Quantum control methods and systems

Abstract

Provided are quantum control methods and systems. The method includes generating a control pulse in a quantum system, determining a measurement of a state of the quantum system in response to the control pulse, determining a control pulse correction based on the control pulse, the measurement of the state of the quantum system, and a model reference state, modifying the control pulse based on the control pulse correction, resulting in a modified control pulse, and generating the modified control pulse in the quantum system.

Description

Attorney Docket No. 08993-2502647 (2024-264)QUANTUM CONTROL METHODS AND SYSTEMSCROSS REFERENCE TO RELATED APPLICATION

[0001] This application claims the benefit of United States Provisional Patent Application No. 63 / 667,302 filed on July 3, 2024, the disclosure of which is hereby incorporated by reference in its entirety.GOVERNMENT RIGHTS NOTICE

[0002] This invention was made with government support under 1730449 awarded by the National Science Foundation (NSF). The government has certain rights in the invention.BACKGROUND1. Field

[0003] This disclosure relates generally to quantum systems and, in non-limiting embodiments, to systems, apparatuses, and methods for controlling a quantum system.2. Technical Considerations

[0004] A pressing issue for realizing useful quantum technologies is the problem of calibrating control sequences on hardware, where parameter drift, noise, and measurement can all pose problems.

[0005] Upon obtaining a set of control pulses, the problem remains of how to properly implement them in an experiment and quantify their performance. In the most immediate sense, this involves converting from pulse amplitudes to the corresponding amplitudes on the control hardware. In general, this will require a transfer function that converts between the two and, depending on the hardware implementation, may also contain a frequency dependence that necessitates calibration.SUMMARY

[0006] According to non-limiting embodiments or aspects, provided is a method comprising: generating a control pulse in a quantum system; determining a measurement of a state of the quantum system in response to the control pulse; determining a control pulse correction based on the control pulse, the measurement of the state of the quantum system, and a model reference state; modifying the control pulse based on the control pulse correction, resulting in a modified control pulse; and generating the modified control pulse in the quantum system. In non-limiting embodiments or aspects, the modified control pulse is generated by at least one of the following: an arbitrary waveform generator, optical pulse shaping, or any combination thereof.65E3815. DOCX Page 1 of 21Attorney Docket No. 08993-2502647 (2024-264)

[0007] In non-limiting embodiments or aspects, the control pulse correction is based on conducting a line search of different step sizes for the control pulse correction. In nonlimiting embodiments or aspects, the model reference state comprises a predicted measurement of the state based on a model of the quantum system. In non-limiting embodiments or aspects, the method includes automatically calibrating the quantum system at regular intervals. In non-limiting embodiments or aspects, wherein determining the measurement of the state of the quantum system comprises selecting a measurement method based on a measurement model and determining the measurement of the state based on the measurement method. In non-limiting embodiments or aspects, the method includes modeling the state of the quantum system in response to the control pulse.

[0008] According to non-limiting embodiments or aspects, provided is a system comprising at least one computing device configured to: generate a control pulse in a quantum system; determine a measurement of a state of the quantum system in response to the control pulse; determine a control pulse correction based on the control pulse, the measurement of the state of the quantum system, and a model reference state; modify the control pulse based on the control pulse correction, resulting in a modified control pulse; and generate the modified control pulse in the quantum system.

[0009] In non-limiting embodiments or aspects, the modified control pulse is generated by at least one of the following: an arbitrary waveform generator, optical pulse shaping, or any combination thereof. In non-limiting embodiments or aspects, the control pulse correction is based on conducting a line search of different step sizes for the control pulse correction. In non-limiting embodiments or aspects, the model reference state comprises a predicted measurement of the state based on a model of the quantum system. In non-limiting embodiments or aspects, the at least one computing device is further configured to: automatically calibrate the quantum system at regular intervals. In non-limiting embodiments or aspects, wherein determining the measurement of the state of the quantum system comprises selecting a measurement method based on a measurement model and determining the measurement of the state based on the measurement method. In non-limiting embodiments or aspects, the at least one computing device is further configured to: model the state of the quantum system in response to the control pulse.

[0010] According to non-limiting embodiments or aspects, provided is a computer program product comprising a non-transitory computer-readable medium including program instructions that, when executed by at least one computing device, cause the computing device to: generate a control pulse in a quantum system; determine a measurement of a state 65E3815. DOCX Page 2 of 21Attorney Docket No. 08993-2502647 (2024-264)of the quantum system in response to the control pulse; determine a control pulse correction based on the control pulse, the measurement of the state of the quantum system, and a model reference state; modify the control pulse based on the control pulse correction, resulting in a modified control pulse; and generate the modified control pulse in the quantum system.

[0011] In non-limiting embodiments or aspects, the modified control pulse is generated by at least one of the following: an arbitrary waveform generator, optical pulse shaping, or any combination thereof. In non-limiting embodiments or aspects, the control pulse correction is based on conducting a line search of different step sizes for the control pulse correction. In non-limiting embodiments or aspects, the at least one computing device is further caused to: automatically calibrate the quantum system at regular intervals. In nonlimiting embodiments or aspects, wherein determining the measurement of the state of the quantum system comprises selecting a measurement method based on a measurement model and determining the measurement of the state based on the measurement method. In nonlimiting embodiments or aspects, the at least one computing device is further caused to: model the state of the quantum system in response to the control pulse.

[0012] Other preferred and non-limiting embodiments or aspects of the present invention will be set forth in the following numbered clauses:

[0013] Clause 1: A method comprising: generating a control pulse in a quantum system; determining a measurement of a state of the quantum system in response to the control pulse; determining a control pulse correction based on the control pulse, the measurement of the state of the quantum system, and a model reference state; modifying the control pulse based on the control pulse correction, resulting in a modified control pulse; and generating the modified control pulse in the quantum system.

[0014] Clause 2: The method of clause 1, wherein the modified control pulse is generated by at least one of the following: an arbitrary waveform generator, optical pulse shaping, or any combination thereof.

[0015] Clause 3: The method of any of clauses 1-2, wherein the control pulse correction is based on conducting a line search of different step sizes for the control pulse correction.

[0016] Clause 4: The method of any of clauses 1-3, wherein the model reference state comprises a predicted measurement of the state based on a model of the quantum system.

[0017] Clause 5: The method of any of clauses 1-4, further comprising: automatically calibrating the quantum system at regular intervals.

[0018] Clause 6: The method of any of clauses 1-5, wherein determining the measurement of the state of the quantum system comprises selecting a measurement method 65E3815. DOCX Page 3 of 21Attorney Docket No. 08993-2502647 (2024-264)based on a measurement model and determining the measurement of the state based on the measurement method.

[0019] Clause 7: The method of any of clauses 1-6, further comprising modeling the state of the quantum system in response to the control pulse.

[0020] Clause 8: A system comprising at least one computing device configured to: generate a control pulse in a quantum system; determine a measurement of a state of the quantum system in response to the control pulse; determine a control pulse correction based on the control pulse, the measurement of the state of the quantum system, and a model reference state; modify the control pulse based on the control pulse correction, resulting in a modified control pulse; and generate the modified control pulse in the quantum system.

[0021] Clause 9: The system of clause 8, wherein the modified control pulse is generated by at least one of the following: an arbitrary waveform generator, optical pulse shaping, or any combination thereof.

[0022] Clause 10: The system of any of clauses 8-9, wherein the control pulse correction is based on conducting a line search of different step sizes for the control pulse correction.

[0023] Clause 11: The system of any of clauses 8-10, wherein the model reference state comprises a predicted measurement of the state based on a model of the quantum system.

[0024] Clause 12: The system of any of clauses 8-11, wherein the at least one computing device is further configured to: automatically calibrate the quantum system at regular intervals.

[0025] Clause 13: The system of any of clauses 8-12, wherein determining the measurement of the state of the quantum system comprises selecting a measurement method based on a measurement model and determining the measurement of the state based on the measurement method.

[0026] Clause 14: The system of any of clauses 8-13, wherein the at least one computing device is further configured to: model the state of the quantum system in response to the control pulse.

[0027] Clause 15: A computer program product comprising a non-transitory computer-readable medium including program instructions that, when executed by at least one computing device, cause the computing device to: generate a control pulse in a quantum system; determine a measurement of a state of the quantum system in response to the control pulse; determine a control pulse correction based on the control pulse, the measurement of the state of the quantum system, and a model reference state; modify the control pulse based65E3815. DOCX Page 4 of 21Attorney Docket No. 08993-2502647 (2024-264)on the control pulse correction, resulting in a modified control pulse; and generate the modified control pulse in the quantum system.

[0028] Clause 16: The computer program product of clause 15, wherein the modified control pulse is generated by at least one of the following: an arbitrary waveform generator, optical pulse shaping, or any combination thereof.

[0029] Clause 17: The computer program product of any of clauses 15-16, wherein the control pulse correction is based on conducting a line search of different step sizes for the control pulse correction.

[0030] Clause 18: The computer program product of any of clauses 15-17, wherein the at least one computing device is further caused to: automatically calibrate the quantum system at regular intervals.

[0031] Clause 19: The computer program product of any of clauses 15-18, wherein determining the measurement of the state of the quantum system comprises selecting a measurement method based on a measurement model and determining the measurement of the state based on the measurement method.

[0032] Clause 20: The computer program product of any of clauses 15-19, wherein the at least one computing device is further caused to: model the state of the quantum system in response to the control pulse.

[0033] These and other features and characteristics of the present disclosure, as well as the methods of operation and functions of the related elements of structures and the combination of parts and economies of manufacture, will become more apparent upon consideration of the following description and the appended claims with reference to the accompanying drawings, all of which form a part of this specification, wherein like reference numerals designate corresponding parts in the various figures. It is to be expressly understood, however, that the drawings are for the purpose of illustration and description only and are not intended as a definition of the limits of the invention.BRIEF DESCRIPTION OF THE DRAWINGS

[0034] Additional advantages and details are explained in greater detail below with reference to the non-limiting, exemplary embodiments that are illustrated in the accompanying figures shown in the separate attachment, in which:

[0035] FIG. 1 is a schematic diagram of a quantum control system according to nonlimiting embodiments or aspects;65E3815. DOCX Page 5 of 21Attorney Docket No. 08993-2502647 (2024-264)

[0036] FIG. 2 is a flow diagram of a quantum control method for iterative learning according to non-limiting embodiments or aspects; and

[0037] FIG. 3 illustrates a schematic diagram of components used in non-limiting embodiments or aspects.DETAILED DESCRIPTION

[0038] It is to be understood that the embodiments may assume various alternative variations and step sequences, except where expressly specified to the contrary. It is also to be understood that the specific devices and processes described in the following specification are simply exemplary embodiments or aspects of the disclosure. Hence, specific dimensions and other physical characteristics related to the embodiments or aspects disclosed herein are not to be considered as limiting. No aspect, component, element, structure, act, step, function, instruction, and / or the like used herein should be construed as critical or essential unless explicitly described as such. Also, as used herein, the articles “a” and “an” are intended to include one or more items and may be used interchangeably with “one or more” and “at least one.” Also, as used herein, the terms “has,” “have,” “having,” or the like are intended to be open-ended terms. Further, the phrase “based on” is intended to mean “based at least partially on” unless explicitly stated otherwise.

[0039] As used herein, the terms “communication” and “communicate” refer to the receipt or transfer of one or more signals, messages, commands, or other type of data. For one unit (e.g., any device, system, or component thereof) to be in communication with another unit means that the one unit is able to directly or indirectly receive data from and / or transmit data to the other unit. This may refer to a direct or indirect connection that is wired and / or wireless in nature. Additionally, two units may be in communication with each other even though the data transmitted may be modified, processed, relayed, and / or routed between the first and second unit. For example, a first unit may be in communication with a second unit even though the first unit passively receives data and does not actively transmit data to the second unit. As another example, a first unit may be in communication with a second unit if an intermediary unit processes data from one unit and transmits processed data to the second unit. It will be appreciated that numerous other arrangements are possible.

[0040] As used herein, the terms “processor” or “computing device” may refer to one or more electronic devices configured to process data. A processor and / or computing device may include, for example, a Central Processing Unit (CPU), a Graphics Processing Unit (GPU), a microprocessor, a controller, a field programmable gate array (FPGA), and / or any other computational device capable of executing logic. A “computer readable medium” may 65E3815. DOCX Page 6 of 21Attorney Docket No. 08993-2502647 (2024-264)refer to one or more memory devices or other non-transitory storage mechanisms capable of storing compiled or non-compiled program instructions for execution by one or more processors. Reference to “a processor” or “a computing device” as used herein, may refer to a previously-recited computing device and / or processor that is recited as performing a previous step or function, a different server and / or processor, and / or a combination of computing devices and / or processors. For example, as used in the specification and the claims, a first computing device and / or a first processor that is recited as performing a first step or function may refer to the same or different computing device and / or a processor recited as performing a second step or function.

[0041] In non-limiting embodiments, the performance of control pulses may be quantified by converting from pulse amplitudes to corresponding amplitudes on the quantum system hardware. This may involve a transfer function that converts between the two and, depending on the hardware implementation, may also contain a frequency dependence that necessitates calibration. The most basic form of an amplitude transfer function is a proportional relationship, as the magnitude of the electronics depend proportionally on the target amplitude. A transfer function may be defined with a single number: the ratio between control electronics amplitude and effective amplitude enacted on the quantum system. However, in practice, the true relationship will often deviate from this ideal. There can be several causes for this that depend primarily on limitations of the control electronics, and the severity of these effects will depend on the nature of the optimized controls. The first cause is the finite resolution of the quantum hardware itself. Some examples of this include the amplitude resolution of a set laser intensity or the bit resolution of an arbitrary waveform generator (AWG) that prevents the generation of arbitrary values. The time resolution of the hardware may be considered, as the optimized pulses will be constant over each of the corresponding time intervals. Many AWGs implement filtering functions to alleviate this effect but, depending on the nature of the optimized controls, the system may still need calibration. Therefore, non-limiting embodiments provide a transfer function by sweeping multiple electronics amplitudes (and times in some examples) with constant pulses and measuring the response on the hardware system to ensure proper input of optimized control pulses. To further ensure accuracy, these effects (including any filtering functions) may be modeled as part of the optimization itself.

[0042] In some examples, a frequency dependence calibration may help with accurately implementing the optimized control pulses. This effect can be caused when wiring, filtering, or other circuit elements between the quantum hardware and the control electronics generate 65E3815. DOCX Page 7 of 21Attorney Docket No. 08993-2502647 (2024-264)a nonconstant frequency dispersion that causes the magnitude of the pulse received by the hardware to be different at different frequencies. Since the optimized control pulses produced with non-limiting embodiments described herein may be non-constant and have some nontrivial Fourier transform with support over a range of frequencies, the accuracy of those pulses may be negatively affected. Consequently, this effect may be calibrated by sweeping the frequency of an input constant amplitude pulse and measuring the response. Even more directly, the optimized pulse may be input while sweeping its carrier frequency and the observed Fourier spectrum may be compared with the ideal one. This allows a determination of whether there is an undesired frequency dependence and allows for the dependence to be calibrated out or included in the modeling and optimization process to ensure the accuracy of the final optimized controls and the ability to implement them.

[0043] Referring to FIG. 1, shown is a quantum control system 1000 according to nonlimiting embodiments or aspects. A computing device 102 is in communication with a control pulse generator 104, which may include but is not limited to an arbitrary waveform generator, a device for optical pulse shaping, and / or any other device or system for producing a pulse capable of controlling and / or affecting a quantum system 100. In operation, a control pulse is generated by the control pulse generator 104 and received by the quantum system 100, which is reactive to the control pulse and results in a state change. The quantum system 100 may include a quantum computing system, a quantum sensor, and / or a model-based simulation of a quantum computing system and / or a quantum sensor. A quantum system may include a physical system implemented using various hardware components, including but not limited to superconducting circuits, spin systems, trapped ions, photonic systems, quantum dots, and / or the like, which may have a state qualified by representation within a Hilbert space.

[0044] With continued reference to FIG. 1, a measurement of the state of the quantum system 100 is measured based on selecting a measurement method (e.g., selecting the protocol for measuring the quantum system, including the specific measurements and parameters thereof). A measurement of the state of the quantum system may include experiments designed to determine properties or observables related to the state of the quantum system. In some non-limiting embodiments, a measurement model may be used to select the measurements based on a model of the quantum system 100. The measurements are obtained in response to the control pulse being received by the quantum system 100 as a signal produced by the quantum system 100 that is digitized and interpreted by the computing device 102. The computing device 102 may store the measurement results in a data storage65E3815. DOCX Page 8 of 21Attorney Docket No. 08993-2502647 (2024-264)device 106. The measurement results may be used to update the model of the quantum system 100 and / or to update the measurement model.

[0045] Still referring to FIG. 1, a control pulse correction may be generated by the computing device 102 based on the state of the quantum system, the initial control pulse that resulted in that state, and a model reference state based on a model of the quantum system (e.g., an expected state of the quantum system based on an output of a model of the quantum system). Determining the control pulse correction involves direct trajectory optimization by optimizing both control pulses and quantum states (e.g., quantum system trajectory), solving for both at the same time. This allows for continual calibration of the control pulses and the quantum system, rather than ad hoc and periodic calibration of only the control pulses. After a control pulse correction is determined, a modified control pulse may be generated by the control pulse generator 104 based on the control pulse correction and the modified control pulse may be received by the quantum system 100. The state of the quantum system may be measured again, and the control pulse modified at intervals with updated control pulse corrections, such as every day, hour, minute, or the like. The model of the quantum system may be updated to learn from the results, including the control pulse correction and the response to the updated control pulse. This allows for the model of the quantum system to become more accurate over time through iterative learning, improving the accuracy of the control pulse corrections by mapping observed deviations to control corrections.

[0046] In non-limiting embodiments, the quantum control method addresses the technical problems of model mismatch and nonlinear measurements that arise in quantum system control. In non-limiting embodiments, the method takes advantage of direct trajectory optimization methods such as, but not limited to, a Pade integrator direct collocation (PICO) pulse generation method to correct for model mismatch encountered when testing on quantum hardware.

[0047] The state of the quantum system on the hardware is denoted by x(t), which in practice is either a state vector in the system Hilbert space or a unitary operator representing the quantum operation implemented via the control pulse. The state is evolved for a time T, from an initial state x0= x(0) to a final state xT= x(T), using a control pulse u(t), which is found to evolve the initial state along a trajectory denoted x9°al(t) in simulation. This reference trajectory can be found using any optimal control or analytic method. In nonlimiting embodiments the reference trajectory corresponds to the quantum states in a solution of the direct optimal control method PICO.65E3815. DOCX Page 9 of 21Attorney Docket No. 08993-2502647 (2024-264)

[0048] State measurements on quantum hardware systems may be time expensive and getting full state information is often not feasible. Measurements may be nonlinear and lower dimensional than the underlying state. For example, a measurement of gate fidelity returns only a single scalar value. To accommodate this, an analytic function may be denoted as g to represent the map from the state to the measurement on the hardware. This allows for formation of reference measurements ygoal= g(xgoal) ∈ ℝm. The analytic form of g is used such that derivatives can be taken of it with respect to reference states.

[0049] In existing quantum control systems, a mismatch between the reference measurement y9°aland the experimental measurement y = $(%) may be observed. This measurement error may be defined as:

[0050] Δy = y - ygoal. (1)

[0051] In the quantum setting this measurement error corresponds to a unitary — vis-a-vis SO(2n) — rotational mismatch as follows:

[0052] x ≈ exp(-G)xgoal,

[0053] ≈ (I - G)xgoal,

[0054] = xgoal+ Δx,

[0055] where G G Skew(n) is skew- symmetric and assumed to be small in norm, x and xgoalare not necessarily members of the same space. In this non-limiting embodiment, the structure of G and x may not be utilized, but instead Δx = -Gxgoalmay be treated as a “free” decision variable.

[0056] Referring now to FIG. 2, shown is a flow diagram for a quantum iterative learning control method (e.g., quantum control algorithm) according to some non-limiting embodiments or aspects. The steps shown in FIG. 2 are for example purposes only. It will be appreciated that additional, fewer, different, and / or a different order of steps may be used in some non-limiting embodiments or aspects. In some non-limiting embodiments or aspects, a step may be automatically performed in response to performance and / or completion of a prior step. At step 200 a signal is received from a direct control system that generates a control pulse. For example, step 200 may include designing a control pulse using PICO direct trajectory optimization. At step 202, the experiment is designed. This may include determining parameters such as, for example, a measurement process (e.g., what type of data to be measured), pulse duration, number of iterations, and / or the like.

[0057] In non-limiting embodiments, a control model may be used to simulate the quantum system to predict how the quantum system will respond under different control65E3815. DOCX Page 10 of 21Attorney Docket No. 08993-2502647 (2024-264)pulses (e.g., different pulse shapes). The control model may be used to compute an optimized control pulse offline that is then optimized further through the iterative learning control method utilizing the quantum system itself. The control model may learn how to generate optimized control pulses for a given experiment based on the control pulse correction generated after the initial control pulse is received by the quantum system.

[0058] At step 204 the experiment designed at step 202 is executed by the quantum system and the results are measured (e.g., as a response to the control pulse, using spectroscopy, photon detection, and / or the like) based on the experiment. In non-limiting embodiments, arbitrary measurement schemes may be used to implement the quantum iterative control method. For example, fidelity measurements and fidelity proxies, such as randomized benchmarking experiments, are nonlinear measurements that may be used to characterize the performance of quantum hardware. Full tomography of the state or process may require many more measurements but can be a practical component in the method for implementing digital twins or other like uses. The measurements may be budgeted to preserve resources and time, and active or adaptive measurement enables closed-loop control. A strong prior (e.g., initial belief) may be imposed on the measurement data in non-limiting embodiments to increase measurement efficiency.

[0059] In non-limiting embodiments, circuits may be designed to accumulate error signals by repeatedly applying or modifying the error channel. For example, the over or under rotation of a gate may be amplified by repeatedly applying the gate. Quantum signal processing may be used to design experiments with specific measurement response functions. The over or under rotation of a gate may be characterized by inverting an appropriate response function. In non-limiting embodiments, if a prior distribution is available, online or adaptive measurements may be used to more efficiently estimate an unknown state or parameter. These measurements may be raw quantum measurement outcomes. Stochastic gradients may be used to differentiate the measurement functions.

[0060] With continued reference to FIG. 2, the measurement result is received by the computing system at step 202. In some examples, step 204 is repeated to obtain several measurements. At step 210, a control pulse correction is determined based on the measurement and the initial control pulse. For example, at an interval (e.g., the z-th iteration) of the method, the control pulsemay be generated and received by the quantum system, which returns a measurement y^ which differs from the ideal measurement ygoal.65E3815. DOCX Page 11 of 21Attorney Docket No. 08993-2502647 (2024-264)

[0061] A control pulse correction Au is calculated that corrects the state of the quantum system to the model reference state (e.g., the predicted state based on a model of the quantum system). The correction can be calculated by coupling the measurement result, y, the state correction, Ar, and the control correction, Au, in a single optimization problem that leverages the PICO method. This problem can be written as:min / (Ax) = ||^(xT+ AxJ - yT||2(2a) Ax, Au z / iTET subject to Ak■ (Axk+, Axk, Auk) = 0 (2b)

[0062] where T c {1,...,1V} is the set of measurement times which correspond to knot points of the reference trajectory. T is a hyperparameter that may be considered in terms of a tradeoff between taking more measurements, which is often costly, and the benefits of getting more trajectory data. In non-limiting embodiments, optimal performance may be achieved with only measuring at the final time, i.e. T = {A}.

[0063] The constraint (2b) enforces a linearization of the dynamics constraints, f(xk+T>xk>uk) = O' about the reference trajectory. This linearization is arrived at via a Taylor expansion of the dynamics constraints about the reference trajectory, xre:f&k+i + &xk+i'xkef+ukef+Aufc) =0(3)

[0064] which yields the condition, with:Ak= (4)

[0065] In this formulation the dynamics / are implemented via Pade integrators. To solve the optimization problem (2), a quadratic model is formed of the nonlinear objective function 7(7?):1 / (Ax) = / (0) + d / Ax + - AxTd2 / Ax (5)= / (0) + 4yTCAx + AxTCTC + Ayj C d2gj Ax (6)\ j /

[0066] where C = d^(xref) and Ay is defined as above. The following quadratic problem (QP) is then setup:min / (Ax) (7) Ax, usubject to Ak■ Axk+1, Axk, Auk~) = 0 (8)

[0067] In non-limiting embodiments, this problem may be solved with any constrained quadratic program solver, for example but not limited to OSQP (Operator Splitting Quadratic Program). The solution contains a control correction Ait* which may be used to calibrate the 65E3815. DOCX Page 12 of 21Attorney Docket No. 08993-2502647 (2024-264)pulse via a line search. In non-limiting embodiments, if there are some other nonlinear constraints c(x, it) which must be satisfied, a general-purpose nonlinear solver such as Interior Point Optimizer (IPOPT) and / or the like may be used instead of a QP solver. The quadratic problem (7) may be augmented to include the constraint in such examples.

[0068] In non-limiting embodiments, after the control correction is computed and before the control pulse is updated for the next iteration, steps 206 and 208 may adjust the step size for the control pulse. For example, a line search to search over a to find the optimal step size for the update to the control pulse may begin with step 206 to test with a step size parameter and continue to step 208 to iteratively update the step size. Steps 206 and 208 may be repeated until the correct step size is reached. In non-limiting embodiments, the line search performed by the quantum hardware may help address noisy hardware measurements. The line search results in a step size a such that the updated pulse u+aAu achieves the best performance on the system. In non-limiting embodiments, a simplified backtracking line search, stochastic versions of a backtracking line search, and / or methods that utilize Gaussian processes to optimally choose test points (e.g., kriging) may be used.

[0069] In non-limiting embodiments, a trust region approach to the step size selection problem may be applied by including a constraint || Alt II < R, such that the magnitude of the update, in a chosen norm, is less than some value, referred to as the trust radius, which is updated based on the performance on the quantum hardware relative to the expected performance given the model. This approach has certain benefits, particularly in the stochastic regime.

[0070] The step size may then be used to update the control pulse at step 210. The method may be iteratively performed until an updated control pulse results in the expected output. The final, updated control pulse may then be used to update and / or train the control model to improve subsequent generations of control pulses. The updated control pulse may also be used to update the measurement model to improve measurement selection for future experiments.

[0071] In non-limiting embodiments, control pulses may be designed to enhance the statistical sensitivity of different measurements. For example, the classical or quantum fisher information can be used to design controls that amplify sensitivity to certain measurements or quantum observables in some non-limiting embodiments.

[0072] In non-limiting embodiments, a digital twin (e.g., simulation model) may be used in conjunction with the quantum control systems and methods described herein. A digital twin of the quantum system may be used for controlled rollouts of modifications and to fix 65E3815. DOCX Page 13 of 21Attorney Docket No. 08993-2502647 (2024-264)errors that come from model approximations between the control model and digital twin. Digital twins allow access to full state measurements that may not be feasible with the hardware. Digital twins may be designed to adapt to known unknowns (e.g. parameter drift from temperature fluctuations). Measurements, which capture information about the timevarying parameter state, may be used to update the parameterized digital twin. The updated digital twin may be used to calibrate the system.

[0073] Due to the way the control pulse is updated, where the internal states x are updated as x <- x + a Ax, the states at later iterations of the quantum control algorithm may leave the group manifold. At some point the algorithm may converge to a minimum, which may not be globally optimal. In the noise-free regime, if the states are projected onto the requisite group manifold once the performance of the algorithm has flatlined and is not significantly improving, the method may be restarted from a new feasible point and achieve further improvements. This augmentation affects the algorithm by adding a fallback to carry out this manifold projection once convergence is reached. Alternatively, the algorithm may be applied directly in the Lie algebra to interact directly with the manifold.

[0074] In non-limiting embodiments, PICO may be used as follows to formulate QOC problems as direct collocation trajectory optimization problems, wherein the focus is on optimizing for SU (n) gates. Using the loss / (■) defined as Z([7):= 1 — | tr(UgOalIf) | for U E SU(n) and the naive dynamics f(Uk+1, Uk,ak, t) = Uk+1— exp(— iH (ak)At) Uk, where At is fixed. In non-limiting embodiments, a direct collocation (DIRCOL) formulation can be written as follows:min l( U„)Ur. NAi-. Nsubject to f(Uk+1, Uk,ak, At = 0U = In

[0075] In non-limiting embodiments, an isomorphic representation for complex vectors and matrices is utilized for complex vectors and matrices to move between complex-valued quantum states and real-valued problem variables. For a complex valued vector ip E Cnand ~ / Rei / A ~ and matrix H E Cn xn, the resulting isomorphic representations are ip = and H =where ip E R2nand H E R2nx2n. in further non-limiting embodiments,VlmH ReH J65E3815. DOCX Page 14 of 21Attorney Docket No. 08993-2502647 (2024-264)G H):= iso(— iH) = f 1™^, is defined, wherein H is linear in a and G is a linear\-ReH \mH / function of H, where 6(a) ■= G(H(a)) = G(H0) +aiG(Hd

[0076] In non-limiting embodiments, the isomorphic dynamics may be in the form of / ([ / k+1, [7k, ak, At) = L / k+1— exp(G(ak)At) L / k. The Pade approximant may be used to approximate the matrix exponential as exp( ) B-1(d)F( / l) where B(d) and F(d) are truncated power series in A whose coefficients can be chosen to match exp(d) up to the desired order. This approximation addresses that the matrix exponential was a costly operation that did not account for how matrix exponentials are numerically computed in practice. In non-limiting embodiments, it is recognized that the leading matrix inverse is computationally expensive and not necessary to compute directly since the dynamics constraints are enforced implicitly. The dynamics constraints may be rewritten using the Pade approximant as P(t / k+1, Uk,ak, At) = B(ak, &t)Uk+1- F(ak, At)Uk

[0077] In non-limiting embodiments, the fourth order diagonal Pade integrator, denoted as is sufficient and is defined by B^4\ak, At) ■= I — ^G(a) + ^-G(a)2andF^4\ak, At):= I + “6(a) + ^-G(a)2. B evolves the state backward a half step in timeand F evolves the state forward a half step. In further non-limiting embodiments, the states of the system ae augmented with the first and second derivatives of the drive parameter, resulting in knot points zk= ([ / k, afc, a’fc, ak)T. The new dynamics for this augmented / P(n)(t7k+1, (7k,ak, At)\problem are given by / (zk, zk+1) = ak+1- ak- ak■ At\ afc+i — ak— ak■ At /

[0078] In non-limiting embodiments, the augmentation allows for the optimal duration of the pulse to be found by the solver, referred to as a free-time problem. A cost term may be added to the objective of the form / mintime(^fi: N-i)=Atkand an inequality constraint F(_UN~) > F may be added on the final state fidelity. This allows for the achievement of minimum-time solutions for a chosen fidelity, which is helpful in realizing higher-fidelity quantum computations in the presence of decoherence.

[0079] Referring now to FIG. 3, shown is a diagram of example components of a device 400 according to non-limiting embodiments or aspects. Device 400 may correspond to at least one of the computing devices referenced herein (e.g., computing device 102). In nonlimiting embodiments or aspects, such systems or devices may include at least one device 400 and / or at least one component of device 400. The number and arrangement of65E3815. DOCX Page 15 of 21Attorney Docket No. 08993-2502647 (2024-264)components shown are provided as an example. In non-limiting embodiments or aspects, device 400 may include additional components, fewer components, different components, or differently arranged components than those shown. Additionally, or alternatively, a set of components (e.g., one or more components) of device 400 may perform one or more functions described as being performed by another set of components of device 400.

[0080] As shown in FIG. 3, device 400 may include bus 402, processor 404, memory 406, storage component 408, input component 410, output component 412, and communication interface 414. Bus 402 may include a component that permits communication among the components of device 400. In non-limiting embodiments or aspects, processor 404 may be implemented in hardware, firmware, or a combination of hardware and software. For example, processor 404 may include a processor (e.g., a central processing unit (CPU), a graphics processing unit (GPU), an accelerated processing unit (APU), etc.), a microprocessor, a digital signal processor (DSP), and / or any processing component (e.g., a field-programmable gate array (FPGA), an application-specific integrated circuit (ASIC), etc.) that can be programmed to perform a function. Memory 406 may include random access memory (RAM), read only memory (ROM), and / or another type of dynamic or static storage device (e.g., flash memory, magnetic memory, optical memory, etc.) that stores information and / or instructions for use by processor 404.

[0081] With continued reference to FIG. 3, storage component 408 may store information and / or software related to the operation and use of device 400. For example, storage component 408 may include a hard disk (e.g., a magnetic disk, an optical disk, a magnetooptic disk, a solid state disk, etc.) and / or another type of computer-readable medium. Input component 410 may include a component that permits device 400 to receive information, such as via user input (e.g., a touch screen display, a keyboard, a keypad, a mouse, a button, a switch, a microphone, etc.). Additionally, or alternatively, input component 410 may include a sensor for sensing information (e.g., a global positioning system (GPS) component, an accelerometer, a gyroscope, an actuator, etc.). Output component 412 may include a component that provides output information from device 400 (e.g., a display, a speaker, one or more light-emitting diodes (LEDs), etc.). Communication interface 414 may include a transceiver-like component (e.g., a transceiver, a separate receiver and transmitter, etc.) that enables device 400 to communicate with other devices, such as via a wired connection, a wireless connection, or a combination of wired and wireless connections. Communication interface 414 may permit device 400 to receive information from another device and / or provide information to another device. For example, communication interface 414 may 65E3815. DOCX Page 16 of 21Attorney Docket No. 08993-2502647 (2024-264)include an Ethernet interface, an optical interface, a coaxial interface, an infrared interface, a radio frequency (RF) interface, a universal serial bus (USB) interface, a Wi-Fi® interface, a cellular network interface, and / or the like.

[0082] Device 400 may perform one or more processes described herein. Device 400 may perform these processes based on processor 404 executing software instructions stored by a computer-readable medium, such as memory 406 and / or storage component 408. A computer-readable medium may include any non-transitory memory device. A memory device includes memory space located inside of a single physical storage device or memory space spread across multiple physical storage devices. Software instructions may be read into memory 406 and / or storage component 408 from another computer-readable medium or from another device via communication interface 414. When executed, software instructions stored in memory 406 and / or storage component 408 may cause processor 404 to perform one or more processes described herein. Additionally, or alternatively, hardwired circuitry may be used in place of or in combination with software instructions to perform one or more processes described herein. Thus, embodiments described herein are not limited to any specific combination of hardware circuitry and software. The term “programmed or configured,” as used herein, refers to an arrangement of software, hardware circuitry, or any combination thereof on one or more devices.

[0083] Although embodiments have been described in detail for the purpose of illustration, it is to be understood that such detail is solely for that purpose and that the disclosure is not limited to the disclosed embodiments, but, on the contrary, is intended to cover modifications and equivalent arrangements that are within the spirit and scope of the appended claims. For example, it is to be understood that the present disclosure contemplates that, to the extent possible, one or more features of any embodiment can be combined with one or more features of any other embodiment.65E3815. DOCX Page 17 of 21

Claims

Attorney Docket No. 08993-2502647 (2024-264)WHAT IS CLAIMED IS1. A method comprising:generating a control pulse in a quantum system;determining a measurement of a state of the quantum system in response to the control pulse;determining a control pulse correction based on the control pulse, the measurement of the state of the quantum system, and a model reference state;modifying the control pulse based on the control pulse correction, resulting in a modified control pulse; andgenerating the modified control pulse in the quantum system.

2. The method of claim 1, wherein the modified control pulse is generated by at least one of the following: an arbitrary waveform generator, optical pulse shaping, or any combination thereof.

3. The method of claim 1, wherein the control pulse correction is based on conducting a line search of different step sizes for the control pulse correction.

4. The method of claim 1, wherein the model reference state comprises a predicted measurement of the state based on a model of the quantum system.

5. The method of claim 1, further comprising:automatically calibrating the quantum system at regular intervals.

6. The method of claim 1, wherein determining the measurement of the state of the quantum system comprises selecting a measurement method based on a measurement model and determining the measurement of the state based on the measurement method.

7. The method of claim 1, further comprising modeling the state of the quantum system in response to the control pulse.

8. A system comprising at least one computing device configured to: generate a control pulse in a quantum system;65E3815. DOCX Page 18 of 21Attorney Docket No. 08993-2502647 (2024-264)determine a measurement of a state of the quantum system in response to the control pulse;determine a control pulse correction based on the control pulse, the measurement of the state of the quantum system, and a model reference state;modify the control pulse based on the control pulse correction, resulting in a modified control pulse; andgenerate the modified control pulse in the quantum system.

9. The system of claim 8, wherein the modified control pulse is generated by at least one of the following: an arbitrary waveform generator, optical pulse shaping, or any combination thereof.

10. The system of claim 8, wherein the control pulse correction is based on conducting a line search of different step sizes for the control pulse correction.

11. The system of claim 8, wherein the model reference state comprises a predicted measurement of the state based on a model of the quantum system.

12. The system of claim 8, wherein the at least one computing device is further configured to:automatically calibrate the quantum system at regular intervals.

13. The system of claim 8, wherein determining the measurement of the state of the quantum system comprises selecting a measurement method based on a measurement model and determining the measurement of the state based on the measurement method.

14. The system of claim 8, wherein the at least one computing device is further configured to:model the state of the quantum system in response to the control pulse.

15. A computer program product comprising a non-transitory computer-readable medium including program instructions that, when executed by at least one computing device, cause the computing device to:generate a control pulse in a quantum system;65E3815. DOCX Page 19 of 21Attorney Docket No. 08993-2502647 (2024-264)determine a measurement of a state of the quantum system in response to the control pulse;determine a control pulse correction based on the control pulse, the measurement of the state of the quantum system, and a model reference state;modify the control pulse based on the control pulse correction, resulting in a modified control pulse; andgenerate the modified control pulse in the quantum system.

16. The computer program product of claim 15, wherein the modified control pulse is generated by at least one of the following: an arbitrary waveform generator, optical pulse shaping, or any combination thereof.

17. The computer program product of claim 15, wherein the control pulse correction is based on conducting a line search of different step sizes for the control pulse correction.

18. The computer program product of claim 15, wherein the at least one computing device is further caused to:automatically calibrate the quantum system at regular intervals.

19. The computer program product of claim 15, wherein determining the measurement of the state of the quantum system comprises selecting a measurement method based on a measurement model and determining the measurement of the state based on the measurement method.

20. The computer program product of claim 15, wherein the at least one computing device is further caused to:model the state of the quantum system in response to the control pulse.65E3815. DOCX Page 20 of 21

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