Deutsches Luft- und Raumfahrtzentrum; 53227 Bonn, Deutschland Method for compilation of a multi-qubit gate in the MAGIC architecture DESCRIPTION The invention refers to a computer-implemented method for optimized compilation of a multi-qubit gate in the MAGIC architecture. The invention refers to a computer- implemented method of compilation. The invention refers to a method of implementing a Gzz gate. The invention refers to a computer program product. The invention refers to a control device. The invention refers to a storage device. The invention refers to a device or system, particularly an ion trap and/or a quantum computer. The invention refers to an electronic signal. An ion trap quantum computer based on the magnetic-gradient induced ion coupling principle (MAGIC) has a native interaction that couples all qubits to each other that can be described by the time evolution operator
In order to implement general quantum logical circuits on such a quantum computer, a classical computer has to devise a set of instructions that convert the MAGIC coupling into the desired circuit. Baßler et al. [Bassler2023a] define a collective quantum gate Gzz that performs the following unitary operation on a register of n qubits:
where A is a real symmetric matrix with zero diagonal. This gate can be implemented on a MAGIC ion trap by modulating the native all-to-all coupling with well-timed alternating spin flips. To determine the required spin flips, they define a linear- programming problem of dimension 2n (“two to the power of n”), that aims to minimize the duration of the pulse sequence required to implement the Gzz gate.
NC-2023-1027 2023/267 2 In a second step, Baßler et al. propose to reorder the segments in their solution such that the amount of spin flips is minimized, in order to reduce the noise impact of the spin flips. This second step requires the solution of a traveling-salesman problem of order n(n - 1)/2 for each Gzz gate in the quantum circuit. The resulting compilation algorithms turn out to be NP-hard as described by Baßler et al. as well [Bassler2023b]. [Bassler2023a] Baßler, P., Zipper, M., Cedzich, C., Heinrich, M., Huber, P., Johanning, M., & Kliesch, M. (2023). "Synthesis of and compilation with time-optimal multi-qubit gates", Quantum 7, 984. [Bassler2023b] Baßler, P., Heinrich, M., & Kliesch, M. (2023). "Time-optimal multi- qubit gates: Complexity, efficient heuristic and gate-time bounds", arXiv preprint arXiv:2307.11160. Ion trap quantum computers based on the magnetic-gradient induced ion coupling (MAGIC) principle are still considered limited in their ability to implement general quantum logical circuits. This is mainly due to the native interaction that couples all qubits to each other and the need for a classical computer to devise a set of instructions to convert the MAGIC coupling into the desired circuit. Therefore, the respective calculations as stated above have to be performed, but as already stated optimal compilation of Gzz gates for the MAGIC architecture is an NP-hard problem scaling exponentially in the number of qubits. An NP-hard problem refers to a class of computational problems that are believed to be difficult to solve efficiently using traditional algorithms, even when the inputs are relatively small. It is an object of the invention to improve the quantum computing capabilities. The object is solved by a computer-implemented method for optimized compilation of a multi-qubit gate in the MAGIC architecture with the features according to claim 1. The invention refers to a computer-implemented method of compilation with the features according to claim 9. The invention refers to a method of implementing a Gzz gate with the features according to claim 11. The invention refers to a computer program product with the features according to claim 12. The invention refers to a control device with the features according to claim 13. The invention refers to a storage device with the features according to claim 14. The invention refers to a device or system, particularly an ion trap and/or a quantum computer, with the
NC-2023-1027 2023/267 3 features according to claim 15. The invention refers to an electronic signal with the features according to claim 16. Advantageous embodiments of the invention are the subject of the dependent claims, the description and the figures. Features, feature combinations, technical effects and advantages described in connection with the computer-implemented methods may also apply to the computer program product, the control device, the storage device, the device and/or system, particularly the ion trap and/or quantum computing system. Features, feature combinations, technical effects and advantages described in connection with any of the computer program product, the control device, the storage device, the device and/or system, particularly the ion trap and/or quantum computing system may also apply with respect to any of the computer-implemented methods accordingly. This also applies the other way around, so that with regard to the disclosure of the individual aspects of the invention, reciprocal reference is or can always be made, particularly independent of the category described and/or claimed. According to an aspect, the object is solved by a method according to claim 1. The computer-implemented method can be configured for a compilation of a multi- qubit ion gate Gzz in the MAGIC architecture. The method may comprise the step of determining a basis matrix B for determining a set of spin-flip indices of the spin flip sets Fsn for a sequence of spin-flip pulses. The spin-flip pulses can be separated by time intervals. The basis matrix B is particularly pair-non-singular. It can have a row dimension at least as large as the number of qubits n. It can have a column dimension of at least n(n+1)/2. The spin flip set may comprise spin-flip indices j. The spin-flip indices may fulfill 1 ≤ j ≤ n. The spin-flip indices which may be comprised by the spin-flip sets Fsn can be set as the indices j for which Bjs ≠ Bjs-1 under the definition that Bj0 = BjK+1, particularly Bj0 = BjK+1=1. The method may comprise the step of optimizing the basis matrix B. B may be optimized such that it minimizes a corresponding total number of spin-flips in the sequence of spin-flip pulses, wherein the spin-flip pulses are separated by the time intervals.
NC-2023-1027 2023/267 4 The method may comprise the step of precalculating a set of spin-flip indices of the spin-flip sets Fsn for a number, particularly each, of qubits n based on the basis matrix B. A computer-implemented method may be configured for compiling a multi-qubit Gzz gate in the MAGIC architecture. It can comprise a step of precalculating a set of spin- flip indices of spin-flip sets Fsn based on a basis matrix B. The method can comprise the step of optimizing the basis matrix B to determine the corresponding set of minimized spin-flip indices of spin-flip sets Fsn (particularly representing a minimized set of spin flips) for a sequence of time intervals. This method streamlines the compilation process by precalculating the necessary information, reducing computational complexity and allowing for faster, more efficient gate implementation in the MAGIC architecture. In the context of quantum computing, one example of an NP-hard problem is the calculation of the minimization of the total length of gate implementation regarding spin-flip operations. This operation can comprise finding the shortest sequence of spin flip pulses, which can interact with the respective qubits to implement a Gzz gate for quantum mechanical operations and which can be used to implement a given set of spin-flip operations on a quantum computer. In order to modulate the magnetic-gradient induced ion coupling
into an implementation of the gate ^^^^^^(^^) = ^^ (^^/2)^^^^^^^^^^^^^^^^^^^^, [4] a classical computer can have to find a set of time intervals ^s, s = 1, ..., K, and sets of spin-flip indices Fs ⊂ {1, ..., n} for s = 1, ..., K+1, such that
Equation [5] particularly describes a sequence of spin flip-pulses which can have the corresponding spin-flip indices contained in the spin flip sets Fs from K+1 of those sets. This can then be translated into a sequence of spin-flip pulses acting on the qubits indexed by Fs, separated by time delays ^s in the quantum computer. Particularly, the
NC-2023-1027 2023/267 5 sum of all time delays may be as short as possible in order to minimize the total length of the gate implementation. Such a solution particularly exists with K = n(n - 1)/2 non- zero time intervals. However, finding the optimal solution may be an NP-hard problem. The provided method allows for an improved compilation of Gzz gates for the MAGIC architecture as it particularly is no longer an NP-hard problem and may not scale exponentially in the number of qubits anymore. It thus may allow to be used for more than 10 qubits and the (optimal) compilation based on the described method reduces the requirements of classical computation time and thus allows for practical real-time usage. Even though, the optimization may still be considered a "hard optimization problem", it allows to exclude an effect onto the performance of the compiler, particularly because optimal results can be calculated once and stored for future use. The method allows implementing stochastic optimization algorithms based on simulated annealing that may provide solutions for all n up to 20. In other embodiments it may provide solutions for all n up to 30 and particularly even further. It is therefore possible to precalculate the spin-slip indices for later use in a method of compilation, particularly based on an determined and optimized basis matrix B. Furthermore, the resulting pulse sequence (resulting from a method of compilation) may take into account the coupling of individual qubits to external magnetic fields, to avoid the induction of additional noise in the quantum circuit. This may in particular be based on minimizing the number of spin flip pulses (by optimizing the basis matrix B, particularly during determination of basis matrix B). Additionally or alternatively, it allows reducing the time of the gate implementation by minimizing the pulse interaction and the time to establish a Gzz gate. For a modular system consisting of multiple ion traps coupled by shuttling operations, the proposed method allows for and may comprise a step of minimizing the length of the longest pulse sequence among the pulse sequences in each ion trap, such that the shuttling operations can be coordinated in time. Therefore, not every single length of each Gzz gate would have to be optimized. The described method allows to perform the most computationally intensive part of the compilation in a one-off optimization calculation that can be performed only once for each number of qubits in the ion trap. The results can be stored and retrieved easily, such that the actual compilation step for specific Gzz gate instances may
NC-2023-1027 2023/267 6 require only polynomial compute time, which greatly enhances the scalability of the method. The ion trap can have a capacity N of for example n=20 ions. The trap can be loaded with different loads from n=1 to n=20. These numbers are just one matter of example. Therefore, one could derive an optimized basis matrix Bn for possible loading states of the ion trap. The method allows to implement means to find a good solution and to precalculate them. The method may introduce a condition on the pulse timings which effectively decouple the qubits from external magnetic fields, which improves the coherence time of the quantum computation. The same decoupling condition also allows for a linear scaling of the time intervals in the pulse sequence, such that the pulse timings for multiple traps may be adjusted. Thereby it allows to coordinate the timing of ion shuttlings. This allows optimizing the pulse sequence lengths for multiple coupled traps simultaneously. The same decoupling condition also allows for a straightforward implementation of a zero-noise extrapolation error mitigation scheme by scaling the time intervals in the pulse sequence to several factors and then extrapolating the results to scaling factor zero. Herein, a computer-implemented method can be a method that is performed by a computer program, comprising one or more steps that are executed by a processor, such as a core / central processor unit (CPU), on a graphic processor unit (GPU). Alternatively or additionally, a computer can be a quantum computer (QC). The compilation of a multi-qubit ion gate Gzz may refer to a process of determining the sequence of instructions needed to implement a multi-qubit gate operation in a MAGIC architecture quantum computer. The Gzz gate is a particular type of multi- qubit gate. It may perform a controlled Z-rotation on a group of qubits. The term MAGIC architecture may refer to a quantum computing architecture based on the use of ion traps and magnetic fields to control the state of ions therein. The MAGIC architecture uses a set of pulse sequences, known as spin-flip pulses, to manipulate the state of the ions and perform gate operations. It is described in detail elsewhere herein.
NC-2023-1027 2023/267 7 The step of a precalculating of a set of spin-flip indices Fsn may refer to determining in advance, usually through computational methods, the respective spin-flip indices Fsn. A basis matrix B may be a mathematical matrix that can represent the possible states of an ion qubit system with n qubits. The basis matrix can be used to calculate the set of spin-flip indices (of spin-flip sets Fsn) for each number of qubits n. This basis matrix B may be optimized as the basis matrix Bn. This optimization may refer to modifying the basis matrix to minimize the number of spin-flip pulses used to implement a multi-qubit Gzz gate operation. This optimization process may involve changing the parameters of the basis matrix, adding and/or removing elements from the basis matrix. Further details of the optimization of basis matrix B are as described elsewhere herein. Spin-flip pulses are a particular type of pulse sequence used in the MAGIC architecture to manipulate the state of ion qubits. Spin-flip pulses are particularly designed to induce transitions between different energy levels of the ions, resulting in changes in the spin state of the ions. Respective ions may be Ytterbium ions wherein a respective Zeeman effect may be used based on a magnetic field to split respective electron spin levels. This allows to address the respective levels individually and use them for quantum mechanical calculations. According to an aspect a basis matrix B may be determined for each value up to the capacity N of a MAGIC ion trap. The method for optimizing compilation of multi-qubit gate in a MAGIC architecture described herein particularly refers to a computer- implemented method for optimizing the compilation of a multi-qubit ion Gzz gate in a MAGIC (Magic-Gradient Induced Coupling) architecture as described elsewhere herein. The method can involve determining a basis matrix B for each value up to the capacity N of the MAGIC ion trap. This particularly means that a different basis matrix will be calculated and optimized for each number of qubits being used in the trap, with N representing the maximum number of qubits that can be used. From the calculated and optimized basis matrix B it is possible to precalculate the set of spin- flip indices (of the spin flip sets Fsn). The Gzz gate is a multi-qubit ion gate that particularly uses spin-flip pulses to manipulate the state of multiple qubits simultaneously. The basis matrix B may refer
NC-2023-1027 2023/267 8 to a mathematical representation of a subset of all possible states for each qubit in the MAGIC architecture, and it can be used to determine the set of spin-flip indices Fsn needed to implement a multi-qubit gate operation. By optimizing the basis matrix B, the method can minimize the number of spin-flip pulses required to implement a Gzz gate, making the compilation process more efficient. The advantages of this method lay in avoiding the need to solve an optimization problem for each Gzz gate instance, as the one-off optimization calculation can be performed only once for each number of qubits in a respective ion trap. The results can then be easily stored and retrieved, making the actual compilation step for specific Gzz gate instances require only polynomial compute time. This greatly enhances the scalability of the method and particularly of respective ion traps for quantum computing. Additionally, the method may introduce a condition on the pulse timings which may effectively decouple the qubits from external magnetic fields, improving the coherence time of the quantum computation. The same decoupling condition may also allow for a linear scaling of the time intervals in the pulse sequence, making it easy to adjust the pulse timings for multiple traps and coordinate the timing of shuttling processes. This allows for the optimization of pulse sequence lengths for multiple coupled traps simultaneously. Finally, the same decoupling condition also enables a straightforward implementation of a zero-noise extrapolation error mitigation scheme, by scaling the time intervals in the pulse sequence to several factors and then extrapolating the results to scaling factor zero. Summarizing this aspect of the invention further, it enhances scalability by providing optimized basis matrices for different numbers of qubits, ensuring efficient gate compilation and improved performance in MAGIC architecture. According to an aspect, the following constraints may be used: - equal-sized time-intervals are used such that the Gzz gate becomes equal to the unity operator, particularly of the form
for any value of t; and wherein
NC-2023-1027 2023/267 9 - for equal-sized time-intervals, the resulting sequence has each qubit flipped half of the time such that the overall orientation of each qubit averages out:
Therein, the magnetic-gradient induced ion coupling DJ(t) can be defined as stated in formulas [1] and [3]. A set of time intervals may be given by ^s, s = 1, ..., K, between (spin flip) pulses Xk and sets of spin-flip indices Fs ⊂ {1, ..., n} for s = 1, ..., K+1 may be given for (spin flip) pulses Xk. Therein, K represents the length of the sequence (in pulse numbers). This may then be translated into a sequence of spin-flip pulses acting on the qubits indexed by Fs, particularly separated by time delays ^s in the quantum computer. Particularly, the sum of all time delays can be as short as possible in order to minimize the total length of the gate implementation. Z may be a vector of type Z = (Z1, …, ZN)T collecting all local Pauli Z operators. The Z’s particularly represent the ZZ coupling interactions in realizing a GZZ-gate, particularly of the type of Ising interactions. Those interactions may be fixed. X may refer to so- called X-layers that interleave a time evolution under the Ising Hamiltonian with respective fixed ZZ interactions to flip the sign of some, at least temporarily, of at least one of the ZZ couplings. These X-layers may be implemented by a respective pulse sequence. Jjk particularly gives a measure for the coupling strength between spin j and spin k. lnstead of solving a minimization problem for each Gzz(A) separately as described elsewhere herein, precalculating a set of flip indices Fsn for each qubit number n, may be subject to further constraints to improve the optimization. The constraints may be based on equal-sized time intervals. These equal-sized time intervals can be defined as the duration of each spin-flip pulse summing up to the sequence duration used to implement the Gzz gate. This constraint allows to define the Gzz gate becoming equal to the unity operator. Furthermore, based on the constraint of equal-sized time intervals the resulting sequence allows that each qubit is flipped half of the time, averaging out the overall orientation of each qubit. In this approach, each qubit can be assigned a spin flip with a certain probability, so that over the respective repetitions of the pulse sequence implementing the gate, the average state of each qubit converges to a unique state. This means that the orientation of each qubit may average out over time, resulting in a Gzz gate that acts
NC-2023-1027 2023/267 10 like the unity operator. By using equal-sized time intervals, the method implements that all addressed qubits may be flipped at the same rate, regardless of their position or coupling strength. One advantage of this approach is that it does not require solving the minimization problem for each individual Gzz gate instance. Instead, the optimization problem can be solved once and optimal results can be stored for future use. This is particularly advantageous in cases where the minimization problem is a hard optimization problem, as it means that the performance of the compiler is not affected by the difficulty of the optimization problem. To further optimize the method, a stochastic optimization algorithm based on simulated annealing has been implemented and has found good solutions for all n up to 20. Other, particularly non-stochastic, optimization algorithms may be implemented. According to an aspect, the spin flip set Fsn may be constructed for a sequence of length K of equidistant time length based on a Hadamard matrix of order K with number of time intervals bigger than n(n+1)/2. Therein, a Hadamard matrix can be an orthogonal matrix whose elements take only the values +1 and -1. Hadamard matrices can exist for all dimensions that are multiples of 4 up to dimension 664, and for most multiples of 4 up to 2000. There may be n(n + 1)/2 conditions that can have to be satisfied. This means that the number of time intervals K can be set to be at least of that size. K can be taken to be an integer value strictly greater than n(n + 1)/2 for which a Hadamard matrix may exist. Using a larger value of K allows for more flexibility in the optimization. Preliminary benchmarks indicate that K ~ n2 gives nearly optimal circuit lengths. Therefore, the scaling would be polynomial in the number of qubits. According to an aspect the basis matrix B may be of dimension n x K comprising n distinct rows of a Hadamard matrix. The rows of B can be mutually orthogonal and all elements of B can thus be either +1 or -1. B can be "decoupled" if all rows are orthogonal to the fully positive vector whose elements are all equal to +1. The Hadamard matrix that may be taken to construct a matrix B, may have to be such that all the prerequisites of B can be fulfilled as described elsewhere herein.
NC-2023-1027 2023/267 11 According to an aspect, the at least one basis matrix B may be pair-non-singular and determined for each value of n. The spin flip indices (can also be labeled flip sets) Fsn may be set as the indices j for which Bjs ≠ Bjs-1 under the definition that Bj0 = BjK+1, wherein particularly Bj0 = BjK+1= 1. A pair matrix PB of dimensions n(n-1)/2 x K can be defined. Its rows may correspond to the product of rows from B, such that ^^ ^^ ^^,^^ = ^^^^^^^^^^^^, [8] where each pair of row indices (j,k), 1 ≤ ^^ < ^^ ≤ ^^, can be mapped onto a unique row p in the matrix PB. The basis (matrix) B can be said to be "pair-non-singular" if the matrix resulting by stacking the pair matrix PB in rows below B, is "non-singular". Stacking the pair matrix PB in rows below B may result in a matrix of the dimension [n+n(n-1)/2] x K. This allows that any permutation of the columns of B can also give a valid set of spin flip indices. According to an aspect, the method may comprise a step of performing a search, particularly exhaustive, over possible subsets, particularly all, of cardinality n, of rows of the Hadamard matrix, and possible permutations, particularly all, of the columns of B in order to find the solution that minimizes the number of spin-flips. This implements a, particularly exhaustive, search over all possible subsets of cardinality n of rows of the Hadamard matrix. Furthermore, particularly all, possible permutations of the columns of B are searched in order to find the solution that minimizes the number of spin-flips. This may be implemented by stochastic or non- stochastic optimization algorithms. In an embodiment it may be implemented as a stochastic optimization algorithm based on simulated annealing which can be implemented providing solutions for all n up to 20. In other embodiments it may provide solutions for n up to 30, particularly even more. A stochastic optimization algorithm based on simulated annealing may start from one subset of rows in the Hadamard matrix. It may comprise removing a row and adding a new row and check if it is better in the meaning of comprising less spin flips. This process is repeated to go to a (local) minimum. For some n there can be multiple Hadamard matrices. For multiples of n=4 they exist. Any one of them could be picked if they allow to fulfill the respective conditions of basis matrix B as described elsewhere herein.
NC-2023-1027 2023/267 12 According to an aspect, at least one of the optimized basis matrix Bn or the corresponding set of minimized spin-flip indices of spin-flip sets Fsn may be stored (particularly representing a set of minimized spin flips). Therefore, the search described above can be performed only once for each value of n. Therefore, even though this is a hard optimization problem, it may not affect the performance of the compiler because optimal results can be calculated once and stored for future use. This way a stochastic optimization algorithm based on simulated annealing can be implemented providing solutions for all n up to 20. In other embodiments it may provide solutions for n up to 30, particularly even more. According to an independent aspect, a computer-implemented method of compilation may comprise the step of compiling a GZZ gate, particularly by solving a linear programming problem using the optimized basis matrix Bn and the corresponding set of minimized spin-flips of spin-flip sets Fsn, particularly obtained by a computer- implemented method as described elsewhere throughout the description. Once the optimized basis matrix Bn and the corresponding set of minimized spin-flips of spin flip sets Fsn have been obtained, any Gzz(A) may be compiled by solving a linear programming problem. According to an aspect, the linear programming problem to be solved can be to minimize the time sum ^^^^^^^^ under the condition that
≥ 0 for all j;
= 0, and
= ^^^^^^ with bijective correspondence between pair index p and pairs of row indices (j, k), particularly as described elsewhere herein. The size of this linear programming problem particularly scales as K, which can be of order n2. Therefore, its solution particularly requires only polynomial time on a classical computer. The solution vector ^ may give a valid set of timings to implement the Gzz(A) gate as a pulse sequence in the quantum computer. The optimal basis solution for equidistant time intervals allows to make sure that the resulting solution for most Gzz(A) gate instances can be close to optimal in terms of pulse sequence length and spin flip count, and the solution allows to (always) decouple the qubits from external magnetic field perturbations. Any other vector obtained by adding a positive constant ^ to all elements of the vector ^^ can still be a valid pulse sequence, but it may have a duration that is longer
NC-2023-1027 2023/267 13 by an amount K^. This allows for a straightforward implementation of a zero-noise extrapolation procedure by performing measurements for circuits compiled with different values of ^ and then extrapolating these to the limit of vanishing pulse circuit length. Finally, one can also use this freedom to adjust compilations for adjacent ion traps for which timings have to be synchronized to allow for ion shuttling between the traps at the right moment in the circuit: it allows for compiling each ion trap independently, and then adding a commensurate value of ^ to the time intervals of the traps with the shorter pulse sequences to extend them to the same length as the longest pulse sequence. According to an independent aspect, a method of implementing a Gzz gate may comprise the step of applying a pulse sequence to an ion trap with n elements of qubits, particularly in a quantum computing device and/or quantum computing system. Particularly, the pulse sequence may be determined based on a computer- implemented method as described elsewhere herein. This allows to implement a Gzz gate in an ion trap, in a quantum computing device and/or quantum computing system. Features, feature combinations, technical effects and advantages described in connection with the computer-implemented methods may also apply to the computer program product, the control device, the storage device, the device and/or system, particularly the ion trap and/or quantum computing system. Features, feature combinations, technical effects and advantages described in connection with any of the computer program product, the control device, the storage device, the device and/or system, particularly the ion trap and/or quantum computing system/device may also apply with respect to any of the computer-implemented methods accordingly. This also applies the other way around, so that with regard to the disclosure of the individual aspects of the invention, reciprocal reference is or can always be made, particularly independent of the category described and/or claimed. According to an independent aspect, a computer program product may comprise instructions which, when the computer program product is executed by a computer, for example a control device as described elsewhere herein, allows to run a method according to any of the preceding claims. A "computer program product" may refer to a set of instructions or data that can be executed by a computer system, typically
NC-2023-1027 2023/267 14 stored on a tangible medium such as a disk, CD-ROM, or memory chip (SSD, CCD). It may include software, firmware, and other digital representations of the instructions or data, particularly to perform a method as described elsewhere herein. According to an independent aspect a control device, particularly executing a computer program product as described elsewhere herein, may be configured to perform a method as described elsewhere herein. The term "control device" or "computer device" may refer to any hardware component that controls the operation of a computer system, such as a central processing unit (CPU), graphics processing unit (GPU), memory, or input/output controllers. The respective term can also refer to the respective computer system. According to an independent aspect a storage device may comprise a computer program product as described elsewhere herein. The storage device may be configured to be read by a control device, particularly as described elsewhere herein. Alternatively or additionally, the computer program product may be executed on the control device. Further in particular such that a method as described elsewhere herein can be performed in case the computer program product can be read by a control device. The term "storage device" may refer to any electronic or magnetic device capable of storing data or instructions, including but not limited to hard drives, solid state drives, CD-ROMs, DVDs, and USB flash drives. According to an independent aspect a device or system, particularly an ion trap and/or a quantum computing device and/or quantum computing system may particularly comprise at least one of a computer program product as described elsewhere herein, a control device as described elsewhere herein or a storage device as described elsewhere herein, particularly configured to apply a pulse sequence to an ion trap with n elements of qubits. The pulse sequence, particularly the timing of the pulses, may particularly be determined by any of the methods as described elsewhere herein. Alternatively or additionally, the pulse sequence may be implemented according to a method of implementing as described elsewhere herein. According to an independent aspect an electronic signal may comprise at least one of the optimized basis matrix Bn, the corresponding set of minimized spin-flips (of spin-flip sets Fsn) or the pulse time sequence based thereon, particularly obtained by a computer-implemented method as described elsewhere herein. An electronic signal
NC-2023-1027 2023/267 15 comprising at least one of an optimized basis matrix Bn and/or a corresponding set of minimized spin-flip indices (of spin flip sets Fsn) in the field of computer-implemented inventions for quantum computing may refer to any digital representation of information that can be transmitted over electrical or optical signals, such as binary code, digital signals, or other electronic representations. This can include data structures, data files, and other digital formats that contain information about optimized basis matrix Bn and/or corresponding set of minimized spin-flip indices (of spin-flip sets Fsn) for quantum computing. Exemplary embodiments of the invention are shown in the drawings and are explained in more detail below. It is shown in: Fig.1 a schematic representation of an embodiment of a method; Fig.2 an exemplary depiction of optimal basic matrices Bn for n= 2, 3, ..., 6; and Fig.3 a schematic representation of an embodiment of a system. Some of the figures contain simplified, schematic representations. In some cases, identical reference signs are used for the same, but possibly not identical, elements. Different views of the same elements might be scaled differently. Directions such as "left", "right", "up" and "down" are to be understood in relation to the respective figure and may vary in the individual representations compared to the object depicted. Fig.1 shows a schematic representation of an embodiment of a method 100. The schematic structural representation of an embodiment of a system 302 is shown in Fig.3. Here, both Figs. are described in parallel to describe a functional connecting of the method 100 and the system 302. The computer-implemented method 100 may be configured for compiling a multi-qubit Gzz gate in the MAGIC architecture 300 (see Fig.3). It particularly comprises a step 120 of precalculating a set of spin-flip indices (of spin-flip sets Fsn) based on a basis matrix B 200 (not shown in Fig.1). The method 100 can comprise the step 130 of optimizing the basis matrix B 200 to determine the corresponding minimized sets of spin-flip indices (of spin flip sets Fsn) for a sequence of time intervals. In an embodiment, the basis matrix B 200 may be determined 110. The basis matrix B 200 may be for determining a set of spin-flip indices (of spin-flip sets Fsn) for a
NC-2023-1027 2023/267 16 sequence of spin-flip pulses 331a, 331b (see Fig.3). The spin-flip pulses 331a, 331b can be separated by time intervals. The basis matrix B 200 can be pair-non-singular. Basis matrix B 200 can have a row dimension at least as large as the number of qubits n and can have a column dimension of at least K = n(n+1)/2. The spin-flip indices with 1 ≤ j ≤ n comprised by the spin-flip sets Fsn can be set as the indices j for which Bjs ≠ Bjs-1 under the definition that Bj0 = BjK+1, particularly Bj0 = BjK+1=1. In a step of optimizing 130 the basis matrix B 200 may be set such that it minimizes a corresponding total number of spin-flips in the sequence of spin-flip pulses 331a, 331b, wherein the spin-flip pulses 331a, 331b can be separated by the time intervals. In a step of precalculating 120 a set of spin-flip indices of the spin-flip sets Fsn may be precalculated for a number, particularly each, of qubits n based on the basis matrix B 200. In embodiments the step of determining 110 the basis matrix B 200 and optimizing 130 the basis matrix B 200 may temporarily overlap, particularly at least partially. In embodiments the step of precalculating 120 the spin-flip indices of the spin-flip sets Fsn may follow at least one of the steps of determining 110 and/or the step of optimizing 130 the basis matrix B 200. This method 100 particularly streamlines the compilation process by precalculating the necessary information (the spin-flip indices, particularly in a method 100 wherein the spin flip indices are precalculated 120, such that in a method of compiling 170 only the respective time intervals solution vector ^ can be determined), reducing computational complexity and allowing for faster, more efficient gate implementation in MAGIC architecture 300. In order to modulate the magnetic-gradient induced ion coupling - the MAGIC -
into an implementation of the gate
a classical computer 250 (see Fig.2) can have to find a set of time intervals ^s, s = 1, ..., K, and sets of spin-flip indices Fs ⊂ {1, ..., n} for s = 1, ..., K+1, such that
NC-2023-1027 2023/267 17
This can then be translated into a sequence of spin-flip pulses 330 acting on the qubits 310 indexed by Fs, separated by time delays ^s in a quantum computer. The definitions of the variables are particularly as given elsewhere herein. Particularly, the sum of all time delays ^s may be as short as possible in order to minimize the total length of the gate implementation. Such a solution particularly exists with K = n(n - 1)/2 non-zero time intervals ^s. However, finding the optimal solution may be an NP-hard problem. The provided method 100 allows for an improved compilation of Gzz gates for the MAGIC architecture 300 as it particularly is no longer an NP-hard problem and may not scale exponentially in the number of qubits 310 anymore. It thus may allow to be used for more than 10 qubits and the (optimal) compilation based on the described method 100 may reduce the requirements of classical computation time and thus allows for practical real-time usage. Even though, the optimization may still be considered a "hard optimization problem", it allows to exclude an effect onto the performance of the compiler, particularly because optimal results can be calculated once and stored for future use as shown and described with respect to Fig.2. The method 100 allows implementing stochastic optimization algorithms based on simulated annealing that may provide solutions for all n up to 20. In other embodiments it may provide solutions for all n up to 30 and particularly even further. Other, particularly non-stochastic, optimization algorithms may be used as well. Furthermore, the resulting pulse sequence 330 may take into account the coupling of individual qubits 320 to external magnetic fields 340 as also shown in Fig.3, to respect the induction of additional noise in the quantum circuit. This may in particular be based on minimizing the number of spin flip pulses 331a, 331b. Additionally or alternatively it allows reducing the time of the gate implementation by minimizing the pulse interaction 331 and the time t to establish a Gzz gate. For a modular system consisting of multiple ion traps 301 coupled by shuttling operations, the proposed method allows for and may comprise a step of minimizing the length of the longest pulse sequence 330 among the pulse sequences 330 in each ion trap 301, such that the shuttling operations can be coordinated in time. Therefore, not every single length of each Gzz gate would have to be optimized.
NC-2023-1027 2023/267 18 The described method 100 allows to perform the most computationally intensive part of the compilation in a one-off optimization calculation that can be performed only once for each number of qubits 320 in the ion trap 301. The results can be stored 140 and retrieved easily, such that the actual compilation step 150 for specific Gzz gate instances may require only polynomial computation time, which may greatly enhance the scalability of the method 100. To derive the basis matrix B 200 (in the first, particularly separate, part of the method to be able to perform the compilation 150 thereafter in a method of compiling 170) one condition may be that all time intervals ^s are equal-sized time intervals ^s as also described elsewhere herein. In case of compiling 170 the specific Gzz gate compiling 150 may be implemented such that a certain program or set of programs can be implemented. Therefore, they may differ in their matrix A resulting also in different sets of solution vectors ^^ The time intervals ^s may not be equal-sized time intervals ^s in this case. Therefore, a respective compiling 150 of the Gzz gate may be performed frequently during operation of a quantum computer and thus it may be important that the respective compiling 150 may only require polynomial computation time to optimize the time intervals. The method 100 may introduce a condition on the pulse timings 331 which can effectively decouple the qubits 320 from external magnetic fields 340, which improves the coherence time of the quantum computation. The same decoupling condition also allows for a linear scaling of the time intervals in the pulse sequence, such that the pulse timings 331 for multiple ion traps 301 may be adjusted. Thereby it allows to coordinate the timing of ion shuttlings. This allows optimizing the pulse sequence lengths for multiple coupled traps simultaneously. The same decoupling condition also allows for a straightforward implementation of a zero-noise extrapolation error mitigation scheme by scaling the time intervals in the pulse sequence to several factors and then extrapolating the results to scaling factor zero. The computer-implemented method 100 can be a method 100 that may be performed by a computer program product 210, comprising one or more steps that are executed by a processor as control device 230, such as a core / central processor unit (CPU), on a graphic processor unit (GPU). A control device 230 in
NC-2023-1027 2023/267 19 form of a classical computer 240 having a storage device 220 holding a respective computer program product 210 is shown in Fig.2. Alternatively or additionally, a computer can be a quantum computer (QC). The compilation of a multi-qubit ion gate Gzz may refer to a process of determining the sequence of instructions needed to implement a multi-qubit gate operation in a MAGIC architecture 300 quantum computer. The Gzz gate is a particular type of multi-qubit gate. It may perform a controlled Z-rotation on a group of qubits. The term MAGIC architecture 300 may refer to a quantum computing architecture based on the use of ion traps 301 and magnetic fields 340 to control the state of ions as qubits 310 therein. The ions as qubits 310 may be held in place using a potential 350 and magnetic fields 340 may be set to implement the MAGIC architecture 300. It uses a set of pulse sequences 330, known as spin-flip pulses 331a, 331b, to manipulate the state of the ions and perform gate operations. The spin-flip pulses 331a, 331b may differ in their respective frequency to address individual ions in the ion trap as individual qubits 320. The step 120 of a precalculating of a set of spin-flip indices Fsn may refer to determining in advance, usually through computational methods, the respective spin- flip indices of spin-flip sets Fsn. A basis matrix B 200 may be a mathematical matrix that can represent a subset of the possible states, possibly all possible states, of an ion qubit system with n qubits 320 in an ion trap 301. The basis matrix can be used to calculate the set of spin-flip indices (of spin-flip sets Fsn) for each number of qubits n. A respective set of optimal basis matrices Bn 202, 203, 204, 205, 206 for n = 2, 3, ... , 6 are shown in Fig.2. They may be stored 140 along with the respective set of minimized spin-flip indices Fsn on a storage device 210 as also shown in Fig.2. They may be provided as electronic signal 240 to transfer the instructions for implementing a pulse sequence as shown schematically in Fig.3 to implement 160 a Gzz gate. This basis matrix B 200 may be optimized as the basis matrix Bn 202, 203, 204, 205, 206. This optimization may refer to modifying the basis matrix to minimize the number of spin-flip pulses 331a, 331b used to implement a multi-qubit Gzz gate operation. This optimization process may involve changing the parameters of the
NC-2023-1027 2023/267 20 basis matrix B 200, adding and/or removing rows and/or columns from the basis matrix B 200. Spin-flip pulses 331a, 331b are a particular type of pulse sequence 330 used in the MAGIC architecture 300 to manipulate the state of ion qubits 320. Spin-flip pulses 331a, 331b are particularly designed to induce transitions between different energy levels of the ions, resulting in changes in the spin state of the ions. Respective ions may be Ytterbium ions wherein a respective Zeeman effect may be used based on a magnetic field 340 to split respective electron spin levels (not shown). This allows to address the respective levels individually and use them for quantum mechanical calculations. A basis matrix B 200 may be determined for each value up to the capacity N of a MAGIC ion trap 301. The method can involve determining a basis matrix B for each value n up to the capacity N of the MAGIC ion trap. This particularly means that a different basis matrix can be calculated and optimized for each number of qubits 320 being used in the ion trap 301 as exemplarily shown in Fig.2, with N representing the maximum number of qubits 320 that can be used. The Gzz gate is a multi-qubit ion gate that particularly uses spin-flip pulses 331a, 331b to manipulate the state of multiple qubits 320 simultaneously. The basis matrix B may refer to a mathematical representation of all possible states for each qubit in the MAGIC architecture 300, and it can be used to determine the set of spin-flip indices (of spin-flip sets Fsn) needed to implement a multi-qubit gate operation. By optimizing 130 the basis matrix B 200, the method 100 can minimize the number of spin-flip pulses 331a, 331b required to implement a Gzz gate, making the compilation process more efficient. The advantages of this method 100 lay in avoiding the need to solve an optimization problem for each Gzz gate instance, as the one-off optimization calculation can be performed only once for each number of qubits 320 in a respective ion trap 301. The results can then be easily stored and retrieved, making the actual compilation step 150 for specific Gzz gate instances to require only polynomial compute time. This may greatly enhance the scalability of the method 100 and particularly of respective ion traps 320 for quantum computing.
NC-2023-1027 2023/267 21 Additionally, the method may introduce a condition on the pulse timings 331 which may effectively decouple the qubits 320 from external magnetic fields 340, improving the coherence time of the quantum computation. The same decoupling condition may also allow for a linear scaling of the time intervals in the pulse sequence 330 (potential optical or magnetic or radio frequency pulses), making it easy to adjust the pulse timings 331 for multiple ion traps 320 and coordinate the timing of shuttling processes. This allows for the optimization of pulse sequence lengths for multiple coupled traps simultaneously. The same decoupling condition may enable a straightforward implementation of a zero-noise extrapolation error mitigation scheme, by scaling the time intervals in the pulse sequence to several factors and then extrapolating the results to scaling factor zero. Summarizing the described method 100 further enhances scalability by providing optimized basis matrices Bn 202, 203, 204, 205, 206 for different numbers n of qubits 320, ensuring efficient gate compilation and improved performance in MAGIC architecture 300.To run the optimization 130 the following constraints may be useful - equal-sized time-intervals can be used such that the Gzz gate becomes equal to the unity operator, particularly of the form
any value of t; and wherein - for equal-sized time-intervals, the resulting sequence may each have qubit 320 flipped half of the time such that the overall orientation of each qubit averages out:
lnstead of solving a minimization problem for each Gzz(A) separately as described elsewhere herein, precalculating a set of flip indices Fsn for each qubit number n, may be subject to further constraints to improve the optimization. The constraints may be based on equal-sized time intervals. These equal-sized time intervals can be defined as the durations between each set of spin-flip pulses summing up to the sequence duration used to implement the Gzz gate. This constraint allows to define the Gzz gate becoming equal to the unity operator. Furthermore, based on the constraint of
NC-2023-1027 2023/267 22 equal-sized time intervals the resulting sequence allows that each qubit is flipped half of the time, averaging out the overall orientation of each qubit. In this approach, each qubit 320 can be assigned a spin flip with a certain probability, so that over the respective repetitions of the pulse sequence 330 implementing the gate, the average state of each qubit 320 converges to a unique state. This means that the orientation of each qubit may average out over time, resulting in a Gzz gate that acts like the unity operator. The spin-flip set Fsn may be constructed for a sequence of length K of equidistant time length based on a Hadamard matrix of order K with number of time intervals bigger or equal to n(n+1)/2. Therein, a Hadamard matrix can be an orthogonal matrix whose elements take only the values +1 and -1. Hadamard matrices can exist for all dimensions that are multiples of 4 up to dimension 664, and for most multiples of 4 up to 2000. There may be n(n + 1)/2 conditions that can have to be satisfied. This means that the number of time intervals K can be set to be at least of that size. K can be taken to be an integer value strictly greater than n(n + 1)/2 for which a Hadamard matrix may exist. Using a larger value of K allows for more flexibility in the optimization. The basis matrix B 200 may be of dimension n x K comprising n distinct rows of a Hadamard matrix (see in Fig.2). The rows of B can be mutually orthogonal and all elements of B can thus be either +1 or -1. B can be "decoupled" if all rows are orthogonal to the fully positive vector whose elements are all equal to +1. The at least one basis matrix B 200 may be pair-non-singular and determined for each value of n. The spin flip indices Fsn may be set as the indices j for which Bjs
Bjs-1 under the definition that Bj0 = BjK+1. A pair matrix PB of dimensions n(n-1)/2 x K can be defined. Its rows may correspond to the product of rows from B, such that ^^ ^^ ^^,^^ = ^^^^^^^^^^^^, where each pair of row indices (j,k), 1 ≤ ^^ < ^^ ≤ ^^, can be mapped onto a unique row p in the matrix PB. The basis B can be said to be "pair-non-singular" if the matrix resulting by stacking the pair matrix PB in rows below B, is "non-singular". This allows
NC-2023-1027 2023/267 23 that other permutations of the columns of basis matrix B 200 can also give a valid set of spin-flip indices (particularly spin-flip sets). The method 100 may comprise a step 120 of performing a search, particularly exhaustive, over possible subsets, particularly all, of cardinality n of rows of the Hadamard matrix, and possible permutations, particularly all, of the columns of B in order to find the solution that minimizes the number of spin-flips. This implements a, particularly exhaustive, search 120 over all possible subsets of cardinality n of rows of the Hadamard matrix. Furthermore, particularly all, possible permutations of the columns of basis matrix B 200 are searched in order to find the solution that minimizes the number of spin-flips. At least one of the optimized basis matrix Bn 202, 203, 204, 205, 206 or the corresponding set of minimized spin-flips of spin-flip sets Fsn may be stored 140. Therefore, the search 120 described above can be performed only once for each value of n (up to the capacity N of the ion trap used). Therefore, even though this is a hard optimization problem, it may not affect the performance of the compiler because optimal results can be calculated once and stored 140 for future use. To summarize, the rows of the basis matrix B 200 can represent the number of qubits n, and the columns can represent the progression in time which can represent an orientation of the qubits relative to their initial orientation. The optimization of Bn allows to implement pulses after compiling a Gzz gate, wherein it is possible to address only the spin flips of interest based on the resonance frequency of the qubits in the potential they sit in in the ion trap. All other qubits might not be addressed as they do not resonate with the respective pulses X (for example microwave pulses or radio frequency pulses). One can also implement combined spin flips addressing for example two qubits with a respective pulse overlapping the respective resonance frequencies. As matter of example, the basis matrix B4204 for n=4 may solely comprise a column dimension of K=12 as there is no Hadamard matrix for K=10. Thus, the next bigger K where a Hadamard matrix exists can be taken, here K=12. Therefore, there can be a certain degree of freedom as it is not mandatory that a spin flip occurs between each column. Therefore, columns may exist where the respective values are identical between the columns. The respective time interval that can be linked to it may thus
NC-2023-1027 2023/267 24 pass when a Gzz gate is compiled without implementing a respective pulse at the respective time interval. Also basis matrix B5205 for n=5 can comprise identical columns. A computer-implemented method 170 of compilation may comprise the step 150 of compiling a GZZ gate, particularly by solving a linear programming problem using the optimized basis matrix Bn 202, 203, 204, 205, 206 and the corresponding set of minimized spin-flips Fsn, particularly obtained by a computer-implemented method 100 as described elsewhere throughout the description. Once the optimized basis matrix Bn 202, 203, 204, 205, 206 and the corresponding set of minimized spin-flips Fsn have been obtained, any Gzz(A) may be compiled 150 by solving a linear programming problem. The linear programming problem to be solved can be to minimize the time sum ^^^^^^^^ under the condition that ^
≥ 0 for all j;
= ^^^^^^ with bijective correspondence between pair index p and pairs of row indices (j, k), particularly as described elsewhere herein. For each program to be implemented in the quantum system there can be a number of matrices ^^^^^^. Therefore, the vector ^ could be different for each of them. This can be optimized by solving the above-mentioned linear programming problem for the vector ^. This operation can be optimized many times during operation of the quantum system. Therefore, it is an advantage that the size of this linear programming problem particularly scales as n2/2. Therefore, its solution particularly requires only polynomial time on a classical computer. The solution vector ^ may give a valid set of timings to implement the Gzz(A) gate as a pulse sequence in the quantum computer. The optimal basis solution for equidistant time intervals allows to make sure that the resulting solution for most Gzz(A) gate instances can be close to optimal in terms of pulse sequence length and spin flip count, and the solution allows to (always) decouple the qubits from external magnetic field perturbations. Any other vector obtained by adding a positive constant ^ to all elements of the vector ^^ can still be a valid pulse sequence, but it may have a duration that is longer by an amount K^. This allows for a straightforward implementation of a zero-noise extrapolation procedure by performing measurements for circuits compiled with
NC-2023-1027 2023/267 25 different values of ^ and then extrapolating these to the limit of vanishing pulse circuit length. Finally, one can also use this freedom to adjust compilations for adjacent ion traps for which timings have to be synchronized to allow for ion shuttling between the traps at the right moment in the circuit: it allows for compiling each ion trap independently, and then adding a commensurate value of ^ to the time intervals of the traps with the shorter pulse sequences to extend them to the same length as the longest pulse sequence. Furthermore, a method of implementing a Gzz gate may comprise the step of applying a pulse sequence 330 to an ion trap 301 with n elements of qubits 320, particularly in a quantum computing device 302 and/or quantum computing system. This allows to implement a Gzz gate in an ion trap 301, in a quantum computing device 302 and/or quantum computing system. In particular, the terms "may" or “can” refer to optional features of the invention. Consequently, there are also further aspects and/or embodiments of the invention which additionally or alternatively have the respective feature or features. All features in feature combinations are also disclosed independently thereof and may also be singled out from the combinations of features disclosed herein and used in combination with other features to specify the subject-matter of the any of the claims, dissolving any structural and/or functional relationship that may exist between the features. Terms like “first”, “second”, “third” may be used to refer to a list of elements but does not necessarily describe these features or elements according to their importance, their order of appearance or order of structure. Therefore, these elements or features may specify the different aspects in any other particular order, unless explicitly expressed otherwise.
NC-2023-1027 2023/267 26 LIST OF REFERENCE SIGNS 100 computer-implemented method 110 step of determining a basis 120 step of precalculating 130 step of optimizing 140 step of storing 150 step of compiling 160 method of implementing 170 method of compiling 200 basis 202, 203, 204, 205, 206 optimized basis 210 computer program product 220 storage device 230 control device 240 electronic signal 250 computer 300 MAGIC architecture 301 ion trap 302 quantum mechanical device 310 qubits 320 ion couplings 330 pulse sequence 331 pulse interaction time points 331a,b spin flip pulses
NC-2023-1027 2023/267 27 ^s time delay t time