Method for operating quantum register

Abstract

According to various embodiments, a method for operating a quantum register (102) may comprise: adapting (300) a first parameter of a first quantum circuit (VQCp) a first time to optimize an objective function by driving (103) the quantum register (102) according to the first quantum circuit (VQCp); first determination of a second quantum circuit (VQCp + 1) comprising a quantum gate and a first quantum circuit (VQCp) having an adapted first parameter derived from the first adaptation (300); adapting (300) a second parameter of the quantum gate for a second time to optimize the objective function by driving (103) the quantum register (102) according to a second quantum circuit (VQCp + 1); a second determination is made as to whether a second quantum circuit (VQCp + 1) having an adapted second parameter derived from the adaptation (300) meets the stored convergence criteria.

Description

Technical Field

[0001] The present disclosure relates to a method for operating a quantum register. Background Art

[0002] Quantum computers and the calculations performed by them (also referred to as quantum computing) are one of the key technologies of the future, as they enable calculations that classical computers cannot or can only perform with difficulty. Quantum computing has a wide range of applications, for example in mathematics (number theory, linear algebra, differential equations) and physics (determining the electronic structure of molecules and materials, real-time simulation of quantum dynamical effects such as protein folding and optical properties). In this context, quantum computers are typically used as co-processors (similar to graphics chips), which complement rather than replace classical computers. Further expectations also relate to the application of quantum computers to time-critical optimization problems, such as those faced by navigation and traffic guidance systems, as well as artificial intelligence (AI), materials research, and medicine. One aspect of materials research is the quantitative exploration of quantum mechanical systems such as molecules (also simply referred to herein as QS). The simulation of molecules (such as proteins) is usually based on quantum mechanics, and its mathematical description is very complex, which makes it difficult to implement quantum mechanics on classical computers and usually at the cost of a discount in terms of accuracy.

[0003] So-called exact diagonalization is one of the most important methods for calculating physical properties in quantum mechanics, especially in condensed matter physics, and is particularly suitable for strongly correlated electron systems such as high-temperature superconductors or manganites. So-called exact diagonalization (ED) provides an exact mathematical solution of the QS by decomposing the complete Hilbert space of all states into eigenstates by means of matrix diagonalization, and these eigenstates can be combined into any thermalized state. However, its resource requirements for ED increase sharply with the size of the QS (e.g., the number of atoms in the QS), so that even modern supercomputers are often not sufficient to obtain a sufficiently accurate solution within an acceptable calculation time. To reduce the calculation time, approximate methods are usually employed, such as including the so-called finite temperature Lanczos method (FTLM), where the ground state and the initially excited states can be calculated very accurately, but the higher excited states are not accurate enough. However, even approximate methods face the problem of exponential growth of the computational amount, so that in practice only relatively small systems (e.g., for the Heisenberg model, on the order of about 40 spins) have been considered so far.

[0004] In contrast, quantum computers use quantum mechanical effects, such as superposition states, to perform calculations, which simplifies simulations and reduces the requirements for accuracy. When using quantum computers, so-called hybrid algorithms, i.e. quantum-classical algorithms, are often used, which outsource only part of the calculation to the quantum computer. Quantum-classical algorithms allow the advantages of both types of computers to be exploited and reduce the requirements on the quantum computer, for example in terms of the number of qubits, circuit depth and gate operation error tolerance (so-called noise or "noise"). A popular representative of hybrid algorithms is the so-called "Variational Quantum Eigensolver" (VQE). VQE uses variational quantum circuits to prepare so-called pseudo-states on the quantum computer. Subsequently, the quantum computer measures the energy of these pseudo-states. The classical (i.e. non-quantum mechanical) computer is responsible for optimizing the quantum circuits so that the pseudo-states converge to the ground state of QS (i.e. at temperature T = 0 Kelvin). One goal is to calculate the states of QS above 0 Kelvin (also called thermalized or excited states), which are not accessible for VQE. Summary of the invention

[0005] According to various embodiments, a method for operating a quantum register (QR) is provided to determine a second quantum circuit (VQC p+l ) satisfies the stored convergence criteria, the method comprising: first adapting the first quantum circuit (e.g., VQC p , where p ≥ 1), to optimize the objective function by driving the quantum register according to the first quantum circuit; first determining the second quantum circuit (e.g., VQC p+l , where l≥1), the second quantum circuit comprises a first quantum circuit having adapted first parameters obtained by the first adaptation and a quantum gate (e.g., connected downstream of the first quantum circuit); adapting the second parameters of the quantum gate for a second time to optimize the objective function by driving the quantum register according to the second quantum circuit (e.g., by); and determining for a second time whether the second quantum circuit having the adapted second parameters obtained by the second adaptation satisfies the stored convergence criterion.

[0006] The method provided herein simplifies the reproducible initialization, simulation, and calculation of quantum mechanical states, for example when the quantum mechanical state is supposed to represent a temperature with a QS above 0 Kelvin. Intuitively, the method helps to form a QS model that can be calculated by means of QR and whose size scales with the number of qubits. The method described herein can be implemented and executed, for example, in a resource-saving and cost-effective manner, since this does not necessarily scale exponentially with the size of the QS and only requires a small amount of resources of the quantum computer. Compared with the popular VQE, the method provided herein does not require approximation (such as an ansatz state), does not require the calculation of the coefficients of the wave function, reduces the complexity of quantum state preparation (for example, does not require preparation), and / or optimizes the parameters of the quantum circuit (instead of the coefficients of the wave function). Compared with VQE, the method provided herein is applicable to a wider range of application fields and allows the calculation of physical properties along with their temperature dependence. Similarly, the method provided herein simplifies the calculation of excited states.

[0007] The following describes different embodiments.

[0008] Embodiment 1 is a method for running QR as described above.

[0009] Embodiment 2 is the method according to Embodiment 1, further comprising: determining the eigenvectors and / or eigenvalues (e.g., with respect to the eigenvectors) of a mapping (such as a matrix and / or Hamiltonian), wherein the objective function is a function of the eigenvalues of the mapping with respect to the eigenvectors of the mapping, wherein the mapping preferably has (e.g., a physical) state model or consists of (e.g., a physical) state model (such as the state of a physical system); and / or wherein the mapping preferably maps the eigenvectors to the eigenvalues. This facilitates more complex quantum physical calculations. Preferably, the method includes the step of determining at least one physical property of a technical system (especially a physical system) based on the determined eigenvectors and / or the determined eigenvalues, wherein the mapping especially represents the Hamiltonian function of the technical system. The physical property can be, for example, the magnetic susceptibility of a solid.

[0010] Embodiment 3 is the method according to Embodiment 1 or 2, wherein the first adaptation is based on first data (such as measurement data or frequency distribution), the first data representing a first state of a quantum register, the first state being directly generated by driving the quantum register according to a first quantum circuit or at least based thereon; and / or wherein the second adaptation is based on second data, the second data representing a second state of the quantum register, the second state being directly generated by driving the quantum register according to a second quantum circuit or at least based thereon. This facilitates the execution of more complex calculations, such as when dealing with a computational physical system.

[0011] Example 4 is the method according to any one of Examples 1 to 3, wherein the objective function: is a function of the first parameter (and / or independent of the second parameter) and / or at least related to the first data during the first adaptation; and / or is a function of the second parameter (and optionally the first parameter) and / or at least related to the second data during the second adaptation. This facilitates performing more complex calculations, for example when a physical system should be calculated.

[0012] Example 5 is the method according to any one of Examples 3 to 4, further comprising: reading out the first data; and / or reading out the second data. This improves the process of the method.

[0013] Example 6 is the method according to any one of Examples 1 to 5, when it is determined for the second time that the second quantum circuit meets the stored convergence criterion, the method further comprises: determining an additional quantum circuit (preferably based on or having the second quantum circuit) for optimizing an additional objective function (e.g., having the objective function or at least a part thereof) by driving a quantum register according to the additional quantum circuit; wherein the additional objective function has the objective function and / or represents a relationship (e.g., overlap) (e.g., is a function thereof) between the result of driving the quantum register according to the second quantum circuit and the result of driving the quantum register according to the additional quantum circuit; wherein for example the relationship penalizes the overlap of these two results (e.g., deviation from orthogonality). This facilitates performing more complex calculations, for example when dealing with calculating excited states. Considering the overlap improves the result, for example in terms of the property of orthogonality.

[0014] Example 7 is the method according to any one of Examples 1 or 6, wherein the second determination is based on the objective function, preferably by determining whether the objective function meets the convergence criterion; and / or wherein when the objective function meets the convergence criterion, the convergence criterion is satisfied. This improves the ratio between accuracy and calculation time.

[0015] Example 8 is the method according to any one of Examples 1 to 7, wherein the second adaptation further comprises: changing one or more of the following to optimize the objective function: the type of quantum gate; the first qubit (also called a quantum bit) of the quantum register, which is affected by the quantum gate; the second qubit of the quantum register, which causes the first qubit to be affected by the quantum gate. This improves the relationship between accuracy and calculation time.

[0016] Embodiment 9 is the method according to any one of Embodiments 1 to 8, further comprising: when it is determined for the second time that the stored convergence criterion is not met, determining a third quantum circuit, which includes a second quantum circuit having an adapted second parameter obtained by the second adaptation and at least one additional quantum gate (e.g., connected downstream of the second quantum circuit); adapting a third parameter of the additional quantum gate for the third time to optimize an objective function by driving a quantum register according to the third quantum circuit; and preferably determining for the third time whether the third quantum circuit having the adapted third parameter obtained by the third adaptation meets the stored convergence criterion. This improves the accuracy of the result.

[0017] Embodiment 10 is the method according to any one of Embodiments 1 to 9, wherein the second adaptation further includes (e.g., re-)adapting the first parameter, and wherein the second determination includes: determining whether the second quantum circuit having the adapted first parameter and the adapted second parameter obtained by the second adaptation meets the stored convergence criterion. This improves the ratio between accuracy and computation time.

[0018] Embodiment 11 is the method according to any one of Embodiments 1 to 10, wherein the quantum register is driven according to the first quantum circuit when the quantum register is placed in an initial state; and / or wherein the quantum register is driven according to the second quantum circuit when the quantum register is placed in an initial state. This improves the accuracy and reduces the computation time.

[0019] Embodiment 12 is the method according to Embodiment 11, further comprising: selecting an initial state from a plurality of (e.g., normalized) basis states of the quantum register, wherein preferably, for each of the plurality of basis states, the selection is performed by driving 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 placed in the basis state. This improves the ratio between accuracy and computation time.

[0020] Embodiment 13 is a control device (e.g., non - quantum - mechanical), which is configured to execute the method according to any one of Embodiments 1 to 12.

[0021] Embodiment 14 is a computer program, which is configured to cause a (e.g., non - quantum - mechanical) processor executing the computer program to execute the method according to any one of Embodiments 1 to 12.

[0022] Embodiment 15 is a computer - readable medium, which stores instructions that are configured to cause a (e.g., non - quantum - mechanical) processor executing the instructions to execute the method according to any one of Embodiments 1 to 12. Embodiments 11 to 13 are conducive to implementing this method. Description of the Drawings

[0023] In the drawings, like reference numerals generally refer to the same parts throughout the different views. The drawings are not necessarily to scale, with emphasis typically placed on showing the principles of the invention. In the following description, various aspects will be described with reference to the following drawings.

[0024] Figure 1 show the structure and operation mode of a quantum processor, Figures 2 to 7 respectively show a method or at least part of the method according to different embodiments, Figure 8 show the magnetic susceptibility in a schematic diagram, Figure 9 show the Swap-Test according to different embodiments in a schematic diagram.

[0025] The following detailed description relates to the drawings, which are used to explain specific details and aspects of the present disclosure, and the present invention can be implemented according to these specific details and aspects. Other aspects can be used, and structural, logical, and electrical changes can be made without departing from the protection scope of the present invention. Various aspects of the present disclosure are not necessarily mutually exclusive, because some aspects of the present disclosure can be combined with one or more other aspects of the present disclosure to form new aspects. Detailed embodiments

[0026] Figure 1 Illustrate in a schematic diagram the structure and operation mode of a quantum processing unit ("quantum processing unit" or simply QPU) according to different embodiments 100. The QPU has a quantum register 102 and a running device 104. The term "quantum register" (QR) refers to a device by means of which multiple (for example, n in number) qubits (exemplarily called q 1 to q n ) can be realized. The term "qubit" refers to a two-state quantum system, which can be mathematically expressed by two basis vectors of a two-dimensional Hilbert space (exemplarily spanned by and ), and the basis vectors represent the two basic states of the qubit. The implementation of qubits can be of different types and different architectures. Examples in this regard include: energy levels in trapped ions, polarization states of photons, spins in quantum dots or silicon, and energy levels of superconducting circuit resonators. Exemplary components of QR 102 include: ion traps, multiple (for example, superconducting) circuit resonators, multiple quantum dots, photon systems, lasers, electromagnetic coils, etc.

[0027] The running device 104 is configured to run QR 102 (for example, apply an action to it), for example, according to a sequence 106 (also called a running sequence). Exemplary processes of the running sequence 106 include: initializing 101 QR (i.e., placing QR in the initial state Z0 in the base state), driving 103 QR 102 and / or reading out 110 QR 102. In the base state of QR 102 (e.g., in the initial state Z 0 ), each qubit can be unentangled and / or in one of the base states of the qubit (exemplarily here), but the base state is not necessarily the same for all qubits. The base state of QR 102 (e.g., in the initial state = Z 0 ) can be normalized to 1, for example. The run sequence 106 (also referred to as the quantum mechanical measurement process or simply measurement) can be repeated multiple times, for example, for each parameter vector and / or each term of the Hamiltonian, for example, in each optimization process sequence, to improve the accuracy of the expected value. The specific implementation of the run device 104 or the run sequence 106 depends on the type and architecture of the qubit. The run device 104 can have one or more converters configured to interact with QR 102 or its environment, for example, at least to change and / or detect its state Z = . Exemplary implementations of the converters of the run device include: actuators (e.g., regulating elements), sensors, transceivers, etc. The exemplary converters of the run device are configured to transmit to and / or detect from QR 102 one or more of the following: optical radiation, magnetic field, electric field, charge, particle radiation. The QPU is a component of a quantum computer, which also has at least the housing environment of the QPU and has control electronics.

[0028] Generally, the drive 103 QR 102 is configured to excite a change in the state of QR 102 (also referred to as a state change). The drive 103 QR 102 is carried out according to one or more quantum gates 108 (see also Figure 5 ). The term "quantum gate" (also referred to as a quantum logic gate or gate) refers to the quantum mechanical analogue of the logic gates of an electronic circuit. However, the quantum gate 108 corresponds to a process that affects QR 102 and changes the state of QR 102 (e.g., the initial state), for example, changing one or more of its qubits. With the help of the quantum gate 108, a unitary function can be implemented similar to a logic gate (intuitively as an analogue of its Boolean function). Mathematically, a quantum gate can be described as a unitary transformation U that transforms a qubit from the state to the state . Examples of quantum gates (also simply referred to as QGs) include: Hadamard gate, rotation gate, T gate, controlled NOT gate (CNOT), swap node, etc. Quantum gates can be configured to form or at least maintain entanglement.

[0029] The term "quantum algorithm" (also known as quantum circuit or simply QSK) generally refers to a set of processes (such as quantum gates) that interact with or are within QR 102, which processes trigger or at least excite the state of QR 102, for example. For better understanding, similar to electronics recording QSK as a circuit diagram, where one axis (here from left to right) represents the time progression. Similar to a circuit, the sum of multiple quantum circuits can itself be regarded as a QSK. From this perspective, the entire operation sequence 106 can be understood as a QSK. However, similarly, its components, such as the process of drive 103 or readout 110, can also be understood as a QSK. From the perspective of data processing, each QSK converts a state of QR 102 (also known as the input state of the quantum circuit) into another state of QR 102 (also known as the output state of the quantum circuit). The output state of drive 103 of QR 102 (also known as drive process 103) is here exemplarily represented as a superposition state, for example having (1 ≤ i, j ≤ n), but does not necessarily have entangled qubits. Generally, when dealing with a single qubit, the superposition state can have a superposition of the states of that single qubit, and / or when dealing with multiple qubits, can have entanglement of multiple qubits and / or a superposition of their states. A QSK can correspondingly be configured to place a qubit in a superposition state, which does not necessarily result in entanglement of that qubit, but is entirely possible.

[0030] A QSK is preferably configured to change the modulus square of the amplitude and the phase of a qubit. A QSK can optionally have one or more parameterized quantum gates (such as multi-qubit quantum gates) or be composed of them (in this case also called a variational circuit or simply VQC) by means of at least one parameter (also called a variational parameter). A QSK configured as a VQC can optionally have one or more non-parameterized quantum gates and / or one or more multi-qubit quantum gates. For simplicity of writing, the parameterization of a QSK is recorded herein in the form of a vector as a tuple (in this case also called a parameter vector), and can but does not necessarily have two or more parameters as components (i.e., can also be a scalar). Thus, the output state of a QSK configured as a VQC (i.e., a parameterized QSK) is a function of the parameter vector of that VQC, where each vector component is a parameter of the VQC.

[0031] The process of readout 110 of QR 102 (also known as readout process 110) includes: detecting the state of QR 102 in a sensor manner (i.e., by means of a sensor), which state can be, for example, a superposition state and / or the result of drive process 103. During readout 110, the state of QR 102 collapses to one of the basis states of the measurement basis of the readout process (here exemplarily Z A). The input state of the readout process 110 can be the output state of the drive process 103. The sensor can be part of a measurement chain that has a corresponding infrastructure (such as having a processor, storage medium, and / or bus system, etc.). The measurement chain can be configured to drive the sensor, process the detected state as an input variable, and output data representing this input variable based on this. The measurement chain can be implemented by means of the operating device 104 or will be implemented by means of the operating device 104. The sensor can have one or more measurement bases, and one of the measurement bases is the measurement base on which the readout process 110 is carried out. Intuitively, the measurement base is the nature of the sensor by which the readout process 110 is carried out. Mathematically, the measurement bases span the Hilbert space of the states onto which the state of the QR is projected during readout. Depending on the technical implementation, the QPU can implement one or more measurement bases. Optionally, the QPU can implement multiple measurement bases, which enables the selection of the measurement bases to be parameterized to replace and / or supplement the parameterized quantum gates.

[0032] Figure 2 A method according to different embodiments 200 is illustrated schematically and can be executed, for example, by a control device 202. The term "control device" can be understood as any type of entity (such as a processor) that enables the processing of data or signals. The data or signals can be processed, for example, according to at least one (i.e., one or more) specific functions executed by the control device or a processor therein. The control device, for example its processor, can have an integrated circuit of analog circuits, digital circuits, logic circuits, microprocessors, microcontrollers, central processing units (CPUs), graphics processing units (GPUs), digital signal processors (DSPs), field-programmable gate arrays (FPGAs), or any combination thereof, or be composed of an integrated circuit of analog circuits, digital circuits, logic circuits, microprocessors, microcontrollers, central processing units (CPUs), graphics processing units (GPUs), digital signal processors (DSPs), field-programmable gate arrays (FPGAs), or any combination thereof. Any other way of implementing the corresponding functions described in more detail herein can also be understood as a processor or logic circuit. One or more of the method steps described in detail herein can be executed (e.g., implemented) by one or more specific functions executed by the processor. The processor of the control device can be, for example, a classical (i.e., non-quantum mechanical, e.g., transistor-based) processor.

[0033] A process sequence 602 (also referred to as an optimization process sequence) of a method for optimizing an objective function is shown. Optimizing the objective function can include: minimizing or maximizing the objective function (e.g., its output value). The method can optionally include: executing the optimization process sequence multiple times (e.g., K times) (also see Figure 3). Next, for simplicity of understanding, the sequence of optimization processes 602 and its components executed in sequence are referred to by index k, where 1 ≤ k ≤ K and / or where the (k + 1)-th optimization process sequence 602 k+1 in the k-th optimization process sequence 602 k is executed after.

[0034] The k-th optimization process sequence 602 k has: at least one (i.e., one or more) k-th run sequence 106 k , which can be implemented by means of a QPU (e.g., according to Embodiment 100), and (e.g., exactly) one k-th update process 213 based on at least one k-th run sequence 106 k (also referred to as the k-th update 213 k ), which can be implemented by means of a control device 202, for example. The QPU has a QR 102, which is configured to implement a plurality of qubits (exemplarily q k to q 1 to q n ). It can be understood that not all components of the update process 213 necessarily have to be executed by a classical processor, but depending on the performance of the QPU, at least some can also be executed by the QPU itself.

[0035] Each (e.g., k-th) run sequence 106k has: a driving process 103 for driving the QR 102 according to a QSK instruction 211 having one or more QGs 108; and an indication 201 for a readout process 110. The readout process 110 can include: determining data 252 (also referred to as state data), which represents 203 the state 204 of the QR 102 (exemplarily Z k = Z(k)). The state 204 can have a superposition state formed by means of the drive 103, for example. The update process 213 k can have a change in the k-th parameter vector of the VQC based on the result of the run sequence 106 k (e.g., having one or more parameters), which will be elaborated in more detail later. The state data can represent an expected value (e.g., EW and / or EW U and / or EW H ) or as a basis for determining the expected value.

[0036] Figure 3 Schematically shows the adaptation 300 of the parameters of the VQC according to different embodiments, which refer to Embodiments 100 and 200. At the start of the first optimization process sequence 602 1 the method can optionally include: initializing (also referred to as starting) the first parameter vector , i.e., setting its starting value. The first parameter vector The starting value can be determined, for example, by means of a random number generator or read out from the memory of the control device 202.

[0037] The adaptation 300 may optionally include repeating the optimization process sequence 602 a plurality of times (e.g., K - 1 times) until a 361 criterion (also referred to as an optimization criterion) is met. For example, when the number of traversals k (also referred to as the traversal counter k) of the optimization process sequence 602 reaches the target number K and / or when the output of the objective function and / or its change is below a threshold, the optimization criterion may be met. The k-th optimization process sequence 602 k may include: based on the run sequence 106 k to update 213 the parameter vector of the VQC . The update 213 of the VQC may include: determining the updated parameter vector and setting = (which is also referred to as the updated parameter vector at this time). The VQC on which the k-th run sequence 106k of the drive 103 is based can be parameterized by means of the k-th parameter vector , i.e., VQC( ) = VQC k . Thus, the k-th run sequence 106 of the drive 103 is performed according to the parameter vector output as the updated parameter vector by the previously executed (k - 1)-th optimization process sequence 602. If the 361 optimization criterion is met, the last updated parameter vector (also referred to as the output vector A) can be output 611.

[0038] Optionally, the optimization process sequence or each optimization process sequence 602 may include: performing the k-th run sequence 106 k a plurality of times, for example, until a 371 criterion (also referred to as a repetition criterion) is met. This improves the statistics. For example, when the number of traversals r of the k-th run sequence 106 k reaches the target number R, the repetition criterion may be met. The update process 213 may then be based on the k-th data of a plurality of k-th run sequences 106 k . The state data may be based on the results of performing the k-th run sequence 106 k a plurality of times.

[0039] After the adaptation 300, it may optionally be determined whether the output vector (e.g., the VQC parameterized thereby) meets the stored convergence criterion, which will be explained in more detail later.

[0040] Figure 4A method according to different embodiments 400 is illustrated schematically, which can be executed, for example, by means of a control device 202 (exemplarily configured as a CPU) and, for example, based on a variational principle (such as the Rayleigh-Ritz principle). Embodiment 400 can be configured as described in the above embodiments, wherein the update process 213 can be based on the objective function 302 (and / or its output 302a), which can, for example, represent the state variables of QS.

[0041] The optimization of the objective function 302 can be carried out by means of a so-called optimization algorithm 304, which is configured to determine the parameter vector based on the output 302a of the objective function 302 or at least its variation, for example, under the boundary conditions of minimizing the output 302a of the objective function 302 (for example, by treating the objective function as a cost function). The minimization can be based, for example, on the history of the parameter vector (1…k…K) 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 calculate its gradient in each pass of the optimization process sequence 602. Exemplary implementations of the optimization algorithm 304 include: the so-called "Constrained optimization by linear approximation" - COBYLA; Conjugent Gradient or "quasi-Newton Optimizer", which can be found, for example, in the Python package scipy.

[0042] The update process 213 can include: changing the value of the parameter vector ( ) based on the output 302a of the objective function 302 by means of the optimization algorithm 304 and outputting the result as the updated parameter vector . These changed values are fed as the actual parameter vector to the subsequent optimization process sequence 602 to drive the QR 102. With each execution of the optimization process sequence 602 (also referred to as an iteration at this time), the parameter vector converges to . This process can be repeated until the output of the objective function 302 converges to a minimum value and / or until another convergence criterion is met.

[0043] Figure 5A method according to different embodiments 500 is shown schematically, which is implemented, for example, by means of a control device 202. Embodiment 500 can, for example, be configured as described in embodiments 100 and 400. Here, for the sake of simplified understanding, the successively executed adaptation processes 300 and their components are referred to by the index p, where 1 ≤ p ≤ P and / or where the (p + 1) - th adaptation process 300 p+1 in the p - th adaptation process 300 p is executed afterwards, for example based on its result.

[0044] The method may include: executing the adaptation 300 (also referred to as the adaptation process 300) a plurality of times (e.g., P times), where the p - th adaptation process 300 p includes: adapting one or more parameters of the p - th VQC p . The p - th adaptation process 300 p and / or the output vector A p of the p - th VQC p can be fed as input to the circuit determination 501. The circuit determination 501 may include: determining the (p + 1) - th VQC p+1 (also referred to as determining 501 VOC). The VQC can be determined as the (p + 1) - th VQC p which has the (p) - th VQC p parameterized by means of A p , i.e., VQC p (A p ), and at least one quantum gate 108 (also referred to as an additional gate) connected thereto. Intuitively, the VQC p is iteratively supplemented here.

[0045] For each VQC p (A p ), it can be determined 503 whether the convergence criterion is met. For example, when the objective function meets the convergence criterion, the convergence criterion can be satisfied.

[0046] Figure 6 A method according to different embodiments 600 is shown schematically, which is implemented, for example, at least in part by means of a control device 202. Embodiment 600 can, for example, be configured as described above. As shown, the determination of the objective function can be performed exemplarily by means of a Hadamard test 511 (e.g., having a plurality of Hadamard gates H).

[0047] A specific implementation of the method for a pre - given QS model (also referred to as a model system) optionally includes: providing (e.g., determining) a unitary matrix U and / or a Hermitian matrix H, which matrices represent the Hamiltonian operator of the model system. The determination of the Hermitian matrix H can be performed, for example, with the aid of a classical computer, e.g., similar to known calculations for exact diagonalization. The determination of the unitary matrix U can be based on the Hermitian matrix H, e.g., by transforming the Hermitian matrix H into the unitary matrix U. It is understood that the unitary matrix U can also be determined in other ways or provided by other stipulations.

[0048] A specific implementation of the method for a pre - given QS model has the following process sequence: initializing 101 a plurality of qubits of QR in the ground state; driving QR in the ground state according to VQC; determining the expectation value (EW U ), e.g., by means of a Hadamard test, of the resulting state (also referred to as the quantum state) of QR; determining the expectation value (EW H ) of H, e.g., by means of an inverse transformation; and adapting 300 the parameters of VQC, e.g., by means of an optimization algorithm (e.g., Powell), to find the lowest eigenvalue of H (which can be performed with the aid of a classical computer). This process sequence can optionally be repeated, e.g., for calculating the eigenvalues of the lowest excited states. For this purpose, for example, one or more penalty terms can be introduced to promote orthogonality with the ground state (also referred to as ).

[0049] A specific implementation of the method for a pre - given QS model can optionally include: determining one or more transition matrix elements; if desired, e.g., when the transition matrix elements contribute to determining the optical conductivity; and / or determining physical variables of the model system based on the determined eigenvalues (which can be performed with the aid of a classical computer).

[0050] Different working examples will be explained below to implement the method, specifically for the case of the magnetic susceptibility of a one - dimensional Heisenberg model (so - called spin chain), which can also be tested or prepared, for example, with the aid of a quantum simulator (e.g., IBM Qiskit).

[0051] Working example 1 involves determining the unitary matrix U based on the Hermitian matrix H representing the Hamiltonian operator of the model system. Thus, it is advantageous to utilize the symmetry of the Hamiltonian operator and correspondingly separately calculate the eigenvalues of matrices for different quantum numbers, which results in significantly smaller matrices (e.g., S z for rotational symmetry, q xetc.). Determining the unitary matrix U simplifies the transformation of the model system into quantum gates (intuitively used to perform unitary operations on a QPU). To this end, it is advantageous to transform the Hermitian matrix H into the unitary matrix U, determine its eigenvalues (and optionally other variables, such as transition matrix elements), and calculate the eigenvalues (and other variables) of H based on the said eigenvalues. This transformation can be carried out according to the following relationship: , where i represents the imaginary number.

[0052] Working Example 2 involves performing the inverse transformation. The inverse transformation can be carried out according to the following relationship: . It can be noted in the inverse transformation that the logarithm ln of a complex number is not uniquely determined, but can be added or subtracted by multiples of 2π. To take this into account, the said energy unit can be selected in a modified energy unit such that the numerical value of H is always less than π. In this modification, the inverse transformation can be carried out according to the following relationship: , and / or the expected value EW can be determined according to the following relationship by means of the inverse transformation H = ln(EW U ) / i.

[0053] Working Example 3 involves initializing the qubits of 101 QR in the ground state. Therefore, the initialization of the qubits of 101 is carried out by means of so-called Amplitude Encoding. This simplifies the initialization of the qubits when initializing more complex states. However, when initializing less complex states (for example, the said state consists only of 0 and 1, such as ), amplitude encoding is complex and not necessarily required.

[0054] Working Example 4 involves driving the quantum register. Therefore, the qubits are driven according to the VQC 1 as simple as possible. For example, VQC 1 can exactly have one quantum gate and can be configured to transform the first qubit of QR into a quantum state according to this quantum gate.

[0055] Working Example 5 involves determining the expected value. Therefore, the determination of the expected value can be carried out by means of the so-called Hadamard test 511, which is particularly reliable and efficient. Intuitively, the Hadamard test 511 refers to a special QR, which is configured to generate the following random variable, and the expected value of this random variable is the real part of the expectation . represents the unitary quantum gate acting on the space of the quantum state . And the modified Hadamard test 511 generates the following random variable, and the expected value of this random variable is the imaginary part of the expectation If the real part of the expected value and the imaginary part of the expected value are determined separately, they can be added to obtain EW U , and based on this, EW can be determined H , for example, according to the relationship EW H = ln(EW U ) / i. It can be understood that the real part and the imaginary part of the expected value can also be determined separately in other ways, that is, it is not necessarily necessary to use the Hadamard test 511 for this purpose

[0056] Working example 6 relates to determining the expected value, in which, alternatively or additionally 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 the eigenvector of a unitary operator. Alternatively or additionally, non-Boolean quantum amplitude amplification and quantum mean estimation can also be used to determine the expected value. This involves a generalization of the algorithms for quantum amplitude amplification and amplitude estimation so that they can work with non-Boolean operators

[0057] Working example 7 relates to a cost function. The cost function can have EW H or be composed of EW H , for example, according to the relationship EW H = ln(EW U ) / i. The cost function can optionally have one or more terms in addition to this relationship, which will be elaborated in more detail later

[0058] Working example 8 relates to determining the ground state of H . For this purpose, it can be utilized that the quantum state with the lowest expected value is the eigenstate (i.e., the ground state) of H. For example, when the expected value reaches a minimum or at least its change EW H (VQC p (A p )) - EW H (VQC p-1 (A p-1 )) is below (for example, a stored) threshold, the convergence criterion can be met

[0059] Once the VQC p (A p ) at which the convergence criterion is met is determined (which optimizes the ground state, for example), this VQC p (A p ) can be output 550 as a result (also called the ground state QSK) (intuitively the result of the p-adaptation process), optionally together with EW H (VQC p (Ap ) Output together.

[0060] Working Example 9 involves determining multiple eigenvalues and / or eigenvectors (also referred to as a set) of H. To this end, for one or more (e.g., each) basis states of QR, the QR in that basis state can be driven according to the basis state QSK, and the resulting QR state can be read out. If the set of eigenvalues and / or eigenvectors thus determined does not meet the requirements, the set can optionally be adapted.

[0061] Working Example 10 involves adapting a set of eigenvalues and / or eigenvectors, for example, to determine the eigenvalue of the lowest excited state (also referred to as the lowest eigenvalue). This can be done by means of an optimization algorithm and / or by means of one or more penalty terms. Similar to the above description, the QR initialized at 101 in the basis state is driven by means of the basis state QSK. The resulting QR state represents the basis state of the lowest expected value. Alternatively, the QR can also be placed in another basis state (e.g., the basis state providing the second lowest expected value), whereupon the process will be initiated using a new VQC.

[0062] Working Example 11 involves determining one or more excited states, for example, orthogonal to GS, by means of penalty terms. Similar to the above description, in the circuit determination 501, the VQC can be determined as the p-th VQC p , which has the (p - 1)-th VQC p-1 parameterized by means of A p-1 , i.e., VQC p-1 (A p-1 ), and at least one (i.e., one or more) additional gates 108 connected thereto. The adaptation of VQC p can be carried out by minimizing the expected value as a cost function according to the following relationship: . Here, the term is used as an exemplary penalty term, which is configured as a function of and is conducive to determining the orthogonal to . The larger the free parameter is chosen, the stronger the effect of the penalty term S. The penalty term intuitively represents the overlap , and can be determined, for example, by means of the so-called Swap-Test 900 shown in Figure 9 . Similarly, for each previously calculated lower excited state , an additional penalty term can be added to optimize each additional excited state , to promote orthogonality. More generally, the penalty term S(x) can be configured to penalize deviations from the orthogonality of the quantum state x (e.g., as part of an objective function, such as a cost function).

[0063] Working example 12 involves determining one or more excited states that are, for example, orthogonal to GS. Thus, the term does not necessarily have to be part of the objective function, but can be used constructively, for example, by means of the Quantum Gram - Schmidt process. The Gram - Schmidt process refers to the process of orthonormalizing a set of vectors in a space with an inner product (standard inner product).

[0064] Working example 13 involves determining one or more excited states by means of a so - called Variational Quantum Thermalizers (VQT), which is intuitively based on the determined GS. The density matrix of the thermalized state (T > 0) of the QS mathematically described by the Hamiltonian operator (also known as the Hamiltonian) can be expressed as: (1) Here, represents the inverse temperature, Tr is the trace, k B is the Boltzmann constant, is the density matrix of the eigenstates of the Hamiltonian operator with energy E i as the eigenvalue. Starting from the determined Gs, the excited states (also known as thermalized states) can be determined by projecting the ground state out of the wave function and / or by assigning an additional penalty term to the ground state. Alternatively or additionally to the VQT, the so - called "Thermofield Double State" method can also be used, where the system is enlarged and then collapsed again by subsequent measurement.

[0065] Working example 14 involves a design of the foregoing. In a less complex first implementation, the adaptation 103 of the p - th VQC p (which has VQC p-1 (A p-1 ) and at least one additional gate 108 connected thereto) can include: setting the parameters of VQC p-1 (A p-1 ) to be invariant. This reduces the number of degrees of freedom because only at least one additional gate 108 (e.g., its parameters, type, the qubit on which the additional gate acts, etc.) is adapted. In a slightly more complex second implementation, the adaptation of the p - th VQC p (which has VQC p-1 (Ap-1 ) and at least one additional gate 108 connected thereto may include: adapting one or more parameters of VQC p-1 (A p-1 ), for example, adapting the alternative or additionally at least one additional gate 108. For example, in a second implementation - in this implementation VQC p-1 (A p-1 ) has VQC p-2 (A p-2 ) and at least one second additional gate 108 connected thereto, including: (e.g., only) adapting at least one second additional gate 108. More generally, the parameters of some or all of the quantum gates of the p-th VQC p can be adapted.

[0066] Working example 15 involves selecting one of a plurality of basis states as the initial state and performing an adaptation process 300 p (p = 1 to p = P) to determine the ground state QSK. Thus, for each of the plurality of basis states, the QR placed in that basis state can be driven according to a VQC 1 having one or more quantum gates, and the expected value representing the resulting QR state can be determined. The basis state that produces the lowest expected value among the plurality of basis states can be selected as the initial state. This improves the starting point, thereby reducing the resource requirements of the method.

[0067] The foregoing will be explained below for an exemplary QR with five qubits, where it can be understood that what is described for this purpose can be similarly applied to any other number of qubits.

[0068] Figure 7 A method according to different embodiments is shown in the form of a schematic diagram 700, in particular the result of the adaptation process 300 p for p = 4 and an exemplary QR 102 with five qubits, where VQC 4 exactly has 4 quantum gates.

[0069] The five qubits of the QR 102 are initialized 101 in the basis state by means of amplitude encoding at the start of the p-th adaptation process 300p, and then driven according to the p-th VQC p , and it is determined by means of a circuit 501 that each time p is incremented, a quantum gate 108 (also called an additional gate) is added (e.g., connected downstream) to the p-th VQC p . The Hadamard test can use, for example, the ancilla qubit (ancilla_0) of the QR 102, if it exists.

[0070] Optionally, one or more properties of the additional gate 108 of the p-th adaptation process 300 are selected from a set of predefined properties (e.g., type, acting between which qubits, and / or parameterization), e.g., by means of an optimization algorithm (e.g., Powell optimizer). The expected value of the p-th VQC p can be used as a cost function, e.g., by minimizing its expected value EW p . More generally, the circuit determination 501 can include determining the additional gate 108 that minimizes the expected value. H

[0071] Working example 16 relates to determining physical variables of a model system. Thus, all desired physical variables, e.g., magnetic susceptibility, can be determined based on the eigenvalues of the determined H (e.g., similar to exact diagonalization). Figure 8 A schematic diagram 800 is shown for the case of magnetic susceptibility based on GZ and the first excited state, in which the determined magnetic susceptibility λ is plotted against temperature T for an exemplary QR with five qubits according to different embodiments, compared to the exact variation curve 811 of the magnetic susceptibility λ that is substantially congruent with the magnetic susceptibility λ obtained by the method set forth herein. As a comparison, case 813 is shown, in which the magnetic susceptibility is based only on GZ (e.g., the ground state in different symmetry sectors). To determine the exact variation curve of the magnetic susceptibility of this model system, rotational symmetry and translational symmetry can be utilized.

[0072] The magnetic susceptibility can be, for example, the Uniform Spin Susceptibility, which can be determined according to the following relationship: . Here, N represents the number of eigenstates, E n represents the energy eigenvalue of eigenstate n, k B represents the Boltzmann constant, T represents the temperature, represents the expected value of the operator (S z )², S z represents the spin operator in the Z component, where ; and .

[0073] This result obtained according to different embodiments can be reproduced, for example, by means of a quantum simulator (e.g., IBM Qiskit). This also shows that the method set forth herein can converge rapidly to the exact variation curve.

[0074] The method provided herein can be identified by the following features: calculating one or more physical variables by means of a quantum computer and by means of the method of exact diagonalization, e.g., based on a combination of an optimization method and eigenvalue calculation.

Claims

1. A method for operating a quantum register (102) to determine whether a determined second quantum circuit (VQC p+1 ) meets a stored convergence criterion, the method Comprising: • First adaptation (300) of the first parameter of the first quantum circuit (VQC p ) to optimize the objective function by driving (103) the quantum register (102) according to the first quantum circuit (VQC p ); • First determination (501) of the second quantum circuit (VQC p+1 ), the second quantum circuit including quantum gates (108) and a first quantum circuit (VQC p ) having adapted first parameters derived from the first adaptation (300); • Second adaptation (300) of the second parameter of the quantum gate (108) to optimize the objective function by driving (103) the quantum register (102) according to the second quantum circuit (VQC p+1 ) • Second determination (501) of whether a second variational quantum circuit (VQC) having an adapted second parameter derived from the second adaptation (300) satisfies a stored convergence criterion. p+1 ​ 2. The method according to claim 1, further Comprising: • Determining eigenvectors and / or eigenvalues of the mapping, wherein the objective function is a function of the eigenvalues of the mapping with respect to the eigenvectors of the mapping.

3. The method according to any 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. According to the method described in any one of claims 1 to 3, when the second determination results in the second quantum circuit (VQC p+1 ) satisfying the stored convergence criterion, further Comprising: • Determining an additional quantum circuit to optimize an additional objective function by driving the quantum register (102) according to the additional quantum circuit; • Wherein the additional objective function represents the relationship between the result of driving the quantum register according to the second quantum circuit (103) and the result of driving the quantum register (102) according to the additional quantum circuit (103).

5. The method according to any one of claims 1 to 4, wherein the second adaptation (300) further Comprising: Changing one or more of the following to optimize the objective function: • The type of the quantum gate (108); • The first qubit of the quantum register (102) affected by the quantum gate (108); • The second qubit of the quantum register (102), which causes the first qubit to be affected by the quantum gate (108).

6. The method according to any one of claims 1 to 5, • Wherein the second adaptation (300) further comprises adapting the first parameter, and • wherein the second determination includes determining whether a second quantum circuit (VQC) having an adapted first parameter and an adapted second parameter derived from the second adaptation (300) satisfies a stored convergence criterion p+1 ).

7. The method according to any one of claims 1 to 6, • wherein the quantum register (102) is driven (103) according to the first quantum circuit (VQC p ) when the quantum register (102) is placed in an initial state; and / or • wherein, according to the second quantum circuit (VQC p+1 ) driving (103) the quantum register (102) is performed when the quantum register (102) is placed in 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 is for each of the plurality of basis states, when the quantum register (102) is placed in the basis state, by driving (103) the quantum register (102) by means of at least a part of the first quantum circuit (VQC p ) to perform.

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 to cause the processor to perform the method according to any one of claims 1 to 7 when executed by the processor.

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

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