Simulation of a quantum computation

Abstract

Disclosed is a method for performing a method for performing a quantum computation, referred to as initial quantum computation, the initial quantum computation 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 having as parameters one or more first rotation angles respectively, each first type quantum operation being a single-qubit rotation around a z- axis in a Bloch sphere by the respective first rotation angle. The method comprises: representing the initial quantum computation by a function f having as arguments the first rotation angles; performing a Taylor expansion of the function f around zero, resulting in foundational derivatives of the function f; for each specific derivative of at least one of the foundational derivatives: using a parameter shift rule for expressing the specific derivative as a linear combination of a set of terms.

Description

1 BUND.223.10EP - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - SIMULATION OF A QUANTUM COMPUTATION - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - FIELD OF THE INVENTION

[0001] The disclosure relates to quantum computing, and particularly to a method for performing a quantum computation. BACKGROUND

[0002] The quantum simulators are based on matrix multiplication. However, since the dimensions of the corresponding matrices grow strongly with the number of qubits, quantum circuits with, e.g., 100 qubits, may not be simulated on the currently available classical computers, since both the demand for main memory and the demand for computing power exceed the currently available capacities. SUMMARY

[0003] Example embodiments provide a method for performing a quantum computation, referred to as initial quantum computation, the initial quantum computation 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 having as parameters one or more first rotation angles respectively, each first type quantum operation being a single-qubit rotation around a z-axis in a Bloch sphere by the respective first rotation angle, the method comprising: representing the initial quantum computation by a function ^ having as arguments the first rotation angles; performing a Taylor expansion of the function ^ around zero, resulting in foundational derivatives of the function ^; for each specific derivative of at least one of the foundational derivatives: using a parameter shift rule for expressing the specific derivative as a linear combination of a set of terms, for each term of the set of terms: the term comprising the function ^ having as arguments a respective set of second rotation angles associated with the first rotation angles respectively, each2 BUND.223.10EP second rotation angle of the set of second rotation angles being a 90-degree angle scaled by a scale factor representing the first rotation angle associated with the second rotation angle, the scale factor being an integer zero or different from zero depending on the order with respect to 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 respectively; in each second set of first type quantum operations, replacing each first type quantum operation at the second rotation angle by a second type quantum operation having as exponent the scale factor of the second rotation angle, the second type quantum operation being a 90-degree rotation around the z-axis, resulting in a set of second type quantum operations; redefining the initial quantum computation by using the sets of second type quantum operations instead of the first set of first type quantum operations, resulting in a redefined quantum computation; simulating the redefined quantum computation on a digital computer.

[0004] Example embodiments provide a computer system for performing an initial quantum computation, the initial quantum computation 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 having as parameters one or more first rotation angles respectively, each first type quantum operation being a single-qubit rotation around a z-axis in a Bloch sphere by the respective first rotation angle, the computer system being configured for: representing the initial quantum computation by a function ^ having as arguments the first rotation angles; performing a Taylor expansion of the function ^ around zero, resulting in foundational derivatives of the function ^; for each specific derivative of at least one of the foundational derivatives: using a parameter shift rule for expressing the specific derivative as a linear combination of a set of terms, for each term of the set of terms: the term comprising the function ^ having as arguments a respective set of second rotation angles associated with the first rotation angles respectively, each second rotation angle of the set of second rotation angles being a 90-degree angle scaled by a scale factor representing the first rotation angle associated with the second rotation angle, the scale factor being an integer zero or different from zero depending on the order with respect to which the associated first rotation angle is differentiated in the specific derivative; wherein3 BUND.223.10EP each of the set of second rotation angles is a parameter of a second set of first type quantum operations respectively; in each second set of first type quantum operations, replacing each first type quantum operation at the second rotation angle by a second type quantum operation having as exponent the scale factor of the second rotation angle, the second type quantum operation being a 90-degree rotation around the z-axis, resulting in a set of second type quantum operations; redefining the initial quantum computation by using the sets of second type quantum operations instead of the first set of first type quantum operations, resulting in a redefined quantum computation; simulating the redefined quantum computation.

[0005] Example embodiments provide a computer program comprising machine executable instructions, wherein execution of the machine executable instructions causes a computer system to: represent an initial quantum computation by a function ^ having as arguments first rotation angles of a first set of one or more first type quantum operations that define the initial quantum computation, each first type quantum operation being a single-qubit rotation around a z-axis in a Bloch sphere by the respective first rotation angle; perform a Taylor expansion of the function ^ around zero, resulting in foundational derivatives of the function ^; for each specific derivative of at least one of the foundational derivatives: use a parameter shift rule for expressing the specific derivative as a linear combination of a set of terms, for each term of the set of terms: the term comprising the function ^ having as arguments a respective set of second rotation angles associated with the first rotation angles respectively, each second rotation angle of the set of second rotation angles being a 90-degree angle scaled by a scale factor representing the first rotation angle associated with the second rotation angle, the scale factor being an integer zero or different from zero depending on the order with respect to 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 respectively; in each second set of first type quantum operations, replace each first type quantum operation at the second rotation angle by a second type quantum operation having as exponent the scale factor of the second rotation angle, the second type quantum operation being a 90-degree4 BUND.223.10EP rotation around the z-axis, resulting in a set of second type quantum operations; redefine the initial quantum computation by using the sets of second type quantum operations instead of the first set of first type quantum operations, resulting in a redefined quantum computation; simulate the redefined quantum computation. The specific derivative may be a derivative of the function ^. The at least one of the foundational derivatives may, for example, be all foundational derivatives of the function ^. BRIEF DESCRIPTION OF THE DRAWINGS

[0006] In the following, examples are described in greater detail making reference to the drawings in which:

[0007] Fig.1 is a flowchart of a method of performing a first set of one or more first type quantum operations in accordance with an example of the present subject matter.

[0008] Fig.2 is a block diagram of an exemplary computer system for implementing at least part of the present method in accordance with an example of the present subject matter.

[0009] Fig.3 is a pseudocode for simulating a redefined quantum computation in accordance with an example of the present subject matter. DETAILED DESCRIPTION

[0010] In 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 illustrative examples that depart from these specific details. In some instances, detailed descriptions of well-known devices and / or methods are omitted so as not to obscure the description with unnecessary detail.

[0011] The present subject matter may enable to simulate quantum computations on classical computers. The classical computer may also be referred to as a digital5 BUND.223.10EP computer. The digital computer may be a machine that can be programmed to carry out sequences of arithmetic or logical operations (computation) automatically. The digital computer may include the hardware such as processor and memory and software such an operating system.

[0012] A quantum system may be defined by a set of qubits. The number of qubits may, for example, be ^ qubits, where ^ ≥ 1. The quantum system may thus be a n- qubit system. The quantum state of the quantum system may be changed by applying one or more quantum operations. This may enable to perform a quantum computation. The quantum computation may refer to the initial quantum computation. The term “initial” is used as a label for naming purpose. The initial quantum computation may be described as an initial set of quantum operations and a measurement. The quantum operation may be a specific class of transformations, mathematically known as unitary transformations. The quantum operation may, for example, be a transformation that may be applied on one or more qubits. The initial set of quantum operations may enable a unitary evolution of the quantum system.

[0013] In one example, the initial quantum computation may comprise a first step of initialization of all qubits in a computational basis state such as the |0> state (denoted |0 >⊗^), a second step of application of the initial set of quantum operations, and a third step of measurement in the computational basis, on one or more of the qubits. Each quantum operation of the initial set of quantum operations may be of a type that belongs to a predefined set of quantum operation types. The predefined set of quantum operations types may, for example, represent a predefined set of gate types respectively. The predefined set of gate types may, for example, be a universal set of gate types respectively. The initial set of quantum operations may be of one or more types of quantum operations. In one example, the initial set of quantum operations may all or in part be first type quantum operations. The first type quantum operation may be a single-qubit rotation around a z-axis in the Bloch sphere, wherein the rotation is by the respective first rotation angle. Hence, starting from the initial state |0 >⊗^of the quantum system, the initial set of quantum operations may define a unitary operator ^(^) ∈ ℂ^^×^^that transforms the initial state so that the state of the quantum system immediately6 BUND.223.10EP before measurement may then be |^(^) >≔ ^(^)|0 >⊗^, where ^ ∈ ℝ^ . The present method may provide the expectation value of the measurement observable, by determining the quantity ^(^)> using the observable ℳ. In one example implementation of the measurement, the measurement may be represented by an observable ℳ which may have two eigenvalues such as −1 and 1 (e.g., ℳthat corresponds to the ℓ-th qubit to be measured in the computational basis, where ^ represents the Pauli-Z gate and ^ the identity operator).

[0014] A quantum circuit may be used to represent the quantum computation. The quantum circuit may, for example, be a computational routine consisting of coherent quantum operations on quantum data, such as qubits. The quantum circuit that represents the quantum computation may comprise a first stage of initialization of all qubits in the |0> state (denoted |0 >⊗^, a second stage of quantum gates, and a third stage of measurements in the computational 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 represent unitary transformations respectively. The quantum gates may comprise gates of the predefined set of gate types. The predefined set of gate types may, for example, include ^ (Hadamard) gate, ^ gate, ^^^^ gate, ^^gate, ^^gate and ^^gate. Thus, the predefined set of quantum gate types may only comprise as non-Clifford gate the rotation ^^gate around z-axis, as the ^^gate and the ^^gate may be defined as function of ^^gate and Clifford gates. Each gate type of the predefined set of gate types may represent a type of quantum operation. The first type quantum operation according to the present subject matter may be the quantum operation represented with the ^^gate. The second type quantum operation according to the present subject matter may be the quantum operation represented with the phase gate, ^ gate (or√^ phase gate). The ^ gate is also known as the Z90 gate, because it represents a 90-degree rotation around the z-axis. The sequential arrangement of quantum gates of the quantum circuit may represent the initial set of quantum operations. The initial set of quantum operations may thus be provided as an ordered list of quantum7 BUND.223.10EP operations. The quantum operation may be implemented with one or more quantum gates.

[0015] However, the quantum circuit may or may not be efficiently simulated with a classical computer depending on the type of quantum operations that it represents. For example, a quantum circuit that consists of only Clifford gates may be efficiently simulated with a classical computer. The present subject matter may make use of this feature for enabling simulation of quantum circuits even when they contain non- Clifford gates of the predefined set of quantum gate types. For that, the present subject matter may identify or select from the initial set of quantum operations the first set of one or more first type quantum operations, wherein the initial set of quantum operations may comprise one or more types of quantum operations including the first type quantum operation. In one example, the first set of one or more first type quantum operations may be a subset of quantum operations of the initial set of quantum operations and the remaining subset of quantum operations may represent Clifford gates. Alternatively, the first set of one or more first type quantum operations may be the initial set of quantum operations respectively. The present subject matter may represent the first set of first type quantum operations with quantum operations which may be represented by Clifford gates and thus may be simulated efficiently on a classical computer.

[0016] For example, the first set of first type quantum operations may comprise a number ^ of first type quantum operations where ^ is higher than or equal to one, ^ ≥ 1. Each first type quantum operation may be a single-qubit rotation around the z-axis in the Bloch sphere by the respective first rotation angle. For example, each first type quantum operation of the first set of first type quantum operations may be a ^^operation that is implemented by the ^^gate. Thus, the first set of first type quantum operations may be associated with a number ^ of first rotation angles respectively. For example, ^^… ^^may be the first rotation angles of the first set of first type quantum operations respectively. The first type quantum operation having first rotation angle ^^may be referred to as ^^(^^), the first type quantum operation having first rotation angle ^^may be referred to as ^^(^^) and so forth. In one example, the unitary operator may be defined as follows: ^(^) = ^^(^^) … ^^(^^) ^^(^^). Alternatively, the unitary operator may be defined as follows: ^(^) =8 BUND.223.10EP ^^^^^^(^^)^^… ^^(^^)^^^^(^^)^^may be unitary operators representing Clifford gates. Alternatively, the unitary operator may be defined as follows: ^(^)= ^^^^^^(^^)^^… ^^(^^)^^^^(^^)^^, where ^^(^^) is ^^(^^) or ^^(^^) or ^^(^^) and ^ ∈ {1, … ,Providing different types of the initial set of quantum operations may enable a wider application of the present subject matter.

[0017] The initial quantum computation may be represented by the function ^(^^, …, ^^) which has as arguments the first rotation angles ^^… ^^respectively. This may enable to use the parameters of the first set of first type quantum operations as variables of the function ^. Based on the above predefined set of gate types, any (potential) other operation of the initial set of quantum operations which has a type different from the first type may have fixed parameters / values e.g., the quantum operation represented by the ^ gate has fixed values. That is, in case the first set of first type quantum operations 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 ^(^^, …, ^^) may, for example, represent the output of the quantum circuit (e.g., the expectation of the observable).

[0018] The Taylor expansion of the function…, ^^) around zero may result in derivatives of the function…, ^^). The Taylor expansion of the function ^(^^, …, ^^) around zero may result in foundational derivatives of the function ^(^^, …, ^^). Each derivative may be a partial or a mixed derivative. The derivative ^ ^|^|^(^) may, for example, be defined as follows: ^ ^(0) where|^|= ^^+⋯ + ^^indicates the order of the derivative and ^^is the order with respect to which the ^-th first rotation angle ^^is differentiated in the derivative, where ^ is an integer that varies between 1 and m, ^ ∈ {1, … ,The Taylor expansion of the function ^ may result in the following sum:. The expansion of the function ^ may result in: ^(^^, …, ^^) ≈^ is the order of the Taylor and ^ may be a multiindex.9 BUND.223.10EP

[0019] The parameter shift rule may be used to express or define each derivative of the function…, ^^). For example, the application of the parameter shift rule on the derivative ^^^(0) may result in an expansion of the derivative as follows:may be smaller than or equal to the order |^|, ^^^is a leading factor e.g., the leading factor ^^^may be equal to 1 or −1 and ^^is the total number of terms e.g., ^^= 2|^|. 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 expansion of the derivative comprises a set of terms. The parameter shift rule may enable to express the partial derivative of the function ^ as a linear combination of the set of terms, wherein each term may comprise the function ^ whith shifted arguments, which may enable to use 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, for example, be ^^= 2|^|. Each summand of the sum comprises the function ^ having as arguments a respective set of second rotation angles associated with the first rotation angles respectively 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 ^ in each summand has ^ arguments: ^^^^^ , …, and ^^^^which are second rotation angles respectively the ^-th second rotation angle may be:Each second rotation angle of ^ 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, for example,for the ^-th first rotation angle ^. The factors are integers,∈ ℤ^. The scale factoris equal zero if the ^- th first rotation angle ^^is not part of the derivative ^^^(0). E.g., if the first rotation angle ^^is not part of the derivative ^^^(0) this means that the order ^^is zero, ^ ^ ^ = 0, which in turn^^means that the corresponding argument ^^^ is zero. However, if the first rotation angle is part of the derivative ^^^(0), the corresponding scale factor is an integer number associated with the order with respect to which the first rotation angle is differentiated in the derivative. For10 BUND.223.10EP example, if the first rotation angle is part of the derivative ^^^(0), the corresponding scale factor may be ^ ^^ ^^^^ . The set of second rotation angles ^^^ , …, and ^^^^^ ^ may be parameters of a second set of first type quantum operations respectively e.g., the second rotation angle ^^^^^ ^ may be the angle of a single-qubit rotation operation around z-axis e.g., it may be the rotation angle of a ^^gate. The second set of first type quantum operations may be defined respectively as follows:The second set of first type quantum operations may comprise at most ^ first type quantum operations. Thus, the parameter shift rule of each derivative ^^^(0) may result in a number ^^(= 2|^|) of second sets of first type quantum operations e.g.,Since thefactors are integers,…, ^ ^ ^ ) ∈ ℤ^, the function ^(^^^^^ ^ , …,of the ^- summand of the derivative ^^^(0) may be evaluated efficiently on a classical computer.

[0020] Thus, in each second set of first type quantum operations, each first type quantum operation at the second rotation angle may be replaced by a product of second type quantum operations based on the scale factor of the second rotation angle. For example, each first type quantum operation at the second rotation angle may be replaced by a second type quantum operation having as exponent the scale factor, where the exponent refers to the number of times the second type quantum operation is multiplied by itself. Since the global phase may not affect the measurement result, the first type operation ^ ^^^^ ^^ ^^ ^ may be replaced by the ^^product of operations ^ , e.g., if= 2, then the first type operation ^^^^ ^ ^ may be replaced by the product ^ × ^ of ^ operations ^ = ^ × ^. If= 0, ^^^^^^^ is the identity operation, ^^^ = ^, which corresponds to dropping the gate from the quantum circuit. Thus, for each ^, where |^| ≤ ^, the second sets of first type quantum operations11 BUND.223.10EP be replaced by the^^^^^, … , ^^^^ …^^^ , … , ^^^^^^^ ^^ ^^^^^ respectively. ^^^_^ ^^^_^^

[0021] The initial quantum computation may be redefined by replacing the first set of first type quantum operations by each set of second type quantum operations respectively. For example, for each ^, where |^| ≤ ^, each set of second type quantum operations of the 2|^|sets of second type quantum operations: may replace the firstset of first type quantum operations in the initial quantum computation. This may result in a number 2|^|of individual redefined quantum computations.

[0022] The (overall) redefined quantum computation may be obtained as a combination of the individual redefined quantum computations. The redefined quantum computation may thus be simulated on a digital computer.

[0023] The initial and redefined quantum computations may, for example, be defined using the following notations. The measurement observable may be definedall (^^, … , ^ ) ∈{^, ^, ^, ^}^. The initial quantum computation may thus be represented as: ^(^) = ∑ ^ ^^^ ^^< ^(^)|(^^^ … ^^^)|^(^) >. The redefined quantum computation may be represented as follows: ,, where ∈ {^, ^, ^, ^}^^is the Pauli operator and ^ = ^^^^ ^ , … ,. In case parameter shift rule as described in the paper A. Mari et al. arXiv:2008.06517, Phys.12 BUND.223.10EP Rev. A 103, 012405 (2021) is used,. The individual redefined quantum operation may be represented by the term. Thus, the simulation of the redefined quantum computation ^^(^) may be performed by simulating the individual quantum computations and combining them e.g., as shown with the equation of ^^^(0) and ^^(^). This may result in redefining the quantum circuit (in a loop or in iterative manner) to represent each individual redefined quantum computation and then execute the quantum circuit. For example, a quantum simulator may be used to simulate the redefined quantum computation. The quantum simulator may be a software program that when executed on a classical computer may run the quantum circuit as if it were being run on a quantum computer.

[0024] The redefinition of the quantum computation may further be improved by reducing the number of second type quantum operations according to the following example. The first type quantum operation at the second rotation angle, ^^^ ^^^^^^ ^, may be redefined by applying the modulo operation on the scale factorwith a fixed number ^ which is a positive integer multiple of four,=In addition, the resulting first type quantum operation ^ ^^^ mod ^) ^may be defined up to a global phase as ^^^ ^), where ^ (^^^ mod ^) ∈{0,1,2, … , ^ − 1}, that is,modIn one example, the fixed number ^ may be four, ^ = 4. This may further reduce the number of the second type quantum operations. This example may be based on a rotational symmetry of a given order e.g., ^, according to which the quantum system may be invariant under operations at certain fixed angles.13 BUND.223.10EP

[0025] According to one example, in case the initial set of quantum operations includes a subset of first type quantum operations and a subset of quantum operations representing single-qubit rotations around x-axis and y-axis in the Bloch sphere, each quantum operation around x-axis or y-axis may be transformed as a function of a first type quantum operation having the same rotation angle; thereby obtaining the first set of first type quantum operations as the subset of first type quantum operations and the transformed subset. For example, the unitary operator may be definedor ^^(^^) and ^For that, the following equalities: ^^(^)= ^^^(^)^ and ^^(^)= ^^^^(^)^^^, may be used to express the unitary operator as function of first type quantum operations and quantum operations representing Clifford gates.

[0026] According to one example, an initial set of first type quantum operations having as parameters initial rotation angles respectively may be part of the initial set of quantum operations. Each initial angle may be expressed 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. This may be proved as follows: for an initial angle ^ ∈ ℝ, the initial angle ^ may be ^ ^ written as ^ =+ ^^for some integer ^ and some first rotation angle ^^∈ (− ^ , ^], then, ^^∈ {^, ^, ^^, ^^}. Thus, each first type quantum operation around the scaled 90-degree angle may be transformed as function of second type quantum operations.

[0027] According to one example, the first set of first type quantum operations are an ordered set of quantum operations. The first set of first type quantum operations define transformations to be applied on the quantum system defined by the set of qubits. ^ ^

[0028] According to one example, the first rotation angle is in a range of (− ^ , ^]. ^ ^ That is, each ^-th first rotation angle ^^is within the range (− ^ , ^], ^^.14 BUND.223.10EP This restriction may be advantageous because the Taylor approximation may get significantly worse when the first rotation angle ^^is far from 0.

[0029] The degree of Taylor expansion is smaller than a predefined maximum degree ^, where ^ ∈ ℤ^^. That is, the Taylor polynomial may be computed up to degree ^, where the order of the derivative |^| is smaller than or equal to the degree ^, |^| ≤ ^.

[0030] According to one example, the scale factor associated with each first rotation angle has a maximum value which is the order with respect to which the first rotation angle is differentiated in the derivative.

[0031] 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 ^^^(0) may result in an expansion of the derivative as follows:than or equal to the order |^|. This indicates that the expansion 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 may be defined by the two sums. The number of terms of the set of terms may for example, be ^^= 2|^|. Each summand of the sum comprises the function ^ having as arguments a respective set of second rotation angles associated with the first rotation angles respectively. For example, each summand may be defined as follows:The function ^ in each summand has ^ arguments: ^^^^,^…, andwhich are second rotation angles respectively 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. The15 BUND.223.10EP scale factor may, for example,for the first rotation angleThe scale factor is equal zero if the first rotation angle is not part of the derivative ^^^(0). If the first rotation angle e.g.,is not part of the derivative ^^^(0) this means that the order ^^is zero, ^^= 0, which in turn means that the corresponding argument ^^^^,^+ ⋯is zero. In general, if ^^= 0, the corresponding summation sign is ^^,^…is replaced by 1 (empty product) and ^^,^+ ⋯ + ^^,^^is replaced by 0 (empty sum). However, if the first rotation angle is part of the derivative ^^^(0), the corresponding scale factor is an integer number associated with the order with respect to which the first rotation angle is differentiated in the derivative. For example, if the first rotation angleis part of the derivative ^^^(0) the corresponding scale factor may be ^^^,^+ ⋯ + ^^,^^^, where the sum over the indices depends on the values of ^^which is the order with respect to which the first rotation angleis differentiated in the derivative. The set of second rotation anglesmay be parameters of a second set of first type quantum operations respectively e.g., the second rotation angle ^^^^,^+ ⋯may be the angle of a single-qubit rotation operation around z-axis e.g., it may be the rotation angle of a ^^gate. The second set of first type quantum operations may be respectively as follows:d ^^^^ ^ … an ^^,^+ ⋯ + ^^,^^^ ^ ^. Each second set of first type quantum operations may comprise ^ first type quantum operations. Thus, the parameter shift rule of each derivative ^^^(0) may result in a number ^^(2|^|) of second sets of first types of quantum operations.

[0032] Hence, the parameter shift rule may enable to obtain terms of the form . Since the scale factors are integers, (^^^, …, ^^) ∈ ℤ , the ^ ^ ^(^^^ , …, ^^^) may be evaluated efficiently on a classical computer, by replacing ^ ^ the first type quantum operation ^^^^^^^ ^^^ ^ as follows ^^^^^^ ^ = ^ , where ^ is the second type quantum operation ^ having as exponent the integer ^^.16 BUND.223.10EP

[0033] In each second set of first type quantum operations, each first type quantum operation at the second rotation angle may be replaced by a product of zero or more second type quantum operations based on the scale factor of the second rotation angle. The product of zero or more second type quantum operations may be ^^^. Since the global phase may not affect the measurement result, the first type operation ^^^^^may be replaced by the product of ^ operations ^^^in the quantum circuit. For exmaple, if ^^= 0, then ^^^= ^, which corresponds to simply "dropping" the gate from the quantum circuit. Since ^^≤ 3, for all ^, the total gate count may increase by at most 2^, e.g., the total gate count may increase by factor at most 3. Furthermore, the resulting quantum circuit contains no other gates than ^, ^, and ^^^^ gates and is hence a Clifford circuit. Thus, according to one example, the first set of first type quantum operations are configured to be applied on 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 second type quantum operations on the digital computer. The second sets of second type quantum operations may be simulated efficiently on the classical computer.

[0034] According to one example, the scale factor associated with each first rotation angle has a maximum value which is the order with respect to which the first rotation angle is differentiated in the derivative. Following the above notation, the first rotation anglethe scale factor, ^^,^+ ⋯ + ^^,^^is smaller than or equal to ^^e.g., ^^,^+ ⋯ + ^^,^^≤ ^^.

[0035] According to one example, the scale factor may be defined as the scale factor modulo a fixed number ^ which is a positive integer multiple of 4, wherein the fixed number is defined such that the number of second type quantum operations is smaller than a threshold. for example, if the scale factor is ^ it may be defined in accordance with this example as ^ = ^ mod ^, e.g., ^ = 4. For example, the derivative may be defined as follows:17 BUND.223.10EP

[0036] For example, by considering the following assumption: for ^ ∈{1, … ,an integer ^^may be defined as follows: ^^≔ ^^mod 4 ∈{0,1,2,3}, then up to global ^ phase the following equality is provided: ^^^^^^ ^ = ^^^. That is, the first type quantum operation ^^^^^may be defined as the product of the second type quantum operations, ^^^, where ^^≔ ^^mod 4 ∈ {0,1,2,3}, that is, ^^^may be the second type quantum operation having as exponent ^^.

[0037] Fig. 1 is a flowchart of a method for performing an initial quantum computation 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 may have as parameters one or more first rotation angles respectively. Each first type quantum operation may be a single-qubit rotation around a z-axis in the Bloch sphere by the respective first rotation angle.

[0038] The initial quantum computation may be represented in step 101 by a function ^ having as arguments the first rotation angles. A Taylor expansion of the function ^ around zero may be performed in step 103. This may result in derivatives of the function ^, wherein each derivative may be a partial derivative.

[0039] For each derivative of the resulting derivatives of the function ^, step 105 may be performed. In step 105, a parameter shift rule may be used for expressing 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 comprises the function ^ having as arguments a respective set of second rotation angles associated with the first rotation angles respectively. 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 equal18 BUND.223.10EP 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 with respect to 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 first type quantum operations respectively.

[0040] The derivative may be referred to as ^^^(0) where ^ is the order of the derivative which is smaller than ^ the order of the Taylor expansion, ^ ∈ ℤ^^and |^| ≤ ^. For example, the number of derivatives obtained from the Taylor expansion may be ^^,…, ^^^^(0)), the application of the parameter shift rule on each derivative of the ^^ derivatives may result into a set of 2|^|terms. That is, the execution of step 105 on all derivatives may result into a total number ^^^ =of terms. Each term of the number ^^^ of terms may be the function ^ with a set of ^ arguments / second rotation angles, wherein ^ may be the number of first rotations angles. Thus, the execution of step 105 may result in a number ^^^ of sets of second rotation angles, each set of second rotation angles comprises ^ second rotation angles. Each set of these ^^^ sets of second rotation angles may be parameters of a respective second set of first type quantum operations respectively. That is, a number ^^^ of second sets of first type quantum operations may be provided.

[0041] In each second set of first type quantum operations of the second sets of first type quantum operations, each first type quantum operation at the second rotation angle may be replaced in step 107 by a second type quantum operation having as exponent the scale factor of the second rotation angle. The second type quantum operation is a 90-degree rotation around the z-axis. The execution of step 107 may result in sets of second type quantum operations e.g., in a number ^^^ of sets of second type quantum operations.

[0042] The quantum computation may be redefined in step 109 by the sets of second type quantum operations instead of the first set of first type quantum operations. This may be advantageous as the sets of second type quantum operations may be evaluated efficiently on a classical computer. The redefinition of the quantum computation may be performed for each set of second type quantum operation by replacing the first set of first type quantum operations in the quantum19 BUND.223.10EP computation by the set of second type quantum operations respectively. This may result in a number ^^^ of individual redefined quantum operations. The individual redefined quantum operations may be combined e.g., by performing a weighted sum of the individual redefined quantum operations using respective weights for obtaining the (overall) redefined quantum computation.

[0043] The redefined quantum computation may be simulated in step 111 on a digital computer. The redefined quantum computation may be executed on the digital computer in step 111.

[0044] Fig.2 is a block diagram of an exemplary computer system for implementing at least part of the present method in accordance with an example of the present subject matter. The computer system of Fig. 2 may provide an example implementation of the digital computer according to the present subject matter.

[0045] The components of the computer system 602 may include, but are not limited to, one or more processors or processing units 603, a storage system 611, a memory unit 605, and a bus 607 that couples various system components including memory unit 605 to processor 603. The storage system 611 may include for example a hard disk drive (HDD). The memory unit 605 may include computer system readable media in the form of volatile memory, such as random access memory (RAM) and / or cache memory.

[0046] The computer system 602 may also communicate with one or more external devices such as a keyboard, a pointing device, a display 613, etc.; one or more devices that enable a user to interact with computer system 602; and / or any devices (e.g., network card, modem, etc.) that enable the computer system 602 to communicate with one or more other computing devices. Such communication can occur via I / O interface(s) 619. Still yet, the computer system 602 can 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 depicted, the network adapter 609 communicates with the other components of the client system 602 via bus 607.20 BUND.223.10EP

[0047] The memory unit 605 is configured to store applications that are executable on the processor 603. For example, the memory unit 605 may comprise an operating system as well as one or more application programs. The application programs comprise instructions that when executed enable to perform the method described with reference to Fig.1.

[0048] An example implementation of the method of Fig. 1 may, for example, be described as follows.

[0049] The quantum computation may be provided with an input. The input may comprise a number of qubits, ^ ∈ ℤ^^. The input may further comprise an ordered list of quantum gates. The gates may be applied in the quantum circuit in order of occurrence in the list, e.g., in reverse-order with respect to matrix multiplication. Each entry of the list contains the gate name, namely ^, ^, ^^^^, ^^and the gate information. For example, if the gate name is ^ or ^, the index of the target qubit may be provided as part of the gate information. If the gate name is ^^^^, the gate information may comprise the indices of the control qubit and of the target qubit. If the gate name is ^^, the gate information may comprise the index of the target qubit and an angle in the range (−. The input may further comprise an index ℓ ∈ {1, …, n} of qubit to be measured in standard basis. The input may further comprise ^ ∈ ℤ^^the order for the Taylor expansion, e.g., the Taylor polynomial may be computed up to degree ^.

[0050] The quantum computation may comprise a measurement ℳ, where ℳ =may be the observable that corresponds to the qubit to be measured in the standard basis. It is assumed that the initial state is |0 >⊗^and that the list of gates from the input defines a unitary ^ ∈ ℂ^^×^^. The state of the quantum circuit immediately before measurement is then |^ >≔ ^|0 >⊗^. The expected value of the measurement observable may be determined e.g., as follows: < ^|^|^ >.

[0051] The observable ℳ has two eigenvalues, namely −1 and 1. When measuring in QISKIT, however, the measurement result may be either 0 (corresponding to Eigenvalue 1) or 1 (corresponding to Eigenvalue −1). Note that, the measurement21 BUND.223.10EP result returned by QISKIT may correspond to the state to which qubit ℓ collapsed as a result of the measurement. That is, the Eigenspace for 1 has a standard orthonormal basis, namelyEigenspace for −1 has a standard orthonormal basis, namely. The disjoint union of these two sets is the standard orthonormal basis forAs a result, the measurement given by ℳ, the state is projected to the Eigenspace corresponding to the measured Eigenvalue of ℳ (and normalized). So, the measurement result returned by QISKIT is ^ ∈ {0, 1}, if and only if the post- measurement state is contained in the C-vector space spanned by

[0052] Having defined the input and the measurement, the quantum computation may be performed in accordance with the following approach. ^ ^

[0053] Let ^ ∈ ℤ^^be the number of ^^gates and∈ (− ^ , ^] be the corresponding angles in order of appearance in the list of quantum gates. The goal may be to evaluateFor that, the Taylor expansion of ^ may be computed around (0,…,0) and subsequentlymay be plugged in. All partials of ^ may be evaluated by simulating Clifford circuits, which can be done efficiently with a classical computer. The following describes how partials of ^ may be evaluated by simulation Clifford circuits.

[0054] In one example, ifthen may be evaluated efficiently on a classical computer. Indeed, for , then, up to global phase, the following equalitymay be provided. Since the global phase may not affect the measurement result,may be replaced byin the quantum circuit. For example, if , then , which may correspond to "dropping" the gate22 BUND.223.10EP from the quantum circuit. Since for all ^, the total gate count may increase by at most 2m, e.g., the total gate count may increase by factor at most 3. Furthermore, the resulting quantum circuit may contain no other gates than ^, ^, and ^^^^ gates and is hence a Clifford circuit which can be simulated efficiently on a classical computer. Using this example, the partials of ^ may be computed as follows.

[0055] For a multiindex the derivative of ^ may be defined as follows:, the derivativemay be computed. In particular, the following equality may be a result of the generalized parameter shift rules e.g., as described in A. Mari et al. arXiv:2008.06517, Phys. Rev. A 103, 012405 (2021):. (cf. section 0031 above)This equation may be referred to as lemma A. If ^^= 0, the corresponding summation sign may be skipped,is replaced by 1 (empty product) andis replaced by 0 (empty sum).

[0057] The sum of the derivative ^^^(0) has 2|^|summands, but the ^-term may not necessarily be distinct between different summands. In one example, one may use some combinatorics and combine several summands into a single summand with a "counting factor". In another example, as described with Fig.3, a list of all ^ integer tuples (^^, … , ^^), for which ^(^^^ , … ,has already been evaluated may be kept. Then, for many summands in the above sum, a previously computed value of ^ may be reused. In fact, given that= ^ ^((^^mod 4) ^ , … , (^^mod 4)a list of tuples with entries in {0, 1, 2, 3} may be stored, which further reduces the number of necessary evaluations for ^. This list may be shared accross all orders |^| = 0, … , ^ of the partials; for example, if ^^, … , ^^are all even, then ^(0, … , 0) may appear in the above sum, no matter how23 BUND.223.10EP large |^| is. The leading factor ^^,^… ^^,^^… ^^,^… ^^,^^is either 1 or −1. It is 1 (resp. −1), if the number of ^^,^which are negative is even (resp. odd).

[0058] Once all these partials have been computed, the approximation of function ^ may be as follows:

[0059] Fig.3 is a pseudocode for simulating the redefined quantum computation in accordance with an example of the present subject matter. The input the pseudocode may comprise a number of qubits, ^ ∈ ℤ^^. The input may further comprise an ordered list of quantum gates. The gates may be applied in the quantum circuit in order of occurrence in the list, e.g., in reverse-order with respect to matrix multiplication. Each entry of the list contains the gate name namely ^, ^, ^^^^, ^^and the gate information. For example, if the gate name is ^ or ^, the index of the target qubit may be provided as part of the gate information. If the gate name is ^^^^, the gate information may comprise the indices of the control qubit and of the target qubit. If the gate name is ^^, the gate information may comprise ^ ^ the index of the target qubit and an angle in the range (− ^ , ^]. The input may further comprise an index ℓ ∈ {1, …, n} of qubit to be measured in standard basis. The input may further comprise ^ ∈ ℤ^^the order for the Taylor expansion, e.g., the Taylor polynomial may be computed up to degree ^.

[0060] An initialization may be performed as follows. A collection “computed_tuples” es (^^, … , ^^) ∈ {0, 1, 2,3^^ ^ of tupl } , for which ^(^^^ , … , ^^^) has already been evaluated may be created. computed_tuples may be initialized as an empty collection. This may provide per derivative and per term of the set of terms a tuple of scale factors representing the second rotation angles of the term respectively. The number ^ ∈ ℤ^^of ^^gates may be determined from the Input. The ^ ^ corresponding angles may be ^^, … , ^^∈ (− ^ , ^] in order of occurrence in the list of quantum gates provided in the Input. In the pseudocode, ^ is defined as in the above text. If ^ = 0, then replace ^ by 0. (If ^ = 0, then ^ is constant, so one may only need the constant term from the Taylor expansion.)24 BUND.223.10EP

[0061] As shown in the pseudocode, the ^-term 301 of each summand may omprise second rotations angles ^^ ^ c ^^,^+ ⋯ +^ , …, and ^^^^,^+ ⋯ +of respectively first type quantum operations …^ ^ ^ and ^ ^ ^^,^+ ⋯ + ^^,^^^ ^ ^ which define the initial quantum operation with the second rotation angles as parameters. This initial quantum computation may be redefined at the line of code indicated by reference 301 by replacing these first type quantum operations as follows:^^= ^

[0062] The pseudocode represents the main body of the algorithm and the output. The algorithm enables the execution of the redefined quantum computation on a classical computer. For that, the derivatives may be sequentially processed term per term. In each iteration, it determines whether the tuple of the current term is already processed. If the tuple of the current term is not already processed, the individual redefined quantum computation of the current term may be evaluated or simulated and the result of evaluation may be associated with the tuple. If the tuple of the current term is already processed, the result of evaluation associated with the stored tuple may be used as a result of the individual redefined quantum computation. The result of the individual redefined quantum computation may be weighted by a respective weight factor. This is indicated by the line of code 303. The weighted results may be combined as indicated by the line of codes 303, 304 and 305.

[0063] Returning the function in addition to “approximated_value” may be useful, since this may allow to evaluate / approximate the expected value of the observable for different rotation angles than (^ , … ,) without having to compute the coefficients again; this may be especially useful e.g., for VQE or QML, since it is common place in these applications to iteratively update the rotation angles of Pauli rotation gates.

[0064] As will be appreciated by one skilled in the art, 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 take25 BUND.223.10EP the form of an entirely hardware embodiment, an entirely software embodiment (including firmware, resident software, micro-code, etc.) or an embodiment combining software and hardware aspects that may all generally be referred to herein as a “circuit,” “module” or “system.” Furthermore, 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".

[0065] The term “computer system” refers to data processing hardware and encompasses all kinds of apparatus, devices, and machines for processing data, including by way of example, a programmable processor, a computer, or multiple processors or computers. The apparatus can also be or further include special purpose logic circuitry, e.g., a central processing unit (CPU), a FPGA (field programmable gate array), or an ASIC (application specific integrated circuit). In some implementations, the data processing apparatus and / or special purpose logic circuitry may be hardware-based and / or software-based. The apparatus can optionally include code that creates an execution environment for computer programs, e.g., code that constitutes processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of them. The present disclosure contemplates the use of data processing apparatuses with or without conventional operating systems, for example LINUX, UNIX, WINDOWS, MAC OS, ANDROID, IOS or any other suitable conventional operating system.

[0066] Any combination of one or more computer readable medium(s) may be utilized. The computer readable medium may be a computer readable storage medium. A ‘computer-readable storage medium’ as used herein encompasses any tangible storage medium which may store instructions which are 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 be26 BUND.223.10EP able to store data which is able to be accessed by the processor of the computing device.

[0067] ‘Computer memory’ or ‘memory’ is an example of a computer-readable storage medium. Computer memory is any memory which is directly accessible to a processor. ‘Computer storage’ or ‘storage’ is a further example of a computer- readable storage medium. Computer storage is any non-volatile computer-readable storage medium. In some embodiments computer storage may also be computer memory or vice versa.

[0068] A ‘processor’ as used herein encompasses an electronic component which is able to execute a program or machine executable instruction or computer executable code. References to the computing device comprising “a processor” should be interpreted as possibly containing more than one processor or processing core. The processor may for instance be a multi-core processor. A processor may also refer to a collection of processors within a single computer system or distributed amongst multiple computer systems. The term computing device should also be interpreted to possibly refer to a collection or network of computing devices each comprising a processor or processors. The computer executable code may be executed by multiple processors that may be within the same computing device or which may even be distributed across multiple computing devices.

[0069] Computer executable code may comprise machine executable instructions or a program which causes a processor to perform an aspect of the present invention. Computer executable code for carrying out operations for aspects of the present invention may be written 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 and compiled into machine executable instructions. In some instances the computer executable code may be in the form of a high level language or in a pre-compiled form and be used in conjunction with an interpreter which generates the machine executable instructions on the fly.27 BUND.223.10EP

[0070] Generally, the program instructions can be executed on one processor or on several processors. In the case of multiple processors, they can be distributed over several different entities. Each processor could execute a portion of the instructions intended for that entity. Thus, when referring to a system or process involving multiple entities, the computer program or program instructions are understood to be adapted to be executed by a processor associated or related to the respective entity.

[0071] While the invention has been illustrated and described in detail in the drawings and foregoing description, such illustration and description are to be considered illustrative or exemplary and not restrictive; the invention is not limited to the disclosed examples.

Claims

28 BUND.223.10EP CLAIMS 1. A method for performing an initial quantum computation, the initial quantum computation 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 having as parameters one or more first rotation angles respectively, each first type quantum operation being a single-qubit rotation around a z-axis in a Bloch sphere by the respective first rotation angle, the method comprising: representing the initial quantum computation by a function ^ having as arguments the first rotation angles; performing a Taylor expansion of the function ^ around zero, resulting in derivatives of the function ^, referred to as foundational derivatives; for each specific derivative of the foundational derivatives: using a parameter shift rule for expressing the specific derivative as a linear combination of a set of terms, for each term of the set of terms: the term comprising the function ^ having as arguments a respective set of second rotation angles associated with the first rotation angles respectively, each second rotation angle of the set of second rotation angles being a 90-degree angle scaled by a scale factor representing the first rotation angle associated with the second rotation angle, the scale factor being an integer equal to zero or different from zero depending on the order with respect to which the associated first rotation angle is differentiated in the specific derivative;29 BUND.223.10EP wherein each of the set of second rotation angles is a parameter of a second set of first type quantum operations respectively; in each second set of first type quantum operations, replacing each first type quantum operation at the second rotation angle by a second type quantum operation having as exponent the scale factor of the second rotation angle, the second type quantum operation being a 90-degree rotation around the z-axis, resulting in a set of second type quantum operations; redefining the initial quantum computation using the sets of second type quantum operations instead of the first set of first type quantum operations, resulting in a redefined quantum computation; simulating the redefined quantum computation on a digital computer.

2. The method of claim 1, the redefining comprising for each derivative of the function ^: for each set of second type quantum operations of the derivative, replacing the first set one or more first type quantum operations by the set of second type quantum operations respectively, resulting in an individual redefined quantum computation; wherein the simulation of the redefined quantum computation comprises evaluating the Taylor expansion using a combination of the individual redefined quantum computations.

3. The method of claim 1 or 2, further comprising: providing an initial set of quantum operations including a subset of first type quantum operations and a subset of quantum operations30 BUND.223.10EP representing single-qubit rotations around x-axis and y-axis in the Bloch sphere; transforming each quantum operation around x-axis or y-axis as a function of a first type quantum operation having the same rotation angle; thereby obtaining the first set of first type quantum operations as the subset of first type quantum operations and the transformed subset.

4. The method of claim 1, 2, or 3, further comprising: providing an initial set of first type quantum operations having as parameters initial rotation angles respectively; 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 around the scaled 90- degree angle as function of second type quantum operations.

5. The method of any of the preceding claims, wherein the first rotation angle ^ ^ is in a range of (− ^ , ^].

6. The method of any of the preceding claims, wherein the degree of Taylor expansion is smaller than a predefined maximum degree.

7. The method of any of the preceding claims, wherein the scale factor associated with each first rotation angle has a maximum value which is the order with respect to which the first rotation angle is differentiated in the derivative.

8. The method of any of the preceding claims, comprising: defining the scale factor as the scale factor modulo a fixed number which is a positive integer31 BUND.223.10EP multiple of four, wherein the fixed number is defined such that the number of second type quantum operations is smaller than a threshold.

9. The method of any of the preceding claims, the first set of first type quantum operations being an ordered set of quantum operations. 10.The method of any of the preceding claims, wherein the first set of first type quantum operations define transformations to be applied on a quantum system defined by a set of qubits. 11.The method of any of the preceding claims 2 to 10, the simulating of the redefined quantum computation comprising: providing per derivative and per term of the set of terms of the derivative a tuple of scale factors representing the second rotation angles of the term respectively; evaluating the derivatives sequentially term per term comprising: determining whether the tuple of the current term is already processed; if the tuple of the current term is not already processed, simulating individual redefined quantum computation of the current term and associating with the tuple the result of simulation; if the tuple of the current term is already processed, using the result of simulation associated with the stored tuple as a result of the current individual redefined quantum computation; weighting the result of the individual redefined quantum computation by a respective weight factor; combining the weighted results. 12.The method of any of the preceding claims, the first type quantum operation being a ^^rotation of ^^gate, and the second type quantum operation being a phase shift ^ of ^ gate.32 BUND.223.10EP A computer program comprising machine executable instructions, wherein execution of the machine executable instructions causes a computer system to: represent an initial quantum computation by a function ^ having as arguments first rotation angles of a first set of one or more first type quantum operations that define the initial quantum computation, each first type quantum operation being a single-qubit rotation around a z- axis in a Bloch sphere by the respective first rotation angle; perform a Taylor expansion of the function ^ around zero, resulting in foundational derivatives of the function ^; for each specific derivative of the foundational derivatives: use a parameter shift rule for expressing the specific derivative as a linear combination of a set of terms, for each term of the set of terms: the term comprising the function ^ having as arguments a respective set of second rotation angles associated with the first rotation angles respectively, each second rotation angle of the set of second rotation angles being a 90-degree angle scaled by a scale factor representing the first rotation angle associated with the second rotation angle, the scale factor being an integer equal to zero or different from zero depending on the order with respect to 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 respectively; in each second set of first type quantum operations, replace each first type quantum operation at the second rotation angle by a second type quantum operation having as exponent the scale factor of the second33 BUND.223.10EP rotation angle, the second type quantum operation being a 90-degree rotation around the z-axis, resulting in a set of second type quantum operations; redefine the initial quantum computation using the sets of second type quantum operations instead of the first set of first type quantum operations, resulting in a redefined quantum computation; simulate the redefined quantum computation. A computer system for performing an initial quantum computation, the initial quantum computation 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 having as parameters one or more first rotation angles respectively, each first type quantum operation being a single-qubit rotation around a z-axis in a Bloch sphere by the respective first rotation angle, the computer system being configured for: representing the initial quantum computation by a function ^ having as arguments the first rotation angles; performing a Taylor expansion of the function ^ around zero, resulting in foundational derivatives of the function ^; for each specific derivative of the foundational derivatives: using a parameter shift rule for expressing the specific derivative as a linear combination of a set of terms, for each term of the set of terms: the term comprising the function ^ having as arguments a respective set of second rotation angles associated with the first rotation angles respectively, each second rotation angle of the set of second rotation angles being a 90-degree angle scaled by a scale factor representing the first rotation angle associated with the34 BUND.223.10EP second rotation angle, the scale factor being an integer equal to zero or different from zero depending on the order with respect to 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 respectively; in each second set of first type quantum operations, replacing each first type quantum operation at the second rotation angle by a second type quantum operation having as exponent the scale factor of the second rotation angle, the second type quantum operation being a 90-degree rotation around the z-axis, resulting in a set of second type quantum operations; redefining the initial quantum computation using the sets of second type quantum operations instead of the first set of first type quantum operations, resulting in a redefined quantum computation; simulating the redefined quantum computation.

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