Method and system for quantum machine learning

Abstract

The invention relates to a method and system for quantum machine learning. In a described embodiment, a hybrid quantum classical computing method is implemented, the method comprising the steps of: receiving time-dependent input data of a dynamic system; transforming a first element of the input data using a plurality of transformation matrices to obtain a set of transformed data; and encoding the transformed data into a quantum circuit by performing a first set of quantum operations, the quantum circuit comprising a plurality of layers, the layers comprising a data encoding layer, a feedback layer, and a random transformation layer; encoding measurement feedback from a previous measurement vector into the quantum circuit is accomplished by performing a second set of quantum operations. The quantum circuit is to generate an updated measurement vector, and to generate a reservoir state based on the updated measurement vector, a previous reservoir state vector, and input data.

Description

Technical Field

[0001] The present application generally relates to methods and systems for quantum computing (QC) and machine learning, and specifically to integrating quantum computing algorithms with machine learning techniques for making predictions in dynamic environments. Background Art

[0002] Recent advances in quantum computing indicate a shift in technology applications, introducing the field of quantum machine learning (QML). QML applies quantum algorithms to enhance traditional machine learning tasks. However, the current stage of technology evolution, known as the noisy intermediate-scale quantum (NISQ) era, is marked by quantum computers with a limited number of qubits and susceptibility to noise. This stage requires the development of methods to maximize the computational efficiency of quantum resources within these constraints.

[0003] To address or circumvent these limitations, hybrid quantum-classical algorithms have been developed to integrate quantum computing subroutines with classical processing capabilities. The goal is to use quantum processors for specific tasks (where quantum processors offer computational advantages such as direct estimation of expectation values ​​through wave function sampling), while relying on classical computers to perform tasks that are currently beyond the scope of quantum devices or are not suitable for quantum devices (such as certain optimization processes).

[0004] Variational Quantum Algorithms (VQA) represent hybrid methods that utilize parameterized quantum circuits combined with classical optimization routines. VQA can be applied to a variety of problems ranging from optimization tasks to simulations in quantum chemistry. However, the implementation of VQA faces challenges, including but not limited to quantum errors or noise, parameter optimization, and effective management of quantum-classical computational interactions.

[0005] In classical machine learning, the following efficacy of reservoir computing (RC) has been recognized: using a network whose internal structure remains unchanged during training to simulate complex temporal dynamics to simplify the learning process. The processing within the RC system involves a reservoir, which maintains a state that reflects the temporal dynamics of the input data. It uses a leakage rate to manage the influx of new information and uses a nonlinear function to update the state based on the new input and the previous state. An autoregressive mechanism is used to calculate the subsequent state of the system, which is then mapped to the output data through a readout layer. This layer utilizes ridge regression, a method that helps generalize the performance of the model and prevents overfitting by adding a regularization term. The optimization of the readout parameters is achieved by solving a linear equation that minimizes the difference between the predicted output and the actual output, and is regularized by a penalty on the size of the readout weight.

[0006] It is therefore desirable to provide a method and system that combines quantum computing with machine learning to address the shortcomings or limitations of existing technologies, or at least provide the public with useful alternatives. Summary of the invention

[0007] The present disclosure aims to provide new and useful methods and systems for quantum machine learning, particularly integrating quantum computing algorithms with machine learning techniques for prediction in dynamic environments.

[0008] Broadly speaking, the present disclosure proposes a hybrid quantum classical computing method, which includes the following steps: receiving time-dependent input data of a dynamic system; using multiple transformation matrices to transform a first element of the input data to obtain a set of transformed data; and encoding the transformed data into a quantum circuit by performing a first set of quantum operations. The quantum circuit may include multiple layers, wherein the layers include a data encoding layer, a feedback layer, and a random transformation layer. One way to use this method is to encode measurement feedback from a previous measurement vector into the quantum circuit by performing a second set of quantum operations. It can be understood from the embodiments that the hybrid quantum classical computing method can operate a quantum circuit to generate an updated measurement vector. A reservoir state can be generated based on the updated measurement vector, a previous reservoir state vector, and the input data.

[0009] In certain embodiments, the hybrid quantum classical computing method may further include determining whether to process additional input data, and forming a reservoir state vector based on one or more of the generated reservoir state, the input data, and a bias term.

[0010] In implementation, the hybrid quantum classical computing method can apply a ridge regression process to a series of formed reservoir state vectors to determine multiple readout parameters.

[0011] This hybrid quantum classical computing method can generate predictions of the future state of a dynamic system based on the readout parameters and the reservoir state vector.

[0012] In implementation, the hybrid quantum classical computing method can use quantum circuits and trained classical processing modules to generate predictions of future states of a dynamic system, wherein the classical processing modules receive as inputs nonlinear transformations of reservoir states and nonlinear transformations of time-dependent input data.

[0013] In some embodiments, the hybrid quantum classical computing method may include initiating entanglement between multiple quantum bits of a quantum circuit in a data encoding layer.

[0014] In an implementation, each of the plurality of transformation matrices may include a plurality of fixed transformation matrices that remain unchanged during the training and prediction phases of the computation process.

[0015] In a specific embodiment, each of the plurality of transformation matrices may include a random weight matrix or a Fourier-like matrix; and the set of transformed data is generated by performing a function transformation on a first element of the input data.

[0016] Each of the first and second sets of quantum operations may include one of the following or a combination of one or more of the following: a single quantum bit rotation around at least one of the X-axis, the Y-axis, and the Z-axis; a controlled phase gate operation; an fSim gate operation; or a 2-qubit XY rotation.

[0017] In an implementation, the hybrid quantum classical computing method may include encoding the transformation data, and encoding the measurement feedback may include using an eigenmapping function for data encoding parameters corresponding to the first set of parameterized layers and the second set of parameterized layers.

[0018] The reservoir circuit layer may include reservoir cells corresponding to a set of quantum gates, wherein parameters of the quantum gates do not depend on measurement feedback, input data, or reservoir states.

[0019] Quantum circuits can be designed to produce stable measurements for integration with classical reservoir computing models to enhance the performance of classical reservoir computing models.

[0020] In some implementations, generation of the reservoir state can be based on a leakage rate parameter and a plurality of activation functions that introduce nonlinearities into operations performed by the quantum circuit.

[0021] The updated measurement vector may be generated by measuring a plurality of quantum data elements on at least one of the X, Y, and Z directional bases.

[0022] In certain embodiments, the updated measurement vector may also include a single-qubit expected value and a multi-qubit correlator, wherein both the single-qubit expected value and the multi-qubit correlator are defined on the measurement map.

[0023] This hybrid quantum-classical computing approach can be used to predict the behavior of chaotic systems and facilitate nonlinear transformations of input data sets through data processing in a quantum framework.

[0024] In an embodiment, the hybrid quantum classical computing method can use the properties of high-dimensional Hilbert space as a reservoir for encoding chaotic dynamics into quantum circuits.

[0025] In implementation, the hybrid quantum classical computing method may include shifting between classical operations and quantum operations, wherein the classical operations pre-process input data for the quantum operations and further post-process measurement results corresponding to the quantum operations.

[0026] In certain embodiments, a previous measurement result corresponding to the measurement feedback may be obtained from a quantum circuit parameterized by a previous iteration of a hybrid quantum classical computing method, wherein the previous measurement result parameterizes the feedback layer.

[0027] Each transformation matrix may be a fixed transformation matrix associated with a reservoir state, a measurement state, and an input state.

[0028] In an embodiment, the time-dependent input data may include a multi-component time series data vector.

[0029] The hybrid quantum classical computing method can also use a classical coprocessor to receive classical output from the quantum circuit and apply post-processing to produce refined data, and repeat the operations that produce the classical output to collect statistics for post-processing refinement.

[0030] The hybrid quantum classical computing method may further include the steps of repeating quantum circuit operations, wherein each repetition is compiled differently to enable post-processing suppression techniques, and augmenting the quantum output with a classical coprocessor for cumulative statistics-based system enhancement.

[0031] These implementations may be represented as a method, or alternatively as a system including one or more computers and one or more storage devices storing instructions that, when executed by one or more computers, cause the one or more computers to perform the method. It may also be represented as a computer program product, such as downloadable program instructions (software) or one or more non-transitory computer storage media storing instructions. When executed by one or more computers, the program instructions cause the one or more computers to perform the method.

[0032] Embodiments described herein may thus provide a method and system for quantum machine learning that implements one or more of the following features:

[0033] Effectively combining quantum and classical computing strategies addresses the challenges of limited qubit availability and quality in current quantum computing hardware.

[0034] Enhance the efficiency and accuracy of machine learning models, especially in processing and predicting complex temporal dynamics within high-dimensional data.

[0035] By taking a hybrid approach, the computational burden associated with training quantum models can be reduced, thereby reducing the need for extensive quantum hardware.

[0036] Improved predictions of chaotic systems, which is essential for complex simulations such as weather modeling or financial markets.

[0037] Compared to its classical counterparts, lower sensitivity to random initialization and hyperparameter variations is achieved, while still accurately reproducing long-term system behavior.

[0038] The above description is provided as an overview of some implementations of the present disclosure. Further descriptions of these implementations and other implementations are described in more detail below. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] Embodiments of the present invention will now be explained for purposes of example only with reference to the following drawings, in which:

[0040] Figure 1 is a functional block diagram of an example of a hybrid quantum classical computing system according to embodiments of the present invention.

[0041] Figure 2 It is to show the use of the embodiment according to this article Figure 1 Flowchart of an example high-level process for a system to predict dynamic environments.

[0042] Figure 3 is a graphical representation of simulation results of a Lorenz 63 chaotic system using a Hybrid Quantum-Classical Reservoir Computing (HQRC) system according to an embodiment of the present invention.

[0043] Figure 4 is a graphical representation of the position of the Lorenz63 attractor over time according to an embodiment of the present invention, which demonstrates the accuracy of the HQRC system in reconstructing long-term qualitative behavior.

[0044] Figure 5is a graphical representation of the Poincare return map of a Lorenz63 chaotic system according to embodiments presented herein, showing an ordered oscillation pattern in the component-wise divergence after approximately 30 time units.

[0045] Figure 6 is a graphical representation of simulation results for a double scroll system using a HQRC system having an 8-qubit structure according to embodiments herein.

[0046] Figure 7 is a graphical representation of the long-term behavior based on the reconstruction of the attractor for a double scroll system using an 8-qubit HQRC configuration according to an embodiment of the present invention.

[0047] Fig. 8A and Figure 8B is a graphical representation of the performance of the HQRC algorithm implemented on an Aspen M-3 chip during and after training when using Clifford Data Regression (CDR) for prediction according to embodiments herein.

[0048] Fig. 9A and Fig. 9B is a graphical representation of the performance of the HQRC algorithm implemented on an Aspen M-3 chip during and after training when making predictions with 1000 attempts without using CDR according to embodiments herein.

[0049] Fig. 10A and Fig. 10B is a graphical representation of the performance of the HQRC algorithm implemented on an Aspen M-3 chip during and after training when making predictions with 10,000 attempts without using CDR according to embodiments herein.

[0050] Fig.11A and Fig. 11B is a graphical representation of the aggregate performance of the HQRC algorithm implemented on an Aspen M-3 chip during and after training when using CDR for prediction and without using CDR at 1000 attempts and 10000 attempts according to an embodiment of the present invention.

[0051] Fig. 12A and Fig. 12B is a graphical representation of the performance of the HQRC algorithm implemented on an 8-bit chip provided by Oxford Quantum Circuits (OQC) during training and when making predictions after training in accordance with embodiments of the present invention.

[0052] Fig.13is a graphical representation of the main layer types used to benchmark Lorenz63 systems according to an embodiment of the present invention, showing quantum circuit designs with varying configurations of Ry and Rz gates and entanglement patterns across multiple qubits, and specific entangled pairs within two different ring groupings.

[0053] Fig.14A , Fig. 14B and Fig. 14C is a graphical representation of a statistical compilation of the Lorenz63 problem according to an embodiment of the present invention, which shows the performance impact of different quantum circuit layer types across the system.

[0054] Fig.15A , Fig. 15B and Fig. 15C is a graphical representation of the performance of the HQRC algorithm on a Rigetti Aspen M-3 chip using 8, 16, and 23 qubits, respectively, according to embodiments of the present invention.

[0055] Fig.16 is a block diagram illustrating an example computer system that may be configured to implement the systems and methods disclosed herein.

[0056] Fig.17 is a schematic diagram of a hybrid quantum classical computing model according to an embodiment of the present invention. DETAILED DESCRIPTION

[0057] Embodiments will now be discussed with reference to the accompanying drawings, which depict one or more exemplary embodiments. These embodiments are described in sufficient detail to enable those skilled in the art to practice them, and it should be understood that mechanical, logical, and other changes may be made without departing from the scope of the embodiments. Thus, the embodiments may be implemented in many different forms and should not be construed as limited to the embodiments set forth herein, shown in the drawings, and / or described below.

[0058] Unless otherwise defined, all terms (including technical and scientific terms) used herein should be interpreted according to common practice in the art. It should also be understood that common terms should also be interpreted according to common practice in the relevant art.

[0059] Figure 1 is a block diagram of an example of a hybrid quantum-classical reservoir system (HQRC) 100 for predicting a dynamic environment according to an embodiment of the present disclosure. Figure 2 The use of the embodiments according to the present disclosure Figure 1 Flowchart of an example high-level process 200 for predicting a dynamic environment using a system. Figure 2 Refer to the steps outlined in Figure 1The components and modules shown in the figure are shown to explain the functions of the components in the process of predicting dynamic environments.

[0060] In an example implementation, the HQRC system 100 is configured to predict a time-dependent dynamic system X t The prediction model of the future state of is used at the sequential time point t 1 ,t 2 …、t n Given a fixed number n of historical observations, the HQRC system 100 learns the inherent patterns of the input data to predict the n+1 ,t n+2 …、t n+p The new state, future time t n+1 ,t n+2 …、t n+p Corresponding to the dynamic system X t In an embodiment, the HQRC system 100 may integrate a classical computing module and a quantum computing module to predict future states.

[0061] In step 202, the data input module 102 receives the classical input data X(X t ), and transform the input data into a quantum compatible format. This step allows the input data X t is transformed into a format suitable for quantum mechanics-based operations on quantum bits in a quantum computer. The data input module 102 may receive input data at time t, which is represented by X t (See Fig.17 , item 1702 is X t ), and using a random transformation matrix To apply a linear transformation (see further Fig.17 , item 1704 is ). Matrix Providing classical data ready for quantum processingX t The initial transformation of .

[0062] The matrices in can have dimensions (dL j, d nput ), where dL j It is layer L j The number of parameters that can be accommodated. For example, in a system with n qubits, if the layer consists of a single qubit X rotation, then dL j =n. Dimension di nput Can correspond to the input vector X t The size of the transformation matrix The elements of can be randomly generated, that is, selected based on a random distribution. Once generated, the matrix can remain unchanged throughout the training and prediction phases of the computation process 200, which provides stability of the encoding process over time.

[0063] In implementation, for a coding layer that performs a Fourier-type transform, the transform matrix The structure of can be non-random and predetermined, meeting the requirements of Fourier transform. In order to ensure that the encoding process of the data encoding module 104 is not affected by disproportionately large values, which may cause computational difficulties, the matrix Normalization. This normalization can be performed by dividing the matrix by its largest singular value to condition the matrix for numerical stability. By using this form of normalization, the model ensures that the encoded quantum data remains at a scale that is conducive to quantum processing.

[0064] In step 204, the quantum data encoding module 104 may convert the transformed data into Encoded into a quantum circuit. Fig.17 , Project 1706 is In an implementation, the quantum circuit may include a quantum data encoding module 104, a quantum feedback module 106, a random transformation module 108, and a quantum measurement module 110. In an example, the quantum data encoding module 104 may include multiple data encoding layers that are parameterized or parameter-free.

[0065] Parameterization

[0066] In step 204, the transformed vector Can be distributed in parameterized layers L j (Y (j) ). Fig.17 , Item 1726 is L j (Y (j) ). Each layer L j It can be expressed as R x , R y and R z A sequence of single-qubit rotation gates corresponding to rotations around the X, Y, and Z axes of the Bloch sphere. The rotation can be determined by Y (j) The elements of are parameterized to facilitate the process of encoding input data into the quantum state of a qubit.

[0067] To ensure the numerical stability of quantum computation, the weight matrix (denoted as W) used to encode the input data into the quantum circuit in ) can be normalized by their largest singular value. This normalization prevents excessively large values ​​that could cause computational problems.

[0068] Entanglement Gate

[0069] In an implementation, the quantum circuit may also include a parameter-free layer of controlled-NOT (CNOT or CX) gates for generating entanglement between qubits.

[0070] Entanglement is a quantum mechanical resource that allows qubits to be correlated in ways that are classically impossible. Example entanglement operations can include controlled phase, iSwap, fSim, and 2-qubit XY rotation gates. These quantum computations are essential to enable quantum circuits to capture correlations and perform computations that exploit quantum mechanics.

[0071] In implementation, the CX gates that provide entanglement between pairs of qubits do not require any external parameters to be set for their entanglement operation. Instead, the CX gates act on pairs of qubits defined by a connectivity graph G. Graph G determines which qubits are linked by the CX gates, effectively mapping out the entanglement structure of the circuit. Graph G can be selected to match the hardware-specific topology of the quantum processor, which helps optimize the circuit for the physical qubit arrangement and minimizes the need for additional operations like SWAP gates. Although SWAP gates can be used to virtually rearrange qubits that are not directly connected, they increase complexity and can introduce errors. By employing a connectivity graph G, the HQRC system 100 minimizes these problems and simplifies the process of qubit interaction.

[0072] Layers of CX gates can be used to generate entanglement within a quantum circuit. The generated entanglement allows the quantum circuit to explore a larger portion of the Hilbert space (the mathematical space that describes quantum states). With a wider exploration of this space, the quantum circuit can represent a more diverse set of solutions, giving it greater expressibility for the task at hand, such as pattern recognition or machine learning processing.

[0073] Feature Mapping

[0074] Feature maps can be used to further encode the parameters of the data into a form that can be used to set parameters of different types of quantum gates. In an example, the feature mapping process may include applying a function represented as The feature map is used by the encoding module 104 to further transform the parameters Transformed into quantum gate parameters. The feature mapping function further enhances the data in the quantum state space Representation.

[0075] Measurement feedback

[0076] After the initial data is encoded into the quantum circuit, in step 206, measurement feedback from the previous measurement vector is further encoded into the quantum circuit by the quantum feedback module 106. The quantum feedback module 106 can obtain the measurement result (denoted as M) from the previous time step. t-1 ) and encode them into the parameters of the quantum gates of the quantum circuit in the current time step. That is, the previous measurement results can be obtained from the quantum circuit parameterized by the previous iteration of the quantum computing process 200. This feedback encoding process can include adjusting the angles or other parameters of the quantum gates based on the values ​​obtained from the previous measurements, effectively incorporating historical data into the current computational state of the circuit. This feedback mechanism is of a time-dependent form, allowing the circuit to retain and utilize information from its previous state, which is necessary for tasks involving storage or prediction over time.

[0077] Since the number of measurements may not match the number of parameters needed in the measurement feedback layer, a distribution mechanism can be used. For example, the single-qubit expectation value (e.g. <xi>) can be used to supplement the single-qubit rotation on the i-th qubit, i.e., RY( <xi>). For parameterized layers of quantum circuits that are not fixed, there may be an additional parameter transformation step after the feature map encoding method. This additional transformation introduces nonlinearities that are essential for learning complex dynamics.

[0078] Similar to the quantum data encoding module 104, the feedback module 106 can also use feature mapping To transform the parameters of the data encoding (corresponding to the feedback measurement) to enhance the representation of the parameters in the quantum state space.

[0079] In an example embodiment, in step 208, a quantum circuit may be executed to generate an updated measurement vector after encoding the measurement feedback. Here, an optional random transformation module 108 may be used to output a quantum state configuration. The quantum state configuration is then further measured by a quantum measurement module 110 to generate an updated measurement vector. For example, the random transformation module 108 may consist of a sequence of quantum gates initialized with random parameters. The random transformation module may include a single-qubit gate that rotates the state of a single qubit around a Bloch sphere by a random angle. These rotations change the superposition state of the qubit in an unpredictable manner. After the single-qubit rotation, a sequence of controlled-not (CX) gates may be applied to generate entanglement between pairs of qubits.

[0080] Following the configuration, the output of the quantum circuit is subjected to a measurement process by the measurement module 110. The quantum measurement module 110 may employ a measurement strategy that maximizes the efficiency of the measurement process while also expanding the types of distinguishable patterns or "problem signatures" that may be detected. To accomplish this, each qubit across the three basic Pauli bases: X, Y, and Z may be measured. The Pauli X, Y, and Z measurements correspond to observing these qubits at different orientations, essentially providing different "views" of the states of these qubits. This measurement approach facilitates the construction of a measurement vector that encapsulates a single qubit state (represented by a single qubit expectation value, such as <x1> 、 <x2> 、…、 <xn> 、 <y1> 、…、 <yn> 、 <z1> 、…、 <zn>) and the relationship between the qubits (called a multi-qubit correlator). Further reference Fig.17 , item 1722 represents the measurement vector that encapsulates the individual quantum bit states.

[0081] These correlators can be determined by a predefined qubit connectivity graph, where the qubits are the vertices of the graph and their connections or edges specify which qubit pairs are involved in each correlator. For example, where the connectivity graph is defined as G = {(1,2), (2,3)}, the correlators can be measured as <x1x2>and <X2X3<, and similar measurements for the Y basis and the Z basis can be performed.

[0082] The quantum measurement module 110 compiles the measurement results into a measurement vector M t . This measurement vector can be a classical representation of the quantum state and can include:

[0083] Single qubit expectation values: These values are the average measurement values of the state of a single qubit when observed in the X, Y, or Z basis (for the i-th qubit, denoted as <xi> 、 <yi>and <zi>). They provide information about the state of the individual qubits.

[0084] Multi-qubit correlators: These correlators are measurements that reflect the relationship between multiple qubits (expressed as <xixj> 、 <xixjxk>wait).

[0085] The quantum measurement module 110 can also convert the vector M from the previous measurement t-1 The measurement result is fed back to the quantum feedback module 106. The feedback module can then encode the measurement result into its parameterized gate at the current time step.

[0086] In generating the updated measurement vector M t Thereafter, in step 210 , the reserve pool state creation module 114 generates a reserve pool state according to the previous cycle state and the input data.

[0087] Using the following relationship, the measurement vector is used to establish the vector called r t The status of the reserve pool:

[0088] r t =(1-α)r t-Δt +α[g(f r (W r ·r t-Δt )+f M (W M ·M t )+f X (W X ·X t ))](See Fig.17 , item 1728 is a representation of the generated reserve pool state).

[0089] In this formula, α is a value between 0 and 1 and is called the leak rate. This rate modulates how much previous state information is kept in the reservoir, balancing the impact of memory retention with new data. Function f r 、f M and f X It is an activation function that introduces data transformation, enriching the system's ability to capture complex dynamics. The weight matrix W r , W M and W X is a random matrix connecting the reservoir state r, the measurement state M and the input state X. A global transformation function g(·) can also be applied, further allowing nonlinear processing of the data.

[0090] The reservoir state equation encapsulates the steps of integrating quantum measurement data into the reservoir state, thereby injecting valuable information from quantum measurements into the system and enabling nonlinear processing. The equation is designed to be general, allowing the contribution of each term to be adjusted or omitted. For example, if f M is set to zero and f r and f X are all identity functions, then the equation is simplified to the classic reserve pool calculation model.

[0091] The weight matrix W can be used r , W M and W X They are normalized by taking the largest singular value of . This normalization can increase the robustness of the system by ensuring that training and prediction operations are performed within a numerically stable range.

[0092] At step 212, process 200 may determine whether all input data has been processed. This decision point is critical to the flow of operations within the quantum reservoir computing system. If there is still unprocessed data, the system loops back to continue transforming, encoding, and integrating the data into the quantum circuit. This iterative process ensures that the measurement vector is updated with each piece of input data, and by extension, the reservoir state is updated. Once all input data has been processed, the process proceeds to step 214, which involves performing a ridge regression on a sequence of reservoir state vectors.

[0093] In step 214, the quantum state is measured and the data is transferred to the reservoir state r t After conversion of the classical information represented, the classical processing module 116 performs classical processing.

[0094] Reservoir state vector R t Formation :

[0095] Reserve pool status t With input state X t can be combined together to form the reservoir state vector R t By transforming the reservoir state f R (r t ) and the transformed version of the input state h X (X t ) appended to a unit value (e.g. 1) to construct this vector. The unit value acts as a bias, similar to the bias in artificial neural networks, which is necessary for learning algorithms because it allows the model to better fit the data. Fig.17 , item 1732 is a representation of the generated reservoir state vector.

[0096] Learning Procedure and Ridge Regression

[0097] In step 214, the assembly vector R may be used in a learning process based on ridge regression implemented by the learning and prediction module 118. t Ridge regression is a variation of linear regression that includes regularization to prevent overfitting and improve the predictive accuracy of the model.

[0098] The learning model can be combined with f R and h X Nonlinear transformations of the representations are applied to the reservoir and input states, respectively, before being used in ridge regression. These nonlinearities are crucial because they allow the model to capture more complex patterns in the data that might be missed by linear methods.

[0099] By introducing nonlinear transformations, the model gains flexibility, enabling it to identify and exploit complex nonlinear relationships within the data. This capability significantly enhances the model’s learning capabilities, allowing for more precise and nuanced predictions.

[0100] In step 214, ridge regression is performed to generate a set of readout parameters or readout weights Read out parameters or read out weights Can be used to make future predictions of dynamic systems.

[0101] In step 216, upon successful completion of determining the read parameters After the ridge regression analysis, process 200 proceeds to generate new predictions for the input data. This is done by applying the readout parameters to the reservoir state vector R generated in the previous step. t The reservoir state vector, which encapsulates the processed information from the quantum circuit and the classical transformation, is now used in conjunction with the readout parameters to produce the prediction vector X t+1 . This vector represents the predicted state of the system at the next time step (t+1).

[0102] The described quantum machine learning system 100 and process 200 emphasize the use of quantum measurements as a proxy for capturing relevant information, rather than relying solely on "physics-inspired" data of Hamiltonian functions as in VQA. The approach treats the measurement operator as a hyperparameter that can be arbitrarily chosen and incorporates randomness into reservoir creation. The technical result of the process 200 is its utilization of the computational power of quantum systems in conjunction with classical reservoir computing techniques. The design strategy provides an adaptive "heuristic function (Ansatz)" for effectively utilizing quantum hardware while efficiently processing complex data representations.

[0103] In some embodiments, process 200 may further employ a classical coprocessor to receive the classical output from the quantum circuit and apply post-processing to produce refined data, and repeat the operation to produce the classical output to collect statistics for post-processing refinement. In an example, process 200 may employ the following steps: repeating the quantum circuit operation, each repetition being compiled differently to enable post-processing suppression techniques, and augmenting the quantum output with a classical coprocessor for system enhancement based on accumulated statistics.

[0104] System 100 and process 200 differ in their measurements not only of their computable quantum properties but also of their role in introducing nonlinearities that are critical to capturing complex data relationships.

[0105] In addition, the method remains flexible, allowing the measurements to be transformed to better fit the specific needs of the problem being solved. This adaptability ensures the applicability of the technique to a wide range of machine learning tasks, from linear to highly nonlinear problems, while maintaining computational efficiency and interpretability.

[0106] Experimental Results

[0107] The following section describes various experiments performed to evaluate embodiments of the present disclosure. Some of these experiments illustrate embodiments of the present disclosure that differ from those described above.

[0108] This section details the application of the hybrid quantum classical computing (HQRC) approach to chaotic systems, focusing on the Lorenz63 model as the main test point for the algorithm. To ensure a thorough evaluation, the approach is also applied to the double scroll model - a three-dimensional chaotic system.

[0109] The simulation of quantum circuits was performed on a classical computing framework to efficiently tune the hyperparameters. In addition, experiments were performed using quantum processing units (QPUs), namely the 79-qubit Aspen-M-3 chip from Rigetti and the 8-qubit Lucy chip from Oxford Quantum Computing (OQC). These QPU experiments were selectively based on the most effective hyperparameter configurations identified during the simulation phase on a classical computer.

[0110] To enhance the results from quantum processing that is susceptible to noise, Clifford Data Regression (CDR) is used as an error mitigation technique. This strategy is confirmed in the study "Error mitigation with Clifford quantum circuit data" by Piotr Czarnik, Andrew Arrasmith, Patrick J. Coles, and Lukasz Cincio (Quantum 5, 592 (2021)) and "Unified approach to data-driven quantum error mitigation" by Angus Lowe, Max Hunter Gordon, Piotr Czarnik, Andrew Arrasmith, Patrick J. Coles, and Lukasz Cincio in Physical Review Research. These cited works support methods for improving the fidelity of quantum computing in the presence of operational noise.

[0111] It should be emphasized that the experiments and results presented herein are used as exemplary illustrations and were performed under specific conditions employing one or more specific embodiments. Therefore, the described experiments and their findings should not be interpreted as limiting the breadth of the disclosure contained in this patent document.

[0112] A. Measurement part

[0113] The valid prediction time (VPT) is the main metric used to evaluate the forecast quality of a network, as defined in Vlachas et al., "Backpropagation algorithms and reservoir computing in recurrent neural networks for the forecasting of complex spatiotemporal dynamics" (2020), arXiv:1910.05266 [eess.SP] and Platt et al., "A systematic exploration of reservoir computing for forecasting complex spatiotemporal dynamics" (2022), arXiv:2201.08910 [cs.NE]. The VPT is the junction point at which the model's predicted output deviates from the actual data measured by the root mean square error (RMSE) by more than a specified threshold:

[0114]

[0115] in, y i (t) are the i-th components of the prediction and true value at time t, respectively, σ i is the ith component of the standard deviation of the true data, and D is the dimensionality of the problem (e.g., D = 3 for Lorenz63 and double scrolls). In our analysis, a threshold ε = 0.3 was chosen, following a systematic review of classical reservoir calculations by Platt et al.

[0116] In the case of chaotic systems, it is clear that uncertain predictions cannot be expected. Therefore, an equally important metric used to benchmark these systems is the long-term attractor prediction, which means that the system stays in its basin of attraction while possibly deviating from the correct component-wise prediction. Therefore, in the analysis, the closeness of the predicted attractor and the true attractor is studied. In addition, the Poincare return mapping is used, as described by Pathak et al. in "Using machine learning to replicate chaotic attractors and calculate Lyapunov exponents from data" (Chaos: An Interdisciplinary Journal of Nonlinear Science 27 (2017), doi: 10.1063 / 1.5010300). This method plots the time series data [z 1 ,z 2 ,...,z m ] relative to each other[z i ,z i+1 ] to sort all local maxima of the forecast and actual time series (for which longer simulations are used).

[0117] B. Classical simulation

[0118] The HQRC framework offers flexibility in choosing the number of qubits, the depth of the circuit, and the number of measurements (which correspond to the size of the reservoir). These choices can be tailored to allow simulations on a classical computer. This section analyzes the results and shows that the framework has sufficient expressive power to predict the behavior of chaotic systems. The predictive performance is comparable to state-of-the-art results obtained with classical reservoir computations.

[0119] 1. Lorentz 63

[0120] As described by Edward N Lorenz in "Deterministic nonperiodic flow" in Journal of atmospheric sciences 20, 130-141 (1963), Lorenz63 is the standard benchmark for classical RC because it is a well-studied chaotic model. The dynamics of the system is governed by the following set of differential equations:

[0121]

[0122]

[0123]

[0124] The coefficients are fixed to match the values ​​commonly used in the literature (such as described in "Next generation reservoir computing" by Daniel J. Gauthier, Erik Bollt, Aaron Griffith, and Wendson AS Barbosa (Nature Communications 12 (2021), 10.1038 / s41467-021-25801-2.)). These equations are similar to a simplified weather model of atmospheric convection that experiences uniform heating and cooling from below and above, respectively.

[0125] Figure 3 Simulation results of a Lorenz63 chaotic system using hybrid quantum classical reservoir computing (HQRC) according to embodiments of this document are shown.

[0126] The Lorenz63 system (a three-dimensional system with a known chaotic solution) is specified with initial conditions of variables x, y and z. The dynamics of the system is discretized with a time step of dt = 0.01, and the simulation involves a leak rate such as 0.7 and a very small regularization term of 10 -8 parameters, which are consistent with other related results in this document. Figure 3 In the experiments described in , a quantum reservoir computation model (HQRC) was trained over 2000 time steps (equal to a simulation time of t=20). An additional 2000-time steps were used to make predictions about the future state of the system. The achieved variational power transfer (VPT) was 10.32. The HQRC model was run on a classical computer using a noiseless simulation method, in which the real expectation values ​​are obtained from an ideal quantum circuit. It has been confirmed that in the case of Lorenz63, significantly better results are obtained for the reservoir in the absence of measurement feedback layers and random circuits, as the addition of the latter can lead to low-quality predictions. The results were collected from a noiseless simulator without shot-noise, which means that the expectation values ​​are calculated from the wave function, rather than with a finite number of shot samples as in the case of the QPU results.

[0127] Figure 4 The position of the Lorenz63 attractor according to an embodiment of the present invention is shown over time, demonstrating the ability of HQRC's system to accurately reconstruct long-term qualitative behavior.

[0128] Figure 5 The Poincare return map of a Lorenz63 chaotic system according to embodiments presented herein is shown, showing an ordered oscillation pattern in component-wise divergence after approximately 30 time units.

[0129] Figure 5 and Figure 6 Follow and Figure 3 The parameters are identical to those defined in , the only difference being that the prediction phase is extended over 10000 steps. This extended sequence allows a comprehensive observation of the long-term dynamics of the system.

[0130] The benchmarked Lorenz63 system consists of a total of 19 data encoding layers, of which 5 are parameter-free layers of CX networks whose underlying graph has ring connectivity, and the remaining 14 layers are single-qubit rotations.

[0131] In this setting, a reservoir size of 127 is used, since the X, Y, Z measurements are used to construct a measurement vector consisting of single qubit expectation values ​​and 2-qubit and 3-qubit correlators between qubits from the fully connected graph. Despite the small size, a VPT of 10.32 is obtained, which is significantly better than the state-of-the-art classical methods with comparable reservoir sizes. However, the reservoir size is not the only important property of these algorithms, whether classical or quantum, although it scales to O(n 3 ), where n is the size of the reserve pool.

[0132] The presented results provide convincing evidence to support that the HQRC method is a viable quantum alternative to classical RC methods. In particular, standard RC methods can be extremely sensitive to various hyperparameters (reservoir size, training length, etc.). Since the proposed method has multiple hyperparameters to be chosen (number and type of layers, type of feature maps, measurement associators, etc.), hyperparameter sensitivity is also observed, yet even though the change between performances may fluctuate, the predictions rarely deviate from stable solutions, which is not the case for classical RC.

[0133] 2. Double scroll

[0134] Another popular benchmark is based on the dynamics of a double-scroll electronic circuit and is given by:

[0135]

[0136]

[0137]

[0138] In dimensionless form, ΔV(t) = V 1 (t)V 2 (t). The above parameters are fixed as: R 1 =1.2, R2=3.44, R4=0.193, β=11.6 and Ir=2.25x10-5. Inspired by the Ansatz entity of the Lorenz63 model, a limited search for suitable hyperparameters was performed.

[0139] Figure 6 Simulation results for a double scroll system using an HQRC system with 8 qubits and a reservoir size of 271 according to embodiments herein are shown. Figure 6 The training results are shown with ground truth overlap in simulation using time increments of dt=0.25 of time units and subsequent prediction steps starting to diverge after approximately 100 time units, with a validation prediction time (VPT) of 107.5, which is competitive with prior art results.

[0140] Figure 7 The attractor-based reconstructed long-term behavior of a double-scroll system using an 8-qubit HQRC configuration according to embodiments herein is shown. Figure 7 Shown with Figure 6 The same experimental arrangement is shown, where the extension is an extended prediction phase consisting of 10,000 time steps.

[0141] C.QPU simulation

[0142] Furthermore, proof-of-concept tests of the HQRC algorithm were run on two superconducting platforms: a Rigetti Aspen M-3 chip with 79 qubits and a Lucy chip with 8 qubits from OQC were performed. As currently available devices suffer from various sources of defects, it is expected that lower quality results will be observed compared to the simulated noise-free results described above. In particular, the first and device-independent errors originate from the finite number of measurements performed to extract the expected value, which is known as shot noise. This seemingly minor feature of the probabilistic interpretation actually has substantial consequences for the predictive power in chaotic systems, where every decimal place in the numerical simulation can lead to a difference between the true value and the prediction.

[0143] The premise of RS stems from the underlying stochasticity, i.e., coherent and incoherent noise present in real devices may have a neutral or negligible effect on performance. While this is the case for weakly uncorrelated noise (e.g., white noise) in classical RC, this is not the case for quantum devices, as noise can exhibit spatial and temporal correlations (crosstalk, 1 / f, etc.). This means that the noise affects each iteration differently, making it more difficult to learn during ridge regression. Even if the training subroutine matches the true value exactly, the predictions diverge rapidly because correlated and non-stationary errors hamper performance.

[0144] 1. Rigetti Aspen M-3

[0145] In the case of the Aspen M-3 chip housing 79 qubits, it is possible to test the implementation of the algorithm in parallel, which means that Ansatz is implemented on three spatially separated regions of the chip, in such a way that each region performs measurements of all their qubits on a different basis (i.e. X, Y and Z). For a fault-tolerant device, this will produce exactly the same results as running three separate circuits on the same qubits. However, since qubits currently exhibit different degrees of error (e.g., different calibrations lead to different fidelity values ​​on the qubit array), a higher degree of fluctuation can be expected. In the experiments, Ansatz with 8, 16 and 23 qubits was used, which, due to the parallel implementation, translated into the simultaneous use of 24, 48 and 69 qubits, respectively. The results deviate significantly from the theoretical noise-free predictions, as well as from simulations with shot noise alone.

[0146] To improve the results and bring them closer to theoretical expectations, Clifford Data Regression (CDR) is implemented - an error mitigation strategy that aims to adjust the expected value based on a linear regression between noise-free and noisy results. In the standard CDR method, an effectively simulated Clifford circuit is used.

[0147] In the tests, the Clifford condition was relaxed, which is equivalent to setting all rotation angles to 0° or π for the Ansatz tested. Although this is not the design implementation of CDR, we use the all angle permitted strategy to demonstrate the feasibility of the technique, as the 8-qubit circuit can be simulated on a personal computer.

[0148] Fig. 8A and Figure 8B is a graphical representation of the performance of the HQRC algorithm implemented on an Aspen M-3 chip during and after training when predicting using Clifford Data Regression (CDR) according to embodiments herein.

[0149] Fig. 9A and Fig. 9B is a graphical representation of the performance of the HQRC algorithm implemented on an Aspen M-3 chip during and after training when making predictions without using CDR with 1000 attempts in accordance with embodiments herein.

[0150] Fig. 10A and Fig. 10B is a graphical representation of the performance of the HQRC algorithm implemented on an Aspen M-3 chip during and after training when predicting without using CDR with 10,000 attempts in accordance with embodiments herein.

[0151] Fig.11A and Fig. 11B is a graphical representation of the collective performance of the HQRC algorithm implemented on an Aspen M-3 chip during and after training when using CDR for prediction and when not using CDR with 1000 attempts and 10000 attempts according to embodiments herein.

[0152] Fig.11A and Fig. 11B (The corresponding prediction focus part) merges the Fig. 8A , Figure 8B , Fig. 9A , Fig. 9B , Fig. 10A and Fig. 10B These results are derived from Fig.13 The six-layer Layer_1 type 8-qubit quantum circuit shown in Figure 1 features the transformation φ(x) = arcsin(tanh(x)) and is implemented on an Aspen M-3 chip. Changes in the setup are distinguished by prediction and data sampling methods:

[0153] Predictions using CDR: This setup uses Clifford Data Regression (CDR) to reduce errors, using a series of 20 circuits in the HQRC configuration to determine the fit coefficients in the CDR method. In addition, 10,000 quantum circuit runs or "attempts" are used to approximate the measurement vector.

[0154] 1k and 10k predictions without CDR: These settings abandon the CDR strategy and are identical except for the number of attempts used to collect data (1000 and 10000 attempts, respectively).

[0155] Noise-free predictions: This refers to the classical simulations used as a baseline for comparison, providing exact expected values ​​for the measurement vectors, assuming an ideal situation with no quantum noise.

[0156] The proof-of-concept findings presented compare different implementations of the same Ansatz. In a noise-free simulation that provided exact expected values, a valid prediction time (VPT) of 4.82 was achieved. However, runs on a quantum processing unit (QPU) did not reach this benchmark. They obtained a VPT of 0.04 when using 1000 attempts with Clifford Data Regression (CDR) and 10,000 attempts without CDR. The VPT for the configuration without CDR dropped further to 0.01 at 1000 attempts. These results highlight the difference between ideal simulation and actual QPU performance under different conditions.

[0157] 2.OQC Lucy 8Q

[0158] A similar setup was tested on an 8-qubit chip provided by Oxford Quantum Circuits (OQC). In this case, the QPU runs needed to be performed sequentially for the different measurement bases, as the chip's architecture was not adequate for a parallel implementation. Similarly, for the results from Aspen M-3, we observed deviations from theoretical predictions.

[0159] Fig. 12A The performance of the HQRC algorithm implemented on an 8-qubit chip provided by Oxford Quantum Circuits (OQC) is shown compared to the HQRC algorithm implemented on a quantum simulator according to embodiments of the present invention. The measurement protocol includes a single-qubit correlator, a 2-qubit correlator, and a 3-qubit correlator, and the data is extracted from 4000 attempts. The circuit operates with consistent global hyperparameters (e.g., leakage rate and initial state) across training (including 900 time steps) and prediction (covering 100 time steps, where the time step dt is 0.01).

[0160] Fig. 12B An extended performance evaluation of the HQRC algorithm when executed on an 8-qubit chip from Oxford Quantum Circuits (OQC) is shown. The iteration uses a training scheme consisting of 1500 training steps, followed by a prediction phase of 500 steps. The measurement protocol includes only single-qubit correlators and 2-qubit correlators. The measurement protocol used for this experiment is restricted to single-qubit correlators and 2-qubit correlators, even though the number of attempts has been increased, which indicates a more detailed data acquisition process. The verification time prediction (VPT) parameter is set to 0.1.

[0161] Tested layer

[0162] Fig.13 The main layer types for benchmarking Lorenz63 systems according to embodiments of the present invention are shown, showing quantum circuit designs with varying configurations of Ry and Rz gates and entanglement patterns across multiple qubits, and specific entangled pairs within two different ring groups. These configurations include different combinations of Ry and Rz rotation gates, controlled NOT gates (CX), and entanglement strategies across qubits. Specifically, it details four main layer types, each with a unique gate arrangement and entanglement pairing, as shown in the corresponding graphical structures. These layers are critical for evaluating the dynamics and performance of quantum systems in simulations.

[0163] Fig.14A , Fig. 14B and Fig. 14C A statistical compilation of the Lorenz63 problem according to embodiments herein is shown, demonstrating the performance impact of different quantum circuit layer types across systems.

[0164] Fig.14A , Fig. 14B and Fig. 14C Three bar graphs are provided summarizing the median valid prediction time (VPT) of classical simulations of the Lorenz63 system performed in the absence of noise and based on exact expectation values. The simulations were performed with 10 random matrix initializations. The layers used (e.g. Fig.13 The figures, detailed in Figure 2, combine various quantum gate configurations and entanglement patterns. The figures illustrate the VPT on systems with 4, 6, and 8 qubits, revealing how the median VPT is affected by the type of circuit layers, the number of layers, and the number of qubits.

[0165] Fig.15A , Fig. 15B and Fig. 15C The performance of the HQRC algorithm according to embodiments herein on a Rigetti Aspen M-3 chip using 8, 16, and 23 qubits, respectively, is shown.

[0166] Fig.15A , Fig. 15B and Fig. 15C Given the results from the 8 ( Fig.15A ), 16( Fig. 15B ) and 23 ( Fig. 15C ) qubits, comparing GPU-based training predictions and noise-free theoretical predictions with actual quantum measurements. These results are based on 1,000 attempts, capturing single-qubit correlations, two-qubit correlations, and three-qubit correlations. The 8-qubit experiment used a complex sequence of rotations and entanglement gates with full ring graph connectivity, while the 16 and 23-qubit setups followed similar but unspecified gate structures, all employing hyperbolic tangent activation functions in the rotations. Experiments were performed in parallel across three regions of the quantum chip to extract different expectation values ​​in the X, Y, and Z measurement bases.

[0167] Feature maps used in the experiment

[0168] In the described experiments, three different methods of feature map encoding were used:

[0169] 1. When combined with the Ry (rotation around y) gate, the arccosine and arcsine functions are used to simulate Chebyshev polynomials, but these functions are also used in combination with other types of quantum rotations.

[0170] 2. The hyperbolic tangent (tanh) function is integrated as a standard machine learning activation function within the quantum circuit, allowing nonlinear transformations in the quantum domain.

[0171] 3. Preprocess the input data using Fourier encoding; this involves scaling the data by integer multiples of 2π, introducing a periodic structure to the data. Variations of this method are explored, including "half-Fourier" encoding, which scales the input data by xkπ, where x is the data point and k is an integer multiplier to adjust the frequency and phase of the encoding.

[0172] As can be seen, HQRC has been tested on chaotic systems, which are generally a good fit for RC methods. Compared to classical RC, HQRC shows promise in making short-term predictions and capturing the long-term behavior of chaotic systems with low-dimensional reservoir states. Unlike traditional RC, HQRC provides a modular structure that allows multiple hyperparameters to be easily adjusted, affecting the performance differently. The robustness of HQRC to hyperparameter variations is such that even random configurations do not produce divergent results, unlike classical RC that may fail in low-dimensional regimes. This shows that despite the variability in performance for different settings, HQRC consistently provides stable predictions.

[0173] Fig.16 Depicted is a schematic representation of a computing device 1600 designed to operate with both classical and quantum computing processes of a hybrid quantum computing system 100 in accordance with an embodiment of the systems and methods disclosed herein. The device includes a central processing unit (CPU) 1622 that interfaces with various data storage and the following memory components: secondary storage 1624; read-only memory (ROM) 1626; and random access memory (RAM) 1628.

[0174] In this configuration, a quantum computing unit (QCU) 1636 is integrated. The QCU 1636 exploits quantum mechanical phenomena to enhance computing. It is designed to execute algorithms that are particularly suitable for quantum computing, such as those involving large-scale digital factorization, quantum simulation, and specific optimization problems, so that computing performance exceeds the capabilities of a standalone CPU.

[0175] Auxiliary storage 1624 may include a dedicated sector 1624a containing instructions executable by CPU 1622 and QCU 1636. These instructions enable device 1600 to perform operations that may be optimized through quantum computing.

[0176] ROM 1626 stores non-volatile code necessary for the initial boot process and normal operation of computing device 1600. RAM 1628 provides volatile storage for immediate access to data by CPU 1622 and QCU 1636 during active tasks.

[0177] Peripheral devices are managed through input / output (I / O) interface 1630, while network connections are achieved through network interface 1632. A graphics processing unit (GPU) 1634 is present to handle parallel processing tasks, which can be separate from or integrated with the quantum computing process.

[0178] The CPU 1622, which is the main processor for general computing tasks, is responsible for executing sequential operations and handling various computing processes in the classical domain. It manages routine tasks with high efficiency and interfaces with system memory (including RAM 1628 and ROM 1626) for data storage and retrieval.

[0179] In this dual-capable system, CPU 1622 typically acts as a coordinator, determining when to use QCU 1636 based on computational requirements. For tasks suitable for quantum, CPU 1622 prepares and relays data to QCU 1636, which then processes the information using its quantum computing capabilities. After QCU 1636 completes quantum processing, it can transmit the results back to CPU 1622, which can perform additional classical processing or output the results.

[0180] The collaboration between CPU 1622 and QCU 1636 effectively extends the computational scope of device 1600, enabling it to switch between classical and quantum operations. This ensures that device 1600 utilizes the most efficient processing method available, whether it is classical computing provided by CPU 1622 or quantum processing provided by QCU 1636.

[0181] Although computing device 1600 is described with reference to a single computer, it should be understood that the computing device can be formed by two or more computers that communicate with each other and collaborate to perform tasks. For example, but not by way of limitation, the application can be divided in a manner that allows concurrent and / or parallel processing of the instructions of the application. Alternatively, the data processed by the application can be divided in a manner that allows different parts of the data set to be processed concurrently and / or in parallel by two or more computers. In one embodiment, computing device 1600 can use virtualization software to provide the functions of multiple servers that are not directly bound to multiple computers in computing device 1600. In one embodiment, the functions disclosed above can be provided by executing applications and / or multiple applications in a cloud computing environment. Cloud computing can include providing computing services via a network connection using dynamically scalable computing resources. A cloud computing environment can be established by an enterprise and / or can be rented from a third-party provider as needed.

[0182] Additional components, such as one or more application specific integrated circuits, neuromorphic computing units, field programmable gate arrays, or other electronic or photonic processing components, may also be included and used in conjunction with or in place of processor 1622 to perform processing operations. The processing operations may include machine learning operations, other operations that support machine learning operations, or a combination thereof.

[0183] Fig.17 A schematic representation of a hybrid quantum classical computing system 1700 is shown according to embodiments herein.

[0184] In an example implementation, the hybrid quantum classical computing system 1700 is a predictive model for predicting future states corresponding to a time-dependent dynamic system.

[0185] In an example embodiment, a fixed transformation matrix 1704 may be used to transform the time-dependent input data 1702 to create transformed data 1706 corresponding to a vector of data encoding parameters. The transformed data 1706 may be encoded into a quantum circuit 1734 by performing multiple sets of quantum operations 1726. The quantum circuit 1734 includes multiple encoding layers 1708, an entanglement layer 1710, feedback layers 1712 and 1714, and a random transformation layer 1716. The multiple sets of quantum operations 1726 include one of the following or a combination of one or more of the following: a single qubit rotation around at least one of the X, Y, and Z axes; a controlled phase gate operation, an fSim gate operation, or a 2-qubit XY rotation. The entanglement layer 1710 may correspond to a controlled NOT (CNOT) gate, indicating a CNOT operation applied without parameters. The entanglement layer 1710 may be used to entangle qubits according to a specific graph G, generating quantum correlations between qubits.

[0186] One or more of the quantum operations 1726 may be performed to encode measurement feedback from a previous measurement vector 1736 into the quantum circuit 1734. These layers take the measurement feedback from the previous time step (t-1) and encode them into the parameterized gates of the current time step. This step allows the system 1700 to have a form of memory because the measurement output of the quantum circuit 1734 is fed back into the system.

[0187] Random transformation layer 1716 may include random quantum gates for outputting quantum state configurations (e.g., single quantum rotations followed by a network of controlled NOT gates). The quantum state configurations may be measured by quantum measurement module 1718 for outputting measurements in X, Y, Z basis 1720 to generate updated measurement vector 1722. Updated measurement vector 1722 may include single qubit expected values ​​and multi-qubit correlators.

[0188] Classical Processing and Ridge Regression

[0189] After generating the updated measurement vector 1722, a reservoir state 1728 is generated based on the updated measurement vector 1722, the previous reservoir state, and the input data 1702. The system 1700 then determines whether additional time-dependent input data 1702 requires processing.

[0190] Once all of the time-dependent input data 1702 has been processed, a reservoir state vector 1732 is generated based on the generated set of reservoir states 1728, input data 1702, and bias terms. In an example embodiment, the process of generating the reservoir state vector 1732 may include a nonlinear transformation of the reservoir state 1728 and the time-dependent input data 1702.

[0191] A ridge regression process is then applied to the series of generated reservoir state vectors 1732 to determine a set of readout parameters 1738. Based on the readout parameters 1738 and the reservoir state vectors 1732, a prediction 1730 of the future state of the dynamic system is generated.

[0192] The technical solutions detailed in the present disclosure may be embodied in the form of a computer program product. The computer program product may be stored in a non-volatile or non-temporary storage medium, which may be a CD-ROM, a USB flash drive, or a removable hard disk. The computer program product includes a plurality of instructions that enable a computer device (a personal computer, a server, or a network device) to execute the methods provided in the embodiments described herein. For example, such execution may correspond to a simulation of a logical operation, including training and aggregation of model updates in a joint learning process, as described herein. The software product may additionally or optionally include a plurality of instructions that enable the computing device 1600 to perform operations for configuring or programming a digital logic device according to an embodiment of the present invention.

[0193] By programming and / or loading executable instructions onto the computing device, at least one of the CPU 1622, RAM 1628, and ROM 1626 is changed, thereby partially converting the computing device into a special-purpose machine or device having novel functions taught by the present disclosure. For the fields of electrical engineering and software engineering, it may be necessary that the functions implemented by loading executable software into a computer can be converted into hardware implementations according to well-known design rules.

[0194] It will be appreciated by those skilled in the art that the foregoing examples and embodiments are exemplary and are not intended to limit the scope of the present disclosure. Although the foregoing description has described exemplary embodiments, it will be appreciated by those skilled in the art that many changes may be made to the embodiments within the scope and spirit of the present invention. It should be noted that the elements of any claim may be arranged differently, including having multiple dependencies, configurations, and combinations.< / xixjxk> < / xixj> < / zi> < / yi> < / xi> < / zn> < / z1> < / yn> < / y1> < / xn> < / x2> < / x1> < / xi> < / xi>

Claims

1. A hybrid quantum classical computing method, the hybrid quantum classical computing method comprising the following steps: receiving time-dependent input data of a dynamic system; transforming the first element of the input data using a plurality of transformation matrices to obtain a set of transformed data; encoding the transformed data into a quantum circuit by performing a first set of quantum operations, wherein the quantum circuit comprises a plurality of layers, wherein the plurality of layers comprises a data encoding layer, a feedback layer, and a random transformation layer; encoding measurement feedback from a previous measurement vector into the quantum circuit by performing a second set of quantum operations; operating the quantum circuit to generate an updated measurement vector; and A reservoir state is generated based on the updated measurement vector, a previous reservoir state, and the input data.

2. The hybrid quantum classical computing method according to claim 1, further comprising: A determination is made whether additional input data is to be processed, and a reservoir state vector is formed based on one or more of the generated reservoir state, the input data, and a bias term.

3. The hybrid quantum classical computing method according to claim 2, further comprising applying a ridge regression process to a series of formed reservoir state vectors to determine a plurality of readout parameters.

4. The hybrid quantum classical computing method according to claim 3, further comprising: Based on the readout parameters and the reservoir state vector, a prediction of a future state of the dynamic system is generated.

5. The hybrid quantum classical computing method according to claim 1, further comprising: A prediction of a future state of the dynamic system is generated using the quantum circuit and a trained classical processing module, wherein the classical processing module receives as input a nonlinear transformation of the reservoir state and a nonlinear transformation of the time-dependent input data.

6. The hybrid quantum classical computing method according to claim 1, further comprising: In the data encoding layer, entanglement is initiated between a plurality of quantum bits of the quantum circuit.

7. The hybrid quantum classical computing method according to claim 1, wherein: Each of the plurality of transformation matrices includes a plurality of fixed transformation matrices that remain unchanged during a training phase and a prediction phase of the computation process.

8. The hybrid quantum classical computing method according to claim 1, wherein: Each of the plurality of transformation matrices includes a random weight matrix or a Fourier-like matrix; and the set of transformed data is generated by performing a function transformation on the first element of the input data.

9. The hybrid quantum classical computing method according to claim 1, wherein: Each of the first set of quantum operations and the second set of quantum operations includes one of the following or a combination of one or more of the following: A single qubit rotation about at least one of the X, Y, and Z axes; Controlled phase gate operation; fSim gate operation; or 2-Qubit XY rotation.

10. The hybrid quantum classical computing method according to claim 1, wherein: The steps of encoding the transformed data and encoding the measurement feedback include transforming data encoding parameters corresponding to the first set of parameterized layers and the second set of parameterized layers using a feature mapping function.

11. The hybrid quantum classical computing method according to claim 1, wherein: The reservoir circuit layer includes reservoir cells corresponding to a set of quantum gates, wherein parameters of the quantum gates do not depend on measurement feedback, input data, or reservoir states.

12. The hybrid quantum classical computing method according to claim 1, wherein: The generation of the reservoir state is based on a leakage rate parameter and a plurality of activation functions that introduce nonlinearity into operations performed by the quantum circuit.

13. The hybrid quantum classical computing method according to claim 1, wherein: The updated measurement vector also includes a single-qubit expected value and a multi-qubit correlator, wherein both the single-qubit expected value and the multi-qubit correlator are defined on the measurement map.

14. The hybrid quantum classical computing method according to claim 1, which utilizes the properties of high-dimensional Hilbert space as a reservoir for encoding chaotic dynamics into the quantum circuit.

15. The hybrid quantum classical computing method according to claim 1, wherein: A previous measurement corresponding to the measurement feedback is obtained from the quantum circuit parameterized by a previous iteration of the hybrid quantum classical computing method, wherein the previous measurement parameterizes the feedback layer.

16. The hybrid quantum classical computing method according to claim 1, wherein: Each of the transformation matrices is a fixed transformation matrix associated with a reservoir state, a measured state, and an input state.

17. The hybrid quantum classical computing method according to claim 1, wherein: The time-dependent input data includes a multi-component time series data vector.

18. The hybrid quantum classical computing method according to claim 1, further comprising: A classical coprocessor is used to receive the classical output from the quantum circuit and apply post-processing to produce refined data, and the operation of producing the classical output is repeated to collect statistics for post-processing refinement.

19. The hybrid quantum classical computing method according to claim 1, further comprising the following steps: The quantum circuit operation is repeated, where each repetition is compiled differently to enable post-processing suppression techniques, and the quantum output is enhanced with a classical coprocessor for systematic enhancement based on accumulated statistics.

20. A quantum computing system, comprising: a data input and transformation module configured to receive time-dependent input data of a dynamic system and transform a first element of the input data into a set of transformed data, wherein the first element corresponds to an initial time; a quantum data encoding module configured to encode the transformed data into a quantum circuit by performing a first set of quantum operations, wherein the quantum circuit comprises a plurality of layers, the plurality of layers comprising a data encoding layer, a feedback layer, and a reservoir circuit layer; a quantum measurement model configured to generate a prior measurement vector; a quantum feedback module configured to integrate measurement feedback from the previous measurement vector into the quantum circuit by performing a second set of quantum operations; a random transformation module configured to output the quantum state configuration measured by the quantum measurement module to generate an updated measurement vector; and a reservoir state creation module configured to generate a reservoir state based on the updated measurement vector, a previous reservoir state, and the input data, wherein the quantum computing system utilizes a plurality of fixed transformation matrices that remain constant during a training phase and a prediction phase of operation of the quantum computing system.

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