Optimization and calibration of quantum operations using sensitivity updates

Abstract

A method for optimizing a set of control parameter of a quantum system comprises obtaining a set of control parameter values and executing at least one measurement process. The at least one measurement process comprises acting with a sequence of operations of a sequence length on the quantum system to measure a sequence fidelity; updating the set of control parameter values by using an optimization routine based on a cost function obtained from the measured sequence fidelity and repeating the aforementioned steps, wherein the sequence length is determined according to an optimal sensitivity criterion.

Description

[0001] WMI24121PCT

[0002] Optimization and calibration of quantum operations using sensitivity updates

[0003] Technical field

[0004] The invention relates to methods, apparatus, computer program and computer-readable data carrier for the optimization of quantum operations acting on a quantum system using sensitivity updates.

[0005] Background

[0006] Achieving fast and high-fidelity quantum operations acting on a quantum system for control and / or measurement is important for unlocking the potential of various quantum technologies including quantum computing, quantum simulations and quantum sensing. Depending on the complexity of the application and the to be controlled quantum system, the space of parameters that are used to control the quantum system can become very large. Therefore, optimization and calibration techniques are an important tool to find an optimal set of such control parameters ensuring high fidelities of the quantum operations. Optimization and calibration techniques known from the prior art either achieve only insufficient fidelities for most quantum applications, are extremely time-consuming or converge to solutions that are not reliable with respect to experimental fluctuations.

[0007] Thus, an object of the invention is to overcome such limitations and provide a method, apparatus, computer program and computer-readable data carrier that improve upon the available methods and devices.

[0008] Summary

[0009] This object of the invention is achieved by a method, apparatus, computer program and computer-readable data carrier for optimizing a set of control parameter of a quantum system as described in the appended independent claims. Advantageous developments and embodiments are described in the dependent claims.

[0010] In a first aspect, the invention relates to a method for optimizing a set of control parameter of a quantum system. The method comprises for an (current) iteration of an iteration loop: obtaining at least one set of control parameter values for the current iteration; executing at least one measurement process. The at least one measurement process comprises acting with a sequence of operations of a sequence length (of the current iteration) on the quantum system to determine a sequence fidelity (of the current iteration), wherein the sequence of operation is parameterized by the set of control parameter (for the current iteration). The method also comprises updating the at least one set of control parameter values (for the current iteration) to obtain the at least one set of control parameter values for the next iteration by using an optimization routine based on a cost function obtained from the measured sequence fidelity of the current iteration. The method further comprises determining the sequence length according to an optimal sensitivity criterion.

[0011] With the proposed method, an updated set of control parameter values can be obtained that enables performing fast, efficient and reliable quantum operations on the quantum system. The method enables an optimization and calibration of the control parameter from initially very crude operation fidelities all the way to highest performance fidelities in a single iteration loop, and therefore in minimal time.

[0012] Optionally, the aforementioned method steps and / or a method step of any aspect of the invention may be carried out for a plurality of iterations of the iteration loop or for each iteration of the iteration loop. The method may also comprise carrying out the next iteration of the iteration loop. The next iteration may comprise the same method steps as the current iteration except that the updated set of control parameter values (i.e., the set of control parameter values of the next iteration) and / or the updated sequence length (i.e., the sequence length of the next iteration) is used to determine the sequence fidelity (of the next iteration).

[0013] In particular, by determining the sequence length (e.g. the initial sequence length or the sequence length of the next iteration) according to an optimal sensitivity criterion, the reliability of the optimization results for the updated control parameter values with respect to experimental fluctuations can be improved. The optimal sensitivity criterion can be chosen to ensure that changes in the sequence fidelity are (predominantly) caused by changes in the fidelity of an operation (i.e., a to-be-optimized (target quantum) operation comprised in a sequence of operations) so that measured changes in the sequence fidelity (e.g., during carrying out the iteration loop) can be attributed to changes in the fidelity of the operation and are less affected by experimental fluctuations.

[0014] Optionally, the determining the sequence length according to an optimal sensitivity criterion may comprise determining the sequence length of the current iteration initially (i.e., during an initialization) before the iteration loop is being carried out according to an optimal sensitivity criterion. Then, the sequence length of the current iteration may be the initially determined sequence length (see further below on how an initial sequence length may be determined).

[0015] Additionally, or alternatively the determining the sequence length according to an optimal sensitivity criterion may comprise updating / adapting the sequence length during the iteration loop according to the optimal sensitivity criterion, e.g., from the current iteration to the next iteration. In this case, the determining the sequence length according to an optimal sensitivity criterion may comprise updating / adapting the sequence length of the current iteration according to an update rule to obtain the sequence length of the next iteration (see further below on how a sequence length may be updated).

[0016] A set of control parameter may comprise a plurality of control parameter. A control parameter may parametrize a control signal for controlling the quantum system. A sequence of operations may be at least partially generated by a control signal parameterized by the set of control parameter. Optionally, a control parameter may parametrize an N-qubit quantum logic gate operation acting on the quantum system. A control signal may be a control signal pulse. The set of control parameter may comprise at least one of a control signal pulse amplitude, a control signal pulse width, a control signal pulse edge width. The set of control parameter may also comprise a single qubit phase of the quantum system (i.e., as controlled by a weak coherent electromagnetic signal).

[0017] Each sequence of operations of a sequence length Ncmay comprise a number of Nc(quantum) operations. The quantum system may be a qubit system comprising a plurality of qubits. For example, a qubit may be a superconducting qubit and the quantum system may be a superconducting circuit.

[0018] The sequence of operations may be a randomized sequence of operations. The (quantum) operations in a randomized sequence of operations may be generated randomly from elements of the Clifford group. A randomized sequence of operations may thus be a Clifford sequence. Each quantum operation in a randomized sequence of operation may correspond to a (randomly selected) Clifford gate / operation.

[0019] A sequence of operations may comprise a predetermined number of (to-be- optimized) target quantum operations / gates. For example, each operation in a sequence of operations may comprise at least one target quantum opera- tion / gate. Optionally, a Clifford operation may comprise at least one target quantum gate. The target quantum gate may be an Nq-qubit quantum gate, wherein Nq is an integer. The parameter Nq may be larger than 1. For example, the Nq-qubit quantum gate may be a two-qubit gate with Nq=2. The two-qubit gate may be a CPHASE gate, a CNOT gate, a controlled-Z gate, or a SWAP gate. Additionally, a quantum operation (or each quantum operation) in a randomized sequence of operation may comprise further quantum operations in addition to the at least one target quantum gate. For example, the further quantum operations may be single-qubit operations. The further quantum operations may be randomized in terms of their number and / or positional order within a quantum operation of the randomized sequence of operations, and / or with respect to the single-qubit rotation angle and / or direction.

[0020] The operation in a sequence of operations acting last on the quantum system may be configured to invert the action of the first Nc— 1 operations of the sequence of operations. The expected final state of the quantum system after acting with a sequence of operations on the quantum system may then correspond to the initial state of the quantum system.

[0021] A sequence fidelity Fseqmay be determined by executing a measurement process. A measurement process may comprise M measurements, wherein M is an integer larger than or equal to 1. Each measurement of the M measurements may comprise the following measurement steps: preparing the quantum system in an initial state; acting with a sequence of operations on the quantum system; measuring the final state of the quantum system (by determining a quantum average for the final state) and comparing the final state with the expected final state.

[0022] Preparing the initial state and measuring the final state may be carried out repeatedly to determine a quantum average for the final state (or any observable thereof, e.g., a final state population). Comparing the final state with the expected final state may comprise determining a measurement sequence fidelity. For example, the measurement sequence fidelity may be obtained from the matrix element overlap of the measured final state of a measurement and the expected final state. A sequence error Eseq= 1 — Fseqmay describe the deviations of the measured final state from the expected final state.

[0023] Each measurement of the M measurements of a measurement process may be carried out with a different randomly generated, i.e., randomized, sequence of operations. The sequence fidelity of a measurement process may then correspond to a random sampling average over the measurement sequence fidelities obtained from the M measurements. In case only one measurement M = 1 is being carried out in each measurement process, the random sampling average over the measurement sequence fidelities is the measurement fidelity itself such that there is no difference between the measurement sequence fidelity and the sequence fidelity.

[0024] It may also be the case that the at least one measurement process comprises a plurality of measurement processes. Each measurement process of the plurality of measurement processes may be carried out for the same sequence length (e.g., of the current iteration), but for a different set of control parameter values. In this case, each measurement process of the plurality of measurement processes yields a sequence fidelity for the sequence length. Another average of the sequence fidelity may then be determined over the plurality of measurement processes, wherein the another average of the sequence fidelity may be obtained by averaging over the sequence fidelities obtained from the plurality of measurement processes. This average may also be referred to as a control parameter sampling average since each measurement process of the plurality of measurement processes is being carried out for a different set of control parameter values.

[0025] In case the at least one measurement process comprises only one measurement process, the control parameter sampling average may be identical with the sequence fidelity determined in said measurement process. Said sequence fidelity may then also correspond to a random sampling average as described further above.

[0026] According to this application, and if not stated otherwise, the term 'sequence fidelity' may also imply that a random sampling average and / or a control parameter sampling average has already been carried out and the result of such averaging has been accounted for in said sequence fidelity.

[0027] The sensitivity may be a sensitivity of the sequence fidelity with respect to a fidelity of an operation (i.e., an operation of a sequence of operations). Optionally, the sensitivity corresponds (also) to a sensitivity of the sequence fidelity with respect to a fidelity of a to-be-optimized target quantum gate. For example, if an operation of a sequence of operations comprises a two- qubit target quantum gate with a two-qubit fidelity and further single-qubit quantum gates with single-qubit fidelities, wherein a single-qubit fidelity is higher than a two-qubit fidelity, the sensitivity of the sequence fidelity with respect to the fidelity of said operation can be largely attributed to the sensitivity of the sequence fidelity with respect to the fidelity of the target quantum gate. A sensitivity may be determined from a sequence fidelity based on a model for the dependence of the sequence fidelity Fseqon the fidelity Fcof an operation of the sequence of operations. The fidelity Fcof an operation of the sequence of operations may be the fidelity of a Clifford operation or may (at least approximately be) the fidelity of a to-be-optimized target quantum gate.

[0028] The model may comprise a power law relationship according to + Bo, wherein 710and Boare fitting constants and p is related to the fidelity Fcof an operation of the sequence of operations via Fc= p + (1 — p) / d, wherein d is a dimensionality of the quantum system. The fitting constants 710and Boand the fidelity Fcmay be obtained by fitting the model to a measured dependence of the sequence fidelity on the sequence length. The measured dependence of the sequence fidelity on the sequence length may be obtained by executing measurement processes for different sequence lengths but a same set of control parameter values.

[0029] The fidelity of an operation Fc, e.g., the fidelity of a Clifford and / or target quantum gate, and / or its sensitivity may then be obtained and extracted from the measured sequence fidelity and the power law model as described further above. Preferably, a sensitivity of the sequence fidelity Fseqwith respect to a fidelity of an operation Fcmay be obtained from a derivative of the sequence fidelity Fseqwith respect to a fidelity of an operation Fcdetermined according to the aforementioned power law model.

[0030] The iteration loop may at least comprise a current iteration (or a first iteration) and a next iteration (or a second iteration) following the current iteration. The iteration loop may comprise more than two iterations. The iteration loop may also correspond to a subset of iterations of a larger iteration loop. The iteration loop may be terminated according to a termina- tion criterion. For example, the iteration loop may be terminated when a predetermined error threshold for a normalized sequence error is achieved. The normalized sequence error may correspond to the sequence error of the current iteration divided by the sequence length of the current iteration.

[0031] Optionally, the method may comprise in an (or in each) iteration of the iteration loop, determining the normalized sequence error of the current iteration; checking whether the normalized sequence error is below a predetermined threshold and if the normalized sequence error is below the predetermined threshold, terminating the iteration loop.

[0032] Optionally, the method comprises determining the sequence length to ensure at least a minimal sensitivity of the sequence fidelity with respect to a fidelity of an operation during an iteration of the iteration loop. This may be achieved by keeping the sequence fidelity in an optimal sequence fidelity range at least initially in a first iteration, in a next iteration or during the whole iteration loop, i.e., in each iteration of the iteration loop.

[0033] The optimal sequence fidelity range may be obtained from a dependence of the sequence fidelity on the sequence length for the quantum system. The measured dependence may be obtained by executing a plurality of measurement processes on the quantum system with varying sequence lengths before carrying out the iteration loop for an initial set of control parameter values.

[0034] Preferably, the sequence length is determined according to the optimal sensitivity criterion by choosing the sequence length such that the sensitivity of the sequence fidelity with respect to a fidelity of an operation is above a predetermined lower bound (for the sensitivity) and / or the sequence fidelity lies within a predetermined optimal sequence fidelity range. The predetermined optimal sequence fidelity range may be corresponding to the predetermined lower bound for the sensitivity.

[0035] Advantageously, the predetermined lower bound for the sensitivity and / or the predetermined optimal sequence fidelity range are determined for the quantum system during an initialization carried out before the performing of the iteration loop. The initialization may comprise determining / measuring a dependence of a sequence fidelity on the sequence length by executing a plurality of measurement processes of varying sequence length on the quantum system and determining the predetermined lower bound for the sensitivity and / or the predetermined optimal sequence fidelity range based on the measured dependence of the sequence fidelity on the sequence length. The plurality of measurement processes may comprise acting with sequences of operations of varying sequence length on the quantum system.

[0036] Preferably, the sequence length can be determined according to an optimal sensitivity criterion by updating / adapting the sequence length during the iteration loop (e.g., according to an update / adaptation rule) to keep the sequence fidelity of a next iteration in a predetermined optimal sequence fidelity range. The updating / adapting of the sequence length may be carried out in an (current) iteration of the iteration loop. Optionally, the updating of the sequence length may be performed to keep the sensitivity of the sequence fidelity of a next iteration with respect to an operation above a predetermined lower bound for the sensitivity. The sequence length may be updated / adapted at least once during the iteration loop, e.g., in at least one iteration of the iteration loop. The sequence length may also be updated / adapted a plurality of times during the iteration loop, e.g., in a plurality of iterations of the iteration loop. The update (or adaptation) of the sequence length in an (current) iteration may be configured to increase the sensitivity of the sequence fidelity in a next iteration with respect to a fidelity of an operation, e.g., with respect to a fidelity of the target quantum gate.

[0037] The updating / adapting the sequence length during the iteration loop according to an update / adaptation rule may comprise in at least one (or in each) iteration of the iteration loop:

[0038] -determining based on the sequence fidelity determined in the current iteration whether an optimal sensitivity criterion is fulfilled; and

[0039] -if the optimal sensitivity criterion is not fulfilled, updating / adapting the sequence length to obtain the sequence length for the next iteration;

[0040] The updating / adapting the sequence length during the iteration loop according to an update / adaptation rule may additionally comprise in at least one (or in each) iteration of the iteration loop: -if the optimal sensitivity criterion is fulfilled, keeping the sequence length of the current iteration to obtain the sequence length for the next iteration, i.e., the sequence length in the current iteration and in the next iteration stays the same and no update / adaptation of the sequence length is performed;

[0041] The optimal sensitivity criterion may be fulfilled when the sequence fidelity (or an average thereof, e.g., a corresponding control parameter sampling average) determined in the current iteration is within a / the predetermined optimal sequence fidelity range.

[0042] Optionally, the updating / adapting the sequence length to obtain the sequence length for the next iteration may comprise increasing the sequence length if the sequence fidelity (e.g. a random sampling average and / or control parameter sampling average of the sequence fidelity) measured in the current iteration exceeds (is larger than) the upper bound / limit of the predetermined optimal sequence fidelity range;

[0043] Additionally, or alternatively, the updating the sequence length to obtain the sequence length for the next iteration may comprise decreasing the sequence length if the sequence fidelity (e.g. a random sampling average and / or control parameter sampling average of the sequence fidelity) measured in the current iteration drops below (is smaller than) the lower bound / limit of the predetermined optimal sequence fidelity range.

[0044] Optionally, the sequence length may be increased and / or decreased in the aforementioned situations by a constant and / or predetermined step size. Optionally, the constant and / or predetermined step size is 1.

[0045] An average of the sequence fidelity may correspond to an average of the sequence fidelities measured in the current iteration by executing a plurality of measurement processes in the current iteration, wherein each measurement process may be executed with the same sequence length, but for another set of control parameter values (control parameter sampling average). If only one measurement process is executed in the current iteration for a certain set of control parameter values, an average of the sequence length may correspond to the sequence fidelity obtained from this (only) measurement process. Optionally, the sequence length may be determined according to an optimal sensitivity criterion at least initially by performing an initialization before carrying out the iteration loop. The initial determination of the sequence length may be performed at least for the first iteration of the iteration loop. An update or adaptation of the sequence length during the iteration loop as explained further above may be performed in addition to the initial determination of the sequence length.

[0046] Preferably, the determining the sequence length according to an optimal sensitivity criterion comprises determining the sequence length according to an optimal sensitivity criterion at least initially by performing an initialization before carrying out the iteration loop to determine the sequence length of the first iteration of the iteration loop. The initialization may comprise measuring a dependence of a sequence fidelity on the sequence length by executing a plurality of initial measurement processes of varying sequence length on the quantum system and determining at least the sequence length of the first iteration of the iteration loop to correspond to a maximum of the sensitivity of the sequence fidelity (with respect to the sequence length) determined during initialization. Optionally, the executing a plurality of initial measurement processes may comprise acting with sequences of operations of varying sequence length on the quantum system.

[0047] The initialization may also comprise:

[0048] -determining an initial set of control parameter values;

[0049] -executing a plurality of (initial) measurement processes by acting repeatedly with sequences of operations of varying sequence length on the quantum system, wherein the sequences of operations are parameterized by the same initial set of control parameter values; optionally, each (initial) measurement process of the plurality of (initial) measurement processes comprises acting with a sequence of operations of a different sequence length but comprising the same initial set of control parameter values on the quantum system;

[0050] -determining a dependence of a sequence fidelity and / or a sensitivity of the sequence fidelity with respect to a fidelity of an operation on the sequence length based on the plurality of (initial) measurement processes;

[0051] -determining an optimal sequence length corresponding to the maximum of the sensitivity of the sequence fidelity (with respect to the sequence length);

[0052] -determining the sequence length at least for the first iteration of the iteration loop as the optimal sequence length.

[0053] The initialization may also comprise determining an initial set of control parameter value ranges. In general, a control parameter range may comprise a range of values around a control parameter value, e.g., sampled from a Gaussian distribution. A set of control parameter value ranges may also be an additional input to the optimization routine to find an update for the set of control parameter values.

[0054] The at least one set of control parameter values for the current iteration may comprise a plurality of sets of control parameter values for the current iteration. Also, the at least one measurement process may comprise a plurality of measurement processes.

[0055] Optionally, an iteration, a plurality of iterations or each iteration of the iteration loop may comprise:

[0056] -obtaining a sequence length of the current iteration;

[0057] -executing the plurality of measurement processes, wherein each measurement process of the plurality of measurement processes comprises acting with a sequence of operations of the (same) sequence length of the current iteration but parameterized by a different set of control parameter values of the current iteration on the quantum system; optionally, the sequence of operations in each measurement process is a randomized sequence of operations, wherein acting with a randomized sequence of operations on the quantum system comprises acting repeatedly with a sequence of operations on the quantum system, wherein in each repetition of the sequence the operations of the sequence may be chosen randomly from the Clifford group, wherein each operation comprises at least one target quantum gate; and wherein in each repetition the sequence length is the same and the set of control parameter values is the same as also described further above;

[0058] -determining a value of the cost function for each one of the sequence fidelities measured in the current iteration; and

[0059] -updating the plurality of sets of control parameter values to obtain the plurality of sets of control parameter values for the next iteration by using an optimization routine based on the values of the cost function determined in the current iteration.

[0060] The optimization routine may be configured to determine an optimized update for the set of control parameter values in each iteration. The optimization routine may be configured to minimize the cost function. The cost function may be the sequence error corresponding to the sequence fidelity. The cost function may comprise or may be a sequence error (or infidelity) as obtained from the sequence fidelity measured in the current iteration or from an average of the sequence fidelity of the current iteration (e.g. a random sampling average and / or control parameter sampling average of the sequence fidelity).

[0061] The optimization routine may comprise an evolutionary optimization algorithm configured to compare only values of the cost function determined in the current iteration in order to obtain at least one (or a plurality of) updated set of control parameter values to be used as the set of control parameter values in the next iteration.

[0062] Optionally an iteration, a plurality of iterations or each iteration of the iteration loop may further comprise: determining a control parameter sampling average of the sequence fidelity by averaging over the sequence fidelities measured for the different sets of control parameter values of the current iteration; and determining the sequence length of the next iteration according to an optimal sensitivity criterion based on the control parameter sampling average of the sequence fidelity. The determining the sequence length of the next iteration according to an optimal sensitivity criterion based on the control parameter sampling average of the sequence fidelity may comprise updating / adapting the sequence length according to an update / adaptation rule as described according to any variant of the first aspect, wherein the sequence fidelity then refers to the control parameter sampling average of the sequence fidelity.

[0063] In a second aspect the invention relates to the use of an updated set of control parameter values determined using the method according to the first aspect or any variant thereof. The updated set of control parameter values may be used to control and / or calibrate the quantum system for performing quantum computations and / or quantum simulations with the quantum system. The updated set of control parameter values to be used may correspond to the updated set of control parameter values obtained in the last iteration of the iteration loop as the final result of the iterations / optimization.

[0064] In a third aspect the invention relates to an apparatus for optimizing a set of control parameter of a quantum system, wherein the apparatus is configured to carry out the method according to the first aspect or any variant thereof. The apparatus may comprise a signal generator unit configured to generate control signals parametrized by the set of control parameter. The apparatus may further comprise a measurement unit configured to execute measurement processes and measure a corresponding sequence fidelity. The apparatus may also comprise an optimization unit configured to carry out the optimization routine and provide updates for the set of control parameter values. The optimization unit may further be configured to determine and update the sequence length according to the optimal sensitivity criterion.

[0065] In a fourth aspect the invention relates to a computer program comprising instructions which, when the program is executed by a computer, cause the computer to carry out the steps of the method according to the first aspect or any variant thereof. The computer program (or a sequence of instructions) may use software means for performing the method for optimizing the control parameter when the computer program runs in a computing unit.

[0066] In a fifth aspect the invention relates to a computer-readable data carrier having stored thereon the computer program according to the fourth aspect. The computer program can be stored directly in an internal memory, a memory unit, or the computer.

[0067] In summary, with the proposed method an efficient and accurate optimization of control parameters of a quantum system can be achieved. In particular, the proposed method enables the optimization from very crude operation fidelities all the way to highest performance in one go, and therefore in minimal time. The proposed method is efficient and can be used to (in-situ) tune and calibrate quantum operations on various quantum systems. This includes application scenarios where the calibration and optimization of a quantum system, e.g., a quantum processor, is taken place in parallel to the processor operation, e.g., during quantum computation and / or simulation. Additionally, applications that employ a variational quantum eigensolver (VQE) algorithm can benefit from continuous sensitivity updates as described further above. Another application example includes optimal metrology algorithms for generating desired target states via optimization.

[0068] Detailed Description

[0069] Exemplary embodiments of the invention are illustrated in the drawings and will now be described with reference to figures 1 to 10.

[0070] In the figures:

[0071] Fig. 1 shows control parameter of the quantum system,

[0072] Fig. 2 shows an embodiment of control signal pulses,

[0073] Fig. 3 shows a quantum logic representation of an operation in a randomized sequence of operations,

[0074] Fig. 4 shows the dependence of the sequence fidelity on the sequence length for three different Clifford operation fidelities,

[0075] Fig. 5 shows a sequence fidelity together with its sensitivity with respect to a Clifford operation fidelity as a function of sequence length,

[0076] Fig. 6 shows a sequence fidelity together with its sensitivity with respect to a Clifford operation fidelity as a function of sequence length and an optimal sequence fidelity range for a first Clifford operation fidelity,

[0077] Fig. 7 shows a sequence fidelity together with its sensitivity with respect to a Clifford operation fidelity as a function of sequence length and an optimal sequence fidelity range for a second Clifford operation fidelity,

[0078] Fig. 8 shows a schematic representation of the steps of an embodiment of the method for optimizing control parameter of a quantum system,

[0079] Fig. 9 shows results of an optimization,

[0080] Fig. 10 shows updated control parameter values.

[0081] Figure 1 shows control parameter of the quantum system. The quantum system comprises two superconducting qubits that are coupled via a tunable coupler. The tunable coupler mediates interactions between the qubits. The strength of the qubit-qubit interaction can be tuned via a qubit-qubit control signal pulse as shown schematically in Figure 1. The qubit-qubit control signal pulse can induce a frequency shift of the tunable coupler and thereby tune, e.g., turn on or turn off, the qubit-qubit interaction. For example, the tunable coupler can be a magnetic flux-tunable coupler and the qubit-qubit control signal pulse can induce a magnetic flux to turn on or turn off the qubit-qubit interaction. In Figure 1 the qubit-qubit control signal pulse has a Gaussian- square shape and is parametrized by a global control signal pulse amplitude A, a control signal pulse width w and a control signal edge widthR. Single-qubit phases <p2(e.g., Z-rotations) and frequencies for both qubits can be controlled independently by additional single-qubit control signal pulses (not shown in Figure 1).

[0082] In the following, an exemplary set of control parameter for the quantum system is given by PGaUss=(w<TR> < >I, Z)- Another exemplary set of control parameter for the quantum system is PPicos=( ,A,w,<p1,<p2>)> wherein Y denotes a vector of pulse amplitude values at discrete knots over the pulse width that parametrize a control signal pulse over time (see also inset in the lower panel of Figure 9). The latter parametrization corresponds to a piecewise-constant-slope (PiCoS) parametrization.

[0083] Recurring features are provided in the following figures with identical reference signs as in Figure 1.

[0084] Figure 2 shows an embodiment of single-qubit control signal pulses for both qubits Q±and Q2and of the qubit-qubit control signal pulse Flux for the tunable coupler as a function of time. The single-qubit control signal pulses and the qubit-qubit control signal pulses as shown in Figure 2 realize singlequbit and two-qubit quantum gates as shown in Figure 3. The set of control parameter can also comprise all control parameters necessary to parametrize the whole sequence of control signal pulses as shown in Figure 2.

[0085] Figure 3 shows a quantum logic representation of an operation Cnin a randomized sequence of operations that is realized with the single-qubit control signal pulses and the qubit-qubit control signal pulses as shown in Figure 2. A single-qubit control signal pulse is represented by a single-qubit quantum logic gate. A qubit-qubit control signal pulse is represented by a two- qubit quantum logic gate. The operation Cnshown in Figure 3 is a Clifford operation selected randomly from elements of the Clifford group.

[0086] A randomized sequence of operations comprises Ncsuch randomized Clifford operations CNc... C3C2C (e.g., mathematically representable as an operator or matrix / tensor product of operations, wherein each operation is represented by a matrix / tensor Cnacting on an initial state of the quantum system representable by a vector), wherein Ncis the sequence length and n = 1... Ncis an index that here runs over the operations in the sequence of operation. Each Clifford operation Cnis randomly chosen from the Clifford group. For example, in Figure 3, a Clifford operation is decomposed into two CPHASE two-qubit gates and X, Y, X90 and Y90 single-qubit gates / rotations. The last operation CNcis chosen such that the previous Nc-1 Clifford operations randomly selected from the Clifford group are being inverted. In that case the to-be-expected final state is the initial state of the quantum system, e.g., the ground state of the two-qubit system.

[0087] Each Clifford operation Cncomprises at least one (to-be-optimized) target quantum gate. In the example shown in Figures 2 and 3, each Clifford operation Cncomprises two CPHASE gates as target quantum gates. The set of control parameter parametrizes these CPHASE gates. The two CPHASE gates are parametrized identically in each Clifford operation by the set of control parameter values.

[0088] The fidelity of a sequence of quantum logic operations can be determined using a (randomized) benchmarking measurement protocol. A (randomized) benchmarking measurement protocol comprises a plurality of measurement processes. Each measurement process comprises preparing the quantum system in an initial state, acting with a randomized sequence of operations of a given sequence length Ncon the quantum system, measuring the final state of the quantum system after acting with the randomized sequence on the quantum system and determining a measurement sequence fidelity as a quantum average for the given sequence length by comparing the final state with the to-be-expected final state. In the aforementioned example, the measurement sequence fidelity corresponds to the population of the final state, i.e., the ground state of the quantum system. This sequence of steps is repeated M times for random sampling, wherein M is an integer, in each measurement process in order to determine the random sampling average of the sequence fidelity by averaging over the M measurement sequence fidelities (as a random sampling average over the M measurement sequence fidelities). Consequently, M different sequences of operations are being generated randomly from the Clifford group to obtain the M randomized sequences of operations of the measurement process and the corresponding random sampling average of the sequence fidelity.

[0089] It is assumed that a sequence fidelity or a random sampling average of the sequence fidelity Fseqbehaves as a power law model according to Fseq=^-oFNc l+ Bo> wherein 40and Boare fitting constants and F is related to the fidelity of a Clifford operation Fcvia Fc= F + (1 — p) / d, wherein d is a dimensionality of the quantum system. For a two-qubit system the dimensionality is d = 4. The fitting constants 40and Bodescribe the errors of state preparation / measurement and the error of the last operation CNc(inversion operation), respectively.

[0090] Figure 4 shows the dependence of the sequence fidelity (or a random sampling average thereof) on the sequence length for three different Clifford operation fidelities Fc. More specifically, figure 4 shows a measured (dependence of the) sequence fidelity (or a random sampling average thereof) Fseqas a function of the sequence length Ncas obtained from a plurality of measurement processes executed for different sequence lengths but the same set of control parameter values. Each curve in Figure 4 corresponds to a different set of control parameter values and thus to a different fidelity Fcof the Clifford operation (and thus also to a different fidelity of the CPHASE gate). The solid points in Figure 4 show that for a fixed sequence length Nc=30 the sequence fidelity scales and increases monotonically with the Clifford operation fidelity Fc.

[0091] Since the measurements of the sequence fidelity (or a random sampling average thereof) Fseqare subject to experimental noise, the sensitivity of the sequence fidelity with respect to changes in the fidelity Fcof a Clifford operation during the measurements shall be as high as possible.

[0092] The sensitivity is defined as a derivative of the sequence fidelity (or a random sampling average thereof) with respect to the fidelity of a (Clifford) operation as follows with the normalization constant D. The normalization constant D is chosen such that the maximum of the sensitivity as a function of sequence length is 1.

[0093] Figure 5 shows a sequence fidelity (or a random sampling average thereof) together with its sensitivity with respect to a Clifford operation fidelity as a function of sequence length. More specifically, figure 5 shows the sensitivity S(Nc) together with the sequence fidelity (or a random sampling average thereof) Fseqas a function of the sequence length Ncfor a fidelity of the Clifford operation Fc= 0.9 corresponding to a certain initial set of control parameter values. The circles in Figure 5 mark the values with maximum sensitivity.

[0094] Changes in the fidelity Fcof a Clifford operation, e.g., caused by changes in the set of control parameter values, should generate large changes in the sequence fidelity Fseq(or a random sampling average thereof) so that experimental fluctuations take only a small influence and measured changes in the sequence fidelity, e.g., measured during optimization, can be assigned to changes in the fidelity of a Clifford operation (and thus also to changes in the fidelity of the target quantum gate, i.e., the CPHASE gate). In principle, experimental fluctuations and noise can also be reduced by increasing the number of measurement repetitions M in a measurement process, but this would result in longer measurement times which ideally is to be avoided.

[0095] It can be shown that the sensitivity S( VC) of the sequence fidelity Fseq(or a random sampling average thereof) has a maximum at an optimal sequence length and decreases for higher as well as smaller values of the sequence length.

[0096] Figure 6 shows a sequence fidelity together with its sensitivity with respect to a Clifford operation fidelity as a function of sequence length and a predetermined optimal sequence fidelity range for a first Clifford operation fidelity. More specifically, figure 6 shows the sensitivity S( VC) together with theran- dom sampling average of the sequence fidelity Fseqas a function of the sequence length Ncfor a fidelity of the Clifford operation Fc= 0.9 as shown in Figure 5. The vertical solid lines mark a predetermined optimal sequence fidelity range and a corresponding predetermined lower bound for the sensitivity. For optimization it is optimal if the sensitivity is kept in a range above the lower bound here chosen to be 0,9 (90 percent) corresponding to S NC> 90%. From that predetermined lower bound of the sensitivity a corresponding predetermined optimal sequence fidelity range can be determined corresponding in Figure 6 to 0,6 < Fseq< 0,8. In terms of a sequence error Eseq= 1 — Fseqthis corresponds to a predetermined optimal sequence infidelity (error) range of 0,2 < Eseq< 0,4. Note, that here it is not the (only) objective to minimize the sequence error to an arbitrarily small number (possible at the expense of a small sensitivity), but rather to minimize a normalized sequence error, i.e., a sequence error normalized / divided by the sequence length while maintaining a high sensitivity. In particular, according to the proposed method it may be advantageous to maintain the sequence error above a lower limit and rather update the sequence length in order to further decrease the normalized sequence error (as defined and shown further below).

[0097] Figure 7 shows a random sampling average of the sequence fidelity together with its sensitivity with respect to a Clifford operation fidelity as a function of sequence length and a corresponding predetermine optimal sequence fidelity range for a second Clifford operation fidelity Fc. More specifically, figure 7 shows the sensitivity S( VC) together with the random sampling average of the sequence fidelity Fseqas a function of the sequence length Ncfor a higher fidelity of the Clifford operation Fc= 0.95, e.g., as obtained from measurement processes executed for another initial set of control parameter values.

[0098] It can be appreciated from a comparison of Figures 6 and 7 that for the same lower bound 0,9 for the sensitivity, the same predetermined optimal sequence fidelity range is obtained independent of the fidelity of the Clifford operation Fc. However, the optimal sequence length has shifted to higher values.

[0099] Therefore, an initial optimal choice of the sequence length according to a lower bound of the sensitivity or a continuous updating / adapting of the sequence length during an optimization in an iteration loop can considerably improve the optimization, because it becomes possible to determine / measure the sequence fidelity with high sensitivity and simultaneously improve the fidelity of the Clifford operation. The two lower and upper limits of a predetermined optimal sequence fidelity range can be clearly assigned to a sequence length being either too high or too low for optimum sensitivity. These considerations can be used to improve available optimization techniques as described further below.

[0100] Figure 8 shows a schematic representation of the steps of an embodiment of the method for optimizing a set of control parameters of a quantum system.

[0101] The method comprises an initialization step SO.

[0102] The initialization comprises determining an initial set of control parameter values and a corresponding initial set of control parameter ranges. Each control parameter range specifies a value range around a control parameter value (here sampled from a Gaussian distribution). The initial set of control parameter values can be guessed based on experience or may be estimated based on an initial simulation model.

[0103] The initialization further comprises executing a plurality of initial measurement processes by acting with randomized sequences of operations of varying sequence length but parameterized with the same initial set of control parameter values on the quantum system in order to determine a dependence of a random sampling average of the sequence fidelity on the sequence length. From said dependence a Clifford operation fidelity for the initial set of control parameter values is determined by fitting the aforementioned power law model to the measured dependence of the random sampling average of the sequence fidelity on the sequence length.

[0104] The initialization then further comprises determining a dependence of the sensitivity of the sequence fidelity with respect to a fidelity of a (Clifford) operation on the sequence length based on the plurality of measurement processes and the aforementioned power law model.

[0105] The initialization further comprises determining an initial value for the sequence length to be used during the first iteration of the optimization (see further below) as the optimal sequence fidelity corresponding to the maximum of the sensitivity of the sequence fidelity, i.e., determining the sequence length in the first iteration of the iteration loop as the optimal sequence length.

[0106] The initialization further comprises determining a predetermined lower bound of the sensitivity and a predetermined optimal sequence fidelity range from the measured dependence of the random sampling average of the sequence fidelity on the sequence length as explained further above.

[0107] After initialization an iteration loop is being carried out. For each iteration i = 1, ... N of the iteration loop wherein N denotes the total number of iterations, the following steps are being carried out:

[0108] Step SI comprises obtaining a plurality of P sets of control parameter values and corresponding control parameter value ranges, wherein P is an integer larger than 1. If the current iteration is the first iteration with i = 1 the plurality of sets of control parameter values are obtained based on the initial set of control parameter values. For the subsequent iterations the plurality of sets of control parameter values are obtained by the optimization routine from the updates based on the cost function determined in a previous iteration (see Step S3).

[0109] Step SI also comprises obtaining a sequence length of / for the current iteration. If the current iteration is the first iteration with i = 1, the sequence length of the current iteration will correspond to optimal sequence length determined during initialization in step SO. For the subsequent iterations, the sequence length of the current iteration is determined based on the optimal sensitivity criterion (see sensitivity update check in Step S3).

[0110] Step S2 comprises executing a plurality of P measurement processes, wherein each measurement process with index p, where p = 1. . P is an integer, comprises acting M times with a randomized sequence of Clifford operations of the sequence length of the current iteration i on the quantum system to determine M measurement sequence fidelities where m = 1. . M is the measurement index running over the M repetitions of acting with the randomized sequence of operations on the quantum system. Each measurement process is executed for another set of the plurality of P sets of control parameter values of the current iteration such that the randomized sequences of Clifford operations of each of the plurality of measurement processes is parameterized by a different set of the plurality of sets of control parameter values of the current iteration. Step S2 further comprises determining for each set of control parameter values of the current iteration a sequence fidelity (as a random sampling average). A random sampling average of the sequence fidelity is obtained for each measurement process p by averaging over the M measurement sequence fidelities, i.e., corresponds to the random sampling average. Thus, a random sampling average of the sequence fidelity is determined for every set p of control parameter values of the current iteration i.

[0111] Step S2 then comprises updating the sets of control parameter values to obtain the sets of control parameter values for the next iteration i + 1 by using an optimization routine based on a cost function obtained from the determined random sampling averages of the sequence fidelity of the current iteration i. More specifically, the optimization routine evaluates in step S2 a cost function based on the random sampling averages of the sequence fidelity obtained from the current iteration and determines updates for the plurality of sets of control parameter values to obtain the plurality of sets of control parameter values for the next iteration (see step SI for the next iteration) based on the values of the cost function determined in the current iteration such that better results, i.e., lower values for the cost function, can be expected from the next iteration.

[0112] The cost function is based on a sequence infidelity or sequence error. The cost function of the current iteration is obtained from the random sampling averages of the sequence fidelity of the current iteration i. In this example, the cost function is identical with the sequence error corresponding to the control parameter sampling average (,Fsleq) of the sequence fidelity. The control parameter sampling average of the sequence fidelity (,Fsleq) is obtained by averaging over the random sampling averages of the sequence fidelity as obtained in the current iteration, i.e., (,Fsleq) = corresponds to the control parameter sampling average. Note that the term control parameter sampling average of the sequence fidelity may also be referred to as sequence fidelity. The cost function is then given by the sequence error Esleq= 1 - <F^q).

[0113] Step S2 also comprises a sensitivity update check. The sensitivity update check comprises determining / obtaining the control parameter sampling average of the sequence fidelity and checking whether the sequence fidelity, i.e., the control parameter sampling average of the sequence fidelity, lies within the predetermined optimal sequence fidelity range.

[0114] If the sequence fidelity or the control parameter sampling average of the sequence fidelity does not lie within the predetermined optimal sequence fidelity range, the sequence length is adapted and updated to obtain the sequence length for the next iteration, i.e., the sequence length of the randomized sequence of operations in step SI of the next iteration.

[0115] The update of the sequence length, i.e., the sequence length of the next iteration, is obtained by increasing the sequence length of the current iteration if the sequence fidelity or control parameter sampling average of the sequence fidelity determined in the current iteration exceeds the upper bound / limit of the predetermined optimal sequence fidelity range (or, equivalently, drops below the lower bound / limit of the corresponding predetermined optimal sequence error range).

[0116] The update of the sequence length, i.e., the sequence length of the next iteration, is obtained by decreasing the sequence length of the current iteration if the sequence fidelity or control parameter sampling average of the sequence fidelity determined in the current iteration drops below the lower bound / limit of the predetermined optimal sequence fidelity range (or, equivalently, exceeds the upper bound / limit of the corresponding predetermined optimal sequence error range).

[0117] If the sequence fidelity or control parameter sampling average of the sequence fidelity lies within the predetermined optimal sequence fidelity range (or, equivalently within the predetermined optimal sequence error range), the sequence length is not adapted and updated, but the sequence length for the next iteration is equal to the sequence length of the current iteration.

[0118] Steps 1-3 are iteratively repeated, i.e., for each iteration in the iteration loop, a predetermined number of times (number of iterations) or until a termination criterion is fulfilled. For example, the termination criterion may specify a target threshold for a normalized sequence error (infidelity) which corresponds to the sequence error (infidelity) divided by the sequence length (see also lower panel of Figure 9).

[0119] Figure 9 shows results of an optimization carried out with an iteration loop as described further above with 260 iterations (evolutions). The upper panel in Figure 9 shows the sequence length during the iteration. Several updates and adaptations of the sequence length during the iteration are shown. The sequence length increases from an initial value of NP(i = 1) = Nc= 2 in the first iteration to an updated value NP(i = 260) = 12 in the last iteration. The middle panel shows the sequence error or infidelity ELeqas obtained from the sequence fidelity. A dot shows the sequence error Eslq= 1 — corresponding to a random sampling average of the (measurement) sequence fidelity as obtained from each measurement process. The black solid line shows the (averaged) sequence error ELeq= 1 — (,Fsleq) corresponding to the control parameter sampling average of the sequence fidelity (,Fsleq) for each iteration. The predetermined optimal sequence error range is 0.2 < Egeq< 0.4. The change in shading in the middle panel indicates changes in the sequence length NP(i) and thus increases in the sensitivity.

[0120] The optimization improves the control parameter sampling average of the sequence fidelity as long as the sequence length stays constant, i.e., decreases the corresponding sequence error ELeqor cost function. Whenever in an iteration the sequence error ELeqdrops below 0.2, the sequence length NPis increased by 1. Since during the optimization, the sequence error ELeqdid not exceed the upper limit, the sequence length was continuously increased during the iteration loop. Note, that the update rule here is described in terms of a predetermined optimal sequence error range but can equivalently be described in terms of a corresponding predetermined sequence fidelity range as explained further above.

[0121] The lower panel shows the normalized sequence error Eporm= Eseq / NP(i), i.e., the (averaged) sequence error of the current iteration divided by the sequence length of the current iteration. The normalized sequence error Enorm 's anormalized sequence error per Clifford operation and can be considered a quality measure of the optimization that continuously decreases during the iteration loop.

[0122] The inset in the lower panel shows a PiCoS parametrization of the qubit-qubit control signal pulse. The set of control parameter comprises a vector of pulse amplitude values Y = (To, representing control signal pulse amplitudes of Nynodes Ykwith equal spacing w / Nyover the pulse width w. The set of control parameter additionally comprises a global amplitude scaling A, the width w, and Z-rotationsX, < >2applied on the respective qubits. While global parameters like A and w seem redundant at first sight due to the presence of the amplitude parametrization with the PiCoS approach, they allow for global adaptations to rescale the control signal pulse and therefore add significant robustness to the optimization / calibration. In another embodiment, a control signal pulse may also parametrized using a Gaussian parametrization or a Fourier decomposition.

[0123] Figure 10 shows the result of the optimization for three control parameter of the PiCoS parametrization shown in the inset of the lower panel of Figure 9. Here, To, Y^, Y16correspond to different amplitude values along the temporal width of the pulse.

[0124] Advantageously, the optimization routine is robust to rescaling of the cost function. In the embodiment of Figure 9, an evolutionary algorithm based on a covariance matrix adaptation evolution strategy (CMA-ES) has been used.

[0125] In this optimization routine only cost function values within an evolutionary step, i.e., within one iteration, are compared to adjust the control parameter values of the sampling distributions used to propose new control parameter values. In particular, in this case the cost function values must not be compared over the entire optimization, i.e., over the entire iteration loop, for example using the global best cost function value, as it would be the case with many other optimization algorithms, such as Nelder-Mead, L-BFGS and Bayesian optimization algorithms.

[0126] In these cases, the optimization would have to be reinitialized as soon as the sequence length is changed. This means that the internally learned algorithmic optimization parameters would have to be determined again after an update of the sequence length which would be a disadvantage.

[0127] By using an evolutionary optimization the internal meta-parameters of the algorithm that are learned during the operation, such as covariance matrices of the control parameters, the shape of the landscape in the form of gradient vectors, and the valid search range of each control parameter, can be retained during the iteration allowing for an overall more efficient optimization.

[0128] However, in principle in another embodiment also a conventional algorithm can be used, provided that the cost function is corrected and rescaled for the increase in sensitivity by rescaling the measured cost function values accordingly.

[0129] In summary, a reliable calibration of complex pulse shapes has been achieved based on closed-loop optimization method with two-qubit Clifford sequences. Optimal high-complexity pulse shapes were found in a time-efficient manner using a method that adapts the sensitivity of the randomized sequences of operation according to an optimal fidelity range of optimized pulses.

[0130] Features of the different embodiments which are merely disclosed in the exemplary embodiments as a matter of course can be combined with one another and can also be claimed individually.

Claims

Claims1. A method for optimizing a set of control parameter of a quantum system, the method comprising in an iteration of an iteration loop: obtaining at least one set of control parameter values for the current iteration; executing at least one measurement process, wherein the at least one measurement process comprises acting with a sequence of operations of a sequence length on the quantum system to determine a sequence fidelity, wherein the sequence of operation is parameterized by the set of control parameter; and updating the at least one set of control parameter values to obtain the at least one set of control parameter values for the next iteration by using an optimization routine based on a cost function obtained from the measured sequence fidelity of the current iteration; wherein the method further comprises determining the sequence length according to an optimal sensitivity criterion.

2. The method according to claim 1, wherein the sequence length is determined according to the optimal sensitivity criterion by choosing the sequence length such that the sensitivity of the sequence fidelity with respect to a fidelity of an operation is above a predetermined lower bound and / or the sequence fidelity lies within a predetermined optimal sequence fidelity range.

3. Method according to claim 2, wherein the predetermined lower bound and / or the predetermined optimal sequence fidelity range are determined for the quantum system during an initialization carried out before the performing of the iteration loop, wherein the initialization comprises determining a dependence of a sequence fidelity on the sequence length by acting with sequences of operations of varying sequence length on the quantum system and determining the predetermined lower bound and / or the predetermined optimal sequence fidel-ity range based on the measured dependence of the sequence fidelity on the sequence length.

4. The method according to any previous claim, wherein determining the sequence length according to an optimal sensitivity criterion comprises: updating the sequence length during the iteration loop according to an update rule to keep the sequence fidelity in a next iteration in a predetermined optimal sequence fidelity range and / or to keep the sensitivity of the sequence fidelity with respect to a fidelity of an operation in a next iteration above a predetermined lower bound and / or to increase the sensitivity of the sequence fidelity in a next iteration with respect to a fidelity of an operation.

5. Method according to claim 4, wherein the updating the sequence length during the iteration loop according to an update rule comprises in at least one iteration of the iteration loop: determining based on the sequence fidelity determined in the current iteration whether the optimal sensitivity criterion is fulfilled, wherein the optimal sensitivity criterion is fulfilled when the sequence fidelity determined in the current iteration is within a predetermined optimal sequence fidelity range; and if the optimal sensitivity criterion is not fulfilled, updating the sequence length to obtain the sequence length for the next iteration.

6. Method according to claim 5, wherein the updating the sequence length to obtain the sequence length for the next iteration comprises: increasing the sequence length if the sequence fidelity determined in the current iteration exceeds the upper limit of the predetermined optimal sequence fidelity range; and / ordecreasing the sequence length if the sequence fidelity determined in the current iteration drops below the lower limit of the predetermined optimal sequence fidelity range.

7. Method according to any previous claim, wherein the determining the sequence length according to an optimal sensitivity criterion comprises determining the sequence length according to an optimal sensitivity criterion at least initially by performing an initialization before carrying out the iteration loop to determine the sequence length of the first iteration of the iteration loop, wherein the initialization comprises determining a dependence of a sequence fidelity on the sequence length by acting with sequences of operations of varying sequence length on the quantum system and determining at least the sequence length of the first iteration of the iteration loop to correspond to a maximum of the sensitivity of the sequence fidelity determined during initialization with respect to the sequence length.

8. Method according to claim 7, wherein the initialization comprises: determining an initial set of control parameter values; executing a plurality of initial measurement processes by acting repeatedly with sequences of operations of varying sequence length on the quantum system, wherein the sequences of operations are parameterized by the same initial set of control parameter values; determining a dependence of a sequence fidelity and / or a sensitivity of the sequence fidelity with respect to a fidelity of an operation on the sequence length based on the plurality of initial measurement processes; determining an optimal sequence length corresponding to the maximum of the sensitivity of the sequence fidelity with respect to the sequence length; determining the sequence length in the first iteration of the iteration loop as the optimal sequence length.

9. The method according to any previous claim, wherein the at least one set of control parameter values for the current iteration comprises a plurality of sets of control parameter values for the current iteration; and the at least one measurement process comprises a plurality of measurement processes; and an iteration of the iteration loop additionally comprises: obtaining a sequence length of the current iteration; executing the plurality of measurement processes, wherein each measurement process comprises acting with a sequence of operations of the sequence length of the current iteration on the quantum system to measure a sequence fidelity, wherein the sequence of operation of each of the plurality of measurement processes is parameterized by a different set of control parameter values of the current iteration; determining a value of the cost function for each one of the sequence fidelities measured in the current iteration; and updating the plurality of sets of control parameter values to obtain the plurality of sets of control parameter values for the next iteration by using an optimization routine based on the values of the cost function determined in the current iteration.

10. The method according to claim 9, wherein the optimization routine comprises an evolutionary optimization algorithm configured to compare only values of the cost function determined in the current iteration in order to obtain the updated plurality of sets of control parameter values.

11. The method according to claim 9 or 10, wherein determining a control parameter sampling average of the sequence fidelity by averaging over the sequence fidelities measured for the different sets of control parameter values of the current iterationand determining the sequence length of the next iteration according to an optimal sensitivity criterion based on the control parameter sampling average of the sequence fidelity.

12. Use of an updated set of control parameter values determined using the method according to any of the claims 1 to 11 in order to control a quantum system for performing quantum computations and / or quantum simulations with the quantum system.

13. An apparatus for optimizing a set of control parameter of a quantum system, wherein the apparatus is configured to carry out the method according to any of the claims 1 to 11.

14. A computer program comprising instructions which, when the program is executed by a computer, cause the computer to carry out the steps of the method according to any of the claims 1 to 11.

15. A computer-readable data carrier having stored thereon the computer program of claim 14.

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