Method for operating a quantum register

Abstract

A method for operating a quantum register (102) can comprise: performance of a first read-out process (201) so that first data (252) representing a first state of the quantum register (102) are determined; first control (205) of the quantum register (102), which is in the first state, in accordance with a quantum gate (VQC2); performance of a second read-out process (207) so that second data (254) representing a second state of the quantum register (102) which results from the first control (205) are determined; adjustment (213) of at least one parameter of the quantum gate (VQC2) using the first data (252) and the second data (254) in order to optimize a target function which is dependent on the first data (252) and on the second data (254); second control (205) of the quantum register (102) in accordance with the quantum gate (VQC2) which results from the adjustment (213).

Description

[0001] Method for operating a quantum register

[0002] State of the art

[0003] The present disclosure relates to methods for operating a quantum register.

[0004] Current-generation quantum computers (also known as "Noisy Intermediate-Scale Quantum" (NISQ) computers) are already being used in fields such as materials research, optimization processes, and artificial intelligence. One aspect of materials research lies in the quantitative exploration of a quantum mechanical system (hereinafter referred to as QS), such as a molecule. Various methods have been developed to gain access to the state of a QS, some of which are explained below as examples.

[0005] The so-called exact diagonalization (ED) offers an exact mathematical solution to QS by decomposing the complete Hilbert space of all states into eigenstates using matrix diagonalization, which can be combined to form the thermalized state. However, its resource requirements increase so rapidly with the size of the QS that even modern supercomputers are rarely sufficient to obtain a solution of sufficient accuracy within a reasonable computing time. As an approximation, the so-called density matrix renormalization group (DMRG) reduces the computing time by neglecting higher energy states. Alternative algorithms are based on the quantum Monte Carlo method (QMC), which integrates over random variables. Examples of QMC algorithms include diffusion QMC, auxiliary field QMC, continuous-time QMC, Green's function QMC, and Hirsch-Fey QMC.What these QMC algorithms have in common is that their convergence towards the exact solution can be improved with more computational effort, but usually only within a small parameter range (e.g. temperature range), outside of which the so-called “Fermionic sign problem” disturbs the convergence.

[0006] These purely classical methods share an exponential increase in computational effort with increasing size of the QS, which often has to be compensated for with significant compromises in accuracy. These compromises are less pronounced when using a quantum computer due to its ability to superposition. While the capabilities of quantum computers are constantly growing, so-called hybrid, i.e., quantum-classical, algorithms are still being used, which outsource only part of the computation to the quantum computer. Quantum-classical algorithms make it possible to exploit the advantages of both computer types and reduce the requirements for the quantum computer, for example, with regard to the number of qubits, circuit depth, and tolerance to errors in gate operations (so-called noise).

[0007] Popular examples of hybrid algorithms are the so-called "Variational Quantum Eigensolver" (VQE) and the "Variational Quantum Thermalizer (VQT)" based on it. The VQE uses a variational quantum circuit to prepare a so-called initial state on the quantum computer. The quantum computer then measures the energy of this initial state. A classical (i.e., non-quantum mechanical) computer optimizes the quantum circuit to converge the initial state to the ground state of the QS (i.e., at a temperature T = 0 Kelvin). One goal is to calculate the state of the QS above 0 Kelvin (also called thermalized or excited state), which, however, is inaccessible to a VQE.

[0008] The density matrix of the thermalized state (T > 0) of a QS, which is mathematically described by a Hamiltonian operator H (also called Hamiltonian), can be expressed as:

[0009] Here, ß = 1 / kßT denotes the inverse temperature, Tr the trace, kß the Boltzmann constant, and pi the density matrix of the eigenstate of the Hamiltonian with energy Ei. The state p is the state of the QS at which the free energy F is minimized. The free energy F can be expressed as a function of the energy E and the entropy S by the following relation:

[0010] F = E' - T - S = Tr(Hp) - r - Tr(pln(p)) (2)

[0011] One way to calculate such excited states is to first find the ground state using VQE and then search for excited states by projecting the ground state out of the waveform or by assigning an additional energetic penalty term to the ground state. Both methods require complex quantum circuits, which complicate the calculation and make it prone to errors. However, with the less complex so-called "thermofield double-state" method, in which the system is enlarged and then collapsed again by a subsequent measurement, a direct measurement of entropy is not possible. To circumvent this, in turn, requires calculating the expectation value of ln(p) on the quantum computer (for example, using a series expansion of the logarithm) and then measuring powers of p, which, however, places high demands on the quantum computer.

[0012] As a generalization of the VQE, the VQT provides access to the thermal state of a system using an additional machine learning (ML) algorithm trained on a classical probability distribution. A sample is taken from this probability distribution and serves as the input state for the VQE. Since the quantum circuit does not change the entropy, the entropy can be calculated from the classical probability distribution. Analogous to the VQE, the variational quantum circuit is optimized and the ML algorithm is trained to minimize the free energy according to relation 2. The functionality of this VQT has already been realized using superconducting qubits, but requires a large number of circuit compilations and a large number of circuit executions.

[0013] Disclosure of the invention

[0014] According to various embodiments, a method for operating a quantum register (QR) is provided, the method comprising: performing a first readout process such that first data are determined which represent a first state (e.g., superposition state) of the QR (e.g., its probability) (e.g., are a function thereof); first controlling the QR, which is in the first state or a state resulting therefrom (e.g., unentangled), according to at least one (i.e., one or more than one) quantum gate (which, for example, forms a variational quantum circuit); performing a second readout process (e.g., comprising reading the QR), such that second data are determined which represent a second state (e.g., superposition state) of the QR (e.g., its probability) resulting from the first controlling (e.g.,one or more than one) 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; secondly controlling the QR according to the at least one quantum gate resulting from the adaptation. In other words, by adapting at least one parameter of the at least one quantum gate, the at least one quantum gate is adapted to use the adapted at least one quantum gate for the second control of the QR.

[0015] This method provided herein simplifies the reproducible initialization, simulation, and calculation of a quantum mechanical state, for example, when it is intended to represent a temperature of the QS above 0 Kelvin. This method described herein facilitates the efficient calculation of the free energy (e.g., according to relation (2)) using a quantum algorithm, for example, using a NISQ computer or a so-called "fully error corrected universal quantum" computer, as well as the preparation of its quantum mechanical state for measurements of other quantities.

[0016] The method described here can be implemented and executed in a resource-saving and cost-efficient manner, as it does not necessarily scale exponentially with the size of the QS and requires only minimal resources of the quantum computer (e.g., comparable to a VQE). Compared to the VQT, the method described here incurs significantly lower compilation costs, enables a better representation of the classical statistical distribution, and is simpler and more hardware-efficient to implement. This is made possible, for example, by the reduced coupling between the CPU and QPU and the high tolerance to noise of the quantum computer. More clearly, the method described here opens up the possibilities of a VQT at a cost comparable to that of a VQE. Various exemplary embodiments are given below.

[0017] Embodiment 1 is a method for operating a QR as specified above. Embodiment 2 is the method according to embodiment 1, wherein the first readout process comprises: reading the QR (e.g., a superposition state thereof); or reading an additional QR (e.g., a superposition state thereof) entangled with the QR (e.g., the first state thereof and / or before the first activation) (e.g., before or simultaneously with the second readout process). The former reduces the resource requirements, e.g., by requiring fewer qubits and / or a smaller circuit depth. The latter allows for the elimination of an intermediate measurement.

[0018] Embodiment 3 is the method according to embodiment 1 or 2, further comprising: additional first control of the QR, preferably when it is in an initial state, according to at least one additional quantum gate (which e.g. forms a variational quantum circuit), wherein the first state of the QR is based on the additional control (e.g. arises directly therefrom); and wherein the adapting preferably comprises adapting or setting invariant at least one parameter of the at least one additional quantum gate using the first data and the second data; additional second control of the QR according to the at least one additional quantum gate, which results from the adapting.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 to use the adjusted at least one additional quantum gate for the additional second control of the quantum gate. This facilitates the simulation of a more complex quantum gate (e.g., a thermalized state thereof) and / or the preparation of a classical probability distribution.

[0019] Embodiment 4 is the 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, according to the at least one additional quantum gate, fewer qubits of the QR are controlled (e.g., brought out of the initial state) than according to the at least one quantum gate. This reduces the resource expenditure (e.g., computing time).

[0020] Embodiment 5 is the method according to one of embodiments 1 to 4, wherein the at least one quantum gate and / or the at least one additional quantum gate are configured to entangle the QR and / or to bring it into a preferably unentangled or at least partially entangled superposition state (which is read out, for example, in the first and / or second readout process). This makes it easier to simulate a more complex QS (e.g., a thermalized state thereof). The at least partially entangled superposition state can, for example, be a result of entangling the QR. The set of entangled superposition states is, for example, a subset of the superposition states.

[0021] Embodiment 6 is the method according to any one of embodiments 1 to 5, wherein the adaptation comprises adjusting one or more parameters per qubit of the QR, according to which a state of the qubit is changed upon activation. This facilitates obtaining a more accurate calculation result.

[0022] Embodiment 7 is the method according to any one of embodiments 1 to 6, wherein the first data and / or the second data represent a superposition state of the QS or a probability of a base state resulting from the superposition state (preferably through the respective readout process). This facilitates the simulation of a more complex QS (e.g., a thermalized state thereof).

[0023] Embodiment 8 is the method according to any one of embodiments 1 to 7, wherein the objective function represents a physical state variable, and / or wherein the adaptation is performed based on an output of the objective function that depends on the first data and second data and / or is minimized. This facilitates simulating a more complex QS (e.g., a thermalized state thereof).

[0024] Embodiment 9 is the method according to one of embodiments 1 to 8, wherein the second data are independent of the first data or depend on a result of the first readout process (or the first data). This makes it easier to simulate a more complex QS or to better take its properties into account and / or to simplify / make the adaptation process more efficient. For example, the second data can depend on the first data, e.g. if the second data has one or more than one piece of information that is a function of the first data (or an piece of information thereof). For example, if the result of the first readout process is that a qubit is in a state |11), the second readout process can deliver a different result than if the result of the first readout process was that the qubit is in a state |0).

[0025] Embodiment 10 is the method according to one of embodiments 1 to 9, further comprising: determining an indication about a physical quantity (e.g., a QS) based on a result of the second control and / or the additional second control, wherein the indication preferably depends on or at least differs from an output of the objective function resulting from the optimization. This makes it easier to evaluate physical quantities (observables) in the calculated thermal state. Embodiment 11 is a (e.g., non-quantum mechanical) control device configured to carry out the method according to one of embodiments 1 to 10.

[0026] Embodiment 12 is a computer program configured to cause a (e.g., non-quantum mechanical) processor executing the computer program to perform the method according to one of embodiments 1 to 10.

[0027] Embodiment 13 is a computer-readable medium storing instructions configured to cause a (e.g., non-quantum mechanical) processor executing the instructions to perform the method according to any one of claims 1 to 10. Embodiments 11 to 13 facilitate the implementation of the method.

[0028] In the drawings, like reference characters generally refer to the same parts throughout the several views. The drawings are not necessarily to scale, emphasis instead generally being placed upon illustrating the principles of the invention. In the following description, various aspects are described with reference to the following drawings.

[0029] Figure 1 shows the structure and operation of a quantum processor according to various embodiments in a schematic diagram, Figures 2 and 3 as well as 6 to 8 and 10 each show a method according to various embodiments in various schematic diagrams, Figures 4A and 4B as well as 5 and 9 each show an operating sequence according to various embodiments in various schematic diagrams.

[0030] The following detailed description refers to the accompanying drawings, which, by way of illustration, show specific details and aspects of this disclosure in which the invention may be practiced. Other aspects may be utilized, and structural, logical, and electrical changes may be made without departing from the scope of the invention. The various aspects of this disclosure are not necessarily mutually exclusive, as some aspects of this disclosure may be combined with one or more other aspects of this disclosure to form new aspects.

[0031] Various examples are described in more detail below.

[0032] Figure 1 illustrates the structure and operation of a quantum processor (“quantum process unit” or QPU for short) according to various embodiments 100 in a schematic diagram. The QPU has a QR 102 and an operating device 104. The term “quantum register” (QR) refers to a device by means of which several (e.g., a number of n) qubits (here exemplarily designated as qi to q n(referred to as an operating sequence). Exemplary components of the QR 102 include: an ion trap, several (e.g., superconducting) circuit resonators, several quantum dots, a photonic system, a laser, an electromagnetic coil, etc. The operating device 104 is configured to operate the QR 102 (e.g., to act on it), for example, according to a sequence 106 (also referred to as an operating sequence). Exemplary processes of the operating sequence 106 include: initiating 101 the QR (i.e., bringing it into an initial state Zo), controlling 103 the QR 102, and / or reading 110 the QR 102. In the initial state Zo of the QR 102, each qubit can be unentangled and / or in a basis state (here, for example, 10), which, however, does not necessarily have to be the same for all qubits.

[0033] The concrete implementation of the operating device 104 or the operating sequence 106 depends on the type and architecture of the qubits. The operating device 104 can have one or more transducers configured to interact with the QR 102 or its environment, e.g., to at least change and / or detect its state Z. Example implementations of a transducer of the operating device include: actuator (e.g., actuator), sensor, transceiver, and the like. Example transducers of the operating device are configured to transmit one or more of the following to the QR 102 and / or detect them from it: optical radiation, a magnetic field, an electric field, electric charge, particle radiation. The QPU is a component of a quantum computer, which further comprises at least the housing environment for the QPU and the control electronics.

[0034] 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, for example, spanned by 10) and 11)). The implementation of a qubit can be of different types and architectures. Examples include: energy levels in trapped ions, polarization states of photons, spins in quantum dots or silicon, and energy levels of superconducting circuit resonators.

[0035] In general, the control 103 of the QR 102 is configured to stimulate a change in the state (also referred to as a state change) of the QR 102. The control 103 of the QR 102 occurs according to one or more than one quantum gate 108 (see also Fig. 5). The term "quantum gate" (also referred to as quantum logic gate or gate) refers to the quantum mechanical analogue to a logic gate of the electrical circuit. However, the quantum gate 108 corresponds to a process of influencing the QR 102, which changes the state Z of the QR 102, e.g., one or more than one qubit thereof. Using the quantum gate 108, a unitary function can be implemented in analogy to a logic gate, illustratively as an analogue to its Boolean function. Mathematically, a quantum gate can be described as a unitary transformation Ü, which transforms a qubit from the state |x) to the state Ü|x).Examples of quantum gates include: Hadamard gates, rotation gates, T-gates, controlled non-gates (CNOTs), exchange nodes, etc. A quantum gate can be configured to form or at least maintain entanglement.

[0036] The term "quantum algorithm" (also referred to as a quantum circuit) generally refers to a set of processes (e.g., quantum gates) interacting with the QR 102 or within the 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 of its qubits). For ease of understanding, a quantum circuit is represented as a circuit diagram, analogous to electronics, with one axis (here, from left to right) representing the temporal progression. Analogous to an electrical circuit, the sum of several quantum circuits can be viewed as a quantum circuit. From this perspective, the entire operating sequence 106 can be understood as a quantum circuit, as can its components, such as the process of controlling 103 or reading 110.From a data processing perspective, each quantum circuit transforms one state of the QR 102 (also referred to as the input state of the quantum circuit) into another state of the QR 102 (also referred to as the output state of the quantum circuit). The output state of the control process 103 (also referred to as the control process 103) is exemplified here as a superposition state, e.g. <i, j<n) aufweisend, muss aber nicht notwendigerweise verschränkte Qubits aufweisen. Ist der Ausgabezustand eines Quantenschaltkreises ein

[0037] If the quantum circuit is in a superposition state (e.g., having entangled qubits), the quantum circuit is also referred to herein as a preparation circuit (or preparation circuit). In general, if it refers to a single qubit, the superposition state can comprise a superposition of states of the single qubit, and / or, if it refers to multiple qubits, an entanglement of the multiple qubits and / or a superposition of their states. The preparation circuit can accordingly be configured to bring a qubit into a superposition state, which does not necessarily have to lead to entanglement of the qubit, but certainly can.

[0038] The approach circuit is preferably configured to change the squared amplitude and phase of a qubit. A quantum circuit can optionally comprise or consist of one or more parameterized quantum gates (e.g., multi-qubit quantum gates), e.g., parameterized by at least one parameter (also referred to as a variational parameter) (in the case of an approach circuit, then also referred to as a variational circuit or VQC for short). The approach circuit can optionally also comprise one or more unparameterized quantum gates and / or one or more multi-qubit quantum gates. For the sake of shorthand, the parameterization of the quantum circuit is notated vectorially herein (then also referred to as a parameter vector) and can, but does not necessarily have to, comprise two or more parameters as components.Accordingly, the output state of the parameterized quantum circuit is a function of the parameter vector, each vector component of which is a parameter of the quantum circuit. The process of reading 110 of a QR 102 (also referred to as readout process 110) comprises sensing (i.e., using a sensor) the state of the QR 102, which may, for example, be a superposition state and / or a result of a control process 103. During readout 110, the state of the QR 102 collapses into a basis state of the measurement basis of the readout process (here, for example, ZA). The readout process 110 (also referred to as quantum mechanical measurement process or, for short, measurement) may, for example, be repeated several times for each parameter vector and / or each term of the Hamiltonian in order to improve the accuracy of the expected value. The input state of the readout process 110 may be the output state of the control process 103.The sensor can be part of a measurement chain having a corresponding infrastructure (e.g., processor, storage medium, and / or bus system, or the like). The measurement chain can be configured to control the sensor, process the detected state as an input variable, and, based thereon, output data representing the input variable. The measurement chain can be implemented, for example, by means of the operating device 104. The sensor can have one or more measurement bases, one of which is the measurement base according to which the readout process 110 is performed. Illustratively speaking, the measurement base is a property of the sensor by means of which the readout process 110 is performed. Mathematically speaking, the measurement base spans the Hilbert space of states onto which the state of the QR is projected during readout. Depending on the technical design, the QPU can implement one or more measurement bases.Optionally, the QPU can implement multiple measurement bases, which allows the choice of measurement basis to be parameterized instead of and / or in addition to a parameterized quantum gate.

[0039] Figure 2 illustrates a method according to various embodiments 200 in a schematic diagram, which may be performed, for example, by a control device 202. The term “control device” may be understood as any type of entity (e.g., a processor) that enables the processing of data or signals. The data or signals may, for example, be treated according to at least one (i.e., one or more than one) specific function performed by the control device or the processor therein. A control device, e.g., its processor, may comprise or be formed from an analog circuit, a digital circuit, a logic circuit, a microprocessor, a microcontroller, a central processing unit (CPU), a graphics processing unit (GPU), a digital signal processor (DSP), an integrated circuit of a programmable gate array (FPGA), or any combination thereof.Any other way of implementing the respective functions described in more detail herein 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 special functions performed by the processor. The processor of the control device can, for example, be a classical (i.e., non-quantum mechanical, e.g., transistor-based) processor. A process sequence 602 of the method (also referred to as optimization process sequence 602) is shown. The process sequence 602 comprises an operating sequence 106, which can be implemented by means of a QPU (e.g., according to embodiments 100), and (e.g., precisely) an adaptation process 213, which can be implemented by means of the control device 202, for example.The QPU has the QR 102, which is configured to store several qubits (here exemplified qi to q. n ). It can be understood that not all of the components of the adaptation process 213 necessarily have to be performed by a conventional processor, but can also, depending on the performance of the QPU, be performed at least partially by the QPU itself. The method can optionally include repeating the optimization process sequence 602 several times (e.g., K times) (see also Fig. 6).

[0040] The operating sequence 106 comprises: first execution 201 of a readout process 110 (also referred to as ms or first readout process 201), such that first data 252 are determined, which represent 203 a first state 204a (here, for example, Zi = Z(ti)) of the QR 102 (e.g., at a first time ti); a control 205 of the QR 102 (also referred to as preparation 205), which is in the first state 204a; second execution 207 of a readout process 110 (also referred to as HIE or second readout process 207), so that second data 254 are determined, which represent 209 a second state 204b (here, for example, Z2 = Z(t2)) of the QR 102 resulting from the control 205 (e.g., at a second time t2 > ti). Depending on the specific implementation (see also Fig.4A and 4B), the first state 204a is a base state (ie, not having a superposition), e.g.if this is the output state of the first readout process 201 or at least based thereon; or a superposition state, e.g., if this is the output state of an additional control 211 of the QR 102 (also referred to as thermalization 211) or at least based on the thermalization 211. If the operating sequence 106 comprises the thermalization 211 of the QR 102, preferably when it is in an initial state and / or before the first time t1, this facilitates initializing the quantum mechanical state at a temperature above 0 Kelvin, since this makes use of the capabilities of the QPU. The readout process 201 and the second readout process 207 can, for example, be carried out according to the same measurement basis of the quantum gate 108.

[0041] The approach preparation 205 can be carried out according to a VQC (also referred to as VQC2) that has one or more than one quantum gate 108 (e.g., per qubit) and / or is parameterized by at least one parameter (0) (e.g., per qubit). The thermalization 211 can be carried out according to an additional VQC (also referred to as VQCi) that has one or more than one quantum gate 108 (e.g., per qubit) and / or is parameterized by at least one parameter (cf>) (e.g., per qubit). It can be understood that what has been described with regard to the quantum gate 108 can also apply analogously to more than one quantum gate per VQC.

[0042] For each optimization process sequence 602, an adjustment 213 of the parameter vector ( <j>, 0) the

[0043] Optimization process sequence 602 (also referred to as adaptation process) is performed using the first data 252 and the second data 254, wherein the parameter vector (cf>, 0) comprises at least one parameter of the VQCi and / or the VQC2. Each optimization process sequence 602 can optionally include executing the operating sequence 106 multiple times, for example, per adaptation process 213 (see also Fig. 6), e.g., without changing the parameter vector (cf>, 0) used and / or to obtain statistics for the first data 252 and second data 254.

[0044] The adaptation process 213 can, for example, be carried out in such a way that an objective function that depends on the first data and the second data is optimized, e.g., by minimizing its output. The objective function (e.g., its output) can, for example, represent a QS, e.g., a state variable of the system, which will be explained in more detail below.

[0045] Figure 3 illustrates the method according to various embodiments 300 in a schematic diagram, which can be carried out, for example, by means of a control device 202, here exemplified as a CPU, for example based on a variational principle (e.g., the Rayleigh-Ritz principle). The embodiments 300 can be configured like the embodiments 200, wherein the output 302a of the objective function 302 can represent, for example, a state variable (e.g., the free energy F) of a thermalized QS, preferably as a function of the entropy S and the energy E of the thermalized QS, further preferably according to relation (2). The first data 252 then clearly represent the entropy and the second data 254 the energy of the thermalized QS, which were determined based on the measured values ​​of the QPU. This is an example of the free energy (e.g.,as output of the objective function) can apply analogously to any other state variable of a (e.g. thermalized) QS.

[0046] 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 (cf>, 0) of the operating sequence 106 or at least its change based on the output 302a of the objective function 302, for example, under the constraint of minimizing the output 302a of the objective function 302 (for example, by treating it as a cost function), for example based on the history of the parameter vector (cf>, 0) and / or the history of the output 302a of the objective function 302. This enables optimization of the parameters of each VQC of the operating sequence 106. In a working example, the input of the operating sequence 106 comprises a molecular Hamiltonian and a parameter vector (cf>, 0) for one or more than one VQC, according to which the state of the QR 102 is prepared to simulate the quantum state of the molecule.In one or another working example, the optimization algorithm 304 is configured to evaluate the objective function 302 as a cost function and to calculate its gradient during each iteration of the optimization process sequence 602. Exemplary implementations of the optimization algorithm 304 include: so-called constrained optimization by linear approximation (COBYLA); conjugate gradient; or "quasi-Newton optimizer," which can be found, for example, in the Python packages qiskit or scipy. The optimization process sequence 602 may include converting its actual parameter vector (cf>, 0) and the output of the objective function 302 into new values ​​for the parameter vector (cf>, 0) using the optimization algorithm 304. These new values ​​are used as the actual parameter vector (cf>, 0) of the subsequent optimization process sequence 602 to control the QR 102.With each execution of the optimization process sequence 602 (also referred to as an iteration), the parameter vector (4>, 0) converges to a vector representing the thermal state of the QS. This process can be repeated until the output of the objective function 302 converges to a minimum or until a convergence criterion is met.

[0047] Figures 4A and 4B each illustrate an operating sequence 106 according to various embodiments 400a, 400b in a schematic diagram, which can be implemented, for example, in embodiments 100 to 300. Common to these is that the operating sequence 106 (then also referred to as a "Full Quantum Variational Thermalizer" or FQVT for short) comprises two process pairs, each of which comprises a VQC (VQCi or VQC2) and a subsequent readout process 110. The VQC of each process pair is configured to provide, as the output state (state of the qubits), a superposition (also referred to as a superposition state) of several quantum states with different amplitudes. The readout process 110 of each process pair can act on a superposition and be configured to convert the superposition into a base state.

[0048] The VQCi and the VQC2 can be executed sequentially. The VQCi depends on a first parameter vector cf>. Example implementations of the VQCi include: "Unitary-Coupled Cluster-Singles-Doubles" (UCCSD) or the so-called "efficient SU2" ("spatial unitary group of rank 2"). Based on a thermalized QS, the VQCi generates classical states with the probabilities pi = exp(-ßEi), and VQC2 generates the corresponding eigenstates \| / i of the Hamiltonian based on this. The VQC2 depends on a second parameter vector 0. Furthermore, embodiments 400a, 400b have in common that the second readout process (ms) comprises detecting (measuring) the output state of the VQC2. The output state of the VQC2 can be converted into the energy state by means of the measurement ms, which represents the energy of the second state 204b.

[0049] According to embodiments 400a, the first readout process (ms) comprises capturing (measuring) the output state of the VQCi, i.e., reading the QR 102 after being controlled according to the VQCi. The output state of the VQCi can be converted into a base state with a probability p (given by the amplitude of this base state) using the measurement ms. For each optimization process sequence 602, the FQVT 106 can be executed multiple times, collecting statistics on the output of the measurements ms. After a frequent repetition of the FQVT 106, the probability pi can be determined from the measurement ms of obtaining the base state i in the measurement ms during a new run. Based on the probabilities pi, the entropy of a quantum mechanical system can be determined, for example, according to the following relation:

[0050] S' = ^i Pi ln(pi).

[0051] Furthermore, the operating sequence 106 according to the embodiments 400a comprises transferring the output state of the first readout process ms into a superposition as a second state 204b by means of the VQC2.

[0052] If the QPU is not capable of performing a readout process 110 between two VQCs (also referred to as an intermediate measurement), or if no intermediate measurement is to be performed for other reasons, the operating sequence 106 according to embodiments 400b can be used. According to embodiments 400b, the available qubits comprise several groups, of which a first group of qubits (also referred to as the logical main register) and a second group of qubits (also referred to as the logical auxiliary register or ancilla qubits) are configured to be interleaved. The main register (qi to q n ) and the auxiliary register (here ai to a n ) are treated herein as (logically) separate QRs 102 for ease of understanding, although it should be understood that these may, but do not necessarily, be part of the same (physical) QR 102. The auxiliary register has at least as many qubits as the main register, e.g., a number of n qubits.

[0053] According to embodiments 400b, only the main register is controlled according to VQCi and VQC2 and read out using DIE. Furthermore, the operating sequence 106 according to embodiments 400b comprises entangling 410 the output state of VQCi (which is, for example, a superposition state) with the auxiliary register (also referred to as entangling process 410). The entangling process 410 may comprise entangling each qubit of the main register with (e.g., exactly) one qubit of the auxiliary register, for example, using at least one controlled not gate. The output of the entanglement process 410 comprises several (e.g., a number of n) pairs of entangled qubits, each pair comprising one qubit of the main register and one qubit of the auxiliary register.

[0054] Furthermore, the operating sequence 106 according to embodiments 400b comprises (e.g., only) converting the state of the main register (which is, for example, a superposition state), resulting from the entanglement process 410, into the superposition as the second state 204b by means of the VQC2. The first readout process ms then comprises detecting (measuring) the state 214b of the auxiliary register, which results from the entanglement process 410. The state 214b of the auxiliary register can be converted into the basis state with a probability p (given by the amplitude of this basis state) by means of the measurement ms. The (e.g., joint) measurement of all qubits at the end of the operating sequence 106 then clearly provides the energy (measurement of q) and entropy (measurement of a). In one example, the first readout process ms (of the auxiliary register) takes place (in time), e.g.,immediately before the second readout process ms or simultaneously with the second readout process DIE or, e.g. immediately after the second readout process HIE.

[0055] Figure 5 illustrates an operating sequence 106 according to various embodiments 500 in a schematic diagram, showing an exemplary implementation of the embodiments 400a in the case of two qubits. R t (0j) denotes a rotation around the t-axis (t=x, y or z) depending on 0j, which, for example, specifies the angle of rotation. The index of cf>i or 0j (i, j GN) references the components of the parameter vector ( <j>, 0). It can be understood that the same may apply analogously to embodiments 400b.

[0056] In a working example, the Hamiltonian for the case of two qubits can be formulated as follows:

[0057] H = -(«o + ni) + 2n0rii + O.5(cjci + CJCQ)

[0058] Where c denotes ! the creation operator, c the destruction operator, and n the occupation number operator. The index of the respective operator indicates the qubit on which it acts.

[0059] Figure 6 illustrates the method according to various embodiments 600 in a schematic diagram, based on which additional exemplary implementations of embodiments 100 to 500 are explained. The method can optionally comprise repeating the optimization process sequence 602 several times (e.g., K times) until a criterion is met, for example, until a number k of runs (also referred to as run counter) reaches a target number K or a change in the output of the target function falls below a threshold value. If the criterion is met, the parameter vector (cf>, 0) = (cf>, 0) output (also referred to as output vector), e.g., (cf>, 0)k = K = (<$>, 0) output, updated in the last performed optimization process sequence 602 can be output 611.

[0060] The or each optimization process sequence 602 may include updating the parameter vector (cf>, 0) based on the result of the adaptation process 213. A VQC (e.g., the VQCi and / or the VQC2) of the k-th (2 <k<K) Optimierungsprozessfolge verwendet dabei den Parametervektor (cf> , 0)ki, which the previously performed (kl)-th optimization process sequence 602 has updated. Accordingly, in the k-th optimization process sequence 602, the QR 102 is controlled according to the VQCi and / or the VQC2, which result from the adaptation process 213 of the (kl)-th optimization process sequence.

[0061] Optionally, the or each optimization process sequence 602 may comprise performing the operating sequence 106 multiple times, for example, until a criterion is met, for example, when a number r of runs reaches a target number R. The adaptation process 213 may then be based on the first data and second data from multiple operating sequences 106. The method may optionally comprise, at the beginning of the optimization process sequence 602, initializing the parameter vector (cf>, 0) in 601, ie, determining its starting values. The starting values ​​of the parameter vector ( <j>, 0) can be determined, for example, by means of a random generator or read from a memory of the control device 202.

[0062] In a first exemplary implementation (illustratively a segmented optimization), the method (e.g., for optimizing free energy) comprises multiple stages, in each stage of which one or more than one optimization process sequence 602 is carried out. See Figure 8, which illustrates the first exemplary implementation in a schematic diagram 800 analogous to embodiments 600. In the first stage 801 of the multiple stages (illustratively a VQCi optimization with a simplified Hamiltonian), the operating sequence 106 comprises only the VQCi parameterized by means of the parameter vector cf> and the first readout process ms. An entropy can be determined from the distribution of the measured values. The energy value can be determined using the same measurement ms. A simplification of the Hamiltonian (also referred to as a model Hamiltonian) is used, which is diagonal in the basis used for the measurement ms.Examples of such a model Hamiltonian include the so-called "atomic limit" of the Hamiltonian, in which all hop terms are set to zero, or a Hamiltonian that assigns the value of its bit string to each measurement. The result of the first stage can be improved by having the model Hamiltonian exhibit as few degeneracies as possible, or by having its degeneracies also occur in real physical QS. In the second stage 803 of the multiple stages (illustratively a VQC2 optimization with a constant probability distribution), the operating sequence 106 exhibits the two process pairs, where the parameter vector cf> output from the first stage is adopted 813 and set invariant. Minimizing the free energy now yields the parameter vector 0, which maps the input states to the correct eigenstates of the physical Hamiltonian.In the third stage 805 of the multiple stages (illustratively a VQCi optimization with a real Hamiltonian), the parameter vector 0 output from the second stage is adopted 811 and kept invariant. Minimizing the free energy with a real Hamiltonian now yields the parameter vector cf>, which assigns the corresponding probabilities to the eigenstates of the Hamiltonian. If necessary, the second and / or third stage can be repeated. This increases the accuracy of the result, since the probability distribution determines the accuracy with which the eigenstates are determined. The first exemplary implementation reduces the resource requirements and the error susceptibility of the method, since an optimization with fewer parameters runs significantly faster and converges to local minima less frequently than finding the global minimum.

[0063] In a second exemplary implementation (illustratively approximate FQVT), the method comprises reducing the size of the VQCi and not changing all qubit variations. See Figure 9, which illustrates the first exemplary implementation in a schematic diagram 900 analogous to embodiments 400a. In other words, the VQCi is configured to change only a portion of the QR 102 and / or to change fewer qubits than the VQC2. This results in a smaller number of states remaining after the entropy measurement. This approximate FQVT therefore only considers the lowest-energy excited states and thus requires significantly fewer parameters, which analogously reduces the resource requirements and the 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.

[0064] Figure 7 illustrates the method according to various embodiments 700 in a schematic diagram in which the output 701 of the objective function (here, for example, the free energy) is plotted as a function of the run counter 703. Line 706 represents the method described herein in comparison to the theoretical value (line 702), to the ideal curve (line 704), in which the probabilities of each state are calculated exactly, and to the FakeVigo simulator (line 708), which uses an error model derived from the real quantum chip "Vigo" from IBM. The solid lines indicate the average of the respective test series, and the flat areas indicate the error interval given by the standard deviation.

[0065] In a first working example, the method (e.g., each optimization process sequence) comprises performing the first readout process and / or the second readout process at least 100 times (e.g., at least 1,000 times, e.g., at least 10,000 times, e.g., at least 30,000 times). In the first or a second working example, the output 701 of the objective function was determined using a "Limited Memory BFGS algorithm" ("L BFGS B") from qiskit as the optimization algorithm, using the Hamiltonian according to relation (3) and the operating sequence 106 according to embodiments 500 with ß = 1. In the second or a third working example, the method according to embodiments 600 was performed at least 20 times, with the parameter vector (eß, 0) being initialized 601 using a random generator. All runs that achieved a free energy less than -2.05 were counted as converged and in all cases accounted for at least 50% of all calculations.

[0066] In a fourth working example, the method explained herein (e.g., its FQVT) is used to solve a so-called quantum impurity problem. Within the framework of the so-called Dynamical Mean Field Theory (DMFT), it can be used to calculate properties of materials with strongly correlated electrons. These include materials that, for example, possess an element with d- or f-electrons and / or have potential for use in superconductors, electrolysis, batteries, (quantum) sensors, or fuel cells. In a fifth or the second working example, the optimization algorithm was configured to process the gradient dF / d(<ß, 0) as input, e.g., the gradients dF / dcß and dF / d0. Accordingly, the adaptation 213 may include determining the gradient dF / d(<ß, 0), for example, using a parameter shift rule (so-called "parameter shift rule").

[0067] Figure 10 illustrates the method according to various embodiments 1000 in a schematic diagram, which can be carried out, for example, by means of a control device 202, here exemplified as a CPU, for example based on a variation principle (e.g., the Rayleigh-Ritz principle). The method can be configured according to one or more of the embodiments 100 to 900, for example, at least according to the embodiments 600 (also referred to as FQVT optimization 600). The QR 102 can be controlled according to the output vector (cf>, 0) outputted by the FQVT optimization 600, for example, by means of an additional operating sequence 1006 (also referred to as evaluation sequence 1006). The evaluation sequence 1006 can be configured like the operating sequence 106 described herein (e.g.,according to embodiments 400a or embodiments 400b), for example, with the difference that the evaluation sequence 1006 has more quantum gates than the operating sequence 1006. Illustratively, this achieves that the calculated thermal state is prepared on the QPU. For example, the evaluation sequence 1006 and the operating sequence 106 of the FQVT optimization 600 may coincide in at least one or more than one (e.g., each) VQC, e.g., in VQCi and / or VQC2. The evaluation sequence 1006 may include, between the VQC2 and the second readout process (not shown) of the evaluation sequence 1006, performing at least one process (also referred to as an operation process) of interaction with the QR 102 or within the QR 102 or between the QR 102 and an additional QR. Examples of the operation process include: driving according to a quantum gate (also called third driving) and / or performing a readout process.This makes it possible to determine one or more physical quantities of the QS (e.g., occupation numbers or Green's functions) in the thermal state. For example, the adaptation process 213 no longer needs to be applied to the evaluation sequence 1006.

[0068] Although specific embodiments have been shown and described herein, it will be recognized by those skilled in the art that the specific embodiments shown and described may be substituted for a variety of alternative and / or equivalent implementations without departing from the scope of the present invention. This application is intended to cover any adaptations or variations of the specific embodiments discussed herein. Therefore, it is intended that this invention be limited only by the claims and the equivalents thereof.< / j> < / j> < / j>

Claims

Claims 1. Method for operating a quantum register (102), comprising: Performing a first readout process (201) so that first data (252) are determined which represent a first state of the quantum register (102); first activation (205) of the quantum register (102) which is in the first state, according to at least one quantum gate (VQC2); Performing a second readout process (207) so that second data (254) are determined, which represent a second state of the quantum register (102) resulting from the first input (205); Fitting (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 second data (254); second driving (205) of the quantum register (102) according to the at least one quantum gate (VQC2) resulting from the fitting (213).

2. Method according to Anspmch 1, wherein the first readout process (201) comprises: reading out the quantum register (102); or reading out an additional quantum register entangled with the quantum register (102).

3. A method according to Annex 1 or 2, further comprising: an additional first input (211) of the quantum register (102) according to at least one additional quantum gate (VQCi), wherein the first state of the quantum register (102) is based on the additional input (211); and wherein the adjustment (213) comprises adjusting or setting invariant at least one parameter of the at least one additional quantum gate (VQCi) using the first data (252) and the second data (254); an additional second input (211) of the quantum register (102) according to the at least one additional quantum gate (VQCi) resulting from the adjustment (213).

4. Method according to Anspmch 3, wherein according to the additional at least one quantum gate (VQCi) fewer qubits of the quantum register (102) are addressed than according to the at least one quantum gate (VQC2).

5. 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 (VQCi) are configured to bring the quantum register (102) into an unentangled superposition state or at least a partially entangled superposition state. A method according to any one of claims 1 to 5, wherein the objective function represents a physical state variable that is minimized. A method according to any one of claims 1 to 6, wherein the adjustment (213) is based on an output (701) of the objective function that depends on and / or is minimized by the first data (252) and second data (254). A control device (202) configured to perform the method according to any one of claims 1 to 7. 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. A computer-readable medium storing instructions configured to cause a processor executing the instructions to perform the method according to any one of claims 1 to 7.

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