Quantum circuit and method for executing a parameterised phase gate in a quantum error correction code

Abstract

The invention relates to a method for executing a parameterised phase gate (RZ, L(φ)) in a quantum error correction code, said method having the following steps: - representing at least one logical qubit by a number of N physical qubits (qb1...N), - representing a logical Pauli operator (XL, YL, ZL) acting on the logical qubit by a tensor product of single-qubit operators acting on the physical qubits (qb1...N), wherein the single-qubit operators are selected from the group consisting of three different Pauli operators (XP, YP, ZP) and an identity operator (IP), and - executing the parameterised phase gate (RZ,L(φ)) on the logical qubit by applying a matrix exponential of the logical Pauli operator (XL, YL, ZL) to the physical qubits (qb1...N), wherein - the matrix exponential of the logical Pauli operator (XL, YL, ZL) is implemented by a gate-based quantum circuit (1) which acts on the N physical qubits (qb1...N). The invention further relates to a computer program, a computer-readable storage medium, a quantum computer, and a quantum circuit.

Description

[0001] P2023,1398 WO N / FOCUS - 2023P66414WO December 18, 2024 - 1 –Description: QUANTUM CIRCUIT AND METHOD FOR IMPLEMENTING A PARAMETERIZED PHASE GATE IN A QUANTUM ERROR CORRECTION CODE. A quantum circuit, a method for executing a parameterized phase gate in a quantum error correction code, a computer program, a computer-readable storage medium, and a quantum computer are provided. At least one object of various embodiments is to provide an efficient quantum circuit and an efficient method for executing a parameterized phase gate in a quantum error correction code. This object is achieved by a method having the steps according to the independent patent claims.Advantageous embodiments and further developments of the quantum circuit, the method for implementing a parameterized phase gate in a quantum error correction code, the computer program, the computer-readable storage medium, and the quantum computer are specified in the dependent claims. According to one embodiment, the method for implementing a parameterized phase gate in a quantum error correction code comprises a step in which at least one logical qubit is represented by a number of N physical qubits. In other words, a coded, fault-tolerant quantum state is initialized to N physical qubits. P2023,1398 WO N / FOKUS - 2023P66414WO December 18, 2024. - 2 –A qubit, specifically the logical qubit or the physical qubit, is a quantum mechanical information unit with two linearly independent basis states, which are denoted by "|0>" and "|1>" below. The basis states |0> and |1>, for example, form a measurement basis of a Hilbert space of the qubit. In the following, the basis states of the logical qubit are denoted by "|0>" L ' and '|1> L “, while the basis states of one of the physical qubits are denoted by “|0> P ' and '|1> PThe logical qubit is, in particular, an abstract information unit of a quantum circuit. For example, the logical qubit represents a quantum mechanical information unit in a quantum circuit or quantum algorithm. For example, the quantum circuit or quantum algorithm performs quantum mechanical operations on the logical qubit. For example, the quantum circuit manipulates the quantum mechanical information represented by the logical qubit. The physical qubit is, in particular, an actual implementation of a qubit in a physical, quantum mechanical system. For example, the physical qubit is a quantum mechanical two-state system with two linearly independent basis states connected to the basis states |0> P and |1> Pthe computational state basis of the physical qubit. For example, the physical qubit is realized by two hyperfine states of an ion or by two energy levels of a superconducting circuit. P2023,1398 WO N / FOCUS - 2023P66414WO December 18, 2024 - 3 –In particular, a physical qubit does not represent a perfect realization of an ideal logical qubit. For example, the quantum mechanical information stored on the physical qubit is lost through decoherence, i.e., through interaction of the physical qubit with its environment, after a characteristic coherence time. The time span during which operations of the quantum circuit can be performed on the physical qubit is limited, for example, by the coherence time of the physical qubit. Quantum error correction codes are used, for example, to at least partially protect the quantum mechanical information stored in a logical qubit from decoherence. In other words, the quantum error correction code, in particular, protects the quantum mechanical information stored in a logical qubit at least partially from decoherence.For example, the quantum error correction code extends the coherence time of the logical qubit compared to the coherence time of a physical qubit, for example, by at least 100%, or the coherence time of the logical qubit is extended to arbitrarily long periods of time. For this purpose, the logical qubit is represented by N physical qubits, for example, by N = 3, 4, 5, 7, 9, or more physical qubits. Two or more logical qubits can also be represented by the N physical qubits. In particular, N is greater than the number of logical qubits represented by the N physical qubits. For example, the basis states |0>L and |1>L of the logical qubit are connected to two entangled states of the NP2023,1398 WO N / FOKUS - 2023P66414WO December 18, 2024. - 4 –physical qubits are identified. Due to this coding of the basis states of the logical qubit by entangled states of the N physical qubits, a logical qubit, for example, is not only assigned to a single physical qubit, but the information of the logical qubit is distributed non-locally across the N physical qubits. Thus, local disturbances or errors, such as unwanted bit flips or phase flips of at least one of the N physical qubits, can be at least partially detected and / or corrected without changing or losing the quantum mechanical information stored in the logical qubit. The quantum error correction code, for example, uses a syndrome measurement to detect the presence and / or type of error, such as a bit flip error or phase flip error, of at least one physical qubit.The quantum mechanical information stored in the logical qubit is not affected. The result of the syndrome measurement can subsequently be used to correct the corresponding error in the physical qubits and thus extend the coherence time of the logical qubit. For example, the syndrome measurement allows at least the detection and / or correction of certain local errors, such as bit flips or phase flips of individual physical qubits. For example, the quantum error correction code is a stabilizer code, in particular a Shor code, a Steane code, a CCS code, or a topological quantum error correction code. In the Shor code, for example, N = 9 physical qubits are used to represent a logical qubit. In the Steane code, P2023,1398 WO N / FOKUS - 2023P66414WO December 18, 2024. - 5 –For example, N = 7 physical qubits are used to represent a logical qubit. In particular, the topological quantum error correction code is a surface code, such as a planar code or a toric code. In the torus code, for example, N ≥ 4 physical qubits are used to represent two logical qubits, where the physical qubits are arranged, for example, on the edges of a regular square lattice that forms a torus. The four basis states of the two logical qubits lie in different topological sectors of a Hilbert space of the N physical qubits. Local perturbations or errors, which, for example, act locally on one of the N physical qubits, cannot therefore cause transitions between the basis states |0> L and |1> Lof each of the two logical qubits. In particular, for each of the two logical qubits, <1|LA |0>L = 0, where A is an arbitrary local operator that acts, for example, only on one of the N physical qubits. The logical qubits in the torus code are thus particularly robust against local error sources. The parameterized phase gate is, in particular, a single-qubit gate that acts on the logical qubit. For example, the parameterized phase gate is a rotation operator that rotates a state of the logical qubit by an arbitrary angle. For example, the parameterized phase gate rotates the state of the logical qubit by an arbitrary angle around a z-axis of a Bloch sphere of the logical qubit. It is also possible that the parameterized phase gate determines the state of the logical qubitP2023,1398 WO N / FOCUS - 2023P66414WO December 18, 2024 - 6 –Alternatively or additionally, the Bloch sphere of the logical qubit is rotated by an arbitrary angle around an x-axis and / or a y-axis. In this case, the parameterized phase gate is referred to, for example, as a U gate, in particular a U3 gate. The Bloch sphere can be used to represent the state of a qubit. For example, the basis states |1> and |0> correspond to a north pole and a south pole of the Bloch sphere, while other points on a surface of the Bloch sphere correspond to complex superpositions of the basis states |0> and |1>. The parameterized phase gate is, for example, a single-qubit gate of a quantum circuit that acts on a plurality of logical qubits. The quantum circuit comprises, for example, a plurality of different single-qubit gates and / or entanglement gates, with single-qubit gates acting on individual logical qubits, while entanglement gates act on at least two logical qubits.For example, each of the logical qubits of the quantum circuit is represented by N physical qubits. For example, the logical qubits of the quantum circuit are represented pairwise by N physical qubits. According to a further embodiment of the method, a logical Pauli operator acting on the logical qubit is represented by a tensor product of single-qubit operators acting on the physical qubits, wherein the single-qubit operators are selected from the group consisting of three different Pauli operators and an identity operator. The tensor product of the ... - 7 – The single-qubit operator acting on physical qubits is also referred to as a Pauli string. The logical Pauli operator is, in particular, one of the three Pauli operators XL, YL, or ZL, which can be represented, for example, by the following 2×2 Pauli matrices. where the basis states |0>L and |1>L of the logical qubits are normalized eigenstates of the Z L -operator with the eigenvalues ​​-1 and +1, ie Z L |0> L = -|0> L and Z L |1> L = |1> L . Accordingly, the Pauli operators X P , Y P and Z P acting on a physical qubit are represented by the same 2×2 Pauli matrices as described above, where the basis states |0> P and |1> P of the physical qubit normalized eigenstates of the Z P-operator with the eigenvalues ​​-1 and +1. The identity operator IP, for example, is represented by the 2×2 identity matrix acting on the two basis states of a physical qubit. For example, the representation of the logical Pauli operator as the tensor product of single-qubit operators acting on the physical qubits is determined by encoding the basis states of the logical qubit using the entangled states of the N physical qubits. For example, the choice of a particular quantum error correction code determines the representation of the logical Pauli operator. The logical Pauli operator ZL can, for example, be represented in the Steane code as a tensor product of seven Z P Pauli-P2023,1398 WO N / FOCUS - 2023P66414WO December 18, 2024 - 8 –Operators that act on one of the seven physical qubits, ZL = ZP⊗ZP⊗ZP⊗ZP⊗ZP⊗ZP⊗ZP. The logical Pauli operator XL can, for example, be expressed in Steane code as a tensor product of seven XP Pauli operators, each acting on one of the seven physical qubits, XL = XP⊗XP⊗XP⊗XP⊗XP⊗XP⊗XP. For example, the logical Pauli operator Z L in the torus code by a tensor product of Z PPauli operators acting on a chain of neighboring physical qubits along a non-contractible loop. In particular, the non-contractible loop encloses one of the two holes of the torus. According to a further embodiment of the method, the parameterized phase gate on the logical qubit is implemented by applying a matrix exponential of the logical Pauli operator to the physical qubits. The parameterized phase gate acting on the logical qubit corresponds, for example, to the rotation operator R Z,L (φ) = exp(-i∙φ∙Z L / 2). It is also possible that the parameterized phase gate acting on the logical qubit corresponds to one of the rotation operators RX,L(φ) = exp(-i∙φ∙XL / 2) orR Y,L (φ) = exp(-i∙φ∙Y L / 2), or any combination of the three operators R X,L (φ), R Y,L (φ) and R Z,L (φ). Where X L, Y L and Z L logical Pauli operators acting on the logical qubit. The rotation operators R X,L (φ), RY, L(φ) or RZ, L(φ) each cause a rotation of the state of the logical qubit by an arbitrary angle φ around the x-, y-, or z-axis of the Bloch sphere of the logical qubit. P2023,1398 WO N / FOCUS - 2023P66414WO December 18, 2024 - 9 – For example, the operator R transforms Y,L (180°) the state |1>L into the state |0>L. Thus, the parameterized phase gate corresponds in particular to a matrix exponential of the logical Z L Pauli operator. The parameterized phase gate can also be assigned to a matrix exponential of one of the logical Pauli operators X Lor YL, or a matrix exponential of any combination of the Pauli operators XL, YL, and ZL. The logical Pauli operator is represented in particular by the Pauli string of single-qubit operators acting on the physical qubits. When the parameterized phase gate is chosen as the rotation operator RZ,L(φ) around the z-axis, the parameterized phase gate corresponds, except for a global phase, to a phase-shifting gate P. L (φ) (English: phase shift gate), ie P L (φ) = R Z,L (φ)∙exp(i∙φ / 2). The global phase factor exp(i∙φ / 2) does not require any further consideration. The phase shift gate can be represented by the 2×2 matrix which are based on the basis states |0> L and |1> Lof the logical qubit. According to a further embodiment of the method, the matrix exponential of the logical Pauli operator is realized by a gate-based quantum circuit that acts on the N physical qubits. For example, the gate-based quantum circuit comprises single-qubit gates and controlled non-gates (CNOT gates), or consists of single-qubit gates and CNOT gates. The CNOT gate is P2023,1398 WO N / FOCUS - 2023P66414WO December 18, 2024 - 10 –In particular, in an entanglement gate applied to two physical qubits. For example, the CNOT gate has inputs and outputs for exactly one physical control qubit and exactly one physical target qubit. In particular, the CNOT gate negates an output state of the target qubit in the computational state basis if the input state of the control qubit is |1>. In other words, an input state |0> of the target qubit is mapped to the output state |1> of the target qubit, while the input state |1> of the target qubit is mapped to the output state |0> of the target qubit if the input state of the control qubit is |1>. In particular, if the input state of the control qubit is |0>, the CNOT gate acts as an identity operation on the target qubit.Here and below, the input state refers to the state of the qubit at the input of a quantum gate or a quantum circuit, while the output state refers to the state of the qubit at the output of a quantum gate or a quantum circuit. According to a preferred embodiment, the method for implementing the parameterized phase gate in the quantum error correction code comprises the following steps: - Representing the at least one logical qubit by the number N of physical qubits, - Representing the logical Pauli operator acting on the logical qubit by the tensor product of the individual qubit operators acting on the physical qubits, wherein the individual qubit operators are selected from the group consisting of the three different Pauli operators and the identity operator. P2023,1398 WO N / FOKUS - 2023P66414WO December 18, 2024. - 11 –- Executing the parameterized phase gate on the logical qubit by applying the matrix exponential of the logical Pauli operator to the physical qubits, where - the matrix exponential of the logical Pauli operator is realized by the gate-based quantum circuit acting on the N physical qubits. The method described here is based in particular on the idea of ​​efficiently executing a parameterized phase gate on a logical qubit of a quantum error correction code, for example, a stabilizer code. Efficient execution here refers to converting the parameterized phase gate into a gate-based quantum circuit acting on the physical qubits and having the smallest possible number of gates. In stabilizer codes, logical Clifford gates can be implemented particularly easily on the physical qubits. Arbitrary Clifford gates can be constructed from phase gates. L(φ=90°) (phase gate), Hadamard gates, and controlled non-gates (CNOT gates). Clifford gates are, in particular, transversal gates. For example, for transversal gates, the logical gate can be implemented by applying the corresponding gate to each of the N physical qubits. For example, the Pauli operator X is a Clifford gate and thus transversal, meaning the logical Pauli operator X L can be realized by a tensor product of N XP Pauli operators acting on the physical qubits,P2023,1398 WO N / FOKUS - 2023P66414WO December 18, 2024 - 12 –XL = XP⊗…⊗XP, as in Shor code or Steane code. The parameterized phase gate is notably neither a Clifford gate nor transversal. Thus, no representation of a logical parameterized phase gate as a simple tensor product of phase gates acting on the physical qubits is known. However, parameterized phase gates are of particular importance for quantum algorithms, since a universal set of elementary quantum gates includes both Clifford gates and parameterized phase gates. For example, quantum circuits that exclusively comprise Clifford gates are easily simulated on classical computers and are therefore of little interest. To implement logical parameterized phase gates in stabilizer codes, the method of magic state distillation can be used. This allows, for example, the phase gate T = P L(φ=45°) can be implemented by gate teleportation. To implement any parameterized phase gate, these can be approximated to a certain accuracy by a sequence of T gates. However, this method has the disadvantage that the accuracy of the approximation of the parameterized phase gate is limited. Furthermore, the number of T gates required can be large and depends on the accuracy of the approximation. This means that the more precisely the parameterized phase gate is to be implemented, the more T gates are required. In contrast, the method described here enables logical parameterized phase gates, particularly in stabilizer codes, to be efficiently implemented on the physical qubits. P2023,1398 WO N / FOKUS - 2023P66414WO December 18, 2024 - 13 –without first having to decode the state of the logical qubit encoded in the physical qubits. Decoding refers, for example, to a step in which the quantum information stored in the logical qubit is transferred to a single physical qubit, with the logical parameterized phase gate subsequently being executed as a single-qubit gate on this qubit. Furthermore, in contrast to magic-state distillation, the method described here requires only a limited number of gates, which does not depend on the required accuracy of the implementation of the parameterized phase gate. Furthermore, the compilation of error-corrected quantum circuits using the method described here requires fewer classical resources than magic-state distillation.According to a further embodiment of the method, the logical Pauli operator is selected from the group consisting of an X Pauli operator, a Y Pauli operator, and a Z Pauli operator. Thus, the parameterized phase gate corresponds to a rotation operator R. X,L (φ) around the x-axis of the Bloch sphere of the logical qubit, a rotation operator R Y,L (φ) around the y-axis of the Bloch sphere of the logical qubit, or a rotation operator R Z,L (φ) around the z-axis of the Bloch sphere of the logical qubit. According to a further embodiment of the method, the tensor product of individual qubit operators acting on the physical qubits is a stabilizer operator. For example, the quantum error correction code is a P2023,1398 WO N / FOCUS - 2023P66414WO December 18, 2024 - 14 –Stabilizer code, in particular a Shor code, a Steane code, or a CSS code. For example, in the stabilizer code, a number N of mutually commuting stabilizer operators is selected, wherein a number of N - 1 of the N stabilizer operators are configured to determine a state space of the logical qubit, and wherein the remaining stabilizer operator is a logical Pauli operator, for example, a ZL Pauli operator of the logical qubit. Each of the N stabilizer operators is in particular a Pauli string, i.e., a tensor product of N single-qubit operators acting on the physical qubits, which are selected from the three different Pauli operators X P , Y P and Z P , and the identity operator. In particular, the state space of the logical qubit is invariant under the N – 1 stabilizer operators. For the N – 1 stabilizer operators Si with i from 1 to N – 1, for example, S i |0> L= |0> L and S i |1> L = |1> L , while for the remaining stabilizer operator, S N = Z L Since the N stabilizer operators commute with each other, their eigenvalues ​​can be measured without affecting the measurement results of the other stabilizer operators. Syndrome measurement is performed, for example, by measuring the eigenvalues ​​of the N - 1 stabilizer operators. If, for example, one of the eigenvalues ​​of the N - 1 stabilizer operators is -1 instead of +1, an error exists that can be corrected by applying an appropriate correction procedure. The P2023,1398 WO N / FOCUS - 2023P66414WO December 18, 2024 - 15 –The state of the logical qubit is not affected due to the commutativity of the stabilizer operators. According to a further embodiment of the method, the matrix exponential of the logical Pauli operator is represented by a matrix exponential of a tensor product of a number of m ZPauli operators, with 1 ≤ m ≤ N, where each of the ZPauli operators acts on an associated physical qubit. In other words, the ZPauli operators are ZPPauli operators. For example, the representation of the logical Pauli operator comprises a tensor product of N single-qubit operators, which, in addition to ZPPauli operators, also include XPPauli operators, YPPauli operators, and / or identity operators. P Pauli operators and Y P Pauli operators can be described in particular by the relations Y P = S + Z P S and X P = H + Z P H to Z PPauli operators, where S is a phase gate and Hein Hadamard gates, which are represented by the 2×2 matrices can be represented, which refer to the basis states |0> P and |1> P of a physical qubit, and where S + and H + denote hermitian conjugate operators of S and H. Since the phase gate S and the Hadamard gate are unitary, i.e. S + S = 1 and H + H = 1, these gates can be extracted from the exponent of the matrix exponential, so that the exponent of the matrix exponential is exclusively Z P Pauli operators and, if applicable, identity operators. Thus, the exponent of the matrix exponential, for example, has a number of m ZP Pauli operators and a number of N – m identity operators, each acting on corresponding physical qubits. P2023,1398 WO N / FOCUS - 2023P66414WO December 18, 2024 - 16 –According to a further embodiment of the method, the parameterized phase gate is represented by where U is a unitary operator consisting of single-qubit operators acting on the physical qubits, U + is a Hermitian conjugate operator of U, ^ is a phase of the parameterized phase operator, and Z P,iZ is a Pauli operator acting on the i-th physical qubit, and where m, with 1 ≤ m ≤ N, denotes those of the N physical qubits on which the logical Pauli operator does not act as an identity operator. In particular, the operator U comprises single-qubit gates selected from the phase gate S, the Hadamard gate H, and the identity operator, or consists of these single-qubit gates. According to a further embodiment of the method, the gate-based quantum circuit has a number of 2∙(m-1)CNOT gates and a parameterized single-qubit phase gate. Here, m, with 1 ≤ m ≤ N, corresponds to the number of Pauli operators in the Pauli string of the representation of the logical Pauli operator. The single-qubit phase gate is, in particular, a parameterized phase gate acting on one of the physical qubits.For example, the single-qubit phase gate corresponds to one of the three rotation operators RX,P(φ) = exp(-i∙φ∙XP / 2), RY,P(φ) = exp(-i∙φ∙YP / 2), or RZ,P(φ) = exp(-i∙φ∙ZP / 2), which rotate the state of one of the physical qubits on its Bloch sphere. According to a further embodiment of the method, the gate-based quantum circuit comprises a first input fan-out gate, a parameterized single-qubit phase gate following the first input fan-out gate, and a parameterized single-qubit phase gate following the parameterized P2023,1398 WO N / FOKUS - 2023P66414WO December 18, 2024. - 17 –The single-qubit phase gate has a second fan-in input gate following the single-qubit phase gate, and the parameterized single-qubit phase gate acts on a common target qubit of the first fan-in input gate and the second fan-in input gate. The fan-in input gate consists, for example, of m - 1 CNOT gates, with 1 ≤ m ≤ N. In particular, the fan-in input gate can be represented exclusively by paired CNOT gates. For example, the fan-in input gate comprises inputs and outputs for exactly one physical target qubit and at least one or more physical control qubits, wherein the target qubit is linked to each of the control qubits via exactly one CNOT gate. According to a further embodiment of the method, the parameterized phase gate is implemented on an auxiliary qubit and teleported into the logical qubit. In particular, the ancilla qubit is an additional logical qubit.For example, the matrix exponential of the logical Pauli operator described above for executing the parameterized phase gate on the logical qubit is executed—in this case, on physical qubits used to represent the auxiliary logical qubit. Subsequently, teleportation into the logical qubit occurs. Teleportation occurs, for example, by executing the parameterized phase gate on the logical qubit. For example, the parameterized phase gate corresponds to the rotation operator RZ,L(φ) = exp(-i∙φ∙ZL / 2). In this case, the rotation operator RZ,L(φ) is applied, for example, to the P2023,1398 WO N / FOCUS - 2023P66414WO December 18, 2024. - 18 – State |+^ = (|0^ + |1^) / √2 of the auxiliary qubit. This results in the rotation operator R Z,L (φ) is programmed into the auxiliary qubit. After that, the rotation operator R Z,L(φ) is teleported into the logical qubit, for example, by applying a CNOT gate and subsequently measuring the Pauli Z-operator on the auxiliary qubit. If the parameterized phase gate is assigned to the rotation operator R X,L(φ) or RY,L(φ), the method described here can be implemented by means of an additional conjugation with the Hadamard gate H or the Hadamard and phase gate S, respectively. For example, the CNOT gate is applied to the auxiliary qubit and the logical qubit on which the parameterized phase gate is to be executed. For example, the auxiliary qubit is the target qubit of the CNOT gate, while the logical qubit is the control qubit of the CNOT gate. This teleportation of the parameterized single-qubit phase gate into the logical qubit is particularly probabilistic due to the measurement and executes the rotation operator RZ,P(φ) with 50% probability and the rotation operator RZ,P(-φ) with 50% probability on the logical qubit, depending on the result of the measurement of the Pauli Z-operator on the auxiliary qubit. If the measurement shows that the rotation operator R Z,P(-φ) has been performed on the physical qubit, the teleportation described above is carried out with a second auxiliary qubit and the rotation operator R Z,P (2φ) is repeated. If, after measuring the second auxiliary qubit, it is determined that the rotation operator RZ,P(-3φ) was executed on the logical qubit instead of RZ,P(φ), the teleportation described above is repeated with a third auxiliary qubit and the P2023,1398 WO N / FOKUS - 2023P66414WO December 18, 2024 - 19 – Rotation operator R Z,P (4φ) is repeated, and so on, until the measurement of one of the auxiliary qubits shows that the rotation operator R Z,P(φ) was executed on the logical qubit. The teleportation of the rotation operator into the logical qubit described above advantageously allows hardware errors to be detected and / or corrected, particularly when executing the parameterized phase gate on the auxiliary qubit. Such errors would, for example, cause the state space of the logical qubit (code space) to be left. For example, by executing the steps syndrome measurement, correction, syndrome measurement before or after executing the parameterized phase gate on the auxiliary logical qubit using the method described here, it can be determined whether such an error has occurred. If such an error is detected, the parameterized single-qubit phase gate is executed again on a new auxiliary qubit and teleported into the logical qubit to correct the error.According to a further embodiment of the method, entanglement gates in the gate-based quantum circuit act exclusively on directly adjacent physical qubits. Entanglement gates, such as the CNOT gate, act in particular on at least two qubits and entangle the states of these at least two qubits. Directly adjacent physical qubits are in particular adjacent physical qubits, such as qubits arranged next to one another. P2023,1398 WO N / FOKUS - 2023P66414WO December 18, 2024. - 20 –By using entanglement gates that act exclusively on directly adjacent physical qubits, the gate-based quantum circuit can be implemented, in particular, on a quantum computer whose physical qubits, for example, exhibit only nearest-neighbor connectivity, such as quantum computers with superconducting physical qubits. For example, swap gates can be inserted into the gate-based quantum circuit so that entanglement gates act exclusively on directly adjacent physical qubits. The swap gate, in particular, swaps the states of two physical qubits. The gate-based quantum circuit can also be optimized to minimize the number of required swap gates. Entanglement gates in the gate-based quantum circuit can also act simultaneously on all physical qubits within a register.This applies, for example, to ion-based physical qubits in an ion trap, where ions trapped in an ion trap form, for example, a register. Furthermore, a computer program is specified. All features of the method are also disclosed for the computer program, and vice versa. According to one embodiment, the computer program comprises instructions that, when the computer program is executed by a control computer, cause the control computer to execute the method described above on a quantum computer. The quantum computer comprises, in particular, the physical qubits. During operation, the quantum computer can, for example, manipulate the states of the physical qubits and / or P2023,1398 WO N / FOKUS - 2023P66414WO December 18, 2024. - 21 –Apply single-qubit gates and entanglement gates to the physical qubits. Furthermore, a computer-readable storage medium is specified. All features of the method are also disclosed for the computer-readable storage medium, and vice versa. According to one embodiment of the computer-readable storage medium, the computer program described above is stored on the computer-readable storage medium. Furthermore, a quantum computer is specified. All features of the method are also disclosed for the quantum computer, and vice versa. According to one embodiment of the quantum computer, the method described above is executed on the quantum computer.According to a further embodiment of the quantum computer, the physical qubits are realized by electronic states of ions in an ion trap, and the gate-based quantum circuit has at least one input fan-out gate, which is converted into single-qubit gates and a generalized Mølmer-Sørensen gate. In ion-based quantum computers, the basis states of a physical qubit are particularly realized by two electronic states of an ion trapped in the ion trap using electric and / or magnetic fields. The generalized Mølmer-Sørensen gate (GMS gate for short) is also called a global Mølmer-Sørensen gate and is a native entanglement gate for ion-based quantum computers. P2023,1398 WO N / FOKUS - 2023P66414WO December 18, 2024 - 22 – Quantum computer. The effect of the GMS gate on N physical qubits can be described by the following unitary operator or the following unitary 2 N ×2 NMatrix can be described: Where ^ ^ ^ with ^ ∈ {^, ^, ^} one of the three 2×2 Pauli matrices X, Y, or Z acting on the i-th physical qubit. In particular, the indices k and l for the i-th and j-th physical qubit are fixed in all summands. Products of Pauli matrices acting on different physical qubits are to be understood as tensor products. The GMS gate thus acts, for example, simultaneously on all pairs of physical qubits in the ion trap. For example, the GMS gate can be realized for ^ = ^ = ^ for all ^ and ^. Furthermore, ^ ^^ a real phase that can be set when implementing the GMS gate. If ^ ^^is different for at least two pairs of physical qubits, the gate is called a non-uniform GMS gate. With non-uniform GMS gates, it is also possible that the phase for certain pairs of physical qubits is set to ^^^ = 0, while the phase for other pairs of physical qubits is different from zero. If the phase ^^^ = ^ is identical for all pairs of physical qubits, the gate is called a uniform GMS gate. Otherwise, it is a non-uniform GMS gate. In particular, the fan-out input gate can be converted into a single-qubit gate and exactly one non-uniform GMS gate, or into a single-qubit gate and two uniform GMS gates. Thus, a complexity P2023,1398 WO N / FOCUS - 2023P66414WO December 18, 2024 - 23 –The gate-based quantum circuit in ion-based quantum computers can be reduced, for example, by reducing the number of necessary native entanglement gates. According to a further embodiment of the quantum computer, the physical qubits are realized by superconducting elements, and entanglement gates in the gate-based quantum circuit act exclusively on directly adjacent physical qubits. For example, the CNOT gate is an entanglement gate. Furthermore, a quantum circuit is specified. All features of the method for implementing a parameterized phase gate in a quantum error correction code are also specified for the quantum circuit, and vice versa. According to one embodiment, the quantum circuit comprises at least one parameterized phase gate implemented according to the method described above.The quantum circuit comprises, for example, a multitude of quantum gates that act on a multitude of logical qubits. In particular, a quantum error correction code is used to represent the logical qubits by physical qubits. The quantum circuit comprises, for example, single-qubit gates and entanglement gates, in particular Clifford gates and the parameterized phase gate, which act on the logical qubits. The quantum gates of the quantum circuit form, in particular, a universal set of quantum gates, so that the quantum circuit can perform arbitrary unitary operations on the multitude of logical qubits. P2023,1398 WO N / FOCUS - 2023P66414WO December 18, 2024. - 24 –Further advantageous embodiments and developments of the quantum circuit and the method for executing a parameterized phase gate in a quantum error correction code emerge from the exemplary embodiments described below in conjunction with the figures. Figure 1 shows a schematic flow diagram of a method for executing a parameterized phase gate in a quantum error correction code according to an exemplary embodiment. Figure 2 shows an implementation of a gate-based quantum circuit according to an exemplary embodiment, which is configured to execute a step of a method for executing a parameterized phase gate in a quantum error correction code. Figure 3 shows a schematic representation of a computer program according to an exemplary embodiment. Figure 4 shows a schematic representation of a computer-readable storage medium according to an exemplary embodiment.Figures 5 and 6 show schematic representations of quantum computers according to various embodiments. Figure 7 shows a step of a method for executing a parameterized phase gate in a quantum error correction code according to another embodiment.P2023,1398 WO N / FOKUS - 2023P66414WO December 18, 2024. - 25 –Figure 8 shows a quantum circuit according to an embodiment. Identical, similar, or similarly acting elements are provided with the same reference numerals in the figures. The figures and the relative sizes of the elements shown in the figures are not to be considered to scale. Rather, individual elements may be exaggerated for clarity and / or clarity. In the first step VS1 of the embodiment shown in Figure 1 of the method for implementing a parameterized phase gate in a quantum error correction code, a logical qubit is represented by a number of N physical qubits. The number of N is not limited. In the present embodiment, N = 4 is used for simplicity and illustration purposes. For this purpose, N - 1 stabilizer operators Si are selected, which commute with one another, i.e., S i S j = S j Si for all 1 ≤ i, j ≤ N – 1. Each of the N – 1 stabilizer operators consists of a tensor product of single-qubit operators acting on the physical qubits. The single-qubit operators are selected from the group consisting of the three Pauli operators X P , Y P and Z P , as well as the identity operator IP. In other words, each stabilizer operator is a Pauli string and has eigenvalues ​​+1 or -1. For example, one of the stabilizer operators has the form S1 = ZP,1⊗IP,2⊗XP,3⊗YP,4, where the index 1 to 4 of the operators in the tensor product on the right-hand side is P2023,1398 WO N / FOCUS - 2023P66414WO December 18, 2024 - 26 –specifies which of the N = 4 physical qubits the respective operator acts on. Each of the N – 1 stabilizer operators divides the Hilbert space of the N physical qubits into two subspaces corresponding to the eigenvalues ​​+1 and -1 of the respective stabilizer operator. In particular, the state space of the logical qubit is defined as the space that is invariant under all N – 1 stabilizer operators. In other words, Si|0>L = |0>L and Si|1>L = |1>L for all 1 ≤ i ≤ N – 1, where |0> L and |1> Ldenote the basis states of the logical qubit. Syndrome measurement for error correction, for example, involves determining the eigenvalues ​​of the N – 1 stabilizer operators. For example, if the measurement of a stabilizer operator on a prepared state of the logical qubit results in the eigenvalue -1, then an error exists in at least one physical qubit, which can be corrected by applying a corresponding operator without affecting the quantum mechanical information stored in the logical qubit.In the second step VS2 of the exemplary embodiment of the method for implementing a parameterized phase gate in a quantum error correction code shown in Figure 1, a logical Pauli operator acting on the logical qubit is represented by a tensor product of single-qubit operators acting on the physical qubits, wherein the single-qubit operators are selected from the group consisting of the three different Pauli operators and the identity operator. For example, another P2023,1398 WO N / FOCUS - 2023P66414WO December 18, 2024. - 27 – Stabilizer operator SN is constructed, which is a Pauli string and commutes with all N – 1 stabilizer operators constructed in the first step VS1 of the method. The further stabilizer operator SN is constructed, for example, using a brute-force search. This further stabilizer operator S Ncorresponds to the logical Pauli operator, which acts on the basis states of the logical qubit. For example, the further stabilizer operator S N with one of the logical Pauli operators X L , Y L , or Z Lidentified. Any other computational steps can be executed on the logical qubits before and / or after the parameterized phase gate. In particular, a quantum algorithm or quantum circuit is executed on the logical qubits, wherein the parameterized phase gate is one of the quantum gates executed on the logical qubits. For example, single-qubit gates and / or entanglement gates, in particular Clifford gates, are applied to the logical qubits before and / or after the parameterized phase gate. The parameterized phase gate can be applied one or more times to one or more logical qubits.In the third step VS3 of the exemplary embodiment of the method for executing a parameterized phase gate in a quantum error correction code shown in Figure 1, the parameterized phase gate is executed on the logical qubit by applying a matrix exponential of the logical Pauli operator, which was represented by the Pauli string in step VS2, to the physical qubits. In doing so, the P2023,1398 WO N / FOCUS - 2023P66414WO December 18, 2024. - 28 – Matrix exponential of the logical Pauli operator is realized by a gate-based quantum circuit acting on the N physical qubits. For example, the logical Pauli operator of the Z L operator and the parameterized phase gate is the rotation operator R Z,L (φ) = exp(-i∙φ∙Z L / 2), by which the state of the logical qubit is rotated by an angle φ around the z-axis of the Bloch sphere. Where ^ ^ =⊗ ^ ^ ^^ ^ ^with the single-qubit operator ^^ ∈ {^^ , ^^ , ^^ , ^^} acting on the i-th physical qubit. The logical Pauli operator can also be X L or Y L , where the parameterized phase operator then corresponds to the rotation operator R X,L (φ) or R Y,L (φ). If the Pauli string of the logical Pauli operator ZL has at least one identity operator IP acting on one of the physical qubits, these identity operators can be extracted from the exponent of the rotation operator RZ,L(φ). Thus, the parameterized phase gate can always be represented as where 1 ≤ m ≤ N. The physical qubits were rearranged or renumbered in such a way that all identity operators I P,j in the logical Pauli operator Z Lwere pushed to the end of the Pauli string. The indices i and j number the physical qubits. Furthermore, the following identities for Pauli operators XP and YP^^ = ^^^^^, ^^ = ^ ^ ^ ^ ^P2023,1398 WO N / FOCUS - 2023P66414WO December 18, 2024 - 29 – used to calculate the Pauli operators X P and Y P in the Pauli string of the logical Pauli operator Z L by the Pauli operator Z P Here, S denotes the phase gate and H the Hadamard gate. Thus, the Pauli string of the logical Pauli operator ZL can be represented in the exponent of the parameterized phase gate as where U is a unitary operator consisting of the single-qubit operators H, S, and / or I. Due to the unitarity of U, this operator can be extracted from the exponent of the deparameterized phase gate, which can thus be represented as follows: This matrix exponential can be realized by a gate-based quantum circuit, as shown in connection with Figure 2. Figure 2 shows a realization of the matrix exponential described in connection with step VS3 in Figure 1 as a gate-based quantum circuit 1, which is based on the N physical qubitsqb 1…N As explained in connection with Figure 1, the operators U + and U from the single-qubit operators S, H and I, and can thus be directly represented as gates in the gate-based quantum circuit 1. Therefore, for the sake of simplicity, the gate-based quantum circuit 1 shown in Figure 2 is limited to the matrix exponential The gate-based quantum circuit 1 in Figure 2 consists of a first input fan-out gate E1 of N - 1 CNOTGats, a subsequent parameterizedP2023,1398 WO N / FOCUS - 2023P66414WO December 18, 2024 - 30 – Single-qubit phase gate ^ (φ) = exp ^^ ^,^ −^ ^ ^ ^,^ ^, which acts on the physical qubit qb1, and a subsequent second input fan-out gate E2 consisting of N - 1 CNOT gates. The parameterized phase gate ^^,^(φ) acts on the common target qubit qb1 of the first and second input fan-out gates E1, E2. A generalization of this gate-based quantum circuit 1 for any number N of physical qubits qb1…N is evident. The proof of the equivalence of the gate-based quantum circuit 1 shown in Figure 2 with the matrix exponential exp^−^ φ / 2 ⊗ ^ ^ ^^ ^ ^,^^, where quantum circuit 1 in Figure 2 is shown as an example for m = 4, can be derived by complete induction. For m = 1, the equivalence of the gate-based quantum circuit in Figure 2 with the matrix exponential is correct by definition. Assuming that the representation in Figure 2 holds for m qubits, it can be shown as follows that this representation is also correct for m + 1 qubits: CNOT refers to m+1,1 a CNOT gate that acts on the physical qubits qb1 and qb m+1 acts, with qb1as target qubit and qb m+1 as a control qubit. The last equation holds because the execution of two identical CNOT gates in succession corresponds to an identity operation, i.e. (CNOT m+1,1 ) 2 = I P,1 ⊗I P,m+1 , and there ^^^^ ^^^,^ ^ ^,^ ^^^^ ^^^,^=^^,^ ⊗ ^^,^^^ holds. The latter can be shown by evaluating matrix elements of both sides of the equation in the computational state basis. P2023,1398 WO N / FOCUS - 2023P66414WO December 18, 2024 - 31 –The computer program 10 according to the embodiment in Figure 3 causes a control computer to execute the method described here for executing a parameterized phase gate in a quantum error correction code on a quantum computer. The computer program 10 described in connection with Figure 3 is stored on the computer-readable storage medium 11 according to the embodiment in Figure 4. The computer-readable storage medium 11 is, for example, an electronic storage medium, e.g., a flash memory, a magnetic storage medium, e.g., a hard disk, or an optical storage medium, e.g., a DVD (digital versatile disc, abbreviated to DVD). The quantum computer 100 according to the embodiment in Figure 5 comprises an ion trap 101, for example, a Penning trap or a Paul trap, which generates electric and / or magnetic fields for confining one or more electrically charged ions 104.Electronic states of the trapped ions 104 are used to realize the physical qubits. Two electronic states of an ion 104 correspond to the two basis states of a physical qubit qb1, qb2. Quantum gates are implemented in particular by laser pulses generated by a laser 102 and / or by microwave pulses, which act on the electronic states of one or more trapped ions 104. The electronic states of the trapped ions 104 are read out, for example, by detecting fluorescence of the trapped ions 104 with a photodetector 103. P2023,1398 WO N / FOKUS - 2023P66414WO December 18, 2024. - 32 – The quantum computer 100 implements the method described in connection with Figure 1 for executing a parameterized phase gate R Z,L(φ) in a quantum error correction code. The input fan-out gates E1, E2 of the gate-based quantum circuit 1 described in connection with Figure 2 are realized by generalized Mølmer-Sørensen gates, which represent an entanglement of several physical qubits qb 1…N by a single gate. The logical parameterized phase gate R Z,L (φ) can thus be executed on ion trap-based quantum computers 100 with constant depth, ie the number of sequentially consecutive operations on the physical qubits qb 1…N acting gates to execute the parameterized phase gate R Z,L (φ) on the logical qubit does not depend on the number N of physical qubits qb 1…Nwhich are used to represent the logical qubit. The quantum computer 100 according to the embodiment in Figure 6 comprises a superconducting circuit with a plurality of superconducting qubits 105, which are designed as superconducting resonant circuits with resonant frequencies in the radio frequency range or in the microwave frequency range. Each superconducting qubit 105 realizes a physical qubit qb1...N. The resonant circuits each have at least one Josephson junction 106 with a nonlinear inductance, so that each resonant circuit realizes an anharmonic oscillator. The basis states |0> P and |1> P the physical qubits qb 1…N are specifically identified with a ground state and an excited state of an associated anharmonic oscillator.P2023,1398 WO N / FOCUS - 2023P66414WO December 18, 2024 - 33 –Single-qubit gates are realized, for example, by applying radio frequency or microwave pulses to the oscillating circuit of a superconducting qubit 105. Entanglement gates can be realized by controlled coupling of neighboring physical qubits 105, in particular directly adjacent oscillating circuits. The architecture or arrangement of the plurality of physical qubits 105 can limit the number of physical qubits 105 that can be directly entangled via gate operations. This limits, for example, the application of CSS codes in which stabilizer operators are applied to a plurality of physical qubits qb. 1…Nact on superconducting quantum computers 100. In contrast, topological quantum error correction codes such as the torus code or the surface code can be realized in superconducting quantum computers 100, since the stabilizer operators of topological quantum error correction codes, for example, only act on directly adjacent physical qubits 105. The logical parameterized phase gate R Z,L (φ) can be executed in a surface code or torus code on superconducting quantum computers 100, for example, with logarithmic depth. In other words, the number of sequentially consecutive qubits qb 1…N acting gates for executing the logical parameterized phase gate R Z,L (φ) proportional to log(N), where N is the number of physical qubits qb 1…Nused to represent the logical qubit. This follows, for example, from the fact that the Z acting on the logical qubit L Pauli operator can be represented as a Pauli string acting on a chain of neighboring physical qubits 105.P2023,1398 WO N / FOCUS - 2023P66414WO December 18, 2024 - 34 – Figure 7 shows a further realization of the matrix exponential described in connection with step VS3 in Figure 1 as a gate-based quantum circuit 1, which is based on the N physical qubitsqb 1…N The gate-based quantum circuit 1 shown in Figure 7 is limited, analogous to the gate-based circuit 1 described in connection with Figure 2, for the sake of simplicity to the matrix exponential In contrast to the embodiment described in connection with Figure 2, the CNOT gates do not act on a common target qubit. Instead, two adjacent physical qubits qb 1…N connected via a CNOT gate. A number of N-1 CNOT gates are arranged in a staircase pattern, with each CNOT gate accessing a neighboring pair of qubits qb 1…N This is followed by a parameterized single-qubit phase gate ^^,^(φ), which acts on the first qubit qb1. This is followed by a number of N-1 further CNOT gates arranged in a staircase pattern. The proof of the equivalence of the gate-based quantum circuit 1 shown in Figure 7 with the matrix exponential ^^,^^ can be proved analogously to the proof shown in Figure 2, where ^^^^ ^^^,^ through ^^^^ ^^^,^must be replaced. The gate-based quantum circuit 1 shown in Figure 7 can, for example, be used to implement the logical parameterized phase gate in a surface code or torus code on a quantum computer 100 with P2023,1398 WO N / FOCUS - 2023P66414WO December 18, 2024 - 35 – superconducting qubits 105, since the CNOT gates are particularly only sensitive to neighboring physical qubits qb 1…N The quantum circuit 1000 according to the embodiment in Figure 8 performs a unitary operation to solve a desired problem on the logical qubits L 1…M Each of the logical qubits L 1…M is by using a quantum error correction code, in particular a CSS code or a surface code, by a plurality of physical qubits qb 1…N(not shown here). The quantum circuit 1000 comprises single-qubit gates 1001 and entanglement gates 1002, which form a universal set of quantum gates for representing arbitrary unitary operations on the logical qubits L 1…M For example, the entanglement gates 1002 are CNOT gates, while the single-qubit gates 1001 are selected from the group consisting of the Hadamard gate, the phase gate, and the parameterized phase gate For example, the CNOT gate, the Hadamard gate, and the phase gate generate arbitrary Clifford gates. The parameterized phase gate is in particular according to the method described in connection with Figure 1 on the physical qubits qb 1…NThis patent application claims the priorities of German patent applications DE 102023135998.3 and DE 102024102565.4, the respective disclosures of which are hereby incorporated by reference. The invention is not limited to the description based on the exemplary embodiments. Rather, the invention encompasses every new feature and every combination of P2023,1398 WO N / FOKUS - 2023P66414WO December 18, 2024 - 36 – Features, which in particular includes any combination of features in the patent claims, even if this feature or combination itself is not explicitly stated in the patent claims or embodiments.

[0002] P2023,1398 WO N / FOCUS - 2023P66414WO December 18, 2024 - 37 – List of reference symbols1 gate-based quantum circuit10 computer program11 computer-readable storage medium100 quantum computer101 ion trap 102 Laser 103 Detector 104 Ion105 superconducting qubit 106 Josephson junction 1000 quantum circuit 1001 single-qubit gate 1002 entanglement gate CNOT CNOT gate VS1…VS3 first to third step of the process qb 1…N physical qubit logical qubit X L , Y L , Z L logical Pauli operator X P , Y P , Z P physical Pauli operator identity operator RZ,L(φ) parameterized phase gate RZ,P(φ) single-qubit phase gate E1, E2 input fan-out gate

Claims

P2023,1398 WO N / FOKUS - 2023P66414WO 18. Dezember 2024 - 38 – 1. Quantum circuit (1000) comprising at least one parameterized phase gate (R Z,L (φ)) in a quantum error correction code, where - zumindest ein logisches qubit (L1…M) durch eine Anzahl von N physikalischen qubits (qb1…N) dargestellt ist, - ein auf das logische qubit (L1…M) wirkender logischer Pauli- Operator (X L , Y L , Z L ) by a tensor product of on the physical qubits (qb 1…N ) acting single qubit operators, where the single qubit operators are selected from the group consisting of three different Pauli Operatoren (XP, YP, ZP) und einem Identitätsoperator (IP), - das parametrisierte Phasengatter (RZ,L(φ)) auf dem logischen qubit (L1…M) ausgeführt wird, indem ein Matrixexponential des logical Pauli operator (X L , Y L , Z L ) on the physical qubits (qb1…N) angewandt wird, und - das Matrixexponential des logischen Pauli-Operators (XL, YL, ZL) durch einen gatterbasierten Quantenschaltkreis (1) which is applied to the N physical qubits (qb 1…N ) acts.

2. Method for implementing a parameterized phase gate (R Z,L (φ)) in a quantum error correction code, comprising the steps: - Darstellen zumindest eines logischen qubits (L1…M) durch eine Anzahl von N physikalischen qubits (qb1…N), - Darstellen eines auf das logische qubit (L1…M) wirkenden logischen Pauli-Operators (XL, YL, ZL) durch ein Tensorproduktfrom to the physical qubits (qb 1…N ) acting single qubit operators, where the single qubit operators are selected from the group consisting of three P2023,1398 WO N / FOKUS - 2023P66414WO 18. Dezember 2024 - 39 – verschiedenen Pauli-Operatoren (XP, YP, ZP) und einem Identitätsoperator (IP), - Ausführen des parametrisierten Phasengatters (RZ,L(φ)) auf the logical qubit (L 1…M ) by applying a matrix exponential of the logical Pauli operator (X L , Y L , Z L ) auf die physikalischen qubits (qb1…N), wobei - das Matrixexponential des logischen Pauli-Operators (XL, YL, ZL) durch einen gatterbasierten Quantenschaltkreis (1) which is applied to the N physical qubits (qb 1…N ) acts.

3. Method according to the preceding claim, wherein the logical Pauli-Operator (XL, YL, ZL) ausgewählt wird aus der Gruppe bestehend aus einem X Pauli-Operator, einem Y Pauli-Operator und einem Z Pauli-Operator.

4. Verfahren nach einem der Ansprüche 2 oder 3, wobei das Tensor product of on the physical qubits (qb 1…N ) wirkenden Einzelqubitoperatoren ein Stabilisator-Operator (S) is.

5. Verfahren nach einem der Ansprüche 2 bis 4, wobei - das Matrixexponential des logischen Pauli-Operators (XL, YL, ZL) durch ein Matrixexponential eines Tensorprodukts einer Anzahl von m Z Pauli-Operatoren (ZP) dargestellt wird, with 1 ≤ m ≤ N, - jeder der Z Pauli-Operatoren (ZP) auf ein zugehöriges physical qubit (qb 1…N ) works.

6. Verfahren nach einem der Ansprüche 2 bis 5, wobei the parameterized phase gate (R Z,L (φ)) is represented by P2023,1398 WO N / FOKUS - 2023P66414WO 18. Dezember 2024 - 40 – wobei U ein aus Einzelqubitoperatoren bestehender unitäreroperator acting on the physical qubits (qb 1…N ) works, U+ ein hermitesch konjugierter Operator von U ist, ^ eine Phase of the parameterized phase operator (R Z (φ)), and ZP,i ein auf das i-te physikalische qubit wirkender Z Pauli- Operator (Z P ), and where m with 1 ≤ m ≤ N those of the N physikalischen qubits (qb1…N) bezeichnet, auf die der logische Pauli operator (X L , Y L , Z L ) does not act as an identity operator (I).

7. Method according to one of claims 5 or 6, wherein the gatterbasierte Quantenschaltkreis (1) eine Anzahl von 2∙(m-1) CNOT gates and a parameterized single-qubit phase gate (R Z,P (φ)).

8. Verfahren nach einem der Ansprüche 2 bis 7, wobei - der gatterbasierte Quantenschaltkreis (1) ein erstes Eingangsfächerungsgatter (E1), ein dem ersten Input fan-out gate (E1) subsequent parameterized Einzelqubitphasengatter (RZ,P(φ)), und ein dem parameterized single-qubit phase gate (R Z,P (φ)) has a second input fan-out gate (E2), and - das parametrisierte Einzelqubitphasengatter (RZ,P(φ)) auf a common target qubit (qb1) of the first input fan-out gate (E1) and the second input fan-out gate (E2) acts.

9. The method according to one of claims 2 to 8, wherein the parametrisierte Phasengatter (RZ,L(φ)) auf einem Hilfsqubit ausgeführt wird und in das logische qubit (L1…M) teleportiert becomes. P2023,1398 WO N / FOKUS - 2023P66414WO 18. Dezember 2024- 41 – 10. Verfahren nach einem der Ansprüche 2 bis 9, wobei Entanglement gates in the gate-based quantum circuit (1) are exclusively directed to directly adjacent physical qubits (qb 1…N ) act.

11. Computer program (10) comprising instructions which, when the computer program (10) is executed by a control computer, cause the control computer to carry out the method according to einem der Ansprüche 2 bis 10 auf einem Quantencomputer (100) to execute.

12. Computerlesbares Speichermedium (11), auf dem das Computer program (10) according to claim 11 is stored.

13. Quantencomputer (100), auf dem das Verfahren nach einem of claims 2 to 10.

14. Quantum computer (100) according to the preceding claim, wherein - die physikalischen qubits (qb1…N) durch elektronische States of ions (104) in an ion trap (101) are realized, and - der gatterbasierte Quantenschaltkreis (1) zumindest ein Input fan-out gate (E1, E2) which is converted into single-qubit gates and a generalized Mølmer-Sørensen gate.

15. Quantencomputer (100) nach Anspruch 13, wobei - die physikalischen qubits (qb1…N) durch supraleitende elements (105) are realized, and - Verschränkungsgatter im gatterbasierten Quantenschaltkreis (1) ausschließlich auf direkt benachbarte physikalische qubits (qb1…N).

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