Method of frequency spectrum shaping of a control pulse, method of generating a control pulse, and corresponding apparatus, quantum computer and quantum sensor

Abstract

Provided is a method for shaping a frequency spectrum of a control pulse with a given control pulse duration. In one aspect, one or more finite frequency ranges to be suppressed in the frequency spectrum are determined, at least one of the finite frequency ranges defined based on one of the predetermined frequencies. Then, a spectral energy of the control pulse over the one or more finite frequency ranges is minimized and at least one of the parameters of the shaped control pulse is determined through the minimizing of the spectral energy. In this manner, the pulse can be shaped not to contain determined finite frequency ranges. Provided is also a method for generating a control pulse with a given pulse duration, said control pulse, an apparatus for generating said control pulse and a quantum computer comprising said apparatus.

Description

[0001]256254 s5 / s32 / sca Method of frequency spectrum shaping of a control pulse, method of generating a control pulse, and corresponding apparatus, quantum computer and quantum sensor TECHNICAL FIELD This present disclosure relates to a method for shaping a frequency spectrum of a control pulse, in particular to suppress one or more predetermined or unwanted finite frequency ranges in the control pulse. The present disclosure also relates to a method and apparatus for generating a control pulse, as well as a quantum computer and a quantum sensor applying the generated control pulse. BACKGROUND A specific and precise form, envelope or shape of a control pulse is required in many technical applications. For example, in gate-based quantum computing, a quantum computation may be performed by applying single- and two-qubit quantum logic gates on an array of qubits in a quantum processing unit (QPU). In many of the quantum computing architectures, these quantum logic gates are implemented by applying electric or magnetic control pulses to the qubits, for example, in the microwave regime. A physical QPU comprises physical qubits that may possess more than two energy levels, in which case two of the energy levels are used as the computational qubit subspace, and the control pulses need to be created in a manner to activate transitions within the computational subspace while avoiding transitions out of it. To reach quantum advantage in a computational problem with a practical application, the quantum logic gates should have sufficiently low error rates and the QPU should incorporate a sufficiently high number of qubits, the desired error rates and number of qubits being notably dependent on the application. In contemporary quantum processing units, the gate errors are mainly caused by incoherent errors related to decoherence, coherent control errors, leakage errors out of the computational qubit subspace, as well as residual interactions between the qubits. In particular, leakage errors describe an error mechanism, in which a physical qubit evolves into a quantum state that is not fully contained within the qubit subspace. Leakage errors occur when a transition between the computational subspace and the non-computational subspace is activated, for example, due to thermal excitations or microwave fields used to implement gate operations. Leakage errors are especially problematic for qubit systems with a low anharmonicity, such as transmons, and for QPU architectures requiring neighbouring qubits to have transition frequencies of similar magnitude, resulting in frequency crowding issues, i.e. issues, for 256254 s5 / s32 / sca example, with addressing individual qubits without perturbing other qubits with close-by transition frequencies. Leakage errors may be classified into different types based on the origin of leakage and the state into which the leakage occurs. Some of the most common types of leakage errors in superconducting QPUs include: Type 1: Leakage from the computational subspace of a qubit to a third state, or higher states, due to its own control signal. Type 2: Leakage from the computational subspace of a qubit to the third state or higher states, due to a control signal of its neighboring qubit mediated by either classical microwave crosstalk or hybridization (quantum) crosstalk. Classical microwave crosstalk may refer to crosstalk arising from the means of exerting control on a qubit, such as for example crosstalk between transmission lines or the presence of stray currents. Here, quantum crosstalk refers to the effective crosstalk arising from the finite coupling strength between the qubits. Type 3: Leakage to a coupler qubit due to imperfect population swapping between the computational qubit and a coupler qubit during a two-qubit gate. Type 4: Leakage to some other parasitic mode(s). Typically, the magnitude of leakage errors increases rapidly as the duration of quantum logic gates is reduced due to an increased spectral width of short control pulses. This sets a lower bound for the duration of quantum logic gates, which both limits the incoherent error per gate as well the clock speed of the QPU, leading to longer runtime of quantum algorithms. To date, a variety of approaches have been proposed with the goal to mitigate leakage errors of different types. Most of the approaches are based on creating the control pulse for the quantum gate to reduce leakage in one way or another. Single-qubit gates are conventionallyimplemented using the DRAG (Derivative Removal by Adiabatic Gate) technique [1] that aimsto counteract unwanted transitions induced by a drive control pulse. Namely, the DRAG technique enables one to suppress the spectrum of the control pulse completely at a single specific frequency that can be set to a known leakage transition frequency. To achieve this suppression, the quadrature component (Q-component) of the control pulse is chosen to be proportional to the derivative of the in-phase component (I-component) of the control pulse. The proportionality constant, known as the DRAG parameter, is obtained for example by means of simulation or experimental optimization. 256254 s5 / s32 / sca Another approach for reducing leakage errors in single- and two-qubit gates is to use a so- called Slepian pulse envelope also known as discrete prolate spheroidal sequence [2] that is also commonly used in signal processing beyond superconducting quantum computation. The Slepian pulse shape helps to reduce the spectral energy of the pulse above a given cutoff frequency, thus reducing leakage due to transitions differing from the drive frequency by more than the cutoff frequency. In addition to the Slepian pulse shape, there exist other pulse shapes that may enable the reduction of either type-1 or type-2 leakage errors under some circumstances. An important example of these is the Wah-Wah pulse [3] that enables the reduction of type-2 leakage errors under certain conditions. It has also been suggested in several past simulation and experimental studies that type-1 and type-2 leakage errors can be reduced using either open-loop or closed-loop optimization of control pulses with an appropriate parametrization. For example, it has been proposed to use a smooth pulse shape based on a Fourier series combined with iterative closed-loop optimization of the gate fidelity in order to reduce the simulated leakage rate of simultaneous single-qubit gates in a two-qubit system by orders of magnitude [4]. It has further been demonstrated that closed-loop optimization of piecewise constant single- qubit pulse shapes can significantly reduce the leakage rate of sub-5-ns single-qubit gates applied to a transmon qubit [5]. One further possibility is to numerically optimize the pulse shape in a simulation based on a known or model Hamiltonian of (a subset of) a QPU, and then transfer the optimal pulse shapes to an actual QPU. This approach is considered in WO 2019 / 152019 A1, where an optimization framework and a cost function for leakage minimization based on the Hamiltonian of the system is disclosed. One further approach to reduce type-2 leakage errors is to use active microwave cancellation to cancel crosstalk-mediated electric fields experienced by a qubit due to a control signal applied to one of its neighboring qubits. SUMMARY Technical Problem Even though there exist several approaches for reducing leakage errors as listed above, all of these approaches have their shortcomings necessitating the development of new methods 256254 s5 / s32 / sca enabling improved leakage suppression, and thus faster and more accurate quantum logic gates. The DRAG framework can be generalized to suppress the spectrum at multiple specified individual frequencies [6], which may be beneficial to suppress multiple harmful transitions. However, this does not allow to overcome leakage errors as well as possible in case that the transition frequencies are not stable in time, for example in the presence of an AC Stark shift, which is a phenomenon causing the energy level of, for example, an atom or artificial atom to be shifted in the presence of an external electric field. In addition, the DRAG framework doesn’t enable any control over the bandwidth of the pulse or the average (or maximum) power of the pulse, both of which may have constraints when implementing a control pulse using actual hardware. Furthermore, the suppression of N individual frequencies with the DRAG framework would require defining a basis envelope function that is N-times differentiable with all the derivatives and the basis envelope itself being preferably smooth functions without discontinuities at the start and end of the pulse. Defining such a function may be challenging and lead to undesirably large peak amplitude / power of the control pulse arising from the higher-order derivatives. The Slepian pulse maximizes the energy in the main lobe, i.e., it minimizes the spectral energy above a given cutoff frequency up to infinity, which helps to reduce leakage errors caused by harmful transitions that differ from the central drive frequency by more than the cutoff frequency. However, this method does not allow to suppress the spectrum across multiple finite frequency ranges. Furthermore, the conventional Slepian pulse is not smooth since it is discontinuous at the start and end of the pulse, which may hamper leakage reduction and cause difficulties in the pulse generation using physical hardware due to sudden sharp variations in the pulse. The problem with the Wah-Wah pulse shape is that type-2 leakage reduction is by default only possible for a certain small range of gate durations. Wah-Wah pulses also do not allow to suppress the spectrum across multiple frequency ranges, and a precise control of the spectrum of Wah-Wah pulses is not possible. The Wah-Wah pulse shape proposed in [3] is also not smooth due to a large discontinuity at the start and end of the pulse resulting in an increased type-1 leakage error. The drawback of iterative optimization methods is that they may be impractical or even infeasible due to the large number of iterations required for convergence and thus a long wall clock time for the calibration. For example, in the approach by Werninghaus et al. [5], the optimization of a single pulse envelope took up to 25 hours. Although the approach of active microwave cancellation has been used successfully in several studies [7, 8, 9], the problem is that the number of calibration measurements and parameters 256254 s5 / s32 / sca scale quadratically in the number of qubits considered. Furthermore, the control pulses for the quantum logic gates are no longer local since they became conditional on the pulses applied to other qubits, as a result of which the approach may not be scalable to implement for a large number of qubits. When considering a specific duration for a pulse, it is also noteworthy that known filter-based methods, such as low-pass filters or band-stop filters, cannot filter out parts of the pulse spectrum without resulting in a distortion of the pulse, and therefore in a potentially significant change in its duration. There is thus a need for new approaches to overcome the above technical disadvantages. Such new approaches should enable lower leakage errors and improve the speed and fidelity of quantum logic operations in quantum computers in a way that is more practical, flexible and efficient. The skilled person understands that such leakage problems can occur in different quantum computer architectures, for example, in superconducting quantum computers. There is also a need for a parametrization of a control pulse shape that would ease the experimental calibration of the pulse parameters: it would be helpful that the pulse parameters would mainly control independent properties of the pulse. For example, the overall amplitude and DRAG coefficient of an on-resonant DRAG pulse can be used to control independently the rotation angle and the rotation axis (phase errors) of the gate, respectively. In general, it would be useful if the overall amplitude would control the rotation angle, DRAG coefficient (or drive frequency or virtual-Z rotations) would control the extent of phase errors, and additional parameters would suppress unwanted transitions in the spectrum of the pulse. If each of the control pulse parameters were to mainly control a single property of the quantum gate, it would then be possible to use error amplification measurements to efficiently minimize coherent errors in a sequential manner by mitigating one coherent error type at a time. Closed-loop optimization using gate fidelity (or infidelity) as an objective function may not be very efficient for mitigating coherent control errors since coherent errors add up quadratically in the infidelity and the number of iterations may be high if a good initial guess is not available. Furthermore, there is a need for a pulse parametrization that would allow the above- described leakage suppression and ease of calibration while abiding to practical hardware constraints, such as limitations on pulse bandwidth and maximum or average power. Thus, it would be beneficial if the pulse parametrization would enable controlling the bandwidth or maximum amplitude of the pulse in order to avoid bandwidth or amplitude limitations introduced by the drive electronics. 256254 s5 / s32 / sca Solution The present disclosure provides a method for shaping a frequency spectrum of a control pulse with a given control pulse duration, the method comprising: determining one or more finite frequency ranges to be suppressed in the frequency spectrum, at least one of the finitefrequency ranges being defined based on one or more respective predetermined frequencies;minimizing a spectral energy of the control pulse over the one or more finite frequency ranges; and determining one or more parameters of the control pulse through the minimizing of the spectral energy. In one aspect, in combination with any embodiment above or below, the at least one of the finite frequency ranges extends across or is located nearby one or more predetermined frequencies. In one aspect, in combination with any embodiment above or below, the control pulse is expressed using a finite weighted sum of basis functions, each of the basis functions being multiplied by respective coefficients in the weighted sum of basis functions and having a duration equal to the given control pulse duration, and the method further comprises: obtaining at least one of the coefficients multiplying each of the basis functions by minimizing the spectral energy of the control pulse across the finite frequency ranges to be suppressed according to a cost function. In one aspect, in combination with any embodiment above or below, the weighted sum of basis functions is implemented as one or a combination of a Fourier series, a weighted sum of trigonometric functions, a Fourier cosine series, a Fourier sine series. In one aspect, in combination with any embodiment above or below, the coefficients in the weighted sum of basis functions are restricted to ensure continuity of the control pulse at the start and end of the control pulse. In one aspect, in combination with any embodiment above or below, the finite frequency ranges to be suppressed are symmetrically located around the center frequency of the pulse and / or the finite frequency ranges to be suppressed are partly or completely overlapping. In one aspect, in combination with any embodiment above or below, the method further comprises: obtaining the coefficients by optimizing the cost function defined with one or more individually definable weights corresponding to each of the one or more finite frequency ranges to be suppressed and wherein the cost function is a weighted sum of a spectral energy across the finite frequency ranges to be suppressed. 256254 s5 / s32 / sca In one aspect, in combination with any embodiment above or below, the weights control an amount by which the spectral energy is reduced within each of the one or more finite frequency ranges to be suppressed. In one aspect, in combination with any embodiment above or below, the optimizing of the cost function is performed under a constraint relating to a predetermined control result of the control pulse and / or the optimizing of the cost function is performed as a quadratic optimization with a linear constraint. In one aspect, in combination with any embodiment above or below, the quadratic optimization is performed by solving a system of linear equations determined by the one or more weights and the one or more finite frequency ranges to be suppressed. In one aspect, in combination with any embodiment above or below, the method further comprises: calibrating amplitude parameters of the control pulse. In one aspect, in combination with any embodiment above or below, the control pulse is summed with another pulse, multiplied with another pulse and / or modulated to convert it to a different frequency and / or summed or multiplied with another pulse after the modulation and / or the control pulse is used alone or as a part of a sequence of control pulses to control a system and / or In one aspect, in combination with any embodiment above or below, the control pulse is used to control one or multiple quantum devices and / or the control pulse is an off-resonant control pulse or a resonant control pulse for implementing a quantum logic gate on one or more physical qubits, and the given control pulse duration is a predefined pulse duration for operating the quantum logic gate. In one aspect, in combination with any embodiment above or below, at least one of the one or more finite frequency ranges to be suppressed respectively correspond to one or more unwanted transitions related to the one or more physical qubits, and wherein the unwanted transitions are leakage transitions beyond a computational subspace or unwanted transitions within the computational subspace. In one aspect, in combination with any embodiment above or below, the one or more physical qubits are one of superconducting qubits, qubits based on trapped ions, spin qubits, qubits based on nitrogen vacancies centers, or qubits based on Rydberg atoms. 256254 s5 / s32 / sca In one aspect, in combination with any embodiment above or below, the superconducting qubits are one of transmon qubits, fluxonium qubits, charge qubits, flux qubits, unimon qubits, or coupling qubits of any type. In one aspect, in combination with any embodiment above or below, a predetermined control result of the control pulse is a rotation angle of the quantum logic gate. In one aspect, in combination with any embodiment above or below, the quantum logic gate is a single-qubit gate or a two-qubit gate. In one aspect, in combination with any embodiment above or below, the control pulse is defined in time domain by an in-phase component and a quadrature component, and wherein the frequency spectrum is shaped for an envelope pulse of the in-phase component, preferably wherein the quadrature component is obtained from a time derivative of the in- phase component. In one aspect, in combination with any embodiment above or below, the method further comprises: measuring or simulating the one or more frequencies of unwanted transitions to determine the one or more finite frequency ranges to be suppressed. In one aspect, in combination with any embodiment above or below, the parameters of the control pulse apart from the coefficients are sequentially calibrated, each of the parameters controlling at least a single property of the quantum logic gate including at least one of a rotation angle, a magnitude of phase errors, and a leakage error. In one aspect, in combination with any embodiment above or below, one or multiple virtual- Z rotations are applied before, during or after the control pulse. In one aspect, in combination with any embodiment above or below, the control pulse applied to each of the physical qubits is constructed by summing the shaped pulses designed for the individual physical qubits multiplied by crosstalk ratios based on measurements of at least one of microwave or quantum crosstalk. In one aspect, in combination with any embodiment above or below, the parameters of the control pulse are further fine-tuned by performing a closed-loop optimization using one or more error metrics and / or fidelity of the quantum logic gate implemented on the one or multiple physical qubits. 256254 s5 / s32 / sca In one aspect, in combination with any embodiment above or below, the method further comprises pre-distorting the control pulse. The present disclosure further provides a method of generating a control pulse, comprising the step of generating a control pulse, wherein properties of the control pulse correspond to parameters of the control pulse determined according to the method of any of the aspects above. The present disclosure further provides a control pulse, wherein properties of the control pulse correspond to parameters of the control pulse determined according to the method of any of the aspects above. The present disclosure further provides an apparatus for generating a control pulse, the apparatus comprising a control pulse generator for generating a control pulse, wherein properties of the control pulse correspond to parameters of the control pulse determined according to the method of any of the aspects above. The present disclosure further provides a method for generating a control pulse with a givenpulse duration comprising: determining a first envelope function for an in-phase componentor a quadrature component of a control pulse such that the envelope function is: continuousin a time domain, continuously differentiable with respect to time, symmetric in the timedomain with respect to a center of the pulse, symmetric in a frequency domain according toan absolute value of a Fourier transform; and minimizing the frequency spectrum of the first envelope function across one or more finite frequency ranges and / or suppressing thefrequency spectrum at one or more specified individual frequencies; obtaining a secondenvelope function for a quadrature component or an in-phase component from a time-derivative of the first envelope function; and generating the control pulse based on the firstenvelope function and the second envelope function. In one aspect, in combination with any embodiment above or below, the method for shaping a frequency spectrum of a control pulse may be applied to minimize the spectrum of the first envelope. In one aspect, in combination with any embodiment above or below, the quadrature component or the in-phase component is scaled with an amplitude factor that results in a zero crossing of the Fourier transform of the control pulse at a frequency that is located at or nearby a suppressed range and / or a suppressed frequency of the in-phase component or quadrature component, respectively. 256254 s5 / s32 / sca In one aspect, in combination with any embodiment above or below, the control pulse is used to control one or multiple quantum devices. In one aspect, in combination with any embodiment above or below, the control pulse is an off-resonant control pulse or a resonant control pulse for implementing a quantum logic gate on one or more physical qubits, and the given control pulse duration is a predefined pulse duration for operating the quantum logic gate. In one aspect, in combination with any embodiment above or below, at least one of the one or more finite frequency ranges to be suppressed respectively correspond to one or more unwanted transitions related to the one or more physical qubits, and wherein the unwanted transitions are leakage transitions beyond a computational subspace or unwanted transitions within the computational subspace. In one aspect, in combination with any embodiment above or below, the one or more physical qubits are one of superconducting qubits, qubits based on trapped ions, spin qubits, qubits based on nitrogen vacancies centers, or qubits based on Rydberg atoms. In one aspect, in combination with any embodiment above or below, the superconducting qubits are one of transmon qubits, fluxonium qubits, charge qubits, flux qubits, unimon qubits, or coupling qubits of any type. In one aspect, in combination with any embodiment above or below, the quantum logic gate is a single-qubit gate or a two-qubit gate. In one aspect, in combination with any embodiment above or below, one or multiple virtual- Z rotations are applied before, during or after the control pulse. In one aspect, in combination with any embodiment above or below, the control pulse applied to each of the physical qubits is constructed by summing the shaped pulses designed for the individual physical qubits multiplied by crosstalk ratios based on measurements of at least one of microwave or quantum crosstalk. In one aspect, in combination with any embodiment above or below, the method of generating a control pulse further comprises pre-distorting the control pulse. The present disclosure further provides a control pulse, wherein the control pulse is generated according to the method of any aspect above. 256254 s5 / s32 / sca The present disclosure further provides an apparatus for generating a control pulse, the apparatus comprising: a control pulse generator for generating a control pulse, wherein the control pulse is generated according to the method of any the aspects above. In one aspect, in combination with any embodiment above or below, the control pulse generator has an in-phase control pulse generator and a quadrature control pulse generator. In one aspect, in combination with any embodiment above or below, the control pulse generator is configured to generate the control pulse or the sequence of control pulses on one or more physical qubits. The present disclosure further provides a quantum computer comprising the apparatus according to the method of any aspect above. In one aspect, in combination with any embodiment above or below, the quantum computer uses one or more superconducting qubits as the one or more physical qubits, and the control pulse or the sequence of control pulses are microwave control pulses. The present disclosure further provides a quantum sensor comprising the apparatus of any according to the method of any aspect above. The method for shaping the frequency spectrum for a control pulse, as described in this specification, can be implemented in particular ways so as to obtain one or more of the following advantageous effects. The method allows for an improved control over a generated pulse shape and its frequency spectrum. In addition, when implementing quantum logic gates with the proposed control pulses, gate infidelity and leakage can be reduced even in the presence of high crosstalk. This advantageous effect can furthermore be obtained without compromising on a desired time characteristic such as a gate duration. Moreover, suppressing the spectrum across unwanted frequency ranges instead of individual frequencies has the advantage to mitigate unwanted transitions even if the transition frequencies were subject to time-dependent AC Stark shifts as is the case, for example, during simultaneous single-qubit gates on a QPU. Furthermore, the present method may help to reduce the spectral energy across a frequency range even in a scenario in which it is not desirable to fully suppress the spectrum at an individual frequency within the frequency range in order to avoid pulses that would not satisfy hardware constraints related to, for example, peak power or bandwidth. As an added benefit of the proposed method, it provides a parametrization that enables controlling the bandwidth and average power of the control pulse through the choice of the suppressed frequency ranges, which may be helpful to satisfy hardware limitations of control electronics while shaping the frequency spectrum. 256254 s5 / s32 / sca The reduced leakage rate provided by the proposed method may enable reducing the gate duration, which is also beneficial to decrease the error limit imposed by incoherent errors and to increase the clock speed of the quantum computer, thus reducing the runtime of quantum algorithms ran on the processor. BRIEF DESCRIPTION OF THE DRAWINGS Fig.1 shows a schematic illustration of a microwave drive implementing a single-qubit rotation(e.g., |0^ → |1^) within the computational subspace of a qubit while also causing leakage outof the computational subspace (|1^ → |2^).Fig.2 illustrates experimentally measured gate infidelity and type-1 leakage error as a function of the quantum gate duration for a standard DRAG pulse with a cosine envelope.Fig. 3A shows an illustration of the energy levels of a system containing two neighboring(uncoupled) qubits, for which simultaneous single-qubit gates are to be performed. Fig.3B shows a representation of a qubit drive affecting a neighboring qubit due to microwave crosstalk during the application of simultaneous single-qubit gates. Fig.4 shows a representation of experimentally measured single-qubit gate error in relation to qubit-qubit detuning for two neighboring qubits when controlling the qubits either individually or simultaneously. Fig.5 shows a method for shaping a frequency spectrum of a control pulse according to an embodiment. Fig.6A shows examples of I-component envelope functions in time domain for the standard cosine pulse shape and for an example pulse shape based on an embodiment. Fig. 6B shows representations of these two I-component envelope functions in frequency domain. Fig.7A shows experimental and simulated single-qubit gate error and leakage resulting from the application of different control pulse shapes depending on a gate duration. 256254 s5 / s32 / sca Fig.7B shows experimentally measured leakage error of a single-qubit gate as a function of the gate duration for different control pulse shapes and different ways of calibrating the DRAG parameter. Fig.8 shows the simulated leakage error as a function of a gate duration for different controlpulse types implementing a single-qubit ^^(^^ / 2) gate.Fig. 9A shows simulated single-qubit gate infidelity and leakage error resulting from the application of different control pulse types depending on a detuning of the ge-transition of one qubit and ef- transition of another qubit, in a two-qubit system, for which simultaneous single-qubit gates are performed. Fig. 9B shows simulated single-qubit gate infidelity and leakage error resulting from the application of different control pulse types depending on a gate duration. Fig.10 shows an aspect of an apparatus for generating a control pulse according to another embodiment. Fig. 11 shows a quantum computer comprising an apparatus for generating a control pulse according to another embodiment. Fig. 12 shows a quantum sensor comprising an apparatus for generating a control pulse according to another embodiment. DETAILED DESCRIPTION While some aspects below discuss shaping a frequency spectrum of a control pulse in order to suppress spectral leakage in quantum logic gates, these aspects can equally be applied to any type of application requiring a control pulse to be applied to a physical system having a set of transition frequencies, such as, for example, applications related to nuclear magnetic resonance (NMR). Where technical features in the drawings, detailed description or any claim are followed by reference signs, the reference signs have been included for the sole purpose of increasing the intelligibility of the drawings, detailed description, and claims. Accordingly, neither the reference signs nor their absence have any limiting effect on the scope of any claim elements. In the following, the term “spectral energy” is used to describe the energy of a pulse within a certain range, or certain ranges, of frequencies based on the Fourier transform of the pulse. 256254 s5 / s32 / sca However, the term “spectral power” can be considered in a similar manner (or used equivalently) given the known relationship between these two quantities. Technical applications requiring precisely defined pulse shapes can benefit from the ability to shape a control pulse such that the spectral energy of the pulse is reduced over unwanted frequency ranges. The present disclosure proposes a frequency domain-based pulse shaping method, presenting the advantage to enable an improved control over the spectral behavior of a pulse to avoid unwanted effects or interactions. For example, to lower error rates of 1- and 2-qubit quantum logic gates caused by spectral leakage to non-computational states, improved pulse shapes and improved control over generating such pulse shapes are needed. Importantly, it may be useful to suppress the frequency spectrum of a control pulse across one or multiple ranges of frequencies instead of individual specific frequencies since the transition frequencies of the qubit system may not be stable and they may vary or fluctuate over time, for example, due to time-dependent AC Stark shifts of the harmful leakage transition frequencies or other transition frequencies. For example, a control pulse implementing a single-qubit rotation can result in a time-dependent change of the transition frequencies of the qubit as a result of an AC Stark shift which can be up to a few tens of MHz for typical drive strengths. Such temporal fluctuations may complicate leakage suppression for fast quantum-logic gates with durations of few or few tens of nanoseconds since it may not be sufficient to suppress the pulse spectrum of the control pulse at a single unwanted frequency to mitigate leakage. Therefore, there is a need for a method able to straightforwardly and flexibly create and shape a control pulse with a given duration, while still being able to suppress specific parts of the spectrum. Fig. 1 represents a physical (transmon) qubit transitioning to another state from its groundstate upon application of the quantum logic gate: the lowest level, namely the |0^ level,corresponds to a ground state (^^) of the qubit. Upon applying a microwave drive (controlpulse) having a frequency corresponding to the resonance frequency of the qubit, the desiredoutcome would be for this qubit to transition to the first excited state (^^), namely to the |1^level. However, the spread of the microwave drive in the frequency domain may also result inthe quantum state partially leaking to the |2^ level, i.e., the ^^ state, which is outside of thecomputational subspace. The computational subspace means a space formed by a linear combination of computational basis states, and typically, it corresponds to the space formedby the two lowest-energy states of a physical qubit, i.e., the states |0^ and |1^ in our example.Here, a physical qubit describes the physical hardware entity on which quantum operations are performed, to change a value of the corresponding logical qubit: more specifically, a 256254 s5 / s32 / sca physical control pulse materializing a quantum logic gate acts on a physical qubit that can encode a logical qubit. For example, a physical qubit may be a superconducting qubit, a qubit based on trapped ions, a spin qubit, a qubit based on nitrogen vacancies centers, or a qubit based on Rydberg atoms. In more detail, a superconducting qubit may be, for instance, a transmon qubit, a fluxonium qubit, a charge qubit, a flux qubit, a unimon qubit or a coupling qubit of any type. Going back to the above example of controlling a physical qubit, the control pulse implementing the gate should have a duration shorter than the coherence time of the qubit in order to minimize incoherent errors caused by environmental noise. The shorter the gate, the lower these incoherent errors are. However, the brevity of the pulse in time domain is associated with a breadth of spectrum in the frequency domain. In other words, the shorter the pulse, the larger the corresponding frequency range, for which the spectrum of the control pulse has non-zero intensity, and the higher the risk of leakage to an undesired state. Thus, there is an inherent tradeoff between the incoherent errors and leakage errors, the first of which decreases as the gate duration is made shorter while the second is rapidly increased below a certain gate duration.This undesired effect is illustrated in Fig.2. This figure shows both simulated and experimentaldata for the gate error and leakage of a transmon qubit with an anharmonicity of -212 MHz demonstrating that longer gate durations, for example, above 10 nanoseconds, lead to a reduced leakage error compared to shorter gate durations. For gate durations shorter than 10 nanoseconds, however, the leakage and gate error exceed the coherence limit of the gate error (which may be considered as the amount of incoherent error, i.e., the gate error caused solely by decoherence). Therefore, when a shorter control pulse is desired, for example to improve the clock speed of the quantum computer and to reduce incoherent errors, the leakage remains to be mitigated. Fig.3A illustrates the predetermined frequencies considering a system of two qubits, Qubit 1(first qubit) and Qubit 2 (second qubit), having their respective transition frequencies: thisschematic illustration shows the energy levels of two neighboring qubits, for whichsimultaneous single-qubit gates are to be performed, with the ge-transition frequency ofQubit 2 being close to the ef-transition of Qubit 1. In this example, the harmful frequencywhen driving Qubit 2 would be the ef-transition frequency of Qubit 1. The transitionfrequencies illustrated in Fig.3A may not be stable in time, as described above, and thereforethe harmful frequency may designate a frequency range rather than a single frequency. When applying a control pulse to Qubit 2, Qubit 1 can be unwantedly excited within or out ofits computational subspace in case of microwave crosstalk between the two qubits: Fig. 3B 256254 s5 / s32 / scaillustrates such a crosstalk. In this figure, Qubit 1 has a resonance frequency ^^1 and Qubit 2has a resonance frequency ^^2. Two drive lines (corresponding to two control pulses)respectively drive each qubit, with a corresponding drive frequency for each drive line. Upon application of the quantum logic gate through the respective control pulses emitted using the drive lines, part of the signal emitted by the Drive line 1 may act on Qubit 2 due to the drive crosstalk, as illustrated with the dashed arrow. In a similar manner, part of the signal emitted by the Drive line 2 may act on Qubit 1, as illustrated with the dotted arrow.In such a situation, Fig. 4 illustrates that the single-qubit gate error, i.e., infidelity, can varydepending on the qubit-qubit detuning: the plots indicate the gate error of a single-qubit gatedepending on a detuning between the ^^^^-transition of the first qubit and the ^^^^-transition ofthe second qubit, when the drive frequency of the second qubit is in the vicinity of the leakagefrequency of the first qubit similarly to Fig 3A. In Fig.4, individual RB means that the gate erroris estimated using standard Randomized Benchmarking performed for one qubit at a time, i.e., not simultaneously, while simultaneous RB means that the randomized benchmarking is performed simultaneously for both qubits, thus characterizing the error for simultaneous 1-qubit gates, which is typically higher than with individual RB. In this example, the ^^^^ leakagetransition frequency of Qubit 1 is at 220 MHz below its ^^^^ frequency, and thus an increasedgate error is observed when the qubit-qubit detuning is around 220 MHz, in which case the^^^^ transition of the second qubit is approximately equal to the ^^^^ transition of the first qubit.Fig. 5 illustrates an aspect of a method for shaping a frequency spectrum of a control pulse ofa given duration, where the method may include the following steps: first, one or more frequencies of the physical system to be controlled may be predetermined in step S110. These frequencies may be predetermined by measurements or by means of simulation of transition frequencies of the physical system upon which the control pulse acts, for example transition frequencies of a physical qubit implemented in a quantum computer. That is, the frequencies determined in advance may depend on the physical system, for example the QPU and potentially the control electronics. As described above, the measured or simulated frequencies may include one or more unwanted frequencies that are related, for example, to leakage transitions that should be avoided. Further, one or more frequency ranges to be suppressed in the frequency spectrum of the control pulse may be determined in step S120. Some (at least one) of these frequency ranges may be derived based on or using at least one of the predetermined frequencies. While some (one or more) such frequency ranges may correspond to specific unwanted frequencies, the rest of the frequency ranges may not correspond to specific unwanted frequencies. For example, these can be frequency ranges that are defined to control the bandwidth of the control pulse (e.g. by setting one of the frequency ranges to an interval between [a cutoff 256254 s5 / s32 / sca frequency, a large frequency]), or a frequency range to control an average power of the control pulse (e.g. by setting one of the frequency ranges to an interval between [0, a large frequency], whereby the frequencies are measured with respect to a center drive frequency). A spectral energy (or spectral power) of the control pulse may then be minimized over the one or more frequency ranges in step S130, and one or more parameters of the control pulse are determined through the minimizing of the spectral energy in step S140. Regarding step S110, a predetermined frequency may be an unwanted frequency of thecontrol pulse and may correspond, for example, to the ^^^^-transition frequency of aneighboring qubit coupled to the qubit being controlled, as exemplified in Fig. 3A. Anotherexample for a predetermined unwanted frequency may be the own ^^^^-transition frequencyof a qubit being controlled. These are merely examples of a predetermined frequency that should be avoided when controlling physical qubits of a quantum computer with a control pulse. The skilled person understands that other applications, such as in NMR, may also involve determining predetermined frequencies that should be suppressed in a control pulse. Regarding step S120, one or more finite frequency ranges may, for example, extend across and / or be located nearby one or more of the predetermined or unwanted frequencies, for example in a symmetric or asymmetric manner below and / or above this frequency. The specific extension may be related to time-dependent frequency fluctuations of the states of the physical system on which the control pulse acts. For example, the time-dependent frequency fluctuations may be due to the AC Stark phenomenon: indeed, a microwave driveapplied on a qubit may modify, for example, its ^^^^-transition frequency. As such, frequencyrange widths of 1 – 50 MHz, preferably 5 – 30 MHz, and even more preferably 10 – 20 MHz may be used. Further, a frequency range located nearby but not overlapping a predetermined frequency may also be determined as a frequency range that should be suppressed in the frequency spectrum of the control pulse. In some implementations, such a frequency range not overlapping a predetermined frequency may be located within 50 MHz, preferably within 30 MHz, and even more preferably within 10 MHz of a predetermined frequency. For example, it is possible that an optimally low leakage error is achieved if the suppressed frequency range is nearby but not overlapping with the unwanted frequency, due for instance to an AC Stark shift during the pulse. Naturally, the frequency ranges to be suppressed in the control pulse may also be chosen otherwise. For example, one of the frequency ranges may be chosen to span from a specific cutoff frequency to a significantly higher (respectively lower) frequency, which may help to reduce the spectral energy above (respectively below) the cutoff frequency, and thus enable controlling the bandwidth of the pulse to abide with the requirements set by the control electronics. A further possibility is to choose one or more of the frequency ranges to span 256254 s5 / s32 / sca across a wide connected range of frequencies, which may be beneficial to limit the average power and maximum amplitude of the control pulse. In one aspect, one or more frequencies of unwanted transitions may be measured or simulated to determine the one or more finite frequency ranges to be suppressed. In this aspect, as an example the obtained frequencies of leakage transitions can be used to define the frequency ranges to be suppressed, for example with a width of frequency range of 5 MHz above and below a frequency of a leakage transition. The suppression of such frequency ranges in the frequency spectrum of a control pulse may limit or inhibit an amount of pulse energy that the control pulse would otherwise, i.e. without such a suppression, exert onto the physical system (e.g. the physical qubit) over these frequency ranges: therefore, a reduced or inhibited excitation over these frequency ranges may result from the application of a control pulse thus shaped. Regarding step S130, the spectral energy designates, for a given frequency, the energy of the signal at this frequency: a spectral energy over a finite frequency range can be computed by integrating the spectral energy over this frequency range. The suppression of one or more frequency ranges in the sense described above is achieved through the minimization or reduction of the spectral energy of the control pulse over these frequency ranges. Regarding step S140, the minimization of the spectral energy may be achieved by identifying parameters of the control pulse that affect the suppression of the spectral energy in the determined (unwanted) frequency ranges. This may be achieved through an optimization (e.g. search and adoption) of parameters values for the control pulse. These parameters may be included in a constraint to abide by when performing the minimizing step S140 or may be calibrated in a subsequent process. The parameters of the shaped control pulse may for instance be its amplitude, drive frequency, basis function coefficients, and / or any other parameters representing or defining the control pulse. The determined one or more parameters of the control pulse define the shape or waveform (e.g. in the time or frequency domain), in particular with regard to a prescribed or given control pulse duration. The parameters may be used by a control pulse generator (see below), e.g. by feeding the parameters (which may be stored, downloaded, or otherwise accessed via a computer system) to an arbitrary waveform generator. 256254 s5 / s32 / sca Based on this method, more control over the frequency spectrum is possible, in particular in case of unstable transition frequencies. Furthermore, the method enables some degree of control over the bandwidth and average power of the control pulse. For example, the control pulse can be a control pulse aiming to implement a quantum logic gate in a quantum processing unit similarly to the above examples. The control pulse may be a microwave pulse with a fixed duration, for example with a duration in the order of few ortens of nanoseconds, as illustrated for example in Fig. 2, preferably with a duration less than40 nanoseconds, more preferably with a duration less than 25 nanoseconds, and even more preferably with a duration less than 10 nanoseconds. The skilled person understands that values of the gate duration may depend on the anharmonicity of the qubit and the qubit type (e.g. transmon qubit). Thus, when it is desired, for example, to solve a crosstalk problem between Qubit 1 and Qubit2, as illustrated in Fig.3B, frequencies capable of unwantedly exciting for example Qubit 1 canbe predetermined, and corresponding frequency ranges may then be defined around the center drive frequency of Qubit 2 as the frequency ranges for which the frequency spectrum of the control pulse for Qubit 2 should be suppressed. In this manner, Qubit 2 may be excited while mostly avoiding exciting Qubit 1 out of its computational subspace, i.e. without leading to a leakage of quantum state population of Qubit 1. In these examples, at least one of the one or more frequency ranges to be suppressed may respectively correspond to one or more unwanted transitions related to one or more physical qubits, and the unwanted transitions may be leakage transitions beyond a computational subspace or unwanted transitions within the computational subspace. Once the frequency ranges to be suppressed have been defined, it is possible to construct the control pulse, the frequency spectrum of which is shaped. According to one aspect, the control pulse to be applied to the physical system may be obtained from the shaped control pulse, e.g. by summing the shaped control pulse with another pulse, by multiplying the shaped control pulse with another pulse and / or by modulating the shaped control pulse to convert it to a different frequency and / or summing or multiplying the shaped control pulse with another pulse after the modulation. For example, in a single-qubit gate example, an envelope function is defined by a shaped control pulse determined as described, above, and the control pulse that is applied to the physical system can be obtained by upconverting the envelope pulse (shaped control pulse).As an example, the control pulse ^^(^^) to be applied to the physical system may be defined inthe time domain by an in-phase component and a quadrature component, and the frequency 256254 s5 / s32 / sca spectrum may be shaped for an envelope pulse of the in-phase component. There, the controlpulse ^^(^^) may be defined as follows with ^^^^(^^) representing the in-phase component and^^^^(^^) representing the quadrature component: with ^^^^ representing the angular drive frequency. In this aspect, the roles of ^^^^(^^) and may be swapped. In one aspect, the quadrature component may be obtained (e.g. derived) from a time derivative of the in-phase component.In this aspect, the quadrature component, as defined in Equation (1), can be expressed asfollows: where ^^ is a parameter called the DRAG parameter, from the eponymous DRAG technique,and the dot denotes a time derivative. As discussed above, the control pulse may be an off-resonant control pulse or a resonant control pulse for implementing a quantum logic gate on one or more physical qubits, and the given control pulse duration is a predefined pulse duration for operating the quantum logicgate. An off-resonant control pulse may designate a pulse, the frequency ^^^^ of which doesnot match the frequency of the qubit transition used to implement the quantum logic gate. An off-resonant drive may be chosen to allow for higher fidelity and lower error in certain applications, however potentially rendering calibration more challenging. In contrast, the drive frequency of a resonant control pulse may correspond to the frequency of the qubit transition. In one aspect, the quantum logic gate may be a single-qubit gate or a two-qubit gate. This terminology applies to the number of qubits a quantum gate may act upon. Although a single- qubit and two-qubit gate are the most frequently used gates, the skilled person understands a quantum gate may control more than two qubits. According to an aspect, the shaped control pulse may be expressed using a finite weighted sum of basis functions, each of the basis functions being multiplied by respective coefficients in the weighted sum of basis functions and being characterized by a duration corresponding 256254 s5 / s32 / scato the given control pulse duration. Following the example in Equation (1), the time-varyingenvelope ^^^^(^^) to be shaped can, for example, be expressed as follows: where ^^ denotes the overall amplitude of the time-varying envelope of the control pulse, {^^^^}denote the ^^ coefficients respectively multiplying the ^^^^ basis functions, ^^ is the finitenumber of basis functions, and ^^^^ designates the given control pulse duration. As will befurther described below, the coefficients are in essence parametrized in terms of suppressed frequency ranges of the spectrum of the control pulse. According to an aspect, the weighted sum of basis functions in Eq. (2) may be implemented as a Fourier series, a weighted sum of trigonometric functions, a Fourier cosine series, a Fourier sine series or any combination of these approaches. According to an aspect, the coefficients in the weighted sum of basis functions may be restricted to ensure continuity of the control pulse at the start and end of the control pulse.For example, a discontinuity problem may arise when the basis functions ^^^^൫^^, ^^^^൯ do not havezero amplitude at ^^ = 0 and at ^^ = ^^^^, which may lead to a slow decay of the spectral energydensity away from the central frequency resulting potentially in significant leakage error.Considering these control requirements, ^^^^(^^) may be expressed as follows: This choice of basis functions, making use of so-called Hanning window functions, ensures the continuity of the control pulse and increases smoothness leading to a lower number of side lobes, compared for example to a rectangular window type of function.The method may further comprise obtaining at least some of the coefficients ^^^^ multiplyingeach of the basis functions by minimizing the spectral energy of the control pulse across the defined finite frequency ranges according to a cost function. It is also possible to set anydesired constraints for the coefficients ^^^^ . The cost function may be constructed with one ormore individually definable weights corresponding to each of the one or more finite frequency ranges to be suppressed, and the cost function may be a weighted sum of a spectral energy 256254 s5 / s32 / sca across the finite frequency ranges to be suppressed. In other words, the cost function can be expressed as follows: ^^^^(^^)^^^^ represents the Fourier transform of the shaped controlpulse, i.e., the time-varying in-phase envelope function in the example of Eq. (1), ^^ denotesthe number of finite frequency ranges to suppress, ^^^^ denotes the weight associated with the^^-th finite frequency range, and and ^^ℎ ,^^ respectively represent the lowest and highestfrequency for the ^^-th finite frequency range to be suppressed. In the light of Eq. (1), the valuesof ^^^^,^^ and ^^ℎ,^^ may correspond to frequency differences measured with respect to the centerdrive frequency ^^^^ / (2^^).According to an aspect, the finite frequency ranges to be suppressed may be symmetrically located around the center frequency of the pulse. This is naturally enforced when using the basis functions defined in Eq. (3) and has the benefit of ensuring that the frequency withmaximal spectral power density is located at ^^^^ / (2^^) independent of the choice of thesuppressed frequency ranges, which can simplify experimental determination of pulseparameters. This also means that the (effective) drive frequency is solely affected by ^^^^. Incontrast, finite frequency ranges to be suppressed that would be asymmetrically located around the center frequency of the pulse may lead to the maximal spectral power densitybeing located away from ^^^^ / (2^^), rendering the effective drive frequency different from ^^^^and thus requiring iterative re-adjustments of the finite frequency ranges to be suppressedand of ^^^^.According to an aspect, the weights may control an amount by which the spectral energy may be individually reduced within each of the one or more finite frequency ranges to be suppressed.For instance, when considering a first frequency range [^^^^,1, ^^ℎ,1] where the spectral energyshould be suppressed and a second frequency range [^^^^,2, ^^ℎ ,2] used for controlling thebandwidth of the pulse and regularizing the pulse shape, the weights may be respectively set, for example, to 100 and 1. Although two frequency ranges are mentioned in this example, the skilled person understands that there is no limitation to the number of frequency ranges to be suppressed. 256254 s5 / s32 / sca As a result, the weights allow for a fine and precise control over the spectral energy, by selectively choosing the amount by which the finite frequency ranges are to be suppressed. In one aspect, the optimizing of the cost function may be performed as a quadratic optimization with a linear constraint. According to this aspect, the optimization problem may thus be formulated as minimizing thefollowing above-defined quantity ^^^^^^^^^^ℎ^^^^^^ : while the below linear constraint given as an example may be fulfilled: In this manner, the coefficients {^^^^} may be straightforwardly calculated to minimize thespectral energy of the pulse over the frequency ranges to be suppressed while satisfying the constraint. Here,^^ represents a predetermined control result resulting from the application of the control pulse. The predetermined control result may be considered as the intended result that is achieved by applying the control pulse to the physical system (for example, the quantum bit). In this example, the optimizing of the cost function may be performed under a constraint relating to the predetermined control result of the control pulse.For example, ^^ may represent any desired outcome from an application of the control pulse,such as the flip angle of a spin resulting from the application of a radiofrequency pulse, when the method is used in an NMR experiment. In another example, the energy of the pulse may be constrained below some given value. Such a constraint would lead to a quadratic optimization problem with a quadratic constraint that is NP-hard and would not have a closed-form solution but may be solved numerically. Whilefidelity may not be used to constrain the coefficient ^^^^ since the constraints should be basedon the pulse properties, fidelity as an objective function may be used for fine-tuning parameters in an experimental closed-loop setting as explained below. 256254 s5 / s32 / sca In another example, the predetermined control result of the control pulse may be a rotationangle of a quantum logic gate, i.e., ^^ in Equation (6) may represent such a rotation angle. Thisquantity determines, on the Bloch sphere, to which extent a quantum state of a qubit is rotated: for example, a NOT quantum logic gate will induce a rotation of 180 degrees along the x-axis of the Bloch sphere, modifying the quantum state of the qubit it is applied to from|0^ to |1^, and reciprocally.According to an aspect, the quadratic optimization may be performed by solving a system of linear equations determined by the one or more weights and the one or more frequency ranges to be suppressed. Accordingly, this optimization problem may be expressed in a matrix form following the subsequent development: =^^^^^^^^ (7.5)where ^^^ (^^) represents the Fourier transform of the ∗^^ ^^-th basis function, ^^ denotes complexconjugate of a complex number ^^, ^^ ∈ ℝ^^×1 is a vector with the basis function coefficients ^^^^as its elements, and ^^ ∈ ℂ^^×^^ is a Hermitian matrix with elements: Importantly, the Fourier transform ^^^^^(^^) has an analytic expression when using the basisfunctions defined in Eq. (3), and hence the elements of matrix ^^ can be efficiently evaluatedusing numerical integration. Consequently, the optimization problem may be rewritten asminimizing ^^^^^^^^ under the constraint: 256254 s5 / s32 / sca where ^^ = (1, … ,1)^^ ∈ ℝ^^×1.The optimization problem may then be readily solved using, for example, the method ofLagrangian multipliers, so that the coefficients {^^^^} can be obtained.The Lagrangian corresponding to the optimization problem may be constructed as L(c, λʹ) =cTAc − λʹ (cT b − ^^ / ^^^^), and the optimality conditions are given by∇c L = (A + ^^^^ )c − λʹ b = 0∇λʹL = cT b − ^^ / ^^^^ = 0These two conditions can be combined into a single matrix equation ^̃^^ ^^ = ^^^ (9)where ^^ =̃ (^^^^, ^^)^^ is the coefficient vector extended by the Lagrangian multiplier, ^^^ = the matrix ^̃^ is defined as^̃^ = ^^^ + ^^^^−^^ ^ ^ ^^^0 The coefficients {^^^^} may thus be derived by solving the above simple matrix equation giventhat the frequency ranges to suppress [^^^^,^^ , ^^ℎ,^^] and the associated weights {^^^^} have beendefined.The coefficients {^^^^} may therefore be easily determined from a closed-form equation tosuppress the Fourier transform of the control pulse across specific frequency ranges that canbe represented for example as , [^^^^,^^, ^^ℎ,^^]} and used as in Equation (4). Ifcontrol signal applied to the physical system is obtained by modulating the shaped controlpulse similarly to Eq. (1), the values of and ^^ℎ,^^ in the above equations may correspondfrequency differences measured with respect to the center drive frequency ^^^^ / (2^^).An example of using the spectrum shaping method is the case of applying fast single-qubitgates on a single qubit. In this case, it may be useful to use two frequency ranges and [^^^^,2,^^ℎ ,2] for the suppression. The first range may correspond to the ef-transition of the qubit, and thus the first frequency range extend across orlocated nearby the anharmonicity of the qubit |α / (2π)| = |^^^^^^ − ^^^^^^ |. The second frequency 256254 s5 / s32 / sca range is preferably used to regularize the pulse shape and to limit the bandwidth of the pulse by forcing the spectral energy to be small at high frequency differences with respect to the central drive frequency. To realize fast (tg< 10 ns), low-leakage gates on a transmon with α / (2π) = −200 MHz, it is a good starting point to use frequency ranges [180, 200] MHz and [500, 1000] MHz with the weights set to, e.g., 10 and 1, respectively. For reducing leakage of simultaneous single-qubit gates, it may be useful to utilize threefrequency ranges ( and [^^^^,3, ^^ℎ,3]) for engineering the spectrum of Qubit2 in the system depicted by Fig. 3A. The first range should cover the detuning δ / (2π) betweenthe ef transition of Qubit 1 and the ge transition of Qubit 2. Based on simulation results, leakage during simultaneous gates due to crosstalk can be reduced by setting the width of the first suppression range to approximately 10 MHz. The second range should cover the own ef-transition of Qubit 2, i.e., |α2 / (2π)| Suppressing the spectral energy across both reduces both type-1 and type-2 leakage errors and thus, enables thepresent method to perform significantly better compared to conventional pulse shapes, such as the raised cosine. The third frequency range is preferably used for regularizing the pulse shape and limiting the bandwidth of the pulse. Improved results may thus be obtained byselecting the weights as w1 = 5, w2 = 100, and w3 = 1 while setting [^^^^,3, ^^ℎ,3] = [150, 1000] MHz.As highlighted by this example, the finite frequency ranges to suppress may be overlapping. Solving an optimization problem by finding a closed-form solution allows for a significantly reduced computational costs (computational load) and wall clock time compared to iterative closed-loop optimization on the actual hardware, which disadvantageously requires to have an initial guess for the coefficients, and then to iterate until an optimal value is found. In contrast, the method according to the present aspect does not necessarily require any iterations to find the values of the coefficients and is therefore faster and more flexible, in addition to straightforwardly taking into account that the (unwanted) transition frequencies may not be stable in time.In one aspect, the amplitude parameters of the control pulse, such as ^^ and ^^ in Eqs. (1’) and(2), may be experimentally calibrated. The calibration may be performed depending on the specific implementation. As an example,the basis function coefficients {^^^^} are determined using the present method based on theunwanted transition frequencies. Then, ^^ (overall amplitude) and ^^ (DRAG parameter) maybe calibrated using a sequence of measurements, for example as follows. First, a Rabimeasurement may be performed, in which an initial estimate for ^^ is obtained by sweepingthe drive amplitude and fitting a periodic model to the resulting Rabi oscillations. Then, Q- scale calibration based on gate sequences (Y180, X90) and (X180, Y90) may be conducted to 256254 s5 / s32 / scamitigate phase errors and to obtain an initial estimate for the DRAG parameter ^^. Next, a fine-tuning estimate of ^^ with an error amplification experiment for the rotation angle as well asa fine-tuning estimate of ^^ with an error amplification experiment for phase errors may beperformed. The above sequence may be iterated upon as needed. For example, the parameters of the control pulse apart from the coefficients may be sequentially calibrated, each of the parameters possibly controlling at least a single property of, for instance, a quantum logic gate including at least one of a rotation angle, a magnitude of phase errors, and a leakage error. In other words, once determined the coefficients may remain fixed or be adjusted, and all the control pulse parameters affecting for instance a rotation angle, phase error and / or leakage error may be calibrated sequentially. However, the coefficients may alternatively be allowed to change for further fine-tuning as explained below. In one aspect, the control pulse may be used alone or as a part of a sequence of control pulses to control a system, for example a quantum system of a plurality of qubits in a quantum computer or a quantum sensor. Here, the sequence of control pulses may represent a plurality of control pulses that have been determined as described above, for example for a same or anumber of different control results (e.g. rotation angles) ^^.According to this aspect, the control pulse shaped with the method described may be used in any application requiring control pulses or sequence of control pulses. For example, NMR applications such as Magnetic Resonance Imaging (MRI), which require specific pulse sequences, may benefit from this method by reducing a pulse energy over frequency ranges that could otherwise lead to artifacts by exciting spins outside a desired slice. In one aspect, the control pulse may be used to control one or multiple quantum devices, in particular one or more physical qubits in a quantum device such as a quantum computer or a quantum sensor. Fig.6A shows an example of a pulse shape in time domain obtained with the present methodto control a quantum device, under the name “FAST”, standing for Fourier Ansatz-basedSpectrum Tuning. The example pulse shape has been obtained by choosing the suppressed frequency ranges as [35, 45] MHz, [190, 210] MHz, and [150, 1000] MHz (w.r.t the center drive frequency) with the corresponding weights equaling 5, 1000, and 1, respectively. In turn, “Cosine” represents a pulse shape obtained by setting the envelope to a cosine function with an additive offset ensuring continuity at the start and end of the pulse. 256254 s5 / s32 / sca Fig.6B shows the Fourier transform of these pulses: for example, the Fourier transform overthe range [35, 45] MHz, which was determined as a frequency range to besuppressed, is indeed suppressed for the “FAST” pulse shape whereas it is not the case for thecosine pulse shape. Fig. 7A shows both experimentally measured and simulated average single-qubit gateinfidelity and leakage error for the gate set {^^, ^^(^^ / 2),^^(^^ / 2)} as a function of the gateduration in the case of controlling a single transmon qubit with an anharmonicity of -212 MHz(see Fig. 1). Here, “FAST” represents the results obtained by using control pulse shapes basedon the present method, while “Cosine” represents the results obtained by applying a controlpulse based on a cosine envelope. For both cases, the DRAG framework is utilized to mitigate phase errors. As shown from the plots, the present method outperforms the conventional method based on the cosine envelope, in particular for gate durations shorter than 13 ns. In one aspect, one or multiple virtual-Z rotations may be applied for the control pulse before, during or after the control pulse (is applied to the physical system).Fig.7B illustrates experimentally measured leakage error as a function of the single-qubit gateduration for two different pulse shapes, each of which is combined with two different ways ofimplementing the DRAG correction. Here, “FAST” represents again a pulse shape based on thepresent method, whereas “Cosine” stands for the conventional cosine envelope. The first wayof implementing DRAG (“DRAG”) is based on selecting the DRAG parameter to mitigate phaseerrors, whereas the second implementation ("DRAG + VZ”) is based on tuning the DRAG parameter to minimize leakage while using an additional Virtual-Z rotation to mitigate phaseerrors. Fig.7B indicates that the pulse shape based on the present method combined with thesecond implementation of DRAG clearly provides the best results and enables a low leakage error down to the gate duration of 6 ns for the studied transmon qubit with an anharmonicity of -212 MHz. In one aspect, the control pulse applied to each of the physical qubits may be constructed by summing the shaped pulses designed for the individual physical qubits multiplied by crosstalk ratios based on measurements of at least one of microwave or quantum crosstalk. In one aspect, the parameters of the control pulse may be further fine-tuned by performing closed-loop optimization using one or more error metrics or the fidelity of the quantum logic gate implemented on the one or multiple physical qubits. According to this aspect, the pulse parameters can be found according to the present method as a starting point, and a fine-tuning of some or all parameters can then be performed by using 256254 s5 / s32 / sca one or more error metrics or fidelity metric of the quantum logic gate as the objective function to be minimized in the fine-tuning. As an example, the overall amplitude of the in-phase component, the overall amplitude of quadrature component, the suppressed frequency ranges, weights, and center frequency can be optimized. Alternatively, the basis function coefficients may be directly used for the optimization. To perform the optimization, the Nelder-Mead optimization method or covariance matrix adaptation evolution strategy may for example be used. The optimization may be performed offline using a simulation of the physical system or in an online closed-loop setting using the actual experimental hardware. Fig.8 shows an example of simulated leakage rate as a function of gate duration for differentpulse shaping methods for the transmon qubit system depicted in Fig 1. In the figure, “FASTDRAG, phase optim. + fine-tuning” represents the pulse obtained by the present method after parameter fine-tuning with Nelder-Mead optimization. The pulse based on the present method increasingly outperforms the other methods the shorter a gate duration becomes for gate durations shorter than 10 ns. In the simulation, the DRAG parameter is chosen to mitigatephase errors apart from the control pulse “Cosine DRAG, leakage optim.”, for which the DRAGparameter is optimized to minimize leakage errors. In Fig. 8, “Extended DRAG” refers to anextension of the DRAG framework up to the third derivative in order to suppress the spectrum at an individual frequency corresponding to the anharmonicity of the qubit, which turns out not to be sufficient to reduce leakage for the shortest gate durations. Simulation results were also obtained for simultaneous single-qubit gates on a system consisting of two uncoupled qubits in the presence of microwave crosstalk for a scenario, in which there is only a relatively small detuning δ / (2π) between an unwanted leakage transitionand the drive frequency of one of the qubits as depicted in Fig.3A. According to the simulationresults, the control pulses generated by the present method can reduce the total leakage rate and gate infidelity by up to few orders of magnitude for certain parameter combinations when applying a Yπ / 2gate to only one of the qubits at a time. When applying simultaneous Yπ / 2gates to both qubits, the control pulses generated by the present method can reduce the total gate infidelity by approximately 50-80%. The exact magnitude of the improvement may be dependent on the detuning between the qubits, the gate duration, the anharmonicities of the qubits and the amount of crosstalk. Fig. 9A shows the simulated total gate infidelity and leakage of simultaneous single-qubit gates as a function of the detuning between the ge-transition of Qubit 2 and the ef-transitionof Qubit 1 for the system depicted in Fig. 3A. Similarly, Fig. 9B illustrates simulated totalinfidelity and leakage of simultaneous single-qubit gates as a function of the gate time. Both 256254 s5 / s32 / scasets of results indicate that the performance of “FAST DRAG + fine-tuning”, representing apulse shape constructed using the present method followed by further parameter optimization, improves significantly upon the conventional approach of using a cosine-shaped pulse envelope with DRAG. Furthermore, the unoptimized version of the present method(“FAST DRAG”) outperforms the conventional cosine-based approach for some of the studiedgate durations and detunings. Importantly, both the unoptimized and optimized version of the present method provide a lower leakage error across most of the studied gate durations and detunings compared to an extension of the DRAG framework using up to the third derivative ("Extended DRAG”). For the system of two uncoupled qubits in the presence of microwave crosstalk, applying the control pulses generated by the present method may provide the most benefit if the detuningto suppress is approximately in the range of δ / (2π) ∈ [25, 60] MHz, since in this regime anincrease in the gate duration reduces the leakage rate only slightly for the gate durations of interest (tg∈ [13, 35] ns), whereas a significant reduction in the leakage rate is obtained with the present method. When parking the other qubit (preserving or maintaining the quantum state of the qubit) in the middle of the straddling regime (separation of quantum qubit energy levels being comparable to fluctuations in external parameters of the control pulse) of theother qubit, i.e., δ / (2π) ≈^^ / (4^^) ≈ 100 MHz, the leakage rate and infidelity are rapidlyreduced as a function of the gate duration, and the benefit provided by present method can be alternatively obtained by increasing the gate duration by a few nanoseconds. The above-described experimental and simulation results provide evidence to show that the control pulses obtained by the present method can reduce the leakage rate and gate infidelity of fast single-qubit gates by up to orders of magnitude compared to conventional leakage- optimized DRAG pulses. In the absence of crosstalk effects, the leakage rate and infidelity ofsingle-qubit gates are improved in particular with gate durations below 4^^ / (^^) correspondingto 10 ns for typical parameters of a transmon qubit. In the presence of crosstalk, the present method may help to reduce leakage rate of (simultaneous) single-qubit gates also for longergate durations on the order of 2^^ / (^^), which may correspond to few tens of nanosecondsfor reasonable parameters. The results indicate that leakage rates on the order of 10-5or below can be attained by the present method. According to a further preferred embodiment, the method for shaping a frequency spectrum of a control pulse with a given control pulse duration may further include the step of pre- distorting the control pulse. For example, when the one or more parameters of the control pulse have been determined through the minimizing of the spectral energy, as discussed above, the control pulse may be further shaped by pre-distorting the control pulse. Pre- distorting the control pulse counter-balances potential distortions of the control pulse(s) that 256254 s5 / s32 / sca occur between the generation of the control pulse and arrival of the control pulse at the physical system (e.g. physical qubit), due to physical interaction of the control pulse with the hardware (e.g. waveguide, impedance mismatches in the control lines) required to propagate the control pulse. Due to these distortions, it may be possible that the arriving control pulse has changed from the generated (and intended) pulse shape. This change may be determined, e.g. during manufacturing or calibration of the physical system, and thus appropriately counter-balanced to partly or fully mitigated by adding the pre-distortion. The pre-distortion may change the duration of the generated control pulse, so that the correctly pre-distorted pulse is distorted along the physical path to the physical qubit and will thus exhibit the intended (targeted) shape and duration at the physical qubit. The above distortion may, for example, be modeled, simulated or otherwise determined as a filter, e.g. an infinite impulse response (IIR) exponential filter, and the inverse filter (coefficients) may be used as a basis for implementing the pre-distortion. The skilled person understands that the IIR exponential filter is a non-limiting example, and other appropriate filters may be applied. In a further preferred embodiment, the control pulse, the sequence of control pulses, the envelope pulse of the in-phase component and / or the envelope pulse of the quadrature component, or the sequence of the envelope pulses of the in-phase and quadrature components may be pre-distorted to mitigate distortions that occur before the control pulse reaches the physical system to be controlled. Fig.10 illustrates an example of an apparatus for generating a control pulse: the apparatus may be configured by appropriate hardware and software components to generate a control pulse according to the features described above, that is to generate a control pulse for which the properties of the control pulse correspond to parameters of the control pulse determined according to the method described in the present disclosure. The apparatus for generating a control pulse may comprise a control pulse generator for generating a control pulse, wherein properties of the control pulse correspond to parameters of the control pulse determined according to any of the above aspects of the present disclosure. That is, the apparatus may acquire the one or more parameters of the control pulse determined as described above and use the control pulse generator to generate the control pulse. The control pulse generator may be an (arbitrary) waveform generator capable of defining arbitrary control pulses / waves. Acquiring the one or more parameters may be performed by reading out the parameters from a memory storage of the apparatus, by inputting the parameters, for example via a graphical user interface, by downloading the parameters from a server or the like. The generated control pulse may then be output to a control target, such as physical qubits of a quantum computer or a quantum sensor, for example via a transmission line, a microwave waveguide, 256254 s5 / s32 / sca a (tunable) microwave coupler or the like. The generated control pulse may be attenuated and / or amplified before directing it to the control target. A control pulse or a series of control pulses thus generated, i.e. having properties corresponding to parameters of the control pulse determined according to the method described in the present disclosure, can be the result of modulating an electrical signal and can be generated, for example, using microwave sources, such as arbitrary waveform generators (AWG), using specialized pulse generators (for example using direct digital synthesis circuits or field-programmable gate arrays), or using a capacitor discharge, and its intensity is measurable. The generated control pulse may have properties identical to properties of a pulse generated using the parameters of the control pulse determined according to the method described in any of the above aspects. In other words, the present method applies an approach in the frequency domain to obtain a shaped control pulse: however, while the determined parameters of the control pulse may also be represented (at least in part) in the time domain, the generated control pulse still has an identical shape. In this example, the parameters specifying a control pulse may be of different nature: according to some aspects developedabove, the parameters pertaining to the present method in the time domain may be the {^^^^}coefficients as for instance described in the sense of Equation (2), whereas in the frequencydomain the control pulse is parametrized in terms of the suppressed frequency ranges andcorresponding weights, from which the {^^^^} coefficients may be derived using Eq. (9).However, these specifying parameters may also be represented (at least in part) in another domain, in another set of basis functions or the like, but the resulting shape of the pulse is still identical. The properties of a control pulse may define the shape of the control pulse and may designate any physical defining quantities, such as the overall amplitude of the control pulse or weights over specific frequency ranges. Despite the different nature of these parameters, the outcome of the control pulse generator, i.e. the control pulse, may be of the same shape and thus have identical properties. In other words, the shape of the control pulse obtained by using parameters calculated according to the present method may be identical to the shape of a control pulse obtained by using corresponding other parameters that still lead to the same control pulse shape. In one aspect, the control pulse generator may have an in-phase control pulse generator and a quadrature control pulse generator. 256254 s5 / s32 / scaFor instance, the control pulse generator can generate components ^^ and ^^ as defined inEquation (1).In one aspect, the control pulse generator may be configured to generate the control pulse or the sequence of control pulses on one or more physical qubits. The present disclosure also pertains to a method for generating a control pulse with a given pulse duration based on the following requirements: The in-phase component of the control pulse is shaped and generated such that the in-phase envelope function and its first time- derivative are continuous in the time domain, the in-phase envelope function is symmetric in the time domain with respect to the center of the pulse, and the in-phase envelope function is symmetric in the frequency domain according to the absolute value of the Fourier transform. Furthermore, one or more pulse parameters of the in-phase component are determined by minimizing the spectral energy of the in-phase component across at least one suppressed frequency range and / or minimizing the spectral energy density across at least one specified individually suppressed frequency. The quadrature component of the control pulse is obtained from a time-derivative of the in-phase component. In some preferred embodiments, the quadrature component may be scaled with an amplitude factor that results in a zero crossing of the Fourier transform of the control pulse at a frequency that is located at or nearby a suppressed range and / or suppressed frequency of the in-phase component. The above example of generating the control pulse is provided on the basis of the in-phase component. The skilled person understands, however, that the role of the in-phase and quadrature component may be interchanged. As such, a method for generating a control pulse with a given pulse duration may comprise the steps of determining a first envelope function for an in-phase component or a quadrature component of a control pulse such that the envelope function is: continuous in a time domain, continuously differentiable with respect to time, symmetric in the time domain with respect to a center of the pulse, and symmetric in a frequency domain according to an absolute value of a Fourier transform; and further comprises the steps of minimizing the frequency spectrum of the first envelope function across one or more finite frequency ranges, in particular one or more suppressed finite frequency ranges, and / or suppressing the frequency spectrum at one or more specified individual frequencies; obtaining a second envelope function for a quadrature component or an in-phase component from a time-derivative of the first envelope function; and generating the control pulse based on the first envelope function and the second 256254 s5 / s32 / sca envelope function. Here, the spectrum shaping method described above may be applied to minimize the spectrum of the first envelope. A control pulse satisfying the above requirements has several advantages. Firstly, the continuity of the in-phase envelope function and its first derivative ensure that the spectral energy density of the control pulse decreases more rapidly as a function of frequency compared to the spectrum of a non-continuous pulse, and thus control pulses satisfying this requirement typically lead to significantly lower leakage errors (the skilled person understands that a spectral energy across a frequency range is obtained by integrating the spectral energy density). The symmetricity of the Fourier transform of the in-phase component ensures that the maximal spectral energy density of the control pulse occurs at the carrier frequency, i.e., drive frequency in case the amplitude of the quadrature component is set to zero. When enforcing the symmetricity requirement, this holds independent of the location of the suppressed frequency ranges and suppressed individual frequencies related to the in-phase component. This has the technical benefit that the condition for minimal phase errors of a single-qubit gate is practically independent of the parameters controlling the suppressed ranges and suppressed frequencies, which can greatly simplify the calibration of the pulse parameters. On the other hand, suppressing certain parts of the spectrum of the in-phase component can significantly reduce leakage errors as already discussed above. Finally, setting the quadrature component as a scaled derivative of the in-phase component is in line with the existing DRAG method and enables controlling phase errors of the quantum logic gate or achieving even stronger spectral suppression around a specified frequency, which can further reduce leakage errors. Importantly, this specific combination of the requirements provides a powerful method for applying high-fidelity quantum logic gates. As an example of the above method, the in-phase component of the pulse may be set as asum of a function ^^0൫^^, ^^^^൯ and its scaled second derivative, i.e., where ^^^^ denotes the gate duration, dot denotes time derivative, and the following notationhas been applied: ^^2൫^^, ^^^^൯. As explained in Ref. [6], this enables a suppression ofthe spectrum of the in-phase component symmetrically at individual frequency differences of This means that the parameter ^^2 of the in-phase component may be determined as ^^2 =1 / (2^^Δ^^)2to fully suppress the Fourier transform of the in-phase component symmetrically 256254 s5 / s32 / scaat individual frequencies of ±Δ^^. As a result, the Fourier transform of the control pulse in Eq.(1) will be suppressed at individual frequencies of ^^^^ / (2^^) ± Δ^^.Alternatively, ^^2 may be determined by using ^^0൫^^, ^^^^൯ and^^2൫^^, ^^^^൯ as a set of basis functionsand minimizing the spectrum across a frequency range to suppress. A skilled person understands that higher-order derivatives could also be included in the sum enabling the suppression of higher number of individual frequencies at the cost of potentially generating a pulse that doesn’t satisfy well all hardware requirements.In the present example, the function ^^0൫^^, ^^^^൯ and its three first derivatives should becontinuous (also at ^^ = 0 and at ^^ = ^^^^) such that the requirement of the continuity of ^^^^(^^)and ^^^^(^^) is satisfied. This may be ensured by expressing ^^0൫^^, ^^^^൯ as a short cosine series,the coefficients of which have been solved to satisfy =This results in the This choice of the function also ensures that the in-phase component ^^^^(^^) issymmetric in time with respect to the center of the pulse. Furthermore, the absolute value ofthe Fourier transform of ^^^^(^^) is an even function in the frequency domain.The skilled person understands that more cosine terms (higher harmonics) can be included inthe function ^^0൫^^, ^^^^൯ to ensure the continuity of derivatives beyond the third order, which isnecessary if ^^^^(^^) contains derivatives beyond the second order.Finally, the quadrature component may be set as a scaled derivative of the in-phase component, i.e., Importantly, the DRAG coefficient ^^ can be tuned to further suppress the spectrum of themicrowave control pulse at a frequency difference Δ^^ = −1 / (2^^^^) (with respect to thecenter drive frequency). A strong suppression can be achieved around an unwanted individualfrequency difference of Δ^^^^ by using ^^ ≈ This maysignificantly reduce the leakage error of a single-qubit quantum logic gate with a shortduration. Importantly, this choice of ^^ may result in phase errors that need to be corrected,for example, by applying virtual-Z durations before, during and / or after the pulse. 256254 s5 / s32 / sca Fig. 11 illustrates an example of a quantum computer comprising the apparatus described above in which a control pulse generator generates a shaped control pulse according to the method(s) described above, for instance with the shape of the control pulse suppressing unwanted frequency ranges. In this example, the pulse thus shaped can be used to act on physical qubits to perform quantum computing operations, for example an application of a quantum gate, through the hardware chain example as follows. The control pulse generator (which also shall be referred to as waveform generator) may generate a shaped control pulse that is applied to a waveguide or transmission line to act on one or more physical qubits. Here, the parameters of the shaped control pulse, determined as described above, may be available in a digital format, may be converted from digital to analog using a digital-to-analog converter to generate a corresponding waveform which may also be appropriately amplified and / or filtered. In other words, the control pulse generator may include an analog-digital converter and for example amplifiers, attenuators and / or filters. The generated shaped control pulse may be directed into one or multiple transmission lines and / or waveguides to direct the signal to the one of more physical qubits to be controlled. The analog signal may be, for example, attenuated or amplified before reaching the physical qubits. The hardware chain may comprise other components, such as for example one or more filters and an I / Q mixer. In one aspect, the quantum computer may use one or more superconducting qubits as the one or more physical qubits, and the control pulse or the sequence of control pulses may be microwave control pulses. Fig.12 illustrates an example of a quantum sensor comprising the apparatus described above. A quantum sensor is a quantum system that utilizes one or more features of quantum physics, for example entanglement or interference enabled by superposition, to perform accurate measurements of physical quantities such as a magnetic field, electric field or electric charge in the surroundings of the sensor. For example, superconducting qubits or other superconducting circuits can be used to sense, e.g., magnetic fields, electric charges or an impact of cosmic rays on the chip hosting the qubit. Using the quantum system as a quantum sensor may require manipulating the quantum state of the sensor to make it sensitive to one or more phenomena. This manipulation may require control pulses which can be shaped to achieve high-precision control of the quantum system, 256254 s5 / s32 / sca through the apparatus and method described above. As illustrated, the quantum sensor includes a control pulse generator as described above, a quantum system or device to be driven by control pulses that are generated by the control pulse generator according to the method described above, as well as a measurement system that evaluates a measurement signal from the quantum system or device for detecting a measurement of the quantum sensor. The quantum system or device may be sensitive to one or multiple physical quantities, such as electric fields, magnetic fields, or electric charges, that are being probed by the quantum sensor. The quantum system or device may include one or multiple physical qubits. The quantum system may also be another quantum system having discrete energy levels and may also include continuous energy levels. Reference List [1] F. Motzoi, J. M. Gambetta, P. Rebentrost, and F. K. Wilhelm. Simple pulses for elimination of leakage in weakly nonlinear qubits. Physical review letters, 103(11):110501, 2009. [2] J. M. Martinis and M. R. Geller. Fast adiabatic qubit gates using only σ z control. Physical Review A, 90(2):022307, 2014. [3] V. Vesterinen, O.-P. Saira, A. Bruno, and L. DiCarlo. Mitigating information leakage in a crowded spectrum of weakly anharmonic qubits. arXiv preprint arXiv:1405.0450, 2014. [4] L. Theis, F. Motzoi, and F. Wilhelm. Simultaneous gates in frequency-crowded multilevel systems using fast, robust, analytic control shapes. Physical Review A, 93(1):012324, 2016. [5] M. Werninghaus, D. J. Egger, F. Roy, S. Machnes, F. K. Wilhelm, and S. Filipp. Leakage reduction in fast superconducting qubit gates via optimal control. npj Quantum Information, 7(1):1–6, 2021. [6] F. Motzoi and F. K. Wilhelm. Improving frequency selection of driven pulses using derivative-based transition suppression. Physical Review A, 88(6):062318, 2013. [7] W. Nuerbolati, Z. Han, J. Chu, Y. Zhou, X. Tan, Y. Yu, S. Liu, F. Yan. Cancelling microwave crosstalk with fixed-frequency qubits. Applied Physics Letters, 120(17), 2022. [8] Y. Sung et al. Realization of High-Fidelity CZ and ZZ-Free iSWAP Gates with a Tunable Coupler. Phys. Rev. X, 11(2), 2021. [9] Google Quantum AI. Suppressing quantum errors by scaling a surface code logical qubit. Nature 614, 676–681 (2023).

Claims

256254 s5 / s32 / sca CLAIMS 1. A method for shaping a frequency spectrum of a control pulse with a given control pulse duration, the method comprising: determining one or more finite frequency ranges to be suppressed in the frequency spectrum, at least one of the finite frequency ranges being defined based on one or more respective predetermined frequencies; minimizing a spectral energy of the control pulse over the one or more finite frequency ranges; and determining one or more parameters of the control pulse through the minimizing of the spectral energy.

2. The method of claim 1, wherein the at least one of the finite frequency ranges extendsacross or is located nearby one or more predetermined frequencies.

3. The method of claim 1, wherein the control pulse is expressed using a finite weightedsum of basis functions, each of the basis functions being multiplied by respective coefficients in the weighted sum of basis functions and having a duration equal to the given control pulse duration, and the method further comprises: obtaining at least one of the coefficients multiplying each of the basis functions by minimizing the spectral energy of the control pulse across the finite frequency ranges to be suppressed according to a cost function.

4. The method of claim 3, wherein the weighted sum of basis functions is implementedas one or a combination of a Fourier series, a weighted sum of trigonometric functions, a Fourier cosine series, a Fourier sine series.

5. The method of claim 3 or 4, wherein the coefficients in the weighted sum of basisfunctions are restricted to ensure continuity of the control pulse at the start and end of the control pulse.

6. The method of any one of claims 1 - 5, wherein the finite frequency ranges to besuppressed are symmetrically located around the center frequency of the pulse and / or256254 s5 / s32 / sca wherein the finite frequency ranges to be suppressed are partly or completely overlapping.

7. The method of any one of claims 3 - 6, further comprising: obtaining the coefficients by optimizing the cost function defined with one or more individually definable weights corresponding to each of the one or more finite frequency ranges to be suppressed and wherein the cost function is a weighted sum of a spectral energy across the finite frequency ranges to be suppressed.

8. The method of claim 7, wherein the weights control an amount by which the spectralenergy is reduced within each of the one or more finite frequency ranges to be suppressed.

9. The method of any one of claims 7 or 8, wherein the optimizing of the cost function is performed under a constraint relating to a predetermined control result of the control pulse and / or wherein the optimizing of the cost function is performed as a quadratic optimization with a linear constraint.

10. The method of claim 9, wherein the quadratic optimization is performed by solving a system of linear equations determined by the one or more weights and the one or more finite frequency ranges to be suppressed.

11. The method of any one of claims 1 - 10, further comprising: calibrating amplitude parameters of the control pulse.

12. The method of any one of claims 1 – 11, wherein the control pulse is summed with another pulse, multiplied with another pulse and / or modulated to convert it to a different frequency and / or summed or multiplied with another pulse after the modulation and / or wherein the control pulse is used alone or as a part of a sequence of control pulses to control a system.

13. The method of any one of claims 1 - 12, wherein the control pulse is used to control one or multiple quantum devices and / or wherein the control pulse is an off-resonant control pulse or a resonant control pulse for implementing a quantum logic gate on one or more physical qubits, and the given control pulse duration is a predefined pulse duration for operating the quantum logic gate.256254 s5 / s32 / sca 14. The method of claim 13, wherein at least one of the one or more finite frequency ranges to be suppressed respectively correspond to one or more unwanted transitions related to the one or more physical qubits, and wherein the unwanted transitions are leakage transitions beyond a computational subspace and / or unwanted transitions within the computational subspace.

15. The method of claim 13 or 14, wherein the one or more physical qubits are one of superconducting qubits, qubits based on trapped ions, spin qubits, qubits based on nitrogen vacancies centers, or qubits based on Rydberg atoms.

16. The method of claim 15, wherein the superconducting qubits are one of transmon qubits, fluxonium qubits, charge qubits, flux qubits, unimon qubits, or coupling qubits of any type.

17. The method of any of claims 13 - 16, wherein a predetermined control result of the control pulse is a rotation angle of the quantum logic gate.

18. The method of any one of claims 13 – 17, wherein the quantum logic gate is a single- qubit gate or a two-qubit gate.

19. The method of any one of claims 1 - 18, wherein the control pulse is defined in time domain by an in-phase component and a quadrature component, and wherein the frequency spectrum is shaped for an envelope pulse of the in-phase component, in particular wherein the quadrature component is obtained from a time derivative of the in-phase component.

20. The method of any one of claims 14 - 19, further comprising: measuring or simulating the one or more frequencies of unwanted transitions to determine the one or more finite frequency ranges to be suppressed.

21. The method of any one of claims 3 – 20, wherein the parameters of the control pulse apart from the coefficients are sequentially calibrated, each of the parameters controlling at least a single property of the quantum logic gate including at least one of a rotation angle, a magnitude of phase errors, and a leakage error.

22. The method of any one of the claims 1 – 21, wherein one or multiple virtual-Z rotations are applied before, during or after the control pulse.256254 s5 / s32 / sca 23. The method of any one of the claims 13 – 22, wherein the control pulse applied to each of the physical qubits is constructed by summing the shaped pulses designed for the individual physical qubits multiplied by crosstalk ratios based on measurements of at least one of microwave or quantum crosstalk.

24. The method of any one of the claims 13 – 23, wherein the parameters of the control pulse are further fine-tuned by performing a closed-loop optimization using one or more error metrics and / or fidelity of the quantum logic gate implemented on the one or multiple physical qubits.

25. The method of any one of the claims 1 – 24, further comprising pre-distorting the control pulse.

26. A method of generating a control pulse, comprising the step of generating a control pulse, wherein properties of the control pulse correspond to parameters of the control pulse determined according to the method of any of claims 1 – 25.

27. A control pulse, wherein properties of the control pulse correspond to parameters of the control pulse determined according to the method of any of claims 1 – 25.

28. An apparatus for generating a control pulse, the apparatus comprising a control pulse generator for generating a control pulse, wherein properties of the control pulse correspond to parameters of the control pulse determined according to the method of any of claims 1 – 25.

29. A method for generating a control pulse with a given pulse duration comprising: determining a first envelope function for an in-phase component or a quadrature component of a control pulse such that the envelope function is: -continuous in a time domain,- continuously differentiable with respect to time,- symmetric in the time domain with respect to a center of the pulse,- symmetric in a frequency domain according to an absolute value of a Fouriertransform; and256254 s5 / s32 / sca minimizing the frequency spectrum of the first envelope function across one or more finite frequency ranges and / or suppressing the frequency spectrum at one or more specified individual frequencies; obtaining a second envelope function for a quadrature component or an in-phase component from a time-derivative of the first envelope function; and generating the control pulse based on the first envelope function and the second envelope function.

30. The method of claim 29, wherein the quadrature component or the in-phase component is scaled with an amplitude factor that results in a zero crossing of the Fourier transform of the control pulse at a frequency that is located at or nearby a suppressed range and / or a suppressed frequency of the in-phase component or quadrature component, respectively.

31. The method of any one of claims 29 or 30, wherein the control pulse is used to control one or multiple quantum devices.

32. The method of any one of claims 29 - 31, wherein the control pulse is an off-resonant control pulse or a resonant control pulse for implementing a quantum logic gate on one or more physical qubits, and the given control pulse duration is a predefined pulse duration for operating the quantum logic gate.

33. The method of any one of claims 29 - 32, wherein at least one of the one or more finite frequency ranges or individual frequencies to be suppressed respectively correspond to one or more unwanted transitions related to the one or more physical qubits, and wherein the unwanted transitions are leakage transitions beyond a computational subspace or unwanted transitions within the computational subspace.

34. The method of any one of claims 29 - 33, wherein the one or more physical qubits are one of superconducting qubits, qubits based on trapped ions, spin qubits, qubits based on nitrogen vacancies centers, or qubits based on Rydberg atoms.

35. The method of claim 34, wherein the superconducting qubits are one of transmon qubits, fluxonium qubits, charge qubits, flux qubits, unimon qubits, or coupling qubits of any type.256254 s5 / s32 / sca 36. The method of any one of claims 32 – 35, wherein the quantum logic gate is a single- qubit gate or a two-qubit gate.

37. The method of any one of the claims 29 - 36, wherein one or multiple virtual-Z rotations are applied before, during or after the control pulse.

38. The method of any one of the claims 32 - 37, wherein the control pulse applied to each of the physical qubits is constructed by summing the shaped pulses designed for the individual physical qubits multiplied by crosstalk ratios based on measurements of at least one of microwave or quantum crosstalk.

39. The method of any one of the claims 29 – 38, further comprising pre-distorting the control pulse.

40. A control pulse, wherein the control pulse is generated according to the method of any of claims 29 – 39.

41. An apparatus for generating a control pulse, the apparatus comprising a control pulse generator for generating a control pulse, wherein the control pulse is generated according to the method of any of claims 29 – 39.

42. The apparatus of claim 28 or claim 41, wherein the control pulse generator has an in- phase control pulse generator and a quadrature control pulse generator.

43. The apparatus of any one of claims 28, 41 or 42, wherein the control pulse generator is configured to generate the control pulse or the sequence of control pulses on one or more physical qubits.

44. A quantum computer comprising the apparatus of any one of claims 28 or 41 - 43.

45. The quantum computer of claim 44, wherein the quantum computer uses one or more superconducting qubits as the one or more physical qubits, and the control pulse or the sequence of control pulses are microwave control pulses.

46. A quantum sensor comprising the apparatus of any one of claims 28 or 41 - 43.

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