SIMULATION OF A QUANTUM COMPUTING

Abstract

A method for performing a quantum computation, called the initial quantum computation, is disclosed, wherein the initial quantum computation is defined by at least a first set of one or more first-type quantum operations, wherein the first set of one or more first-type quantum operations each has one or more first rotation angles as parameters, and wherein each first-type quantum operation is a one-qubit rotation about a z-axis in a Bloch sphere by the respective first rotation angle.The procedure includes: representing the initial quantum computation by a function f that takes the first rotation angles as arguments; performing a Taylor expansion of the function f around zero, which leads to fundamental derivatives of the function f; for each specific derivative of at least one of the fundamental derivatives: using a parameter shift rule to express the specific derivative as a linear combination of a set of terms.

Description

FIELD OF THE INVENTIONThe disclosure relates to quantum computing, and more particularly to a method of performing quantum computation.BACKGROUNDQuantum simulators are based on matrix multiplication. However, since the dimensions of the respective arrays grow strongly with the number of qubits, quantum circuits with e.g. 100 qubits cannot be simulated on the currently available classical computers, since both the need for main memory and the need for computing power exceed the currently available capacities.SUMMARYExample embodiments provide a method for performing a quantum calculation, referred to as an initial quantum calculation, the initial quantum calculation defined by at least a first set of one or more first type quantum operations, the first set of one or more first type quantum operations each having as parameters one or more first rotation angles, each first type quantum operation being a one-qubit rotation about a z-axis in a Bloch sphere about the respective first rotation angle, the method comprising: representing the initial quantum calculation by a function f having as arguments the first rotation angles; performing a Taylor expansion of the function f about zero, resulting in basic derivatives of the function f; for each specific derivative of at least one of the basic derivatives: using a parameter shift rule to express the specific derivative as a linear combination of a set of terms, for each term of the set of terms: wherein the term comprising the function f has as arguments a respective set of second rotation angles associated with the first rotation angles, respectively, wherein each second rotation angle of the set of second rotation angles is a 90-degree angle scaled by a scale factor representing the first rotation angle associated with the second rotation angle, wherein the scale factor is an integer that is zero or different from zero, depending on the order in which the associated first rotation angle is differentiated in the specific derivative; wherein each of the set of second rotation angles is a parameter of a second set of first type quantum operations; in each second set of first type quantum operations, replacing each first type quantum operation at the second rotation angle with a second type quantum operation having as an exponent the scale factor of the second rotation angle, wherein the second type quantum operation is a 90 degree rotation about the z-axis, resulting in a set of second type quantum operations; redefining the initial quantum calculation by using the sets of second type quantum operations instead of the first set of first type quantum operations, resulting in a redefined quantum calculation; simulating the redefined quantum calculation on a digital computer.Example embodiments provide a computer system for performing an initial quantum calculation, the initial quantum calculation defined by at least a first set of one or more quantum operations of the first type, the first set of one or more quantum operations of the first type each having as parameters one or more first rotation angles, each quantum operation of the first type being a one-qubit rotation about a z-axis in a Bloch sphere about the respective first rotation angle, the computer system being configured to: represent the initial quantum calculation by a function f having as arguments the first rotation angles; perform a Taylor expansion of the function f about zero, resulting in basic derivatives of the function f; for each specific derivative of at least one of the basic derivatives: using a parameter shift rule to express the specific derivative as a linear combination of a set of terms, for each term of the set of terms: wherein the term comprising the function f has as arguments a respective set of second rotation angles associated with the first rotation angles, respectively, wherein each second rotation angle of the set of second rotation angles is a 90-degree angle scaled by a scale factor representing the first rotation angle associated with the second rotation angle, wherein the scale factor is an integer that is zero or different from zero, depending on the order in which the associated first rotation angle is differentiated in the specific derivative; wherein each of the set of second rotation angles is a parameter of a second set of first type quantum operations; in each second set of first type quantum operations, replacing each first type quantum operation at the second rotation angle with a second type quantum operation having as an exponent the scale factor of the second rotation angle, wherein the second type quantum operation is a 90 degree rotation about the z-axis, resulting in a set of second type quantum operations; redefining the initial quantum calculation by using the sets of second type quantum operations instead of the first set of first type quantum operations, resulting in a newly defined quantum calculation; simulating the newly defined quantum calculation.Example embodiments provide a computer program comprising machine executable instructions, wherein execution of the machine executable instructions causes a computer system to perform the following: representing an initial quantum calculation by a function f having, as arguments, first rotation angles of a first set of one or more quantum operations of the first type defining the initial quantum calculation, wherein each quantum operation of the first type is a one-qubit rotation about a z-axis in a Bloch sphere by the respective first rotation angle; performing a Taylor expansion of the function f about zero, resulting in basic derivatives of the function f; for each specific derivative of at least one of the basic derivatives: use of a parameter shift rule to express the specific derivative as a linear combination of a set of terms, for each term of the set of terms: wherein the term comprising the function f has as arguments a respective set of second rotation angles associated with the first rotation angles, respectively, wherein each second rotation angle of the set of second rotation angles is a 90-degree angle scaled by a scale factor representing the first rotation angle associated with the second rotation angle, wherein the scale factor is an integer that is zero or different from zero, depending on the order in which the associated first rotation angle is differentiated in the specific derivative; wherein each of the set of second rotation angles is a parameter of a second set of first type quantum operations; in each second set of first type quantum operations, replacing each first type quantum operation at the second rotation angle with a second type quantum operation having as an exponent the scale factor of the second rotation angle, wherein the second type quantum operation is a 90 degree rotation about the z-axis, resulting in a set of second type quantum operations; redefining the initial quantum calculation by using the sets of second type quantum operations instead of the first set of first type quantum operations, resulting in a newly defined quantum calculation; simulating the newly defined quantum calculation. The specific derivative may be a derivative of the function f. The at least one of the basic derivatives may be, for example, all basic derivatives of the function f.BRIEF DESCRIPTION OF THE DRAWINGSExamples are described in more detail below with reference to the drawings, in which: FIG. 1 is a flow diagram of a method for performing a first set of one or more quantum operations of the first type, according to an example of the present subject matter. FIG. 2 is a block diagram of an example computer system for performing at least a portion of the present method, according to an example of the present subject matter. FIG. 3 is pseudo code for simulating a newly defined quantum calculation according to an example of the present subject matter.DETAILED DESCRIPTIONIn the following description, for purposes of explanation and not limitation, specific details are set forth, such as particular architectures, interfaces, techniques, etc., in order to provide a thorough understanding of the examples. However, it will be apparent to those skilled in the art that the disclosed subject matter may be practiced in other examples that depart from these specific details. In some instances, detailed descriptions of known devices and / or methods are omitted so as not to obscure the description with unnecessary detail.The present subject matter may allow to simulate quantum computations on classical computers. The classical computer may also be referred to as a digital computer. The digital computer may be a machine that can be programmed to automatically perform sequences of arithmetic or logical operations (computations). The digital computer may include the hardware such as processor and memory, and software such as an operating system.A quantum system may be defined by a set of qubits. The number of qubits can be e.g. n qubits, where n ≥ 1. The quantum system can thus be an n-qubit system. The quantum state of the quantum system may be varied by employing one or more quantum operations. This may allow quantum calculation to be performed. The quantum calculation may refer to the initial quantum calculation. The term "initial" is used as a label for naming. The initial quantum calculation may be described as an initial set of quantum operations and a measurement. The quantum operations may be a particular class of transformations that are mathematically referred to as unitary transformations. The quantum operation may be, for example, a transform that may be applied to one or more qubits. The initial set of quantum operations may allow for unitary development of the quantum system.In one example, the initial quantum calculation may include a first step of initializing all qubits in a computational base state (computational base state) such as the |0> state (denoted as |0> ⊗n), a second step of applying the initial set of quantum operations, and a third step of measuring in the computational base on one or more of the qubits. Each quantum operation of the initial set of quantum operations may be of a type belonging to a predefined set of quantum operation types. The predefined set of quantum operation types may represent, for example, a predefined set of gate types. The predefined set of gate types may be, for example, a universal set of gate types. The initial set of quantum operations may consist of one or more types of quantum operations. In an example, the initial set of quantum operations may consist wholly or partly of quantum operations of the first type. The quantum operation of the first type can be a one-qubit rotation about a z-axis in the Bloch sphere, which rotation takes place about the respective first rotation angle. Thus, starting from the initial state |0> ⊗n of the quantum system, the initial set of quantum operations may define a unitary operator U(θ) ∈ C 2n×2n that transforms the initial state such that the state of the quantum system immediately prior to measurement may be | ψ(θ)>:=U(θ)|0> ⊗n where θ ∈ R m. The present method may provide the expectation value of observable measurement by determining the quantity using the observable. In an example implementation of the measurement, the measurement may be represented by an observable that may have two eigenvalues such as -1 and 1 (e.g., corresponding to the l-th qubit to be measured in the computation base, where Z represents the Pauli-Z gate and / or the identity operator).A quantum circuit may be used to represent the quantum calculation. The quantum circuit may be, for example, a computation routine consisting of coherent quantum operations on quantum data such as qubits. The quantum circuit representing the quantum calculation may include a first stage of initializing all qubits in the state |0> (denoted as |0> ⊗n), a second stage of quantum gates, and a third stage of measurements in the calculation basis on one or more of the qubits. The quantum gates comprise a sequential arrangement of quantum gates in different qubits. The quantum gates may each represent unitary transformations. The quantum gates may consist of gates of the predefined set of gate types. The predefined set of gate types may include, for example, H (Hadamard) gates, S gates, CNOT gates, R Z- gates, R X- gates, and R Y- gates. Thus, the predefined set of quantum gate types as non-Califord gates may only include rotation of the R Z- gate about the z-axis, as the R X- gate and the R Y- gate may be defined as a function of the R Z- gate and Califord gate. Each gate type of the predefined set of gate types may represent a type of quantum operation. The first type of quantum operation according to the present subject matter may be the quantum operation represented by the R z- gate. The second type of quantum operation within the meaning of the present subject matter may be the quantum operation represented by the phase gate, the S gate (or). The S gate is also known as a Z90 gate because it represents a 90 degree rotation about the z axis. The sequential arrangement of the quantum gates of the quantum circuit may represent the initial set of quantum operations. The initial set of quantum operations can thus be provided as an ordered list of quantum operations. The quantum operations may be implemented with one or more quantum gates.However, depending on the type of quantum operations that the quantum circuit represents, this quantum circuit may or may not be efficiently simulated using a classical computer. For example, a quantum circuit consisting of only Cififord gates can be simulated efficiently with a classical computer. The present subject matter may utilize this property to enable the simulation of quantum circuits even if they include non-Clipfford gates of the predefined set of quantum gate types. To this end, the present subject matter may identify or select from the initial set of quantum operations the first set of one or more quantum operations of the first type, wherein the initial set of quantum operations may include one or more types of quantum operations including the quantum operation of the first type. In an example, the first set of one or more quantum operations of the first type may be a subset of quantum operations of the initial set of quantum operations and the remaining subset of quantum operations may represent Clipfford gates. Alternatively, the first set of one or more quantum operations of the first type may be the initial set of quantum operations. The present subject matter may represent the first set of quantum operations of the first type with quantum operations that may be represented by Cififord gates and thus may be efficiently simulated on a classical computer.For example, the first set of quantum operations of the first type may comprise a number m of quantum operations of the first type, where m is greater than or equal to one, m≥1. Each quantum operation of the first type may be a one-qubit rotation about the z-axis in the Bloch sphere about the respective first rotation angle. For example, each first type quantum operation of the first set of first type quantum operations may be an R Z- operation implemented by the R z- gate. Thus, the first set of quantum operations of the first type may each be associated with a number m of first rotation angles. For example, θ 1... θ m may be the first rotation angles of the first set of quantum operations of the first type. The first type quantum operation having the first rotation angle θ 1 may be referred to as R Z( θ 1) the first type quantum operation having the first rotation angle θ 2 may be referred to as R Z( θ 2) and so forth. In one example, the unitary operator may be defined as: U(θ)=R Z( θ m)... R Z( θ 2) R Z( θ 1) · Alternatively, the unitary operator may be defined as follows: U(θ)=C m+1 R Z( θ m) C m... R Z( θ 2) C 2 R Z( θ 1) C 1 where C 1... C m+1 may be unitary operators representing Clipfford gates. Alternatively, the unitary operator may be defined as follows: U(θ)=C m+1 R m( θ m) C m... R 2( θ 2) C 2 R 1( θ 1) C 1, where R j( θ j) is either R Z( θ j) or R X( θ j) or R Y( θ j) and j ∈ {1,...,m}. Providing various types of the initial set of quantum operations may enable a broader application of the present subject matter.The initial quantum calculation may be represented by the function f(θ 1,..., θ m) having, as arguments, the first rotation angles θ 1... θ m respectively. This may allow the parameters of the first set of quantum operations of the first type to be used as variables of the function f. Based on the above-mentioned predefined set of gate types, each (potential) other operation of the initial set of quantum operations having a different type than the first type may have fixed parameters / values, e.g. the quantum operation represented by the H-gate has fixed values. That is, in the case that the first set of quantum operations of the first type is a subset of the initial set of quantum operations, the parameters of the remaining subset of quantum operations may not be variables. The function f(θ 1,..., θ m) may represent, for example, the output of the quantum circuit (e.g., the expected value of observables).Taylor evolution of the function f(θ 1,..., θ m) around zero can lead to derivatives of the function f(θ 1,..., θ m). Taylor evolution of the function f(θ 1,..., θ m) around zero can lead to fundamental derivatives of the function f(θ 1,..., θ m). Each derivative may be a partial or a mixed derivative. The derivative may be defined, for example, as follows: wherein |α|=α 1+... + α m indicates the order of the derivative, and α j is the order according to which the jthfirst rotation angle θ j in the derivative is differentiated, where j is an integer varying between 1 and m, j ∈{1,..., m}. Taylor evolution of function f may result in the following sum: Taylor evolution of function f may result in where L is the order of Taylor evolution and α may be a multi-index.The parameter shift rule may be used to express or define each derivative of the function f(θ 1,..., θ m). For example, applying the parameter shift rule to the derivative Dαf(0)may result in an extension of the derivative as follows: =wherein and α j may be less than or equal to the order |α|, SC g is a leading factor, e.g., the leading factor SC g may be equal to 1 or -1, and Nαis the total number of terms, e.g., Nα=2|α|. In one option, the derivative may be defined as a weighted sum of the set of terms using weighting factors associated with the set of terms, respectively. This indicates that the extension of the derivative comprises a set of terms. The parameter shift rule may allow the partial derivative of the function f to be expressed as a linear combination of the set of terms, where each term may include the function f with shifted arguments, which may allow the use of the same quantum circuit. The set of terms may be the summands of the sum. The number of terms of the set of terms may be, for example, Nα=2|α|. Each summand of the sum comprises the function f having as arguments a respective set of second rotation angles associated with the first rotation angles, e.g. the second rotation angle may be obtained by shifting the corresponding first rotation angle. For example, each summand may be defined as follows: The function f has m arguments in each summands: which are respectively second rotation angles, e.g. the m-th second rotation angle may be: Each second rotation angle of the set of second rotation angles is a 90-degree angle scaled by a scale factor representing the first rotation angle associated with the second rotation angle. The scale factor may be j for the j-th first rotation angle θ, for example. The scale factors are integers, the scale factor is zero when the jth first rotation angle θ j is not part of the derivative Dαf( 0). For example, if the first rotation angle θ 1 is not part of the derivative Dαf( 0), this means that the order α 1 is zero, α 1= 0, which in turn means that the corresponding argument is zero. However, if the first rotation angle is part of the derivative D a f(0), the corresponding scale factor is an integer that is associated with the order according to which the first rotation angle is differentiated in the derivative.For example, if the first rotation angle θ 1 is part of the derivative Dαf( 0), the corresponding scale factor may be. The set of second rotation angles may be parameters of a second set of quantum operations of the first type, e.g. the second rotation angle may be the angle of a one-qubit rotation about the z-axis, e.g. it may be the rotation angle of an R Z- gate. The second set of quantum operations of the first type may each be defined as follows: The second set of quantum operations of the first type may comprise at most m quantum operations of the first type. Thus, the parameter shift rule of each derivative Dαf( 0) may result in a number Nα(=2|α|) of second sets of quantum operations of the first type, e.g., since the scale factors are integer, the function of the gth summand of the derivative Dαf( 0) may be efficiently evaluated on a classical computer.Thus, in each second set of quantum operations of the first type, each quantum operation of the first type at the second rotation angle may be replaced with a product of quantum operations of the second type based on the scale factor of the second rotation angle. For example, each quantum operation of the first type at the second rotation angle may be replaced by a quantum operation of the second type having the scale factor as an exponent, the exponent referring to how often the quantum operation of the second type is multiplied by itself. Since the global phase cannot affect the measurement result, the first type operation may be replaced with the product of operations, for example, if so, the first type operation may be replaced with the product S×S of S operations S 2= S×S. If so, the identity operation is that corresponding to omitting the gate from the quantum circuit. Thus, for each α, where |α|≤L, the second sets of first type of derivative quantum operations Dαf( 0) may be replaced with respective sets of second type quantum operations.The initial quantum calculation may be re-defined by replacing the first set of quantum operations of the first type with a set of quantum operations of the second type, respectively. For example, for each α, where |α|≤L, each set of second type quantum operations of the 2^(|α|) sets of second type quantum operations may: replace the first set of first type quantum operations in the initial quantum calculation. This may result in a number 2|α| of individual newly defined quantum computations.The (total) re-defined quantum computation may be obtained as a combination of the individual re-defined quantum computations. The newly defined quantum calculation can thus be simulated on a digital computer.The initial and newly defined quantum computations may be defined using the following notation, for example. The observable of the measurement can be defined as where a t ∈Rfor all (P 1,..., P n) ∈{I,X,Y,Z} n. The initial quantum calculation can thus be represented as follows: The newly defined quantum calculation can be represented as follows: wherein, where the Pauli operator is and if the parameter shift rule is as described in Paper A. Mari et al. arXiv:2008.06517, Phys. Rev. A 103, 012405 (2021), The individual newly defined quantum operation may be represented by the term. Thus, the simulation of the newly defined quantum calculation f≅(θ) can be performed by simulating the individual quantum calculations and combining them, for example, as shown in the equation of Dαf(0) and f≅(θ). This may result in a redefinition of the quantum circuit (in a loop or in an iterative manner) to represent each individual redefined quantum calculation and then to perform the quantum circuit. For simulating the newly defined quantum calculation, a quantum simulator may be used, for example. The quantum simulator may be a software program which, when executed on a classical computer, runs the quantum circuit as if it were executed on a quantum computer.The redefining of the quantum calculation can be further improved by reducing the number of quantum operations of the second type according to the following example. The first type quantum operation at the second rotation angle may be newly defined by applying the modulo operation to the scale factor with a fixed number F being a positive integer multiple of four. Moreover, the resultant first type quantum operation up to a global phase may be defined, that is, In one example, the fixed number F may be four, F=4. Thereby, the number of the second type quantum operations may be further reduced. This example may be based on a rotational symmetry of a certain order, e.g. F, according to which the quantum system may be invariant under operations at certain fixed angles.According to an example, in the case that the initial set of quantum operations comprises a subset of quantum operations of the first type and a subset of quantum operations representing one-qubit rotations about the x-axis and the y-axis in the Bloch sphere, each quantum operation about the x-axis or the y-axis may be transformed as a function of a quantum operation of the first type with the same rotation angle; thereby obtaining the first set of quantum operations of the first type as the subset of quantum operations of the first type and the transformed subset. For example, the unitary operator may be referred to as U(θ)=R m( θ m)... R 2( θ 2) R 1( θ 1) where R j( θ j) is either R Z( θ j) or R X( θ j) or RY(θj), and j ∈ {1,...,m}. To do this, the following equations: R X( θ)=HR Z( θ)H, and R Y( θ)=SHR Z( θ)NR †, may be used to express the unitary operator as a function of quantum operations of the first type and quantum operations representing Clipfford gates.According to one example, an initial set of quantum operations of the first type, the initial rotation angles each having as parameters, can be part of the initial set of quantum operations. Each initial angle may be expressed as the sum of a first angle of rotation and a 90 degree angle scaled by an integer, wherein the first type quantum operation of the initial angle is the product of the first type quantum operation of the scaled 90 degree angle and the first type quantum operation of the first angle of rotation, thereby obtaining the first set of first type quantum operations. This can be proved as follows: For an initial angle ∈R, the initial angle can be written as for an integer k and a first rotation angle so: Thus, each quantum operation of the first type can be transformed by the scaled 90-degree angle as a function of quantum operations of the second type.According to an example, the first set of quantum operations of the first type is an ordered set of quantum operations. The first set of quantum operations of the first type defines transformations to be applied to the quantum system defined by the set of qubits.According to one example, the first rotation angle is in a range of That is, each j-th first rotation angle θ j is in the range of This constraint may be advantageous because the Taylor approximation may deteriorate significantly when the first rotation angle θ j is far from 0.The rank of Taylor evolution is less than a predefined maximum rank L, where L ∈ Z ≥0. That is, the Taylor polynomial can be calculated up to rank L, where the order of the derivative |α| is less than or equal to rank L, |α|≤L.According to one example, the scale factor associated with each first angle of rotation has a maximum value corresponding to the order according to which the first angle of rotation is differentiated in the derivative.For example, the application of the parameter shift rule as described in the reference A. Mari et al. arXiv:2008.06517, Phys. Rev. A 103, 012405 (2021), on the derivative Dαf(0), the derivative is developed as follows: wherein and α j may be less than or equal to the order |α|. This indicates that the evolution of the derivative comprises a set of terms. The set of terms may be the summands of the sum. The number of terms of the set of terms can be defined by the two sums. The number of terms of the set of terms may be, for example, Nα=2|α|. Each summand of the sum includes the function f having a respective set of second rotation angles respectively associated with the first rotation angles as arguments. For example, each summand may be defined as follows: The function f has m arguments in each summands: and which are respectively second rotation angles, e.g. the second rotation angle may be: Each second rotation angle of the set of second rotation angles is a 90-degree angle scaled by a scale factor representing the first rotation angle associated with the second rotation angle. The scale factor may be, for example, (i 1,1+... + i1, α1for the first rotation angle θ 1. The scale factor is equal to zero when the first rotation angle is not part of the derivative Dαf( 0). For example, if the first rotation angle θ 1 is not part of the derivative Dαf( 0), this means that the order of α 1 is zero, α 1= 0, which in turn means that the corresponding argument is zero. Generally, when α j= 0, the associated summation sign i j,1... ij,αjis replaced by 1 (empty product) and i j,1+... + ij,αjis replaced by 0 (empty sum). However, if the first rotation angle is part of the derivative Dαf(0), the corresponding scale factor is an integer associated with the order by which the first rotation angle in the derivative is differentiated. For example, if the first rotation angle θ 1 is part of the derivative Dαf( 0), the corresponding scale factor i may be 1,1+...+ i1, α1, the sum over the indices depending on the values of α 1 i.e. on the order in which the first rotation angle θ 1 is differentiated in the derivative. The set of second rotation angles and may be parameters of a second set of quantum operations of the first type, e.g. the second rotation angle may be the angle of a one-qubit rotation about the z-axis, e.g. the rotation angle of an R Z- gate. The second set of quantum operations of the first type may be as follows: Each second set of quantum operations of the first type may comprise m quantum operations of the first type. Thus, the parameter shift rule of each derivative Dαf( 0) may result in an Nα(2|α|) anzhal of second sets of quantum operations of the first type.Therefore, the parameter shift rule may allow terms of the shape to be obtained. Since the scale factors are integers (k 1,..., k m) ∈ Z m, the function can be efficiently evaluated on a classical computer by replacing the quantum operation of the first type as follows, where S kj is the quantum operation of the second type S, whose exponent is the integer k j.In each second set of quantum operations of the first type, each quantum operation of the first type at the second rotation angle may be replaced with a product of zero or more quantum operations of the second type based on the scale factor of the second rotation angle. The product of zero or more quantum operations of the second type may be S lj. Since the global phase cannot affect the measurement result, the quantum operation of the first type can be replaced with the product of S operations S lj in the quantum circuit. For example, if l j= 0, then S lj= I, which corresponds to a "omission" of the gate from the quantum circuit. Since l j ≤ 3 for all j, the total number of gates may increase by at most 2m, e.g. the total number of gates may increase by at most the factor 3. In addition, the resulting quantum circuit does not include gates other than S, H, and CNOT gates, and thus is a Clipfford circuit. Thus, according to one example, the first set of quantum operations of the first type is configured to be applied to a quantum system defined by a set of qubits, wherein the performing step comprises: simulating the quantum system on a digital computer and executing the sets of quantum operations of the second type on the digital computer. The second sets of quantum operations of the second type can be simulated efficiently on the classical computer.According to one example, the scale factor associated with each first angle of rotation has a maximum value corresponding to the order according to which the first angle of rotation is differentiated in the derivative. According to the above notation, the first rotation angle θ 1, is the scale factor, i 1,1+... + i1, α1is smaller than or equal to α 1 e.g., i 1,1+ ···+i1, α1≤ α 1.According to an example, the scale factor may be defined as the scale factor modulo a fixed number F that is a positive integer multiple of 4, the fixed number defined such that the number of quantum operations of the second type is less than a threshold. For example, if the scale factor is k, it may be defined as k=kmod F, e.g., F=4, according to this example.For example, considering the following assumption: for j ∈ {1,..., m}, an integer l j may be defined as follows: l j:= k j mod 4 ∈ {0,1,2,3}, then except for a global phase, the following equality is provided: That is, the quantum operation of the first type may be defined as the product of the quantum operations of the second type, S lj where l j:= k j mod 4 ∈ {0,1,2,3}, that is, S lj the quantum operation of the second type with the exponent l may be j.FIG. 1 is a flow diagram of a method for performing an initial quantum calculation defined by at least a first set of one or more quantum operations of the first type. The first set of one or more quantum operations of the first type may each have one or more first rotation angles as parameters. Each quantum operation of the first type may be a one-qubit rotation about a z-axis in the Bloch sphere by the respective first rotation angle.The initial quantum calculation can be represented in step 101 by a function f with the first rotation angles as arguments. In step 103, a Taylor expansion of the function f about zero may be performed. This may result in derivatives of the function f, wherein each derivative may be a partial derivative.For each derivation of the resulting derivatives of the function f, step 105 may be performed. In step 105, a parameter shift rule may be used to express the current derivative as a linear combination of a set of terms. Each term of the set of terms may be defined as follows. The term includes the function f having, as arguments, a respective set of second rotation angles respectively associated with the first rotation angles. Each second rotation angle of the set of second rotation angles is a 90-degree angle scaled by a scale factor representing the first rotation angle associated with the second rotation angle. The scale factor is equal to zero if the first rotation angle is not part of the derivative, otherwise the scale factor is an integer number associated with the order by which the first rotation angle is differentiated in the derivative. Each of the set of second rotation angles is a parameter of a second set of quantum operations of the first type.The derivative may be referred to as Dαf( 0), where α is the order of the derivative that is less than the order L of the Taylor evolution, L ∈ Z ≥0 and |α|≤L. For example, the number of derivatives obtained from the Taylor evolution may be nd (D 1 f( 0),..., D nd f( 0)), the application of the parameter shift rule to each derivative of the nd derivatives may result in a set of 2|α| terms. That is, the execution of step 105 on all derivatives may result in a total number of terms. Each term of the set of terms totmay be the function f with a set of m arguments / second rotation angles, where m may be the number of the first rotation angles. Thus, the execution of step 105 may result in a number tot of sets of second rotation angles, each set of second rotation angles comprising m second rotation angles. Each set of these dead sets of second rotation angles may be parameters of a respective second set of quantum operations of the first type. That is, a total of a number tot of second sets of quantum operations of the first type may be provided.In each second set of quantum operations of the first type of the second sets of quantum operations of the first type, in step 107, each quantum operation of the first type at the second rotation angle may be replaced by a quantum operation of the second type whose exponent is the scale factor of the second rotation angle. The quantum operation of the second type is a 90 degree rotation about the z-axis. Execution of step 107 may result in sets of quantum operations of the second type, e.g., a number tot of sets of quantum operations of the second type.The quantum calculation may be re-defined in step 109 by the sets of quantum operations of the second type instead of the first set of quantum operations of the first type. This may be advantageous because the sets of quantum operations of the second type can be efficiently evaluated on a classical computer. The redefinition of the quantum calculation may be performed for each set of quantum operations of the second type by replacing the first set of quantum operations of the first type with the set of quantum operations of the second type, respectively, in the quantum calculation. This may result in a number of individual redefinitions of quantum operations. The individual newly defined quantum operations may be combined, for example, by a weighted sum of the individual newly defined quantum operations using the respective weights, in order to obtain the (overall) newly defined quantum calculation.The newly defined quantum calculation can be simulated on a digital computer in step 111. The newly defined quantum calculation may be performed on the digital computer in step 111.FIG. 2 is a block diagram of an example computer system for implementing at least a portion of the present method according to an example of the present subject matter. The computer system of FIG. 2 may provide an example implementation of the digital computer in accordance with the present subject matter.The components of computer system 602 may include, but are not limited to, one or more processors or processor units 603, a memory system 611, a memory unit 605, and a bus 607 that connects various system components, including memory unit 605, to processor 603. The storage system 611 may include, for example, a hard disk drive (HDD). The storage unit 605 may include computer readable media in the form of volatile memory, such as random access memory (RAM) and / or cache memory.Computer system 602 may also communicate with one or more external devices such as a keyboard, a pointing device, a display 613, etc.; with one or more devices that allow a user to interact with computer system 602; and / or with any devices (e.g., network card, modem, etc.) that allow computer system 602 to communicate with one or more other computing devices. Such communication may be via I / O interface(s) 619. Nevertheless, the computer system 602 may communicate with one or more networks such as a local area network (LAN), a general wide area network (WAN), and / or a public network (e.g., the Internet) via a network adapter 609. As shown in the figure, network adapter 609 communicates with the other components of client system 602 via bus 607.The storage unit 605 is configured to store applications executable on the processor 603. The storage unit 605 may include, for example, an operating system as well as one or more application programs. The application programs include instructions that when executed enable the method described in FIG. 1 to be performed.An example implementation of the method of FIG. 1 may be described, for example, as follows.An input can be provided to the quantum calculation. The input may include a number of qubits, n ∈ Z ≥1. The input may further comprise an ordered list of quantum gates. The gates may be applied in the quantum circuit in the order of their occurrence in the list, e.g. in the reverse order with respect to matrix multiplication. Each entry of the list contains the gate name, namely H, S, CNOT, R Z and the gate information. For example, if the gate name is H or S, the index of the destination qubit may be provided as part of the gate information. If the gate name is CNOT, the gate information may include the indices of the control qubit and the destination qubit. If the gate name R is Z, the gate information may include the index of the target qubit and an angle in the range. The input may further comprise an index l ∈ {1,..., n} of the qubit to be measured on a standard basis. The input may further comprise L ∈ Z ≥0. the order for Taylor evolution, e.g. the Taylor polynomial may be calculated to rank L.The quantum calculation may comprise a measurement, wherein the observable corresponding to the qubit to be measured in the standard basis may be. Assume that the initial state is |0> ⊗n and that the list of gates from the input defines a unitary U ∈ C 2n×2n. The state of the quantum circuit immediately before the measurement is then | ψ >:=U|0 > ⊗n. The expected value of the observables of the measurement can be determined, for example, as follows: < ψ|M| ψ>.The observable M has two eigenvalues, namely -1 and 1. However, in the measurement in QISKIT, the measurement result may be either 0 (corresponding to the eigenvalue 1) or 1 (corresponding to the eigenvalue -1). Note that the measurement result returned from QISKIT may correspond to the state in which the qubit l is collapsed as a result of the measurement. That is, the eigenspace for 1 has a standard orthonormal basis, namely, B 0= {| i 1... i ℓ-1 〉 |0 〉 |i ℓ+1... i n ≅ : i 1,... i ℓ-1, i ℓ+1,..., i n ∈ {0,1}}; the eigenspace for -1 has an orthonormal standard basis, namely B 1= {| i 1... i ℓ-1 ≅ ⊗ |1 〉 |i ℓ+1... i n ≅ : i 1,... i ℓ-1, i ℓ+1,..., i n ∈ {0,1}}. The disjunctive union of these two sets is the standard orthonormal basis for (C 2)⊗n. As a result of the measurement given by M, the state is projected onto the eigenspace corresponding to the measured eigen value of M (and normalized). The measurement result returned by QISKIT is thus i ∈{0,1}, and this only when the state after the measurement is contained in the C vector space spanned by B i.After the input and the measurement are defined, the quantum calculation may be performed according to the following approach.Let m ∈ Z ≥0 the number of R z- gates and the associated angles in the order of their appearance in the list of quantum gates. The objective may be to evaluate f(θ 1,..., θ m). For this purpose, the Taylor evolution of f around (0,...,0) can be calculated and subsequently (θ 1,..., θ m) can be used. All subfunctions (partials) of f can be evaluated by simulating Clipfford circuits, which can be carried out efficiently with a classic computer. The following describes how subfunctions of f can be evaluated by the simulation of Clipfford circuits.In one example, if (k 1,..., k m) ∈ Z m then it is possible to evaluate efficiently on a classical computer. Indeed, for j ∈ {1,...,m} let l j:= k j mod 4 ∈ {0,1,2,3}, apart from a global phase, the following equality R Z can be provided. Since the global phase cannot affect the measurement result, S lj in the quantum circuit can be substituted. For example, if l j= 0, then S lj= I, which may correspond to the "omission" of the gate from the quantum circuit. Since l j ≤ 3 for all j, the total number of gates may increase by at most 2m, e.g., the total number of gates may increase by a factor of at most 3. In addition, the resulting quantum circuit must not contain any gates other than S, H, and CNOT gates, and is thus a Clichford circuit that can be simulated efficiently on a classical computer. Using this example, the sub-functions of f can be calculated as follows.For a multi-index, the derivative of f may be defined as follows:For all, the derivative Dαf(0) can be calculated. In particular, the following equation may be a result of the generalized parameter shift rule, e.g., as described in A. Mari et al. arXiv:2008.06517, Phys. Rev. A 103, 012405 (2021): (cf. paragraph 0031 above) This equation may be referred to as Lemma A. If α j= 0, the associated summation character can be skipped, i j,1... ij,αjis replaced by 1 (empty product) and i j,1+... + ij,αjis replaced by 0 (empty sum).The sum of the derivative Dαf(0) has 2|α|Summanden but the f term does not necessarily have to be distinguishable between different summands. In one example, combinatorics may be applied and multiple summands may be merged into a single summands having a "count factor.". In another example, as described in FIG. 3, a list of all integer tuple (k 1,..., k m) for which has already been evaluated can be maintained. Then, for many summands in the above sum, a previously calculated value of f can be reused. As such, a list of tuple with entries in {0,1,2,3} can be stored, which further reduces the number of evaluations required for f. This list can be used in common for all orders |α|=0,..., L of the sub-functions; for example, if α 1,..., α m are all even, then f(0,..., 0) can appear in the above sum no matter how large |α| is. The leading factor i 1,1... i1, α1... i m,1... im,αmis either 1 or -1. It is 1 (or -1) when the number of i j,k which are negative, is even (or odd).After all these subfunctions have been calculated, the function f can be approximated as follows:FIG. 3 is pseudo code for simulating the redefining of quantum computation according to an example of the present subject matter. The input of the pseudo code may include a number of qubits, n ∈ Z ≥1. The input may further comprise an ordered list of quantum gates. The gates may be applied in the quantum circuit in the order of their occurrence in the list, e.g. in the reverse order with respect to matrix multiplication. Each entry of the list contains the gate name, namely H, S, CNOT, R Z and the gate information. For example, if the gate name is H or S, the index of the destination qubit may be provided as part of the gate information. If the gate name is CNOT, the gate information may include the indices of the control qubit and the destination qubit. If the gate name R is Z, the gate information may include the index of the target qubit and an angle in the range. The input may further comprise an index l ∈ {1,..., n} of the qubit to be measured on a standard basis. The input may further comprise L ∈ Z ≥0. the order for Taylor evolution, e.g. the Taylor polynomial may be calculated to rank L.Initialization may be performed as follows. A collection "computed_tuples" of tuple (l 1,..., l m) ∈{0, 1, 2, 3} m, for which has already been evaluated, can be created. computed_tuples can be initialized as an empty collection. This may provide for each derivative and for each term of the set of terms a tuple of scale factors representing the second angles of rotation of the term, respectively. The number m ∈ Z ≥0 of R Z- gates can be determined from the input. The respective angles may be in the order of their occurrence in the input provided list of quantum gates. In pseudo code, f is as defined above. When m=0, L is to be replaced by 0. (If m=0, then f is constant, so that only the constant term from Taylor evolution is required.)As shown in pseudo code, the f term 301 of each summands may include second rotation angles of respective quantum operations of the first type, which define the initial quantum operation with the second rotation angles as parameters defining the initial quantum operation with the second rotation angles as parameters. This initial quantum calculation can be re-defined in the code line labeled 301 by replacing these quantum operations of the first type as follows:Pseudo code represents the major part of the algorithm and the output. The algorithm enables the newly defined quantum calculation to be performed on a classical computer. For this purpose, the derivatives can be processed term by term sequentially. At each iteration, it is determined whether the tuple of the current term has already been processed. If the tuple of the current term has not yet been processed, the individual newly defined quantum calculation of the current term can be evaluated or simulated and the result of the evaluation can be assigned to the tuple. If the tuple of the current term has already been processed, the evaluation result associated with the stored tuple can be used as a result of the individual newly defined quantum calculation. The result of the individual newly defined quantum calculation can be weighted with a corresponding weighting factor. This is indicated by the line of code 303. The weighted results may be combined as indicated in code lines 303, 304 and 305.Returning the function in addition to "approximated_value" may be useful because it may allow evaluation / approximation of the expected value of observables for rotation angles other than (θ 1,..., θ m) without having to recompute the Taylor coefficients; this may be particularly useful, for example, for VQE or QML because it is common in these applications to iteratively update the rotation angles of Pauli rotation gates.As one skilled in the art will appreciate, aspects of the present invention may be embodied as an apparatus, method, computer program, or computer program product. Accordingly, aspects of the present invention may take the form of a hardware-only embodiment, a software-only embodiment (including firmware, resident software, microcode, etc.), or a combination of software and hardware aspects, which may be referred to generally herein as a "circuit," "module," or "system.". Moreover, aspects of the present invention may take the form of a computer program product embodied in one or more computer readable medium(s) having computer executable code embodied thereon. A computer program comprises the computer executable code or "program instructions".The term "computer system" refers to data processing hardware and includes all types of apparatus, apparatus, and machines for processing data, including, for example, a programmable processor, a computer, or multiple processors or computers. The device may also be or further comprise a special purpose logic circuit, e.g. a central processing unit (CPU), an FPGA (field programmable gate array) or an ASIC (application specific integrated circuit). In some implementations, the computing device and / or the dedicated logic circuitry may be hardware-based and / or software-based. The apparatus may optionally include code that provides an execution environment for computer programs, e.g., code that represents processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of these systems. The present disclosure contemplates the use of computing devices with or without conventional operating systems, e.g., LINUX, UNIX, WINDOWS, MAC OS, ANDROID, IOS, or any other suitable conventional operating system.Any combination of one or more computer readable medium(s) may be used. The computer readable medium may be a computer readable storage medium. A "computer readable storage medium" as used herein includes any tangible storage medium that can store instructions executable by a processor of a computing device. The computer readable storage medium may be referred to as a computer readable non-transitory storage medium. The computer readable storage medium may also be referred to as a tangible computer readable medium. In some embodiments, a computer readable storage medium may also be capable of storing data accessible by the processor of the computing device."Computer Memory" (computer memory) or "Memory" (memory) is an example of a computer readable storage medium. Computer memory is any memory that can be accessed directly by a processor. Another example of a computer readable storage medium is computer storage ("computer storage") or storage ("memory"). Computer memory is any non-transitory computer readable storage medium. In some embodiments, computer memory may also be computer memory or vice versa.A "processor" as used herein includes an electronic component capable of executing a program or machine-executable instruction or code. References to the computing system comprising "a processor" should be understood to possibly include more than one processor or processing core. The processor may be, for example, a multi-core processor. A processor may also refer to a collection of processors within a single computer system or distributed among multiple computer systems. The term "computing device" should also be interpreted as referring, where applicable, to a collection or network of computing devices, each of which includes one or more processors. Executable computer code may be executed by multiple processors located in one and the same computing device or even distributed across multiple computing devices.Computer-executable code may include machine-executable instructions or a program that causes a processor to execute an aspect of the present invention. Computer-executable code for performing operations for aspects of the present invention may be written and compiled into machine-executable instructions in any combination of one or more programming languages, including an object-oriented programming language such as Java, Smalltalk, C++ or the like and conventional procedural programming languages, such as the "C" programming language or similar programming languages. In some cases, the computer executable code may be in the form of a high level language or precompilation form and may be used in conjunction with an interpreter that generates the machine executable instructions in operation.Generally, the program instructions may be executed on one processor or on multiple processors. In the case of multiple processors, these may be distributed among multiple different units. Each processor could execute a portion of the instructions intended for that unit. Thus, when referring to a system or process involving multiple entities, the computer program or program instructions should be understood to be executable by a processor associated with or connected to the respective entity.Although the invention has been illustrated and described in detail in the drawings and the foregoing description, such illustrations and descriptions are to be considered as illustrative or exemplary and not restrictive; the invention is not limited to the disclosed examples.References included in the specificationThis list of documents cited by the applicant has been produced in an automated manner and is only included for the better information of the reader. The list is not part of the German patent application or utility model application. The DPMA does not take any adhesion for any faults or omissions.Cited Non-Patent LiteratureA. Mari et al. arXiv:2008.06517, Phys. Rev. A 103, 012405 (2021 [0021, 0029, 0054]

Claims

A method for performing an initial quantum calculation, the initial quantum calculation being defined by at least a first set of one or more first type quantum operations, the first set of one or more first type quantum operations each having as parameters one or more first rotation angles, each first type quantum operation being a one-qubit rotation about a z-axis in a Bloch sphere about the respective first rotation angle, the method comprising: representing the initial quantum calculation by a function f having the first rotation angles as arguments; performing a Taylor evolution of the function f about zero, resulting in derivatives of the function f being referred to as fundamental derivatives; for each specific derivative of the basic derivatives: using a parameter shift rule to express the specific derivative as a linear combination of a set of terms, for each term of the set of terms: wherein the term comprising the function f has as arguments a respective set of second rotation angles associated with the first rotation angles, respectively, wherein each second rotation angle of the set of second rotation angles is a 90-degree angle scaled by a scale factor representing the first rotation angle associated with the second rotation angle, wherein the scale factor is an integer equal to zero or different from zero depending on the order in which the associated first rotation angle is differentiated in the specific derivative; wherein each of the set of second rotation angles is a parameter of a second set of quantum operations of the first type; in each second set of quantum operations of the first type, replacing each quantum operation of the first type at the second rotation angle with a quantum operation of the second type having, as an exponent, the scale factor of the second rotation angle, wherein the quantum operation of the second type is a 90-degree rotation about the z-axis, resulting in a set of quantum operations of the second type; redefining the initial quantum calculation using the sets of quantum operations of the second type instead of the first set of quantum operations of the first type, resulting in a redefining quantum calculation; simulating the redefining quantum calculation on a digital computer.The method of claim 1, wherein redefining for each derivative of the function f comprises: for each set of quantum operations of the second type of the derivative, replacing the first set of one or more quantum operations of the first type with the set of quantum operations of the second type, resulting in an individual redefinition of the quantum calculation; wherein simulating the redefining quantum calculation comprises evaluating Taylor evolution using a combination of the individual redefining quantum calculations.The method of claim 1 or 2, further comprising: providing an initial set of quantum operations including a subset of quantum operations of the first type and a subset of quantum operations representing one-qubit rotations about the x-axis and the y-axis in the Bloch sphere; transforming each quantum operation about the x-axis or the y-axis as a function of a quantum operation of a first type with the same rotation angle; thereby obtaining the first set of quantum operations of the first type as the subset of quantum operations of the first type and the transformed subset.The method of claims 1, 2, or 3, further comprising: providing an initial set of first type quantum operations having as parameters respective initial rotation angles; expressing each initial angle as the sum of a first rotation angle and a 90 degree angle scaled by an integer, wherein the first type quantum operation of the initial angle is the product of the first type quantum operation of the scaled 90 degree angle and the first type quantum operation of the first rotation angle, thereby obtaining the first set of first type quantum operations; transforming each first type quantum operation by the scaled 90 degree angle as a function of the second type quantum operations.The method according to any one of the preceding claims, wherein the first rotation angle is in an interval ( - π 4, π 4 ].The method of any preceding claim, wherein the rank of Taylor evolution is less than a predefined maximum rank.The method of any preceding claim, wherein the scale factor associated with each first angle of rotation has a maximum value corresponding to the order by which the first angle of rotation is differentiated in the derivative.The method of any preceding claim, comprising: defining the scale factor as the scale factor modulo a fixed number that is a positive integer multiple of four, wherein the fixed number is defined such that the number of quantum operations of the second type is less than a threshold.The method of any preceding claim, wherein the first set of quantum operations of the first type is an ordered set of quantum operations.The method of any preceding claim, wherein the first set of quantum operations of the first type defines transformations to be applied to a quantum system defined by a set of qubits.The method of any of the preceding claims 2 to 10, wherein simulating the newly defined quantum calculation comprises: providing a tuple of scale factors per derivative and per term of the set of terms of the derivative, each representing the second angles of rotation of the term; sequentially evaluating the derivatives term by term comprising: determining whether the tuple of the current term has already been processed; if the tuple of the current term has not yet been processed, simulating the individual redefinition of the quantum calculation of the current term and assigning the simulation result to the tuple; if the tuple of the current term has already been processed, using the simulation result associated with the stored tuple as a result of the current individual newly defined quantum calculation; weighting the result of the individual newly defined quantum calculation with a respective weighting factor; combining the weighted results.The method of any preceding claim, wherein the quantum operation of the first type is an R Z rotation of an R Z gate, and the quantum operation of the second type is a phase shift S of an S gate.A computer program including machine executable instructions, wherein execution of the machine executable instructions causes a computer system to perform: representing, by a function f having as arguments first rotation angles of a first set of one or more quantum operations of the first type defining the initial quantum calculation, wherein each quantum operation of the first type is a one-qubit rotation about a z-axis in a Bloch sphere by the respective first rotation angle; performing a Taylor evolution of the function f about zero, resulting in basic derivatives of the function f; for each specific derivative of the basic derivatives: using a parameter shift rule to express the specific derivative as a linear combination of a set of terms, for each term of the set of terms: wherein the term comprising the function f has as arguments a respective set of second rotation angles associated with the first rotation angles, respectively, wherein each second rotation angle of the set of second rotation angles is a 90-degree angle scaled by a scale factor representing the first rotation angle associated with the second rotation angle, wherein the scale factor is an integer equal to zero or different from zero depending on the order in which the associated first rotation angle is differentiated in the specific derivative; wherein each of the set of second rotation angles is a parameter of a second set of quantum operations of the first type; in each second set of quantum operations of the first type, replacing each quantum operation of the first type at the second rotation angle with a quantum operation of the second type having, as an exponent, the scale factor of the second rotation angle, wherein the quantum operation of the second type is a 90-degree rotation about the z-axis, resulting in a set of quantum operations of the second type; redefining the initial quantum calculation using the sets of quantum operations of the second type instead of the first set of quantum operations of the first type, resulting in a redefining quantum calculation; simulating the redefining quantum calculation.A computer system for performing an initial quantum calculation, the initial quantum calculation being defined by at least a first set of one or more quantum operations of the first type, the first set of one or more quantum operations of the first type each having as parameters one or more first rotation angles, each quantum operation of the first type being a one-qubit rotation about a z-axis in a Bloch sphere about the respective first rotation angle, the computer system being configured to: represent an initial quantum calculation by a function f having as arguments first rotation angles of a first set of one or more quantum operations of the first type defining the initial quantum calculation, wherein each quantum operation of the first type is a one-qubit rotation about a z-axis in a Bloch sphere about the respective first rotation angle; performing a Taylor development of the function f about zero, resulting in basic derivatives of the function f; for each specific derivative of the basic derivatives: using a parameter shift rule to express the specific derivative as a linear combination of a set of terms, for each term of the set of terms: wherein the term comprising the function f has as arguments a respective set of second rotation angles associated with the first rotation angles, respectively, wherein each second rotation angle of the set of second rotation angles is a 90-degree angle scaled by a scale factor representing the first rotation angle associated with the second rotation angle, wherein the scale factor is an integer equal to zero or different from zero depending on the order in which the associated first rotation angle is differentiated in the specific derivative; wherein each of the set of second rotation angles is a parameter of a second set of quantum operations of the first type; in each second set of quantum operations of the first type, replacing each quantum operation of the first type at the second rotation angle with a quantum operation of the second type having, as an exponent, the scale factor of the second rotation angle, wherein the quantum operation of the second type is a 90-degree rotation about the z-axis, resulting in a set of quantum operations of the second type; redefining the initial quantum calculation using the sets of quantum operations of the second type instead of the first set of quantum operations of the first type, resulting in a redefining quantum calculation; simulating the redefining quantum calculation.

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