Method for operating quantum register

Abstract

A method for operating a quantum register (102) may have: performing a first readout process (201) such that first data (252) is determined, the first data representing a first state of the quantum register (102); performing first driving (205) on the quantum register (102) in the first state according to a quantum gate (VQC2); performing a second readout process (207) such that second data (254) is determined, the second data representing a second state of the quantum register (102) derived from the first drive (205); adjusting (213) at least one parameter of the quantum gate (VQC2) using the first data (252) and the second data (254) to optimize an objective function dependent on the first data (252) and the second data (254); the quantum register (102) is second driven (205) in accordance with the quantum gate (VQC2) derived from the adjustment (213).

Description

Technical Field

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

[0002] The current generation of quantum computers (also called "noisy intermediate-scale quantum" (NISQ) computers) has been mainly applied in the fields of materials research, optimization processes and artificial intelligence. One aspect of materials research is the quantitative exploitation of quantum mechanical systems (also referred to herein as QS) (such as molecules). So far, various methods have been developed to gain access to QS states, some of which are exemplarily described below.

[0003] The 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, which can be combined into thermalized states. However, its resource requirements increase dramatically with the size of the QS, so that even modern supercomputers are rarely sufficient to obtain a solution of sufficient accuracy within a reasonable computing time. As an approximation to it, the so-called density matrix renormalization group ("Density Matrix Renormalisation Group" - DMRG) reduces the computing time by ignoring higher energy states. Alternative algorithms are based on quantum Monte Carlo methods (QMC), in which integration is performed over random variables. Exemplary QMC algorithms include so-called "Diffusion-QMC", "Auxillary-Field-QMC", "Continuous-time QMC", "Greens-Function-QMC" or "Hirsch-Fey-QMC". What these QMC algorithms have in common is that, although their convergence to the exact solution can be improved with more computational effort, this is usually only possible within a small parameter range (e.g. temperature range), outside of which the so-called "Fermionic-Sign-Problem" destroys the convergence.

[0004] What these purely classical methods have in common is that as the size of the QS increases, the computational effort increases exponentially, which usually has to be compensated with significant losses in accuracy. When using quantum computers, these losses are smaller due to their superposition capabilities. Although the capabilities of quantum computers are constantly increasing. However, so-called hybrid algorithms, i.e. quantum classical algorithms, are used, which only outsource part of the calculation to the quantum computer. Quantum classical algorithms allow the advantages of both types of computers to be utilized and reduce the requirements on quantum computers, such as the number of qubits, circuit depth, and gate operation error tolerance (so-called noise or "Noise"). Popular representatives of hybrid algorithms are the so-called "Variational Quantum Eigensolver (VQE)" and the "Variational Quantum Thermalizer (VQT)" based on it. VQE uses variational quantum circuits to prepare the so-called starting state on the quantum computer. The quantum computer then measures the energy of this starting state. A classical (i.e. non-quantum mechanical) computer is responsible for optimizing the quantum circuit so that this starting state converges to the ground state of QS (i.e. temperature T=0 Kelvin). One goal is to compute QS states above 0 Kelvin (also called thermalizations or excitations), which, however, are not accessible to VQE.

[0005] The density matrix of the thermalized state (T>0) of QS is mathematically expressed by the Hamiltonian operator (also called Hamiltonian), which can be expressed as:

[0006]

[0007] Here, β = 1 / k B T represents inverse temperature, Tr represents trajectory, k B represents the Boltzmann constant, and ρ i Energy is E i The density matrix of the eigenstates of the Hamiltonian operator. Here, state ρ is the QS state where the free energy F is minimized. The free energy F can be expressed as a function of energy E and entropy S by the following relationship:

[0008]

[0009] One way to calculate such excited states is to first find the ground state with the help of VQE and then search for excited states by projecting the ground state outside the wave function or applying an additional energy penalty term to the ground state. Both require complex quantum circuits, which make the calculation more difficult and prone to errors. However, with the less complex so-called "thermofield-double-state" method, i.e., the system is amplified and collapsed again by subsequent measurements, it is impossible to measure the entropy directly. In order to solve this problem, it is necessary to calculate the expected value of ln(ρ) on a quantum computer (for example, with the help of a logarithmic series expansion) and then measure the power of ρ, which, however, places high demands on the quantum computer.

[0010] As a generalization of VQE, VQT provides access to the thermal state of the system with the help of an additional machine learning (ML) algorithm trained on a classical probability distribution. Samples are extracted from this probability distribution, which are used as input states for the VQE. Since quantum circuits do not change the entropy, the entropy can be calculated based on the classical probability distribution. Similar to VQE, the variational quantum circuits are optimized and the ML algorithm is trained to minimize the free energy according to equation 2. The working principle of this VQT has been realized with the help of superconducting qubits, but requires a large number of necessary circuit compilations and large-scale implementations of these circuits. Summary of the invention

[0011] According to various embodiments, a method for operating a quantum register (QR) is provided, the method comprising: performing a first readout process so as to determine first data, which first data represent a first state (e.g., a superposition state) of the QR (e.g., its probability) (e.g., a function of the QR); performing a first drive on the QR in the first state or a (e.g., non-entangled) state derived therefrom according to at least one (i.e., one or more) quantum gate (the quantum gate, for example, forms a variational quantum circuit); performing a second readout process (the second readout process, for example, has a readout QR) so as to determine second data, which second data represent a second state (e.g., a superposition state) of the QR (e.g., its probability) derived from the first drive (e.g., a function of the QR); adjusting at least one (i.e., one or more) parameter of the at least one quantum gate using the first data and the second data to optimize a target function that depends on the first data and the second data; performing a second drive on the QR according to the at least one quantum gate derived from the adjustment. That is, in other words, by adjusting at least one parameter of the at least one quantum gate, the at least one quantum gate is adjusted so as to use the adjusted at least one quantum gate to perform a second driving on the QR.

[0012] The method provided herein simplifies the reproducible initialization, simulation and calculation of quantum mechanical states, for example when the quantum mechanical state is intended to represent a QS temperature above 0 Kelvin. The method described herein makes it easier to efficiently calculate free energy with the help of quantum algorithms (for example according to relation (2)), for example with the help of NISQ computers, or so-called "Fully Error Corrected Universal Quantum" computers, and to prepare their quantum mechanical states for the measurement of other parameters.

[0013] The method described herein can be implemented and executed in a resource-saving and cost-efficient manner, for example, since it does not necessarily scale exponentially with the size of the QS and requires only a small amount of resources of the quantum computer (e.g. comparable to VQE). Compared to VQT, the method described herein results in significantly lower compilation costs, enables better characterization of classical statistical distributions, and can be implemented more easily and hardware-efficiently. This can be achieved, for example, because there is less coupling between the CPU and the QPU and a high tolerance to the noise of the quantum computer is achieved. More clearly, the method described herein opens up the possibility of VQT at a cost comparable to VQE.

[0014] Various embodiments are described below.

[0015] Example 1 is a method for operating a QR as described above.

[0016] Embodiment 2 is a method according to embodiment 1, wherein the first readout process has: reading out a QR (e.g., a superposition state of the QR); or (e.g., before the second readout process or simultaneously with the second readout process) reading out an additional QR (e.g., a superposition state thereof) entangled with the QR (e.g., its first state and / or before the first drive). The former reduces resource costs, for example, by requiring fewer qubits and / or a smaller circuit depth. The latter allows the ability to omit intermediate measurements.

[0017] Embodiment 3 is a method according to embodiment 1 or 2, the method further comprising: preferably when the QR is in an initial state, performing an additional first drive on the QR according to at least one additional quantum gate (the additional quantum gate, for example, forms a variational quantum circuit), wherein the first state of the QR is based on the additional drive (for example, directly generated therefrom); and wherein the adjustment preferably comprises: adjusting at least one parameter of the at least one additional quantum gate or setting the at least one parameter to be constant while using the first data and the second data; performing an additional second drive on the QR according to the at least one additional quantum gate derived from the adjustment. That is, in other words, by adjusting at least one parameter of the at least one additional quantum gate, the at least one additional quantum gate is adjusted so as to perform an additional second drive on the QR using the adjusted at least one additional quantum gate. This makes it easier to simulate more complex QS (for example, its thermalized state) and / or prepare classical probability distributions.

[0018] Embodiment 4 is a method according to embodiment 3, wherein the at least one additional quantum gate has fewer parameters than the at least one quantum gate; and / or wherein fewer qubits of the QR are driven (e.g., out of an initial state) according to the at least one additional quantum gate than according to the at least one quantum gate. This reduces resource expenditure (e.g., computation time).

[0019] Embodiment 5 is a method according to any one of embodiments 1 to 4,

[0020] Wherein, the at least one quantum gate and / or the at least one additional quantum gate are set up to entangle the QR and / or bring the QR into a superposition state, preferably a non-entangled or at least partially entangled superposition state (the superposition state is read out, for example, in the first and / or second readout process). This makes it easier to simulate more complex QS (for example, its thermalized state). The at least partially entangled superposition state can, for example, be the result of QR entanglement. The set of entangled superposition states is, for example, a subset of superposition states.

[0021] Embodiment 6 is a method according to any one of embodiments 1 to 5, wherein the adjustment comprises: for each qubit of the QR, adjusting one or more parameters, and changing the state of the qubit according to the parameter when driving. This makes it easier to obtain more accurate calculation results.

[0022] Embodiment 7 is a method according to any one of embodiments 1 to 6, wherein the first data and / or the second data represent the superposition state of the QR or the probability of a ground state generated from the superposition state (preferably by a corresponding readout process). This makes it easier to simulate more complex QS (e.g., its thermalized state).

[0023] Embodiment 8 is a method according to any one of embodiments 1 to 7, wherein the objective function represents a physical state parameter, and / or wherein the adjustment is performed based on an output of the objective function, the output depending on the first data and the second data and / or being minimized. This makes it easier to simulate more complex QS (e.g., its thermalized state).

[0024] Embodiment 9 is a method according to any one of embodiments 1 to 8, wherein the second data is independent of the first data or depends on the result of the first readout process (or the first data). This makes it easier to simulate more complex QS or better consider the characteristics of the QS and / or simplify the adjustment process / design the adjustment process to be more efficient. The second data can, for example, depend on the first data, for example when the second data has one or more indications that are a function of the first data (or the indications of these first data). For example, when the result of the first readout process is that the qubit is in state |1>, the second readout process can provide a different result than when the result of the first readout process is that the qubit is in state |0>.

[0025] Embodiment 10 is a method according to any one of embodiments 1 to 9, further comprising: determining an indication about a physical quantity (e.g., QS) based on the result of the second drive and / or the additional second drive, wherein the indication preferably depends on the output of the objective function derived from the optimization or at least differs from the output. This makes it easier to evaluate the physical quantity (observable) under the calculated thermal state.

[0026] Embodiment 11 is a (eg non-quantum mechanical) control device which is configured to carry out the method according to any one of embodiments 1 to 10.

[0027] Embodiment 12 is a computer program configured to cause a (eg, non-quantum mechanical) processor executing the computer program to perform a method according to any one of embodiments 1 to 10.

[0028] Embodiment 13 is a computer-readable medium storing instructions, which are configured to cause a (eg non-quantum mechanical) processor executing the instructions to perform the method according to any one of claims 1 to 10. Embodiments 11 to 13 make it easier to implement the method. BRIEF DESCRIPTION OF THE DRAWINGS

[0029] In the accompanying drawings, like reference numerals generally refer to the same parts in all the different views. These drawings are not necessarily to scale, with emphasis instead generally being placed on presenting the principles of the invention. In the description below, various aspects are described with reference to the following drawings.

[0030] Figure 1The structure and working principle of a quantum processor according to various embodiments are schematically shown; Figure 2 and Figure 3 as well as Figures 6 to 8 and Fig.10 The methods according to various embodiments are respectively shown in different schematic diagrams; Figure 4A and Figure 4B as well as Figure 5 and Fig. 9 The operation sequences according to various embodiments are respectively shown in different schematic diagrams. DETAILED DESCRIPTION

[0031] The detailed description below refers to the accompanying drawings, which show, for the purpose of illustration, the specific details and aspects of the present disclosure in which the present invention may be implemented. Other aspects may be used and structural changes, logical changes, and electrical changes may be performed without departing from the scope of protection of the present invention. Different aspects of the present disclosure are not necessarily mutually exclusive, as some aspects of the present disclosure may be combined with one or more other aspects of the present disclosure to form new aspects.

[0032] Various examples are described in more detail below.

[0033] Figure 1 The schematic diagram illustrates the structure and operating principle of a quantum processor ("quantum processing unit" or QPU for short) according to various embodiments 100. The QPU has a QR 102 and an operating device 104. The term "quantum register" (QR) refers to a register that can be used to implement a plurality of (e.g., n) qubits (here exemplarily referred to as q1 to q n ). Exemplary components of the QR 102 include: an ion trap, a plurality of (e.g. superconducting) circuit resonators, a plurality of quantum dots, a photonic system, a laser, an electromagnetic coil, etc. An operating device 104 is configured to operate (e.g. exert influence on) the QR 102, e.g. according to a sequence 106 (also referred to as an operating sequence). An exemplary process of the operating sequence 106 includes: starting 101 the QR (i.e., bringing the QR into an initial state Z0), driving 103 the QR 102 and / or reading out 110 the QR 102. In the initial state Z0 of the QR 102, each qubit may be non-entangled and / or in a ground state (here exemplarily |0>), but the ground state does not necessarily have to be the same for all qubits.

[0034] The specific implementation of the operation device 104 or the operation sequence 106 depends on the type and architecture of the qubit. The operation device 104 can have one or more transducers, which are configured to interact with the QR 102 or its environment, such as at least changing and / or detecting the state Z of the QR. Exemplary implementations of the transducers of the operation device have: actuators (such as actuators), sensors, transceivers, etc. Exemplary transducers of the operation device are configured to transmit one or more of the following to the QR 102 and / or detect one or more of the following from the QR: optical radiation, magnetic field, electric field, charge, particle radiation. The QPU is a component of a quantum computer, which also has at least a housing environment for the QPU and control electronics.

[0035] The term "qubit" refers to a two-state quantum system that can be mathematically expressed by two basis vectors of a two-dimensional Hilbert space (here exemplarily extended by |0> and |1>). The implementation of a qubit can be of different types and in different architectures. Examples of this are: the energy levels of trapped ions, the polarization states of photons, spins in quantum dots or silicon, and the energy levels of superconducting circuit resonators.

[0036] In general, the driving 103 of the QR 102 is designed to induce a change in the state (also called state change) of the QR 102. The driving 103 of the QR 102 is performed according to one or more quantum gates 108 (see also Figure 5 ). The term "quantum gate" (also called quantum logic gate or gate) refers to the quantum mechanical analog of a circuit logic gate. However, quantum gate 108 corresponds to a process that affects QR 102, which changes the state Z of QR 102, such as the state of one or more of its qubits. With the help of quantum gate 108, unitary functions can be implemented similarly to logic gates, clearly as an analog of the Boolean function of such logic gates. Mathematically, a quantum gate can be described as a unitary transformation This unitary transformation transforms the qubit from the state |x> to the state Examples of quantum gates include: Hadamard gate, rotation gate, T-gate, controlled NOT gate (CNOT), switch node, etc. Quantum gates can be set up to form or at least maintain entanglement.

[0037] The term "quantum algorithm" (also called quantum circuit) generally refers to a set of processes (e.g. quantum gates) interacting with or within a QR 102, which, for example, trigger or at least stimulate a change in the state Z of the QR 102 (e.g. at least one qubit thereof). For a better understanding, a quantum circuit is recorded similarly to an electronic device as a circuit diagram, in which an axis (here from left to right) represents the change process over time. Similar to an electric circuit, the sum of a plurality of quantum circuits can again be regarded as a quantum circuit. From this point of view, the entire sequence of operations 106 can be understood as a quantum circuit, but also as components of this sequence of operations, such as the processes of driving 103 or reading out 110. From the point of view of data processing, each quantum circuit transforms one state of the QR 102 (also called the input state of the quantum circuit) into another state of the QR 102 (also called the output state of the quantum circuit). Here, the output state of the process of driving 103 (also called the driving process 103) is exemplarily indicated as a superposition state, for example with |x| i >|x j >(1≤i, j≤n), but it does not necessarily have to have entangled qubits. If the output state of the quantum circuit is a superposition state (for example, with entangled qubits), the quantum circuit is also referred to as a starting circuit (or a preparation circuit) in this article. In general, if it is related to a single qubit, the superposition state can have a superposition of the state of the single qubit, and / or, if it is related to multiple qubits, the superposition state can have the entanglement of multiple qubits and / or the superposition of the states of these qubits. Correspondingly, the starting circuit can be set up to put the qubit into a superposition state, which does not necessarily have to, but can definitely cause entanglement of the qubit.

[0038] The starting circuit is preferably set up to change the square of the amplitude of the qubit and the phase of the qubit. Optionally, the quantum circuit can have or consist of one or more parameterized quantum gates (e.g., multi-qubit quantum gates), for example, parameterized by means of at least one parameter (also called variational parameter) (in the case of the starting circuit, then also called variational circuit or VQC for short). Optionally, the starting circuit can also have one or more non-parameterized quantum gates and / or one or more multi-qubit quantum gates. In order to simplify the notation, the parameterization of the quantum circuit is recorded in the form of a vector (hereinafter also called a parameter vector) and can, but does not necessarily have to have two or more parameters as components. Therefore, the output state of the parameterized quantum circuit is a function of the parameter vector, each vector component in the parameter vector being a parameter of the quantum circuit.

[0039] The process of reading out 110 of QR 102 (also referred to as readout process 110) comprises: sensing (i.e., by means of a sensor) detecting the state of QR 102, which state can be, for example, a superposition state and / or a result of the driving process 103. During the readout 110, the state of QR 102 collapses to the ground state (here exemplarily Z) of the measurement basis of the readout process. A ). The readout process 110 (also called quantum mechanical measurement process or simply measurement) can be repeated multiple times for each parameter vector and / or each term of the Hamiltonian, for example, in order to improve the accuracy of the expected value. 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, which has a corresponding infrastructure (the infrastructure has, for example, a processor, a storage medium and / or a bus system, etc.). The measurement chain can be set up to: drive the sensor; process the detected state as an input parameter; and output data representing the input parameter based on this. The measurement chain can be implemented, for example, with the aid of an operating device 104. The sensor can have one or more measurement bases, one of which is the measurement basis according to which the readout process 110 is performed. To be clear, the measurement basis is a characteristic of the sensor used to perform the readout process 110. Mathematically speaking, the measurement basis expands the Hilbert space of states to which the state of the QR is projected when reading. According to the technical design, the QPU can implement one or more measurement bases. Optionally, the QPU may implement multiple measurement bases, which enables parameterization of the selection of the measurement basis instead of and / or in addition to parameterized quantum gates.

[0040] Figure 2 The method according to various embodiments 200 is illustrated in a schematic diagram, which can be performed by a control device 202, for example. The term "control device" can be understood as any type of entity (e.g., a processor) capable of processing data or signals. For example, these data or signals can be processed according to at least one (i.e., one or more than one) special function, which is performed by the control device or the processor therein. The control device, such as its processor, can have an analog circuit, a digital circuit, a logic circuit, a microprocessor, a microcontroller, a central 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, or be formed by these. Any other means for implementing the corresponding functions described in more detail herein can also be understood as a processor or a logic circuit. One or more of the method steps described in detail here can be implemented (e.g., realized) by one or more special functions, which are performed by the processor. The processor of the control device can be, for example, a classical (i.e., non-quantum mechanical, such as transistor-based) processor.

[0041] The process sequence 602 of the method (also referred to as optimization process sequence 602) is shown. The process sequence 602 has an operation sequence 106, which can be implemented by means of a QPU (for example according to the embodiment 100); and (for example exactly one) adjustment process 213, which can be implemented by means of a control device 202, for example. The QPU has a QR 102, which is set up to implement a plurality of quantum bits (here exemplarily q1 to q n ). It is understood that all components of the adjustment process 213 do not necessarily have to be executed by a classical processor, but can also be at least partially executed by the QPU itself according to the functional capabilities of the QPU. Optionally, the method can include: repeating the optimization process sequence 602 multiple times (for example, K times) (see also Figure 6 ).

[0042] The operation sequence 106 has: first execution 201 of the read process 110 (also referred to as m S or first readout process 201), so that first data 252 are determined, which first data represent 203 a first state 204a (here exemplary Z1=Z(t1)) of the QR 102 (for example at a first time point t1); the QR 102 in the first state 204a is driven 205 (also called start preparation 205); the readout process 110 is performed 207 for the second time (also called m E or a second readout process 207), so that second data 254 are determined, which represent 209 the second state 204b (here exemplary Z2=Z(t2)) of the QR 102 (for example at the second time t2>t1) resulting from the drive 205. Depending on the specific implementation (see also Figure 4A and Figure 4B ), for example when the first state 204a is the output state of the first readout process 201 or at least based on the output state, the first state is the ground state (i.e. without superposition); or for example when the first state is the output state of the additional drive 211 (also called thermalization 211) of the QR 102 or at least based on the thermalization 211, the first state is a superposition state. If, preferably when the QR 102 is in the initial state and / or before the first time point t1, the operation sequence 106 has a thermalization 211 of the QR, this makes it easier to initialize the quantum mechanical state at a temperature above 0 Kelvin, because in this case the capabilities of the QPU are used. For example, the readout process 201 and the second readout process 207 can be performed according to the same measurement basis of the quantum gate 108.

[0043] The start preparation 205 may be performed according to a VQC (also referred to as VQC2) having one or more quantum gates 108 (eg, for each qubit) and / or with the aid of at least one parameter (eg, for each qubit) Thermalization 211 may be performed according to an additional VQC (also referred to as VQC1) having one or more quantum gates 108 (eg, for each qubit) and / or by means of at least one parameter (eg, for each qubit). It is understood that what is described about the quantum gate 108 can also be applied by analogy to more than one quantum gate per VQC.

[0044] For each optimization process sequence 602, using the first data 252 and the second data 254, the parameter vector of the optimization process sequence 602 may be (eg, just) adjusted 213. (also called the tuning process), where the parameter vector At least one parameter of VQC1 and / or VQC2. Optionally, each optimization process sequence 602 may have: for example, for each adjustment process 213, multiple executions of the operation sequence 106 (see also Figure 6 ), i.e., without changing the parameter vector used And / or to obtain statistical data for the first data 252 and the second data 254 .

[0045] For example, the adjustment process 213 can be performed in the following manner: optimize the target function that depends on the first data and the second data, for example, minimize the output of the target function. The target function (for example, its output) can represent QS, for example, a state variable of the QS, which is explained in more detail below.

[0046] Figure 3 A method according to various embodiments 300 is illustrated schematically, which method can be executed, for example, by means of a control device 202 (here exemplarily designed as a CPU), for example based on a variational principle (for example the Rayleigh-Ritz principle). Embodiment 300 can be set up like embodiment 200, wherein the output 302a of the objective function 302 can, for example, represent a state variable of the thermalized QS (for example the free energy F), preferably as a function of the entropy S and the energy E of the thermalized QS, further preferably according to the relation (2). Then, the first data 252 clearly represent the entropy of the thermalized QS, and the second data 254 clearly represent the energy of the thermalized QS, which entropy and energy are determined based on the measured values ​​of the QPU. What is described here exemplarily for the free energy (for example as the output of the objective function) can be applied by analogy to any other state variable of the (for example thermalized) QS.

[0047] The objective function 302 can be optimized by means of a so-called optimization algorithm 304, which is designed to determine a parameter vector of the operation sequence 106 based on an output 302a of the objective function 302. Or at least determine the change of the parameter vector, such as the boundary condition: for example, based on the parameter vector The output 302a of the objective function 302 is minimized by treating the output 302a as a cost function. This enables the optimization of the parameters of each VQC of the sequence of operations 106. In a working example, the input of the sequence of operations 106 has a numerator Hamiltonian operator and a parameter vector for one or more VQCs. According to the parameter vector, the state of the QR 102 is prepared in order to simulate the quantum state of the molecule. In this working example or another working example, the optimization algorithm 304 is set up to evaluate the objective function 302 as a cost function and to calculate the gradient of the objective function each time the optimization process sequence 602 is traversed. Exemplary implementations of the optimization algorithm 304 include: so-called linear approximate constrained optimization ("Constrained optimization by linear approximation" - COBYLA); conjugate gradient ("Conjugent Gradient") or "quasi-Newton optimizer (quasi-Newton Optimizer)", which can be found in the Python package qiskit or scipy, for example.

[0048] The optimization process sequence 602 may include: with the help of the optimization algorithm 304, the actual parameter vector of the optimization process sequence and the output of the objective function 302 is transformed into a parameter vector These new values ​​are used as the actual parameter vectors for the subsequent optimization process sequence 602. In order to drive the QR 102. Each time the optimization process sequence 602 (also referred to as iteration) is performed, the parameter vector All converge to a vector representing the thermal state of QS. This process may be repeated until the output of objective function 302 converges to a minimum value or until a convergence criterion is met.

[0049] Figure 4A and Figure 4BThe schematic diagrams illustrate operation sequences 106 according to various embodiments 400a, 400b, which can be implemented, for example, in embodiments 100 to 300 and which have in common that the operation sequence 106 (hereinafter also referred to as "Full Quantum Variational Thermalizer" or FQVT for short) has two process pairs, each of which has a VQC (VQC1 or VQC2) and a subsequent readout process 110. The VQC of each process pair is designed to provide a superposition of multiple quantum states with different amplitudes (also called superposition state) as an output state (state of the qubit). The readout process 110 of each process pair can influence the superposition and is designed to convert the superposition into a ground state.

[0050] VQC1 and VQC2 can be performed sequentially. VQC1 depends on the first parameter vector An exemplary implementation of VQC1 has: "Unitary-Coupled Cluster-Singles-Doubles" (UCCSD) or the so-called "efficient SU2" ("spatial unitary group of rank 2"). In relation to the thermalized QS, VQC1 is generated with probability p i =exp(-βE i ), and VQC2 generates the corresponding eigenstate ψ of the Hamiltonian operator based on this i . VQC2 depends on the second parameter vector In addition, the common point of the embodiments 400a and 400b is that the second readout process (m E ) has the following features: detecting (measuring) the output state of VQC2. The output state of VQC2 can be measured by measuring m E is converted into an energy state which represents the energy of the second state 204b.

[0051] According to the embodiment 400a, the first readout process (m S ) has: detecting (measuring) the output state of VQC1, that is, reading QR 102 after driving according to VQC1. The output state of VQC1 can be measured by measuring m S is transformed into the ground state with probability p (given by the magnitude of the ground state). For each optimization process sequence 602, FQVT 106 may be performed multiple times, wherein data about the measurements m are collected. S After frequent repetitions of FQVT 106, the output statistics of m S To determine when measuring m SThe probability p of obtaining the ground state i in i Based on the probability ρ i , for example, the entropy of a quantum mechanical system can be determined according to the following relationship:

[0052] S=∑ i p i ln(p i ).

[0053] Furthermore, the operation sequence 106 according to the embodiment 400a comprises: by means of VQC2, the first readout process m S The output state of is transformed into a superposition as the second state 204b.

[0054] If the QPU is not able to perform a readout process 110 (also referred to as an intermediate measurement) between two VQCs, or if an intermediate measurement should not be performed for other reasons, then an operation sequence 106 according to embodiment 400b may be used. According to embodiment 400b, the available qubits have a plurality of groups, and a first group of qubits (also referred to as a logical main register) and a second group of qubits (also referred to as a logical auxiliary register or auxiliary qubit) in these groups are set up to be able to be entangled with each other. For easier understanding, the main registers (q1 to q n ) and auxiliary registers (here a1 to a n ) are treated herein as (logically) separate QRs 102, wherein it is understood, however, that the main register and the auxiliary register may be part of the same (physical) QR 102, but this is not necessarily required. The auxiliary register has at least as many qubits as the main register, for example n qubits.

[0055] According to the embodiment 400b, only the main register is driven according to VQC1 and according to VQC2 and is controlled by means of m E is read out. In addition, the operation sequence 106 according to the embodiment 400b has: entangle 410 the output state of VQC1 (the output state is, for example, a superposition state) with the auxiliary register (also referred to as entanglement process 410). The entanglement process 410 can have: for example, by means of at least one controlled NOT gate, each qubit of the main register is entangled with (for example, exactly) one qubit of the auxiliary register. The output of the entanglement process 410 has a plurality of pairs (for example, n pairs) of qubits entangled with each other, each of which has a qubit of the main register and a qubit of the auxiliary register.

[0056] Furthermore, the operation sequence 106 according to the embodiment 400b includes: by means of VQC2, (for example, only) the state of the master register resulting from the entanglement process 410 (which state is, for example, a superposition state) is transformed into a superposition as the second state 204b. SThe state 214b of the auxiliary register resulting from the entanglement process 410 is detected (measured). The state 214b of the auxiliary register can be measured by measuring m S is transformed into the ground state with probability p (given by the amplitude of the ground state). Then, at the end of the sequence of operations 106, the (e.g. joint) measurement of all qubits clearly provides the energy (q i measurement of) and entropy (a i In one example, (in time) during the second readout process m E before, for example, immediately before, or with the second readout process E At the same time, or in the second readout process m E Afterwards, for example, immediately after the second readout process, a first readout process (of the auxiliary register) is performed. S .

[0057] Figure 5 The operation sequence 106 according to various embodiments 500 is schematically shown, which shows an exemplary implementation of the embodiment 400a in the case of two qubits. t (θ j ) represents the angle θ indicating the rotation according to, for example j Rotation about the t-axis (t = x, y or z). φ i or The index references the parameter vector It can be understood that the same situation can be applied to implementation 400b by analogy.

[0058] In a worked example, the Hamiltonian for the two-qubit case can be expressed as follows:

[0059]

[0060] Here, represents the creation operator, c represents the destruction operator, and n represents the occupation number operator. The index of the corresponding operator indicates the qubit on which the operator acts.

[0061] Figure 6 A method according to various embodiments 600 is shown in a schematic diagram, based on which additional exemplary implementations of embodiments 100 to 500 are explained. Optionally, the method can include: repeating the optimization process sequence 602 multiple times (for example K times) until a criterion is met, for example until the number of traversals k (also called traversal counter) reaches a target number of times K or the change in the output of the objective function is below a threshold. If the criterion is met, the parameter vector updated in the most recently executed optimization process sequence 602 can be output 611 (also called the output vector), e.g.

[0062] The or each optimization process sequence 602 may have: based on the results of the adjustment process 213, updating the parameter vector Here, the VQC (eg, VQC1 and / or VQC2) of the kth (2≤k≤K) optimization process sequence uses the parameter vector φ(k-1) output by the previously executed (k-1)th optimization process sequence 602. Therefore, in the kth optimization process sequence 602, the QR 102 is driven according to the VQC1 and / or VQC2 obtained from the adjustment process 213 of the (k-1)th optimization process sequence.

[0063] Optionally, the or each optimization process sequence 602 may include: executing the operation sequence 106 multiple times, for example until a criterion is met, for example when the number of iterations r reaches the target number R. Then, the adjustment process 213 may be based on the first data and the second data from the plurality of operation sequences 106. At the start of the optimization process sequence 602, the method may optionally include: at 601, adjusting the parameter vector Initialize, that is, specify the starting value of the parameter vector. The starting value of may be determined, for example, by means of a random generator or read out from a memory of control device 202 .

[0064] In a first exemplary implementation (clearly a piecewise optimization), the method (for example for optimizing free energy) has a plurality of stages, in each of which one or more optimization process sequences 602 are performed. Figure 8 , which illustrates a first exemplary implementation with a schematic diagram 800 similar to embodiment 600. In the first stage 801 of a plurality of stages (clearly a VQC1 optimization with a simplified Hamiltonian), the sequence of operations 106 has only the parameter vector Parameterized VQC1 and the first readout process m S From the distribution of the measured values, the entropy can be determined. S Here, a simplification of the Hamiltonian (also called the model Hamiltonian) is used, which is used to measure m Sdiagonal in the basis of . Examples of such model Hamiltonians are: the so-called "atomic limit" of the Hamiltonian, where all jump terms are set to zero; or a Hamiltonian operator that assigns to each measurement the value of its bit string. The results of the first stage can be improved in the following way: the model Hamiltonian has as little degeneracy as possible, or the degeneracy of this Hamiltonian also appears in the real physical QS. In the second stage 803 of this plurality of stages (clearly a VQC2 optimization with a constant probability distribution), the operation sequence 106 has two process pairs, in which the parameter vector output by the first stage is adopted 813 and is set unchanged. Now, the minimization of the free energy provides the parameter vector This parameter vector maps the input state to the correct eigenstate of the physical Hamiltonian operator. In the third stage 805 of this multi-stage process (clearly a VQC1 optimization using the real Hamiltonian), the parameter vector output by the second stage is adopted as 811 and remains unchanged. Now, minimization of the free energy using the real Hamiltonian provides the parameter vector It assigns corresponding probabilities to the eigenstates of the Hamiltonian operator. If necessary, the second stage and / or the third stage can be repeated. This improves the accuracy of the result, because the probability distribution determines the accuracy with which the eigenstates are determined. The first exemplary implementation reduces the resource requirements and error susceptibility of the method, because the optimization with fewer parameters runs significantly faster and converges less frequently to a local minimum than to finding a global minimum.

[0065] In a second exemplary implementation (clearly an approximation to FQVT), the method has the following features: VQC1 is reduced and not all qubits are varied in a variational manner. See for this purpose Fig. 9 , which illustrates a first exemplary implementation with a schematic diagram 900 similar to embodiment 400a. In other words, VQC1 is set up to change only a portion of QR 102 and / or to change fewer qubits than VQC2. This results in a smaller number of states remaining after the entropy measurement. Therefore, this approximate FQVT only considers the lowest energy excited states and therefore requires significantly fewer parameters, which in turn reduces the resource requirements and error susceptibility of the method. Especially for large systems, this modification can significantly reduce the resource requirements (e.g., computing time) without a significant loss of accuracy.

[0066] Figure 7The method according to various embodiments 700 is illustrated in a schematic diagram in which an output 701 (here exemplary free energy) of an objective function depending on a traversal counter 703 is shown. Line 706 represents the method described herein compared to a theoretical value (line 702), an ideal course of change in which the probability of each state is accurately calculated (line 704), and a FakeVigo simulator (line 708) using an error model derived from IBM's real quantum chip "Vigo". The solid line indicates the average value of the corresponding test series, and the flat area indicates the error interval given by the standard deviation.

[0067] In the first working example, the method (e.g., each optimization process sequence) has: executing the first readout process and / or the second readout process at least 100 times (e.g., at least 1000 times, e.g., at least 10,000 times, e.g., at least 30,000 times). In the first or second working example, the output 701 of the objective function is determined with the aid of qiskit's "Limited-Memory-BFGS-Algorithmus" ("L_BFGS_B") as the optimization algorithm, wherein the Hamiltonian operator is used according to relation (3), and the operation sequence 106 is used according to implementation 500 when β=1. In the second or third working example, the method according to implementation 600 is executed at least 20 times, wherein the parameter vector was initialized 601 with the aid of a random generator. All runs that achieved a free energy less than -2.05 were considered to have converged and in all cases accounted for at least 50% of all calculations.

[0068] In a fourth working example, the method described in this article (e.g., FQVT of the method) is used to solve the so-called quantum impurity problem (also called "quantum impurity problem"). With this, within the framework of the so-called dynamical mean field theory (DMFT), the properties of materials with strongly correlated electrons can be calculated. These materials include, for example, materials with elements with d or f electrons; and / or materials with potentials for application in superconductors, electrolysis, batteries, (quantum) sensor devices or in fuel cells. In the fifth or second working example, the optimization algorithm is set up to: process the gradient For example, the gradients dF / dφ and dF / dθ are used as input. Correspondingly, the adjustment 213 can include, for example, determining the gradients by means of a parameter shift rule (so-called “Parameter-Shift-Rule”).

[0069] Fig.10 The schematic diagram illustrates a method according to various embodiments 1000, which can be executed, for example, by means of a control device 202 (here exemplarily designed as a CPU), for example based on a variational principle (for example the Rayleigh-Ritz principle). The method can be set up according to one or more of the embodiments 100 to 900, for example at least according to the embodiment 600 (also referred to as FQVT optimization 600). The QR 102 can be set up according to the output vector output from the FQVT optimization 600. To be driven, for example, by means of an additional operation sequence 1006 (also referred to as evaluation sequence 1006). The evaluation sequence 1006 can be set up like the operation sequence 106 described herein (for example, set up according to embodiment 400a or embodiment 400b), for example, with the difference that the evaluation sequence 1006 has more quantum gates than the operation sequence 1006. Clearly, this enables: the calculated thermal state to be prepared on the QPU. For example, the evaluation sequence 1006 and the operation sequence 106 of the FQVT optimization 600 can be matched in at least one or more than one (for example, each) VQC, for example in VQC1 and / or VQC2. The evaluation sequence 1006 can have: between VQC2 and the second readout process (not shown) of the evaluation sequence 1006, at least one process (also referred to as operation process) that interacts with the QR 102 or within the QR 102 or between the QR 102 and the additional QR is executed. Examples of operating procedures include: driving according to a quantum gate (also referred to as a third drive) and / or performing a readout procedure. This enables one or more physical quantities of QS in a thermal state (e.g., an occupation number, or a Green's function) to be determined. For example, the adjustment procedure 213 for the evaluation sequence 1006 is no longer required.

[0070] Although specific embodiments are presented and described herein, it is recognized by those skilled in the art that the specific embodiments shown and described may be replaced with a variety of alternative and / or equivalent implementations without departing from the scope of protection of the present invention. The present application should encompass any adjustments or changes to the specific embodiments discussed herein. It is therefore intended that the present invention be limited only by the claims and their equivalents.

Claims

1. A method for operating a quantum register (102), the method comprising: Performing a first readout process (201) so as to determine first data (252), the first data representing a first state of the quantum register (102); According to at least one quantum gate (VQC2), the quantum register (102) in the first state is first driven (205); performing a second readout process (207) so as to determine second data (254) representing a second state of the quantum register (102) resulting from the first drive (205); adjusting (213) at least one parameter of the at least one quantum gate (VQC2) using the first data (252) and the second data (254) to optimize an objective function that depends on the first data (252) and the second data (254); The quantum register (102) is secondly driven (205) according to the at least one quantum gate (VQC2) obtained from the adjustment (213).

2. The method according to claim 1, wherein: The first reading process (201) comprises: reading out the quantum register (102); or An additional quantum register entangled with the quantum register (102) is read out.

3. The method according to claim 1 or 2, further comprising: According to at least one additional quantum gate (VQC1), the quantum register (102) is additionally driven (211), wherein: The first state of the quantum register (102) is based on the additional drive (211); and The adjusting (213) includes: adjusting at least one parameter of the at least one additional quantum gate (VQC1) or setting the at least one parameter unchanged when using the first data (252) and the second data (254); The quantum register (102) is additionally driven (211) in accordance with the at least one additional quantum gate (VQC1) resulting from the adjustment (213).

4. The method according to claim 3, wherein: Fewer qubits of the quantum register (102) are driven according to the at least one additional quantum gate (VQC1) than according to the at least one quantum gate (VQC2).

5. The method according to any one of claims 1 to 4, wherein: The at least one quantum gate (VQC2) and / or the at least one additional quantum gate (VQC1) are configured to bring the quantum register (102) into a non-entangled superposition state or an at least partially entangled superposition state.

6. The method according to any one of claims 1 to 5, wherein: The objective function represents the physical state variable to be minimized.

7. The method according to any one of claims 1 to 6, wherein: The adjusting (213) is performed based on an output (701) of the objective function, the output being dependent on the first data (252) and the second data (254) and / or being minimized.

8. A control device (202) configured to carry out the method according to any one of claims 1 to 7. 9 . A computer program, the computer program being configured to cause a processor executing the computer program to perform the method according to claim 1 . 10 . A computer-readable medium storing instructions, wherein the instructions are configured to cause a processor executing the instructions to perform the method according to claim 1 .

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