physical system is to be calculated. Embodiment 5 is the method according to one of embodiments 3 to 4, further comprising: reading out the first data; and / or reading out the second data. This improves the flow of the method.Embodiment 6 is the method according to one of the embodiments 1 to 5, the method, if the second determination shows that the second quantum circuit satisfies the stored convergence criterion, further comprising: determining an additional quantum circuit (preferably based on the second quantum circuit or comprising it) for optimizing an additional objective function (e.g. comprising the objective function or at least parts thereof) by controlling the quantum register according to the additional quantum circuit; wherein the additional objective function comprises the objective function and / or represents a relation (e.g. overlap) between a result of controlling the quantum register according to the second quantum circuit and a result of controlling the quantum register according to the additional quantum circuit (e.g. is a function thereof); wherein, for example, the relation represents an overlap (e.g.A deviation from the orthogonality of the two results is penalized. This makes it easier to perform more complex calculations, for example, when excited states are to be calculated. Taking the overlap into account improves the result, for example, with regard to the properties of orthogonality. Embodiment 7 is the method according to one of embodiments 1 or 6, wherein the second determination is based on the objective function, preferably by determining whether the objective function satisfies the convergence criterion; and / or wherein the convergence criterion is satisfied if the objective function satisfies the convergence criterion. This improves the ratio of accuracy to computing time.Embodiment 8 is the method according to one of the embodiments 1 to 7, wherein the second adaptation further comprises changing one or more of the following to optimize the objective function: a type of quantum gate; a first qubit (also referred to as quantum bit) of the quantum register, which is influenced by means of the quantum gate; a second qubit of the quantum register, which determines the influencing of the first qubit by means of the quantum gate. This improves the ratio of accuracy to computing time. Embodiment 9 is the method according to one of the embodiments 1 to 8, further comprising, if the second determination shows that the stored convergence criterion is not met: determining a third quantum circuit that corresponds to the second quantum circuit with the adapted second parameter resulting from the second adaptation, and (e.g.downstream thereof) has at least one additional quantum gate; third adaptation of a third parameter of the additional quantum gate for optimizing the objective function by controlling the R.401124 - 5 - quantum register according to the third quantum circuit; and preferably third determination of whether the third quantum circuit, with an adapted third parameter resulting from the third adaptation, satisfies the stored convergence criterion. This improves the accuracy of the result. Embodiment 10 is the method according to one of embodiments 1 to 9, wherein the second adaptation further comprises adapting the first parameter (e.g. again), and wherein the second determination comprises determining whether the second quantum circuit, with the adapted first parameter and the adapted second parameter resulting from the second adaptation, satisfies a stored convergence criterion.This improves the ratio of accuracy to computing time. Embodiment 11 is the method according to one of embodiments 1 to 10, wherein the quantum register is controlled according to the first quantum circuit when the quantum register is brought into an initial state; and / or wherein the quantum register is controlled according to the second quantum circuit when the quantum register is brought into the initial state. This improves the accuracy and reduces the computing time. Embodiment 12 is the method according to embodiment 11, further comprising: selecting the initial state from a plurality of (e.g., normalized) basis states of the quantum register, wherein the selection, for each basis state of the plurality of basis states, preferably takes place by controlling the quantum register according to at least a part of the first quantum circuit (e.g., according to the first quantum circuit) when the quantum register is brought into the basis state.This improves the ratio of accuracy to computing time. Embodiment 13 is a (e.g., non-quantum mechanical) control device configured to perform the method according to any one of embodiments 1 to 12. Embodiment 14 is a computer program configured to cause a (e.g., non-quantum mechanical) processor executing the computer program to perform the method according to any one of embodiments 1 to 12. Embodiment 15 is a computer-readable medium storing instructions configured to cause a (e.g., non-quantum mechanical) processor executing the instructions to perform the method according to any one of embodiments 1 to 12. Embodiments 11 to 13 facilitate implementation of the method. In the drawings, similar reference numerals generally refer to the same parts throughout the several views.The drawings are not necessarily to scale, emphasis instead being generally placed upon illustrating the principles of the invention. In the following description, various aspects will be described with reference to the following drawings. Figure 1 shows the structure and operation of a quantum processor, Figures 2 to 7 each show a method according to various embodiments or at least parts of the method, Figure 8 shows the magnetic susceptibility in a schematic diagram, and Figure 9 shows a swap test according to various embodiments in a schematic diagram. The following detailed description refers to the accompanying drawings, which, by way of illustration, show specific details and aspects of this disclosure in which the invention may be practiced.Other aspects may be used and structural, logical and electrical changes may be made without departing from the scope of the invention. The various aspects of this disclosure are not necessarily mutually exclusive, as some aspects of this disclosure may be combined with one or more other aspects of this disclosure to form new aspects. Figure 1 illustrates the structure and operation of a quantum processor (“quantum process unit” or QPU for short) according to various embodiments 100 in a schematic diagram. The QPU has a quantum register 102 and an operating device 104. The term “quantum register” (QR) refers to a device by means of which several (e.g., a number of n) qubits (here exemplarily designated as q1 to q. nThe term "qubit" refers to a two-state quantum system, which can be mathematically expressed by two basis vectors of a two-dimensional Hilbert space (here exemplified by |0^ and |1^), which represent the two basis states of the qubit. The implementation of a qubit can be of different types and different architectures. Examples include: energy levels in trapped ions, polarization states of photons, spins in quantum dots or silicon, and energy levels of superconducting circuit resonators. Example components of the QR 102 include: an ion trap, several (e.g., superconducting) circuit resonators, several quantum dots, a photonic system, a laser, an electromagnetic coil, etc. The operating device 104 is configured to operate the QR 102 (e.g.,to act on it), for example according to a sequence 106 (also referred to as operating sequence). Example processes of the operating sequence 106 include: initiating 101 the QR (i.e. bringing it into a basis state as initial state Z0), controlling 103 the QR 102 and / or reading 110 the QR 102. In the basis state (e.g. in the initial state Z0) of the QR 102, each qubit can be unentangled and / or in one of the basis states (here exemplarily |0^) of the qubit, which, however, does not necessarily have to be the same for all qubits. The basis state (e.g. in the initial state |^. ^^ = Z0) of the QR 102 can, for example, be normalized to 1. The operating sequence 106 (also referred to as a quantum mechanical measurement process or, for short, as a measurement) can, for example, be repeated several times for each parameter vector and / or each term of the R.401124 - 7 - Hamiltonian, e.g., per optimization process sequence, in order to improve the accuracy of the expected value. The concrete implementation of the operating device 104 or the operating sequence 106 depends on the type and architecture of the qubits. The operating device 104 can have one or more than one transducer configured to interact with the QR 102 or its environment, e.g., to change and / or detect at least its state Z = |^^. Example implementations of a transducer of the operating device include: actuator (e.g., final control element), sensor, transceiver, and the like.Exemplary transducers of the operating device are configured to transmit one or more of the following to and / or detect from the QR 102: optical radiation, a magnetic field, an electric field, electric charge, particle radiation. The QPU is a component of a quantum computer, which further comprises at least the housing environment for the QPU and the control electronics. In general, the control 103 of the QR 102 is configured to stimulate a change in the state (also referred to as a state change) of the QR 102. The control 103 of the QR 102 occurs according to one or more than one quantum gate 108 (see also Fig. 5). The term "quantum gate" (also referred to as quantum logic gate or gate) refers to the quantum mechanical analogue of a logic gate of the electrical circuit. However, the quantum gate 108 corresponds to a process of influencing the QR 102, which changes the state |^^ of the QR 102 (e.g. the initial state), e.g.one or more than one qubit thereof, is changed. By means of the quantum gate 108, a unitary function can be implemented in analogy to a logic gate, clearly as an analogue to its Boolean function. Mathematically, a quantum gate can be described as a unitary transformation U, which transfers a qubit from the state |^^ to the state U|^^. Examples of quantum gates (also referred to as QGs for short) include: Hadamard gates, rotation gates, T-gates, controlled-not gates (CNOT), exchange nodes, etc. A quantum gate can be configured to form or at least maintain entanglement. The term "quantum algorithm" (also referred to as quantum circuit or QSK for short) generally refers to a set of processes (e.g., quantum gates) of interaction with the QR 102 or within the QR 102, which, for example, cause a change in the state |^^ of the QR 102 (e.g.,at least one of its qubits). For better understanding, a QSK is notated as a circuit diagram in analogy to electronics, with one axis (here from left to right) representing the temporal progression. Analogous to an electrical circuit, the sum of several quantum circuits can in turn be understood as a QSK. From this perspective, the entire operating sequence 106 can be understood as a QSK, but so can its components, such as the process of controlling 103 or reading 110. From the perspective of data processing, each QSK converts one state of the QR 102 (also referred to as the input state of the quantum circuit) into another state of the QR 102 (also referred to as the output state of the quantum circuit). The output state of the controlling 103 of the QR 102 (also referred to as the controlling process 103) is given here as an example as the superposition state R.401124 - 8 -, e.g. (1≤i, j≤n), but does not necessarily have to have entangled qubits. In general, if related to a single qubit, the superposition state can comprise a superposition of states of the single qubit, and / or, if related to multiple qubits, an entanglement of the multiple qubits and / or a superposition of their states. The QSK can accordingly be configured to bring a qubit into a superposition state, which does not necessarily have to lead to entanglement of the qubit, but certainly can. The QSK is preferably configured to change the squared amplitude and phase of a qubit. A QSK can optionally comprise or consist of one or more quantum gates (e.g., multi-qubit quantum gates) parameterized by at least one parameter (also referred to as a variational parameter) (then also referred to as a variational circuit or VQC for short).A QSK configured as a VQC can optionally have one or more unparameterized quantum gates and / or one or more multi-qubit quantum gates. For the sake of simplicity, the parameterization of the QSK is notated herein as a vectorial tuple (then also referred to as a parameter vector) and can, but does not necessarily have to, have two or more parameters as components (i.e., can also be a scalar). Accordingly, the output state of the QSK configured as a VQC (i.e., parameterized QSK) is a function of the parameter vector of the VQC, each vector component of which is a parameter of the VQC. The process of reading 110 of a QSK 102 (also referred to as readout process 110) comprises sensing (i.e., using a sensor) the state of the QSK 102, which can, for example, be a superposition state and / or a result of a control process 103.During readout 110, the state of the QR 102 collapses into a base state of the measurement basis of the readout process (here, for example, Z. A). The input state of the readout process 110 can be the output state of the control process 103. The sensor can be part of a measuring chain that has a corresponding infrastructure (e.g., processor, storage medium, and / or bus system, or the like). The measuring chain can be configured to control the sensor, process the detected state as an input variable, and, based thereon, output data that represent the input variable. The measuring chain can be implemented, for example, by means of the operating device 104. The sensor can have one or more measurement bases, one of which is the measurement base according to which the readout process 110 is carried out. Illustratively speaking, the measurement base is a property of the sensor by means of which the readout process 110 is carried out. Mathematically speaking, the measurement base spans the Hilbert space of the states onto which the state of the QR is projected during readout.The QPU can, depending on the technical implementation, implement one or more than one measurement basis. Optionally, the QPU can implement multiple measurement basis, which makes it possible to parameterize the choice of measurement basis instead of and / or in addition to a parameterized quantum gate. R.401124 - 9 - Figure 2 illustrates a method according to various embodiments 200 in a schematic diagram, which can be performed, for example, by a control device 202. The term "control device" can be understood as any type of entity (e.g., a processor) that enables the processing of data or signals. The data or signals can, for example, be treated according to at least one (i.e., one or more than one) special function performed by the control device or the processor therein. A control device, e.g.The control device, or its processor, may comprise or be formed from an analog circuit, a digital circuit, a logic circuit, a microprocessor, a microcontroller, a central processing unit (CPU), a graphics processing unit (GPU), a digital signal processor (DSP), an integrated circuit of a programmable gate array (FPGA), or any combination thereof. Any other way of implementing the respective functions described in more detail herein may also be understood as a processor or logic circuit. One or more of the method steps described in detail herein may be carried out (e.g., implemented) by one or more special functions performed by the processor. The processor of the control device may, for example, be a classical (i.e., non-quantum mechanical, e.g., transistor-based) processor.Shown is a process sequence 602 (also referred to as an optimization process sequence) of the method for optimizing the objective function. Optimizing the objective function may involve minimizing or maximizing the objective function (e.g., its output value). The method may optionally include performing the optimization process sequence multiple times (e.g., K times) (see also Fig. 3). For ease of understanding, the successively performed optimization process sequences 602 and their components are referenced below with the index k, where 1≤k≤K and / or where the (k+1)th optimization process sequence 602. k+1 after the k-th optimization process sequences 602 k is carried out. The k-th optimization process sequence 602 k comprises: at least one (ie one or more than one) k-th operating sequence 106 k, which can be implemented by means of a QPU (for example according to embodiments 100), and (for example exactly) one on the at least one k-th operating sequence 106 k based k-th update process 213 k (also known as k-th update 213 k ), which can be implemented, for example, by means of the control device 202. The QPU has the QR 102, which is configured to control several qubits (here, for example, q1 to q n ). It can be understood that not all components of the update process 213 necessarily have to be performed by a classical processor, but can also, depending on the performance of the QPU, be performed at least partially by the QPU itself. Each (e.g., k-th) operating sequence 106 kcomprises: instructing 211 a control process 103 of the QR 102 according to the QSK, which has one or more than one QG 108; and instructing 201 a readout process 110. The readout process 110 may comprise determining data 252 (also referred to as state data) which represents a state 204 (here, exemplarily, Z k = Z(k)) of the QR 102 represent 203. The R.401124 - 10 - state 204 can, for example, have a superposition, which is formed by means of the control 103. The update process 213 k can have a k-th parameter vector ^ ϕ ^ k (e.g., having one or more than one parameter) of the VQC based on a result of the operating sequence 106 k , as will be explained in more detail later. The state data can, for example, be an expected value (e.g. EW U and / or EW H) or serve as a basis for determining the expected value. Figure 3 illustrates the adaptation 300 of a parameter of a VQC according to various embodiments in a schematic diagram, which refers to the embodiments 100 and 200. The method can optionally comprise, at the beginning of the 1st optimization process sequence 6021, the 1st parameter vector ^ ϕ ^ 1 (also called initiation), ie, to determine its starting values. The starting values ​​of the 1st parameter vector ^ ϕ ^1 can be determined, for example, by means of a random generator or read from a memory of the control device 202. The adaptation 300 can optionally comprise repeating the optimization process sequence 602 several times (e.g., K-1 times) until a criterion (also referred to as optimization criterion) is met 361. The optimization criterion can be met, for example, when a number k of runs (also referred to as run counter k) of the optimization process sequence 602 reaches a target number K and / or when an output of the objective function and / or its change falls below a threshold value. The k-th optimization process sequence 602 k can have the parameter vector ^ ϕ ^ to update the VQC 213 based on the result of the operating sequence 106 k , for example, on the status data. Updating 213 the VQC may include an updated ^ ϕ ^ Uto determine the parameter vector and ^ ϕ ^ U = ^ ϕ ^ k (then also referred to as updated parameter vector). The VQC, according to which the control 103 of the k-th operating sequence 106 k can be determined using the k-th parameter vector ^ ϕ ^ k be parameterized, ie VQC( ^ ϕ ^ k ) = VQC k . The control 103 of the k-th operating sequence 106 is thus carried out according to that parameter vector ^ ϕ ^ , which the previously performed (k-1)-th optimization process sequence 602 has output as an updated parameter vector. If the optimization criterion is met 361, the last updated parameter vector ^ ϕ ^ k=K = ^ ϕ ^ Ausgabe(also referred to as output vector A) are output 611. Optionally, the or each optimization process sequence 602 may comprise the k-th operating sequence 106 k several times, for example, until a criterion (also called the repetition criterion) is met. 371 This improves the statistics. The repetition criterion can be met, for example, when a number r of runs of the k-th operating sequence 106 k a target number R is reached. The update process 213 can then be based on the k-th data of several k-th operating sequences 106 k The status data can be based on the result of the kth operating sequence 106 k R.401124 - 11 - After the adjustment 300, it can optionally be determined whether the output vector ^ ϕ ^ Ausgabe(e.g., the VQC parameterized thereby) fulfills a stored convergence criterion, as will be explained in more detail later. Figure 4 illustrates the method according to various embodiments 400 in a schematic diagram, which can be carried out, for example, by means of a control device 202, here exemplary designed as a CPU, for example based on a variational principle (e.g., the Rayleigh-Ritz principle). The embodiments 400 can be configured as described for the above embodiments, wherein the update process 213 can be based on an objective function 302 (and / or its output 302a), which can represent, for example, a state variable of a QS. The optimization of the objective function 302 can be performed by means of a so-called optimization algorithm 304, which is configured to determine the parameter vector ^ ϕ ^or at least to determine its change based on the output 302a of the objective function 302, for example, under the constraint of minimizing the output 302a of the objective function 302 (for example, by treating it as a cost function). The minimization can be based, for example, on the history (1…k…K) of the parameter vector ^ ϕ ^and / or the history of the output 302a of the objective function 302. In a working example, the optimization algorithm 304 is configured to evaluate the objective function 302 as a cost function and to calculate its gradient during each iteration of the optimization process sequence 602. Exemplary implementations of the optimization algorithm 304 include: the so-called constrained optimization by linear approximation (COBYLA); conjugate gradient (Conjugent Gradient) or "quasi-Newton Optimizer", which can be found, for example, in the Python package scipy. The update process 213 may include updating the values ​​of the parameter vector ( ^ ϕ ^ ) based on the output 302a of the objective function 302 by means of the optimization algorithm 304 and the result thereof as an updated parameter vector ^ ϕ ^ UThese changed values ​​are fed as actual parameter vectors to the subsequent optimization process sequence 602 in order to control the QR 102. With each execution of the optimization process sequence 602 (then also referred to as iteration), the parameter vector converges. ^ ϕ ^ against ^ ϕ ^ AusgabeThis process can be repeated until the output of the objective function 302 converges to a minimum and / or until another convergence criterion is met. Figure 5 illustrates the method according to various embodiments 500 in a schematic diagram, which is implemented, for example, by means of a control device 202. The embodiments 500 can, for example, be configured as described for embodiments 100 and 400. Herein, the successively performed adaptation processes 300 and their components are referenced with the index p for easier understanding, where 1≤p≤P R.401124 - 12 - and / or where the (p+1)-th adaptation process 300 p+1 after the p-th adjustment process 300 p is carried out, e.g., based on a result thereof. The method may comprise carrying out the adaptation 300 (also referred to as adaptation process 300) several times (e.g., P times), wherein the p-th adaptation process 300p has one or more than one parameter of a p-th VQC p The output vector A p of the p-th adaptation process 300 p and / or the p-th VQC p may be supplied as input to a circuit determining circuit 501. The circuit determining circuit 501 may include determining the (p+1)-th VQC p+1 (also called determining 501 a VOC). As a (p+1)-th VQC p A VQC can be determined, which is the result of A p parameterized (p)-th VQC p , i.e. VQC p (A p ) and at least one quantum gate 108 connected to it (also called an additional gate). This illustrates the VQC p supplemented iteratively. For each VQC p (A p) it can be determined 503 whether the convergence criterion is met. The convergence criterion can be met, for example, if the objective function meets the convergence criterion. Figure 6 illustrates the method according to various embodiments 600 in a schematic diagram, which is implemented, for example, at least partially by means of a control device 202. The embodiments 600 can, for example, be configured as described above. As shown, the objective function can be determined, for example, by means of a Hadamard test 511 (e.g., having a plurality of Hadamard gates H). A concrete implementation of the method for a given model of a QS (also referred to as a model system) optionally comprises: providing (e.g., determining) a unitary matrix U and / or a hermetic matrix H, which represent the Hamiltonian of the model system.The determination of the hermetic matrix H can be carried out, for example, using a classical computer, for example analogous to the known calculation for exact diagonalization. The determination of the unitary matrix U can be based on the hermetic matrix H, for example by transforming the hermetic matrix H into the unitary matrix U. It is understood that the unitary matrix U can also be determined in another way, or using a differently provided specification. The concrete implementation of the method for the given model of the QS has the following process sequence: Initializing 101 several quantum bits of a QR with a basis state; Driving the QR, which is in the basis state, according to a VQC; Determining an expectation value (EW). U ) of the resulting state of the QR (also called quantum state), e.g. by means of a Hadamard test; determining the expected value (EW H) of H, e.g., by means of an inverse transformation; and adjusting 300 the parameters of the VQC, e.g., by means of an optimization algorithm (e.g., Powell), to find the lowest eigenvalue of H (which can be done using a classical computer). This sequence of processes can optionally be repeated, e.g., to calculate R.401124 - 13 - the eigenvalues ​​of the lowest excited states. For this purpose, one or more penalty terms can be introduced, for example, to favor orthogonality to the ground state (also denoted |^^^) and previously calculated eigenstates.The concrete implementation of the method for the given QS model can optionally include: determining one or more transition matrix elements; if desired, for example, if this facilitates the determination of the optical conductivity; and / or determining the physical quantities for the model system (which can be done using a classical computer) based on the determined eigenvalue. Various working examples are explained below to implement the method, using the specific case of magnetic susceptibility for a 1-dimensional Heisenberg model (a so-called spin chain), such as can be tested or prepared using a quantum simulator (e.g., IBM Qiskit). Working example 1 concerns the determination of the unitary matrix U based on a hermetic matrix H, which represents the Hamiltonian operator of the model system.Accordingly, it is advantageous to exploit the symmetries of the Hamiltonian and to calculate the eigenvalues ​​of the matrices for the different quantum numbers separately, which leads to significantly smaller matrices (e.g. p. z for rotational symmetry, q x for translational symmetry in the x-direction, etc.). Determining the unitary matrix U facilitates the implementation of the model system into quantum gates (illustratively for performing a unitary operation on the QPU). To do this, it is advantageous to transform the hermetic matrix H into a unitary matrix U, determine its eigenvalues ​​(and optionally other quantities, such as transition matrix elements), and from these, calculate the eigenvalues ​​(and other quantities) of H. This transformation can be performed according to the following relation: ^ = ^ ^∙^, where i denotes the imaginary number. Working example 2 concerns the implementation of the inverse transformation. The inverse transformation can be performed according to ^^(^) the following relation: ^ = . When performing the inverse transformation, it should be noted that the logarithm ln of complex numbers is not uniquely defined, but adds or subtracts multiples of 2π. To take this into account, the energy unit can be chosen in a modification such that the absolute value of H is always smaller than π. In this modification, the inverse transformation can be carried out according to the following relation: ^ = ^^ and / or the expected value can be determined by means of inverse transformation according to the following relation: EWH = ln(EWU) / i. Working example 3 concerns the initialization 101 of the quantum bits of the QR with a basis state. Accordingly, the initialization 101 of the quantum bits is carried out using so-called amplitude encoding. This facilitates the initialization 101 of the quantum bits when more complex states are initialized. However, amplitude encoding is complex and not absolutely necessary when R.401124 - 14 - less complex states are initialized, which, for example, consist only of 0 and 1, e.g., |00010^. Working example 4 concerns the control of the quantum register. Accordingly, the quantum bits are controlled according to a VQC1, which can be as uncomplex as possible.For example, the VQC1 can have exactly one quantum gate and be configured to convert a first quantum bit of the QR into a quantum state |^^ according to the quantum gate. Working example 5 concerns the determination of the expected value. Accordingly, the determination of the expected value can be carried out using the so-called Hadamard test 511, which is particularly reliable and efficient. Illustratively, the Hadamard test 511 refers to a special QR that is configured to generate a random variable whose expected value is the expected real part Re^^|^|^^. Û represents a unitary quantum gate that acts on the space of the quantum state |^^. A modified Hadamard test 511, on the other hand, generates a random variable whose expected value is the expected imaginary part Im^^|^|^^. If the real part of the expected value and the imaginary part of the expected value are determined individually, they can be added EW. U based on which EW Hcan be determined, for example according to the relation EW H = ln(EW U) / i. It can be understood that the real part of the expected value and the imaginary part of the expected value can also be determined individually in another way, i.e., they do not necessarily have to be done using the Hadamard test 511. Working example 6 concerns the determination of the expected value, in which, alternatively or in addition to the Hadamard test 511, a quantum phase estimation algorithm is used to determine the expected value. The quantum phase estimation algorithm is a QSK for estimating the phase (or eigenvalue) of an eigenvector of a unitary operator. Alternatively or additionally, non-Boolean quantum amplitude amplification and quantum mean value estimation can also be used to determine the expected value. This is a generalization of the quantum amplitude amplification and amplitude estimation algorithms to work with non-Boolean operators.Working example 7 concerns the cost function. The cost function can be EW. H have or consist of, for example, according to the relation EW H = ln(EW U ) / i. The cost function can optionally have one or more terms in addition to the relation, as will be explained in more detail later. Working example 8 concerns the determination of the ground state |^^^=|^ ^^ ^ of H. For this purpose, it can be exploited that the quantum state |^^, which has the lowest expectation value, is an eigenstate (ie the ground state) of H. For example, the convergence criterion can be met if the expectation value has reached a minimum or at least its change EW H (VQC p (A p ))- EW H (VQC p-1 (A p-1 )) falls below a (e.g. stored) threshold value. R.401124 - 15 - As soon as a VQC p (A p) is determined where the convergence criterion is met (which, for example, optimizes the ground state), the VQC p (A p ) as a result (also called ground state QSK) 550 (illustratively the result of the p-th fitting process), optionally together with EW H (VQC p (A p)). Working example 9 concerns the determination of several eigenvalues ​​and / or eigenvectors for H (also referred to as a set). To do this, for one or more than one (e.g., each) basis state of the QR, the QR that is in the basis state can be controlled according to the ground state QSK and the resulting state of the QR can be read out. If the set of eigenvalues ​​and / or eigenvectors determined in this way does not meet the requirements, it can optionally be adapted. Working example 10 concerns the adaptation of the set of eigenvalues ​​and / or eigenvectors, for example to determine the eigenvalues ​​of the lowest excited states (also referred to as lowest eigenvalues). This can be done using an optimization algorithm and / or using one or more than one penalty term. Analogous to the description above, the QR that is initialized 101 in the basis state is controlled using the ground state QSK.The resulting state of the QR represents the ground state with the lowest expectation value. Alternatively, the QR can also be placed in a different base state (e.g., one that provides the second-lowest expectation value), in which case the process is started with a new VQC. Working example 11 concerns the determination of one or more excited states, which are, for example, orthogonal to the GS, using a penalty term. Analogous to the above, the circuit can be used to determine 501 as the p-th VQC. p a VQC can be determined, which is determined by means of A p-1 parameterized (p-1)-th VQC p-1 , i.e. VQC p-1 (A p-1 ), and at least one (ie one or more than one) additional gate 108 connected thereto. Adapting the VQC p can be done by minimizing the expected value as a cost function according to the following relation: ^^|^|^^ + ^ ∗^^^|^^. The term S(x = ^^)|^^ = ^^^|^^ acts here as an exemplary penalty term, which is set up as a function of |^^^ and favors the determination of a |^^ that is orthogonal to |^^^. The larger the free parameter α > 0 is chosen, the stronger the effect of the penalty term S. The penalty term clearly represents the overlap ^^^|^^ and can be determined, for example, using the so-called swap test 900, which is shown in Fig. 9. Analogously, each additional excited state |^ ^ ^ can be optimized by using for each previously calculated lower excited state |^ ^^^ ^ an additional penalty term S(x = |^ ^^^ ^)|^ ^^ is added to favor orthogonality. More generally, the penalty term S(x) can be configured to penalize a deviation from the orthogonality to the quantum state x (for example, as part of the objective function, e.g., the cost function). R.401124 - 16 - Working Example 12 concerns the determination of one or more excited states that are, for example, orthogonal to the GS. Accordingly, the term S(x)|^^ = ^^|^^ does not necessarily have to be part of the objective function, but can be used constructively, e.g., by means of a quantum Gram-Schmidt process. The Gram-Schmidt process denotes a process for orthonormalizing a set of vectors in a space with an inner product, with the standard inner product.Working Example 13 concerns the determination of one or more excited states using a so-called variational quantum thermalizer (VQT), which is based on the determined GS. The density matrix of the thermalized state (T > 0) of a QS is given by a Hamiltonian ^. ^ (also called Hamiltonian) can be described mathematically as: Here, β = 1 / kBT denotes the inverse temperature, Tr the trace, kB the Boltzmann constant, and ρi the density matrix of the eigenstate of the Hamiltonian with the energy Ei as the eigenvalue. Starting from the determined Gs, excited states (also called thermalized states) can be determined by projecting the ground state out of the wavefunction and / or assigning the ground state the additional penalty term. Alternatively or in addition to the VQT, the so-called "thermofield double-state" method can be used, in which the system is enlarged and collapsed again by a subsequent measurement. Working example 14 concerns an embodiment of the above description. In a less complex first implementation, the adaptation 103 of the p-th VQCp, which has VQCp-1(Ap-1) and at least one additional gate 108 connected to it, can comprise setting the parameters of VQCp-1(Ap-1) invariant.This reduces the number of degrees of freedom, since only the at least one additional gate 108 (e.g., its parameters, type, qubits it acts on, etc.) is adjusted. In a somewhat more complex second implementation, adjusting the p-th VQCp, which comprises VQCp-1(Ap-1) and at least one first additional gate 108 connected to it, may comprise adjusting one or more than one parameter of VQCp-1(Ap-1), e.g., alternatively or in addition to adjusting the at least one additional gate 108. For example, in the second implementation, in which VQCp-1(Ap-1) comprises VQCp-2(Ap-2) and at least one second additional gate 108 connected to it, may comprise (e.g., only) adjusting the at least one second additional gate 108. More generally, the parameters of some or all of the quantum gates of the p-th VQCp may be adjusted.Working Example 15 concerns the selection of one of several basis states as the initial state, on the basis of which the adaptation processes 300p (p=1 to p=P) are carried out to determine the ground-state QSK. Accordingly, for each basis state of the several basis states, the QR brought into the basis state can be controlled according to a VQC1 having one or more R.401124 - 17 - quantum gates, and the expectation value representing the resulting state of the QR can be determined. The basis state of the several basis states that yields the lowest expectation value can be selected as the initial state. This improves the starting point and thus reduces the resource requirements of the method. The above description will be explained below for an exemplary QR with five qubits, although it should be understood that the description can also apply analogously to any other number of qubits.Figure 7 illustrates the method, in particular the result of the adaptation process 300. p for p=4 and the exemplary QR 102 with five qubits, according to various embodiments, in a schematic diagram 700, in which the VQC4 has exactly 4 quantum gates. The five qubits of the QR 102 are used at the start of the p-th adaptation process 300 p initialized with a base state using amplitude encoding 101 and subsequently according to the p-th VQC p controlled, to which a quantum gate 108 (also referred to as an additional gate) is added (e.g., connected downstream) by means of the circuit determination 501 with each increment of p. The Hadamard test can, for example, use an ancilla bit (ancilla_0) of the QR 102, if present. Optionally, one or more than one property of the additional gate 108 of the p-th adaptation process 300 p(e.g., type, between which qubits it acts, and / or parameterization) from a set of predefined properties, for example, using an optimization algorithm (e.g., Powell optimizer). The cost function can be the expected value of the p-th VQC p serve, for example, by its expected value EW His minimized. More generally, the circuit determination 501 may comprise determining an additional gate 108 which minimizes the expectation value. Working Example 16 relates to the determination of the physical quantities for the model system. Accordingly, based on the determined eigenvalues ​​of H (e.g., analogous to the exact diagonalization), all desired physical quantities, such as the magnetic susceptibility, can be determined. Figure 8 illustrates a schematic diagram 800 in which a magnetic susceptibility λ determined in this way is plotted against the temperature T for the exemplary QR with five qubits according to various embodiments, in comparison to the exact curve 811 of the susceptibility λ, which is essentially identical to the susceptibility λ obtained by means of the method explained herein, in the case that the susceptibility is based on the GZ and the first excited states.For comparison, case 813 is shown, in which the susceptibility is based only on the GZ (e.g., the ground states in the various symmetry sectors). Rotational symmetry and translational symmetry can be used to determine the exact course of the magnetic susceptibility for this model system. R.401124 - 18 - The magnetic susceptibility can, for example, be a uniform spin susceptibility, which can be determined according to the following relation: λ = < (Sz)2> / ( N ∙ k. B ∙ T). Where N denotes the number of eigenstates, E n the energy eigenvalue of eigenstate n, k B the Boltzmann constant, T the temperature, < (Sz)2> the expectation value of operator (Sz)2, Sz the spin operator in Z-component, where < (Sz)2> = 1 / Z ∑ ^ ^ ^^ (^ ^ ^ ) ^ ∙ ^^^(−^ ^ / (^ ^ ∙ ^)) ; and Z = ∑ ^ ^^^(−^ ^ / (^ ^∙ ^) ). This result, obtained according to various embodiments, can be reproduced, for example, using a quantum simulator (e.g., IBM Qiskit). This also demonstrates that the method explained herein quickly converges to the exact curve. The method provided herein can be recognized by the fact that one or more physical quantities are calculated using a quantum computer and using methods of exact diagonalization, for example, based on a combination of optimization methods and eigenvalue calculation.

Claims

R.401124 - 19 - Claims 1. A method for operating a quantum register (102) to determine whether a detected second quantum circuit (VQC p+l ) satisfies a stored convergence criterion, the method comprising: ^ first adjusting (300) a first parameter of a first quantum circuit (VQC p ) for optimizing a target function by controlling (103) the quantum register (102) according to the first quantum circuit (VQC p ); ^ first determining (501) the second quantum circuit (VQC p+l ), which created the first quantum circuit (VQC p ) with an adapted first parameter resulting from the first adaptation (300), and having a quantum gate (108); ^ second adaptation (300) of a second parameter of the quantum gate (108) for optimizing the objective function by controlling (103) the quantum register (102) according to the second quantum circuit (VQC p+l); ^ second determining (501) whether the second quantum circuit (VQC p+l ) with an adapted second parameter resulting from the second adaptation (300) satisfies the stored convergence criterion.

2. The method according to claim 1, further comprising: ^ determining an eigenvector and / or an eigenvalue of a mapping, wherein the objective function is a function of the eigenvalue of the mapping relative to the eigenvector of the mapping.

3. The method according to one of claims 1 to 2, wherein the objective function: ^ is a function of the first parameter during the first adaptation (300); and / or ^ is a function of the first parameter and the second parameter during the second adaptation (300).

4. The method according to one of claims 1 to 3, if the second determination shows that the second quantum circuit (VQC p+l) satisfies the stored convergence criterion, further comprising: ^ determining an additional quantum circuit for optimizing an additional objective function by controlling (103) the quantum register (102) according to the additional quantum circuit; ^ wherein the additional objective function represents a relation between a result of controlling (103) the quantum register according to the second quantum circuit and a result of controlling (103) the quantum register (102) according to the additional quantum circuit. R.401124 - 20 - 5. The method according to any one of claims 1 to 4, wherein the second adjusting (300) further comprises changing one or more of the following to optimize the objective function: ^ a type of the quantum gate (108); ^ a first qubit of the quantum register (102) which is influenced by the quantum gate (108); ^ a second qubit of the quantum register (102) which determines the influencing of the first qubit by the quantum gate (108).

6. The method according to any one of claims 1 to 5, ^ wherein the second adjusting (300) further comprises adjusting the first parameter, and ^ wherein the second determining comprises determining whether the second quantum circuit (VQC p+l) with the adjusted first parameter and the adjusted second parameter resulting from the second adjustment (300), satisfies the stored convergence criterion.

7. The method according to one of claims 1 to 6, ^ wherein the control (103) of the quantum register (102) according to the first quantum circuit (VQC p ) occurs when the quantum register (102) is brought into an initial state; and / or ^ wherein the driving (103) of the quantum register (102) according to the second quantum circuit (VQC p+l ) occurs when the quantum register (102) is brought into the initial state; the method further comprising: ^ selecting the initial state from a plurality of basis states of the quantum register (102), ^ wherein the selection, for each basis state of the plurality of basis states, is carried out by controlling (103) the quantum register (102) according to at least a part of the first quantum circuit (VQC p) when the quantum register (102) is brought into the base state.

8. A computer program configured to cause a processor executing the computer program to perform the method according to any one of claims 1 to 7.

9. A computer-readable medium storing instructions configured, when executed by a processor, to cause the processor to perform the method according to any one of claims 1 to 7.

10. A control device comprising one or more processors configured to perform the method according to any one of claims 1 to 7.

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