Quantum-assisted verification of electronic circuit functionality

Abstract

The present subject matter relates to a method for proving a statement descriptive of a functionality of an electronic circuit using a quantum computer and a proof assistant. The method comprises: encoding a current proof state into a vector of real numbers of a fixed length, the current proof state defining a task for proving at least part of the statement, encoding the vector into a quantum state of a quantum system of the quantum computer, using the quantum state as an input quantum state by a quantum machine learning model for providing an output quantum state whose measurement represents a proof step for the defined task, measuring the output quantum state, thereby obtaining the proof step for the defined task, providing the proof step to the proof assistant, in response to providing the proof step, receiving a next proof state from the proof assistant.

Description

[0001] QUANTUM-ASSISTED VERIFICATION OF ELECTRONIC CIRCUIT FUNCTIONALITY

[0002] Technical Field

[0003] Various example embodiments relate to computer systems, and more particularly to a method for proving a statement describing a functionality of an electronic circuit.

[0004] Background

[0005] The evolution of technology has led to the development of complex electronic circuits that perform a variety of functions. These circuits are integral components of numerous devices, ranging from simple household appliances to advanced computing systems. The design and operation of these circuits often involve the use of mathematical proofs to ensure their correctness and reliability. However, there is a need for improving these proofs.

[0006] Summary

[0007] Example embodiments provide a method for proving a statement descriptive of a functionality of an electronic circuit using a quantum computer and a proof assistant, the method comprising: encoding a current proof state into a vector of real numbers of a fixed length, the current proof state defining a task for proving at least part of the statement, encoding the vector into a quantum state of a quantum system of the quantum computer, using the quantum state as an input quantum state by a quantum machine learning model for providing by the quantum machine learning model an output quantum state whose measurement represents a proof step for the defined task, measuring the output quantum state, thereby obtaining the proof step for the defined task, providing the proof step to the proof assistant, in response to providing the proof step, receiving a next proof state from the proof assistant.

[0008] Example embodiments provide a computer system for proving a statement descriptive of a functionality of an electronic circuit using a quantum computer and a proof assistant, the computer system comprising the proof assistant, the computer system being configured for: encoding a current proof state into a vector of real numbers of a fixed length, the current proof state defining a task for proving at least part of the statement, controlling the quantum computer for encoding the vector into a quantum state of a quantum system of the quantum computer, controlling the quantum computer for using the quantum state as an input quantum state by a quantum machine learning model for providing by the quantum machine learning model an output quantum state whose measurement represents a proof step for the defined task, controlling the quantum computer for measuring the output quantum state for obtaining the proof step for the defined task, providing the proof step to the proof assistant, in response to providing the proof step, receiving a next proof state from the proof assistant.

[0009] Example embodiments provide a computer program product comprising instructions for causing a computer system for performing at least the following: encoding a current proof state into a vector of real numbers of a fixed length, the current proof state defining a task for proving at least part of a statement descriptive of a functionality of an electronic circuit, controlling the quantum computer for encoding the vector into a quantum state of a quantum system of the quantum computer, controlling the quantum computer for using the quantum state as an input quantum state by a quantum machine learning model for providing by the quantum machine learning model an output quantum state whose measurement represents a proof step for the defined task, controlling the quantum computer for measuring the output quantum state for obtaining the proof step for the defined task, providing the proof step to the proof assistant, in response to providing the proof step, receiving a next proof state from the proof assistant.

[0010] Brief Description of the Drawings

[0011] The accompanying figures are included to provide a further understanding of examples and are incorporated in and constitute part of this specification. In the figures:

[0012] Fig. 1 is a diagram illustrating a computer system in accordance with an example of the present subject matter. Fig. 2 is a flowchart of a method for proving a statement descriptive of a functionality of an electronic circuit in accordance with an example of the present subject matter.

[0013] Fig. 3 is a block diagram of an exemplary computer system for implementing at least part of the present method in accordance with an example of the present subject matter.

[0014] Detailed Description

[0015] In the following description, for purposes of explanation and not limitation, specific details are set forth such as particular architectures, interfaces, techniques, etc., in order to provide a thorough understanding of the examples. However, it will be apparent to those skilled in the art that the disclosed subject matter may be practiced in other illustrative examples that depart from these specific details. In some instances, detailed descriptions of well-known devices and / or methods are omitted so as not to obscure the description with unnecessary detail.

[0016] The present subject matter may enable the verification of electronic circuits. For example, the present subject matter may reduce prototyping iterations by allowing an accurate verification of the functionality of the electronic circuits. This may significantly reduce development time and cost associated with multiple iterations. By verifying the circuit's functionality, one may identify potential issues and make necessary adjustments before physically building the circuit. This may save time and resources that would otherwise be spent on building and testing multiple physical prototypes.

[0017] The present subject matter may prove the statement of an electronic circuit using the quantum computer and the proof assistant. The statement may be descriptive of a functionality of the electronic circuit.

[0018] The electronic circuit may be composed of individual electronic components, such as resistors, transistors, capacitors, inductors and diodes, connected by conductive wires or traces through which electric current may flow. The components of the electronic circuit may or may not comprise a processor such as a microprocessor. The electronic circuit may, for example, be a discrete circuit. The components of the electronic circuit may, for example, comprise active components. The combination of components and wires may allow various operations to be performed such as signal amplification, computations, and data management.

[0019] The statement may be descriptive of a functionality of the electronic circuit. The statement may, for example, be a specification of what the electronic circuit is expected to do. This specification may include properties, requirements, or invariants that the electronic circuit should satisfy. The specification may serve as a precise description of the desired behavior of the electronic circuit. The statement may be written in the formal language of the proof assistant. The statement may be stored as a plain text file or stored in a database. In one example, the statement may be a compound statement which may contain a sequence of statements. The sequence of statements may, for example, be referred to as intermediate statements or sub-statements of the compound statement. The combination of the intermediate statements may represent (or equivalent to) the compound statement of the electronic circuit. Each intermediate statement may, for example, describe a functionality of a respective part of the electronic circuit.

[0020] The proof assistant may be a software tool that assists in the development of formal proofs. The proof assistant may also be referred to as an interactive theorem proven The proof assistant may provide a formal language for expressing proofs and also provide a set of rules for manipulating these formal expressions. The proof assistant may generate, e.g., sequentially, proof states during a proof process. The proof state may represent the current state of a proof in the proof assistant. The proof state may provide a snapshot of the progress made in constructing the proof and guide the subsequent application of proof steps. The proof state may, for example, include a goal or a subgoal that needs to be proven. The goal may represent a specific statement to be proven, where the specific statement may be the (whole) statement of the electronic circuit or an intermediate statement to be proven. The proof state may optionally further comprise at least one of: a definition, a property or an assumption that may be associated with the goal defined in the proof state. The proof state may be written in the formal language of the proof assistant using at least one of: one or more predicates, one or more symbols that represent logical connectives, one or more quantifiers, or proposition. The proof state may be updated as proof steps are applied. The proof state may be represented as a structured data object.

[0021] A proof method may be used to prove the statement of the electronic circuit using the quantum computer and the proof assistant. The proof method comprises the step of encoding a current proof state into a vector of real numbers of a fixed length. The current proof state may define a task for proving at least part of the statement of the electronic circuit. For example, at start of the proof method, the proof assistant may be provided with the statement of the electronic circuit and a description of the electronic circuit. In response, the proof assistant may generate the current proof state which may define the task of proving the whole statement. For example, the statement of the electronic circuit and the description of the electronic circuit may be formalized or expressed using the programming language or syntax of the proof assistant. For example, the encoding may be performed such that each unique poof state is represented by a respective unique vector of real numbers of fixed length. The current proof state may be encoded into a vector of real numbers. The vector has a predefined length. By encoding the proof state into a vector, it may become amenable to manipulation and processing by the quantum computer.

[0022] In one example, the encoding of the current proof state into the vector of real numbers of fixed length may be achieved by representing different aspects of the proof state as specific values within the vector. For example, the vector may include elements that represent the goal or any other relevant information which is included in the proof state. The vector may for example be an embedding in a vector space. For example, each unique combination of symbol (e.g., variables, constants, function symbols) and logical operation (e.g., conjunction, disjunction, negation, equality) that forms a proof state may be mapped to a unique vector in a high-dimensional vector space. This mapping may be arbitrary or might be optimized through machine learning processes.

[0023] In one example, the proof state may be structured to comprise different parts, where a part may comprise a declaration or definition of variables, another part may comprise assumptions and a part may comprise the statement to prove given the other parts. These parts may be provided in a formal language of the proof assistant. In this case, the vector may include elements that, for example, represent these parts respectively. Considering, for simplification purpose, an example where the proof state may require to prove that: if a and b are both even, then the sum a+b is also even. The proof state may, for example, be provided as follows: (a b : N) (ha : even a) (hb : even b) : even (a + b), where ha and hb are assumptions that can be used to prove the goal a+b being even. This proof state may, for example, be processed by considering that it comprises a definition part “(a b : N)”, an assumption part “(ha : even a) (hb : even b)” and the goal part “even (a + b)”. These types of proof state parts may, for example, be represented as a vector of real numbers in a systematic way since they use predefined symbols and terms of the proof assistant.

[0024] The proof method further comprises the step of representing the vector into a quantum state of a quantum system of the quantum computer. For example, before the encoding, the vector of real number may be normalized so that the values of the vector may be between zero and one. For example, once the vector representation of the proof state is obtained, the quantum computer may be controlled to encode this vector into a quantum state of a quantum system. This encoding process may leverage the unique properties of quantum systems, such as superposition and entanglement, to represent the proof state in a quantum form. The quantum state serves as an input to a quantum machine learning model.

[0025] The proof method further comprises the step of using the quantum state as an input quantum state by the quantum machine learning model for providing by the quantum machine learning (QML) model an output quantum state whose measurement represents a proof step for the defined task. For example, the quantum machine learning model may utilize the input quantum state to perform computations and generate the output quantum state. The output quantum state represents a proof step for the defined task. The measurement of the output quantum state provides specific combination of data (e.g., classical bits) that represent the proof step and that can be manipulated and interpreted by a classical computer. This combination of data may be associated with a respective proof step in a predefined mapping. For example, the combination of data may be associated with a specific tactic or command of the proof assistant. The command may, for example, provide instructions that may be used by the classical computer to build the poof step. The tactic may, for example, refer to a high-level strategy or rule that may be used to construct the proof step. The tactic may be used by the classical computer to generate the proof step. The proof step may refer to the individual actions or operations taken during the proof construction process. The proof step may, for example, include the application of logical rules, the introduction or elimination of variables, and any other transformations or deductions made during the proof process. By leveraging quantum machine learning, the apparatus can explore and exploit the computational advantages offered by quantum systems to enhance the efficiency and accuracy of the proof process. This may enable an automated proof method.

[0026] The proof method further comprises the step of inputting the proof step to the proof assistant. For example, the proof step may be written in a language supported by the proof assistant before being provided to the proof assistant.

[0027] The proof method further comprises the step of receiving a next proof state from the proof assistant. The proof assistant generates that next proof state based on the input proof step. The next proof state may represent the updated state of the proof task after applying the proof step. For example, the proof assistant may analyze the proof step and provide the next proof state. The next proof state may indicate the completion of the proof. Alternatively, the next proof state may include a goal, corrections to be made, or any other relevant information to advance the proof process. The updated proof state may then be used as the current proof state for a subsequent repetition of the proof method.

[0028] In one example, the proof method may be repeated based on a status of the proof. For that, it may be determined whether the proof of the statement of the electronic circuit is completed. For example, to determine whether the proof is completed, the received proof state may be examined. The proof state may include indicators or flags that signify the completion of the proof task. For example, the proof state may contain a specific value or a combination of values that indicate the successful completion of the proof. By analyzing the received proof state, it may be determined whether the proof is completed or if further iterations are required. In case the received proof state does not indicate that the proof is completed, the proof method may be repeated using the last received proof state as the current proof state for the repetition of the proof method. For this repetition of the proof method, the proof state may, for example, define a task to prove a sub-statement of the statement. This iterative process may allow to gradually progress towards completing the proof task by encoding, processing, and analyzing the proof states using the quantum computer and the proof assistant. In case the received proof state indicates that the proof is completed, the method may stop indicating that the statement is proved.

[0029] The functionality of the electronic circuit may involve a basic operation or a combination of basic operations. The basic operation may, for example, include any one of: AND operation, OR operation and NOT operation. Proving the functionality of the electronic circuit may comprise proving each of the basic operations or the combination of basic operations that is involved in the functionality.

[0030] According to one example, the proof method may be a step of a quality assurance process for electronic circuits. This may enable efficient and accurate electronic circuit manufacturing ensuring that the products meet the highest standards of quality and reliability.

[0031] According to one example, the electronic circuit is a chip card of an identity document, wherein the functionality of the electronic circuit is an authentication functionality and / or an encryption functionality. The identity document may be any document that may be used to prove a person's identity. The identity document may be provided as an identity card in a standard credit card size form, or as a passport.

[0032] The authentication functionality may involve the comparison of correct sensitive data which is stored in the electronic circuit with input data that is provided by a user as being the correct sensitive data. The sensitive data may include biometric data such as fingerprints or identification data such as Personal Identification Numbers (PINs). The authentication functionality may include, for example, at least one of: fingerprint recognition or PIN-based authentication. For example, the chip card may store fingerprint data of the authorized user, and the authentication process may require the user to place their finger on a fingerprint scanner connected to the chip card for comparison.

[0033] Configuration parameters may be set to control the authentication functionality. The configuration parameters may be crucial for maintaining the security and integrity of authentication in the chip card. And properly setting these parameters may help to protect against unauthorized access, ensure user privacy, and enhance the overall security of the transaction or access control system.

[0034] For example, for the PIN-based authentication on the chip card, the configuration parameters may include the PIN format and the minimum and maximum number of digits allowed in the PIN. The PIN format may determine the acceptable characters (numeric only or alphanumeric) and any patterns or sequences that are prohibited (like consecutive numbers). For the biometric based authentication, the configuration parameters may include biometric thresholds to adjust sensitivity or acceptance rates to reduce false positives or negatives.

[0035] Following the non-limiting example of the PIN-based authentication, the proof method may be performed as follows. The authentication functionality may, for example, be based on the comparison of Personal Identification Numbers (PINs) in the chip card. This may involve verifying the identity of the cardholder by matching the PIN entered by the user against the PIN stored on the chip card. The chip card may contain a secure element (SE) that stores the correct PIN. Proving the authentication functionality may comprise proving the correctness of a comparison logic on the chip card, e.g., for comparing Personal Identification Numbers (PINs). The statement descriptive of the authentication functionality may, for example, be defined as a function to compare two PINs. The function may represent the comparison logic being used by the chip card. The comparison logic may, for example, include the comparison of an entered PIN directly with the stored PIN, or a hashed comparison where both the stored PIN and the entered PIN are hashed (converted into a fixed-size string of characters, typically through a cryptographic hash function), and the resulting hash values are compared. Following the first example of the comparison logic, the compared PINs may be represented as two lists of digits with specific lengths as supported by the chip card. The function may, for example, take as input two lists of natural numbers representing the entered PIN and the stored PIN and return a true value if they are equal and false value otherwise. The function may zip the two lists together, and compare each pair of digits, and accumulate the result, ensuring all corresponding digits must match for the PINs to be considered equal. The proof method may prove that the function correctly identifies when two PINs are the same and when they are different. For that, the statement may represent two properties. The first property may state that if the function returns a true value for two input pins pin 1 and pin2 then pin 1 and pin2 are the same list. The second property may state that if the function returns a false value for two input pins pin1 and pin2 then pin1 and pin2 are different lists. These properties may be expressed in a formal language of the proof assistant and provided to the proof assistant. In response, the proof assistant may provide a proof state e.g., to prove the whole statement. The proof state may comprise the statement. The proof state may be encoded into a vector of real numbers and then provided to the QML. The QML may provide as output an indication of a proof step. The resulting proof step that is provided by the QML may, for example, involve induction on the lists and showing that if all corresponding elements are equal then the lists themselves are equal. The proof step may be expressed in a formal language of the proof assistant and then provided to it. This interaction may be performed until the proof is completed. With the hash-based comparison, the statement to be proven may state that if two PINs are equal, their hashed values are also equal, given the hash function's properties.

[0036] The encryption functionality of the chip card may be utilized to secure communication between the identity document and an external device. For instance, the chip card may generate a unique encryption key that is used to encrypt data transmitted between the identity document and a card reader. This may ensure that sensitive information stored on the chip card, such as personal identification numbers or biometric data, remains confidential during data transfer. If, for example, the encryption functionality of the electronic circuit uses an RSA algorithm, then the statement descriptive of this functionality may for example be to prove the correctness of RSA decryption. The statement may, for example include: given prime numbers p,q, n=pq, e,d such that ed =1 mod <t>(n) and a message m such that 0<m<n, proving that (mAe)Ad = m (mod n). The statement may be provided to the proof assistant. In response, the proof assistant may provide a proof state e.g., to prove the whole statement. The proof state may comprise the statement. The proof state may be represented into a vector of reals numbers and then provided to the QML. The resulting proof step that is provided by the QML may, for example, include the application of Euler's Theorem or using the properties of modular arithmetic. This interaction may be performed until the proof is completed.

[0037] According to one example, the method further comprises: receiving a description of the electronic circuit in the form of a hardware description language code and using the description for determining the statement descriptive of the functionality of the electronic circuit. For example, the description of the electronic circuit may be provided as a hardware description language code such as Very High-Speed Integrated Circuit Hardware Description Language (VHDL) code, where the statement descriptive of the functionality of the electronic circuit may be a statement descriptive of a functionality of the hardware description language code. For example, the proving of the statement comprises proving the correctness of the hardware description language code. Proving the correctness of the hardware description language code may involve demonstrating that the hardware description language code behaves as intended and does not have any logical or functional errors. For example, the hardware description language code may be represented in a format of the proof assistant e.g., using a formal language, of the proof assistant. Then a functionality may be defined in the form of one or more properties about the code. These properties may be about the functionality of the hardware description language code design (e.g., correctness of an algorithm implemented in VHDL), its temporal behavior (e.g., absence of deadlock), or other aspects like power consumption or timing constraints. These properties may be expressed in proof assistant’s formal language. The proof may involve using assistant’s proof tactics to establish that the code satisfies the specified properties. This may involve construction of logical arguments within the proof assistant that demonstrate the truth of the properties, leveraging the interactive theorem proving environment. According to one example, the functionality of the electronic circuit represents at least one of: a functional correctness of one or more circuit components, an expected behaviour of the circuit, or a desired architecture of the electronic circuit.

[0038] According to one example, the functionality of the electronic circuit represents an expected behaviour of the electronic circuit. The expected behaviour of the circuit may refer to the functionality provided by the electronic circuit such as authentication, encryption or any other functionality.

[0039] According to one example, the functionality of the electronic circuit represents a desired architecture of the electronic circuit. The architecture of the electronic circuit may refer to the design or arrangements of the electronic components of the electronic circuit. The desired architecture may, for example, be the architecture that fulfills a target performance value such as a target power consumption or target speed. For example, to prove this functionality for target power consumption, the modelling of each component's behavior in the electronic circuit regarding its power consumption may be determined. For resistors, capacitors, inductors, transistors, and other components, mathematical models that describe their power usage may be provided. The laws of electrical circuits that apply to the current design, such as Ohm's Law, Kirchhoff's Voltage Law (KVL), and Kirchhoffs Current Law (KCL) may be formalized. These laws may be used to derive the currents and voltages at various points in the circuit, which may be used for calculating power consumption. The target power consumption may be defined by setting a maximum power consumption level under specific operational conditions, which would be represented as a formal property.

[0040] Using the models and laws formalized along with the specification of the threshold, a statement may be provided. The statement may be is built based on the above models and laws. The statement may be provided to the proof assistant. In response, the proof assistant may provide a proof state e.g., to prove the whole statement. The proof state may comprise the statement. The proof state may be encoded into the vector of real numbers and provided to the QML to provide a proof strategy. The proof strategy may be a prove of the electronic circuit whether it meets the target power consumption specification may be performed. This proof may involve mathematical reasoning about the electronic circuit's behavior including induction over the structure of the circuit. If the target power consumption is not fulfilled, another architecture of the circuit may be used e.g., keeping the same components. Indeed, even if two different circuit designs may use exactly the same types and number of electronic components to perform one functionality; however, they may not provide the same performances.

[0041] According to one example, the functionality of the electronic circuit represents at least one of: a functional correctness of one or more circuit components. The functional correctness of a component of the electronic circuit may, for example, require that the component has the power consumption as expected e.g., below a certain threshold. For that, the above models and laws formalized along with the specification of the threshold, a prove of the specific component (as placed in the circuit) whether it meets the target power consumption specification may be performed.

[0042] According to one example, the method further comprises: in response to determining that the proof is not completed and failed to fulfill the desired functionality, the electronic circuit may be adapted accordingly, and the method may be repeated for proving the statement of the adapted electronic circuit.

[0043] This example may provide a feedback mechanism. This feedback mechanism is triggered when the proof is not completed and fails to fulfill the desired functionality. The adaptation process may involve modifying the code used in the chip card, configurations, or settings of the chip card.

[0044] According to one example, adapting the electronic circuit comprises at least one of: changing the architecture of the components of the electronic circuit, or adapting configuration parameters that are used to perform the functionality of the electronic circuit.

[0045] According to one example, encoding the current proof state into the vector comprises: providing a recurrent neural network (RNN) that is trained using proof states as sequential data in order to encode the proof state into a fixed-size vector and inputting the current proof state into to the recurrent neural network for obtaining the vector. Indeed, during a proof, the proof state of the proof assistant may be given by a sequence of signs. Since the length of said sequence may change from proof step to proof step, the sequence may be encoded into a vector of real numbers of fixed length. One way of doing this may be by using the recurrent neural network to encode the proof state into a so-called "context". The recurrent neural network may be specifically designed and trained to process sequential data, such as proof states, in order to encode them into fixed-size vectors. The training of the recurrent neural network may involve using a large dataset of proof states, allowing it to learn patterns and relationships within the sequential data.

[0046] In an alternative example, instead of using a recurrent neural network, other machine learning techniques, such as convolutional neural networks (CNNs) or transformers, may be used to encode the proof state into a fixed-size vector. These alternative techniques may be more suitable for certain types of proof states or provide different advantages in terms of computational efficiency or accuracy.

[0047] Once the current proof state is encoded into a vector, it may be further utilized for various purposes. For instance, the encoded vector can be used for comparison with other proof states to identify similarities or differences. This can help in detecting patterns or recurring vulnerabilities in the electronic circuit.

[0048] According to one example, encoding the vector into the quantum state is performed using amplitude encoding or angle encoding. The choice of encoding may, for example, depend on the quantum machine learning model of consideration.

[0049] Amplitude encoding may involve representing the vector as a linear combination of basis states, where the amplitudes of the states may correspond to the elements of the vector. This encoding technique may allow for efficient manipulation and processing of the quantum state. On the other hand, angle encoding may represent the vector by encoding the angles between the basis states. The angles may be determined based on the elements of the vector, and the quantum state is then prepared accordingly. Angle encoding may offer advantages in terms of robustness against certain types of errors and noise.

[0050] In an example where the vector to be encoded is [1 , 0], the prepared quantum state may be provided by the amplitude encoding as a linear combination of the basis states |0>, and 11 >, with amplitudes corresponding to the elements of the vector. For instance, the resulting quantum state could be represented as a|0) + |3|1 >, where a, and [3 are complex numbers determined by the vector elements.

[0051] Alternatively, if angle encoding is employed, the angles between the basis states may be determined based on the vector elements. If, for example, the angles are 01 , and 02 are provided for the basis states |0>, and |1 >, respectively, the quantum state may then be prepared by applying appropriate quantum gates to create the desired angles between the basis states.

[0052] While amplitude encoding and angle encoding are described here as examples, other quantum encoding techniques may also be utilized. These alternative techniques could involve different mathematical representations or encoding schemes, depending on the specific requirements and constraints of the system.

[0053] According to one example, the method is automatically performed in response to receiving the statement of the electronic circuit. This may speed up the proof process and reduce user interventions in the proof process.

[0054] According to one example, the proof assistant is configured to be executed on a classical computer, wherein the method is implemented by a hybrid classical algorithm a quantum algorithm. The execution of the classical algorithm on the classical computer causes the classical computer to perform the method comprising controlling, according to the quantum algorithm, the quantum computer to perform the encoding of the vector, the provision of the output quantum state and the measurement of the output quantum state. The present method may be hybridized, meaning that it can contain a combination of classical components and quantum components. According to one example, the quantum system is defined by a number of qubits which is fixed based on the number of elements of the vector.

[0055] The quantum system may consist of a specific number of qubits which is determined based on the number of elements present in the vector. This determination may ensure that the quantum system is appropriately sized to handle the proof process. By keeping the number of qubits constant, this may ensure consistency and reliability in the proof process.

[0056] For example, the quantum system may comprise a set of qubits, wherein the number of qubits in the set is defined by the size of the vector. For example, if the size of the vector is N, the quantum system may comprise a number n of qubits where N = 2n. The quantum state may represent the values of the vector. In case the size of the vector is not a power of two, the vector may, for example, be padded with predefined values such as zeros to a dimension that is a power of 2.

[0057] According to one example, the size of the vector is fixed to a value dependent on available number of qubits in the quantum computer.

[0058] The fixed size of the vector is determined based on the number of qubits available in the quantum computer. This may ensure that the vector is optimized for the specific quantum computing resources being utilized. For example, the fixed size of the vector may be adjusted based on the specific requirements of the quantum computer being used. For example, if a different quantum computer with a different number of qubits is utilized, the size of the vector may be modified accordingly. This flexibility may allow the method to be adaptable to various quantum computing resources.

[0059] According to one example, the method comprises before the proving of the statement of the electronic circuit: training the quantum machine learning model. The training comprises: providing a training dataset. The training dataset comprises entries, wherein each entry of the training dataset comprises a proof state and corresponding tactic or command to build a proof step. Alternatively, the training dataset may be provided so that each entry of the training dataset may comprise a proof state and corresponding proof step that is derived from a tactic. Alternatively, each entry of the training dataset comprises a proof and corresponding statement. The tactic may, for example, refer to a high-level strategy or rule used to construct the proof step, wherein the proof step may refer to the individual actions or operations taken during the proof construction process. The quantum machine learning model may be trained for providing a tactic for a specific proof state.

[0060] In one training data generation example, in order to generate the training dataset, a proof assistant may be used to generate a set of proof states. The proof states may represent intermediate steps in the process of proving statements of the electronic circuit. Each proof state is associated, in the respective entry in the training dataset, with a corresponding tactic, which may be a specific strategy or rule used to build the proof step.

[0061] According to one example, the training dataset is obtained using the proof assistant used by the proof method and / or one or more other proof assistants. This may allow for the collection of a comprehensive set of data that can be used for training the quantum machine learning model. Alternatively, the training dataset may be obtained using a different proof assistant than the one used by the proof method. This proof assistant may have different features or capabilities, allowing for a diverse range of proof states and tactics to be collected. This variation in the training dataset may help improve the robustness and generalization of the quantum machine learning model.

[0062] In addition to the proof assistant, alternative methods for obtaining the training dataset can be explored. For example, the dataset may be generated through the use of multiple automated theorem provers or formal verification tools. These tools may assist in generating proof states and corresponding tactics or statements that can be used as entries in the training dataset.

[0063] In one training data generation example, the method may generate the training dataset by directly providing a set of pre-existing proofs along with their corresponding statements. These proofs serve as examples for the quantum machine learning model to learn from. Once the training dataset is obtained, the method proceeds to train the quantum machine learning model. This training process may involve feeding the training dataset into the model and iteratively adjusting its parameters to optimize its performance in proving statements of the electronic circuit.

[0064] In one training data generation example, the training dataset may be provided based on the field or domain of the electronic circuit being proved. For example, where the method is used for proving sorting algorithms, the training of the quantum machine learning model may be performed to prove the correctness of sorting algorithms. The training dataset may be generated using the proof assistant. The proof assistant may generate a set of proof states, each representing a step in the proof of the correctness of a specific sorting algorithm. These proof states are then paired with the corresponding tactics, which outline the specific strategies used to build each proof step. Alternatively, if a set of pre-existing proofs is provided for the domain, the method may directly use these proofs along with their corresponding statements as the training dataset. For instance, a set of proofs demonstrating the correctness of various sorting algorithms can be used as the training dataset.

[0065] In one training data generation example, the training dataset as described above may be updated. For example, in order to further enhance the training process, the method may incorporate various techniques for data augmentation. The update may include manipulating the existing training dataset to create additional variations of the proof states and corresponding tactics or statements. This may be achieved through techniques such as data perturbation, where small modifications are made to the existing entries to create new instances. The updated training dataset may comprise the manipulated entries and optionally the existing training dataset. Another approach may involve generating synthetic data by using generative models or simulation techniques. The updated training dataset may comprise the synthetic data and optionally the existing training dataset.

[0066] In another example, alternative approaches may be used to generate the training dataset. For example, the method may incorporate techniques from natural language processing to generate statements that describe the desired properties of the electronic circuit using the description of the electronic circuit. In one training data generation example, at least part of the training dataset may be generated using a machine learning model to generate true statements and proofs. The at least part of the training dataset may be the training dataset. Alternatively, the at least part of the training dataset may be a portion of entries of the training dataset. The other portions of the training dataset may, for example, be provided using one or more of the above training data generation examples. For example, the true statements and proofs may be generated using classical reinforcement learning or hybrid reinforcement learning or quantum reinforcement learning or other classical machine learning techniques or hybrid machine learning techniques or quantum machine learning techniques. These generated statements and proofs may, for example, be used to enrich an existing database of statements and proofs for the proof assistant, providing enough data points to train a quantum large language model "from scratch".

[0067] For example, the machine learning model (to generate true statements) may be trained on a large corpus of existing true statements and proofs. This corpus can be sourced from various reliable and verified databases or repositories. The machine learning model learns the patterns, structures, and logical reasoning behind these true statements and proofs. Once the machine learning model is trained, it may generate new true statements and proofs based on the learned patterns and reasoning. These generated statements and proofs may then be used to create at least a portion of the training dataset. Each entry in the training dataset may comprise a generated true statement or proof and its corresponding proof step or tactic. Alternatively, the machine learning model may be trained on different subsets of existing true statements and proofs to create specialized training datasets. This may allow the quantum machine learning model to be trained for specific domains or problem types, enhancing its performance and accuracy in those areas. These generated true statements and proofs, along with their corresponding proof steps or tactics, form a part of the training dataset. The quantum machine learning model can then utilize this dataset to learn how to prove statements of the electronic circuit using quantum computing techniques. In one example, the machine learning model used for generating the training dataset may be fine-tuned and optimized based on the specific requirements of the quantum machine learning model being trained. Different machine learning algorithms and architectures can be explored to improve the quality and diversity of the generated statements and proofs.

[0068] In addition to using a machine learning model, alternative methods for generating the training dataset can also be considered. For example, instead of relying solely on a machine learning model, human experts may manually create a set of true statements and proofs that cover a wide range of scenarios. These manually created statements and proofs may then be combined with the machine-generated ones to form a comprehensive training dataset.

[0069] During the training process, the quantum machine learning model may learn to recognize patterns and relationships between the proof states and the corresponding tactics or statements. By iteratively adjusting its parameters, the model may become increasingly proficient at providing proof steps.

[0070] In one training example, the quantum machine learning model may comprise an encoding layer. The encoding layer may be configured to encode the vector of fixed size into a quantum state using a set of qubits. The quantum machine learning model may further comprise a learning layer having one or more trainable or free parameters. The learning layer may be configured to change the quantum state by applying one or more unitary transformations. The learnable parameters may, for example, be the rotation angles of single qubit Pauli rotation gates e.g., the Pauli rotation angle may be applied on qubits of the set of qubits after the quantum state has been created. The learnable parameters may, for example, comprise a number of rotation angles which are applied to the set of qubits respectively. The quantum machine learning model may further comprise a measurement layer for measuring the output quantum state. A loss function may be evaluated using the proof step represented by the measurement and the corresponding proof step in the training dataset. The quantum machine learning model may be trained by backpropagation using the loss function and an optimization technique that is performed by the classical computer to check a convergence criterion. The backpropagation may enable to update of the learnable parameters using gradient descent. The convergence criterion may, for example, require that the loss function exceeds a threshold.

[0071] In one training example, and in case the vector is provided by the recurrent neural network as described herein, the quantum machine learning model and the recurrent neural network may be jointly trained. For example, in each iteration of the training, the proof state may be input to the recurrent neural network to generate the vector, the vector is provided as input to the quantum machine learning model, and the resulting measurement may provide an indication of the proof step. A loss function may be evaluated using the measurement and the proof step in the training dataset. In case the loss function does not fulfill a convergence criterion, the backpropagation is performed in order to update both the learnable parameters of the quantum machine learning model as well as the weights of the recurrent neural network. The update of the recurrent neural network weights and the learnable parameters may be performed using gradient descent. In case the loss function fulfills the convergence criterion, the resulting trained recurrent neural network and quantum machine learning model may be provided. The convergence criterion may, for example, require that the loss function exceeds a threshold.

[0072] According to one example, the quantum machine learning model is a quantum reinforcement learning model, wherein before the proving of the statement of the electronic circuit, the method comprises training the quantum reinforcement learning model using the proof assistant or another proof assistant as environment. Reinforcement learning is a type of machine learning where an agent learns to make decisions by interacting with an environment and receiving feedback in the form of rewards or penalties. The quantum reinforcement learning model utilizes quantum algorithms to optimize the decision-making process and improve the learning efficiency. This quantum reinforcement learning model may address the limitations of classical methods in efficiently and purposefully searching through large 'Action Spaces". During the training process, the quantum reinforcement learning model may interact with the proof assistant or another proof assistant, which serves as the environment. The model may learn to make decisions based on the current proof state provided by the proof assistant and the available tactics, aiming to reach the desired proof outcome. Through iterative training, the model may improve its ability to provide proof step(s) for a proof state.

[0073] In addition to the described example, there are alternative approaches that can be employed in the training process. For example, instead of using a proof assistant as the environment, the method may utilize a simulated environment specifically designed for training the quantum reinforcement learning model. This simulated environment may mimic the behaviour of a proof assistant and provide a controlled environment for training the model.

[0074] According to one example, the encoding of the current proof state into the vector is performed using a recurrent neural network, wherein the training of the quantum machine learning model comprises training the quantum machine learning model and the recurrent neural network in unison.

[0075] The RNN may be a type of artificial neural network that is capable of processing sequential data by maintaining an internal memory. This memory allows the RNN to capture and utilize information from previous proof states when encoding the current proof state. For example, the RNN can be implemented using various architectures such as Long Short-Term Memory (LSTM) or Gated Recurrent Unit (GRU). These architectures may enable the RNN to effectively capture long-term dependencies and handle vanishing or exploding gradient problems that may arise during training.

[0076] For example, the unison training may be performed as follows. The training dataset consists of entries, each entry containing a proof state and a corresponding tactic to build a proof step. The RNN encodes the proof state of a current entry into a vector representation, which is then fed into the quantum machine learning model. The model utilizes this encoded information to generate the appropriate tactic for the given proof state. Specifically, the training in unison between the RNN and the QML model may, for example, be performed as follows. The training dataset, in particular the roof states, may be prepared in a sequential format suitable for the RNN component. The training dataset may be compatible with both the RNN and the QML components. In each iteration of the training, a forward pass through the joint model may be performed, where the proof state is fed into the RNN component, and the output vector of the RNN is passed to the QML component as a quantum state. The QML component processes the quantum state and generates the output quantum state that represents a proof step. The loss between the predicted proof step and the target proof step of the training dataset is determined. A backpropagation may be performed through the QML component to compute the gradients of the parameters with respect to the loss. The parameters of the QML component may be updated e.g., using an optimization algorithm. A backpropagation through time in the RNN component may be performed to compute the gradients of the RNN weights with respect to the loss. Backpropagation Through Time (BPTT) may involve unfolding the RNN over time and propagating the gradients through each time step. The RNN weights may be updated using the optimization algorithm. The repetition may be for a specified number of iterations or until convergence is reached.

[0077] According to one example, the quantum machine learning model is a quantum neural network a Quantum Support Vector Machine (QSVM), a quantum large language model (LLM) or a quantum reinforcement learning model.

[0078] QSVM is a quantum version of the classical Support Vector Machine (SVM) algorithm, which is a supervised learning method used for classification and regression tasks. The QSVM utilizes quantum algorithms to perform classification tasks on quantum data. These quantum machine learning models may address the limitations of classical methods in efficiently. In this case, the RNN and the QSVM may, for example, be trained in unison as described in the document arXiv:2308.08467.

[0079] The quantum LLM model may be designed to process and understand natural language text using quantum algorithms. It can be trained on a large corpus of text data and used for tasks such as language translation, sentiment analysis, and text generation. The proof assistant may, for example, be the LEAN proof assistant or any other proof assistant that may enable proving the statements according to the present subject matter.

[0080] Fig. 1 is a diagram illustrating a computer system in accordance with an example of the present subject matter. The computer system 100 comprises a classical computer 101. The computer system 100 may further comprise a quantum computer 102. An example implementation of the classical computer 101 is described with reference to Fig. 3. The quantum computer 102 may comprise qubits. For example, qubits may be part of quantum registers 105.1 through 105.L. The classical computer 101 may be configured to control operation of the quantum computer 102. The classical computer 101 may use an interface 103 with the quantum computer 102 to control operation of the quantum computer 102 in accordance with an example of the present subject matter. The classical computer may comprise a proof assistant 107. The proof assistant may, for example, be the LEAN proof assistant or any other proof assistant that may enable proving the statements according to the present subject matter.

[0081] FIG. 2 is a flowchart of a method for proving a statement descriptive of a functionality of an electronic circuit using a quantum computer and a proof assistant according to an example of the present subject matter. For the purpose of explanation, the method described in FIG 2 may be implemented in the system illustrated in FIG. 1 , but is not limited to this implementation. The method may, for example, be performed by the computer system 100.

[0082] A current proof state may be encoded in step 201 into a vector of real numbers of a fixed length. The current proof state defines a task for proving at least part of the statement. The vector may be encoded in step 203 into a quantum state of a quantum system of the quantum computer (e.g., 102). For example, the classical computer 101 may control the quantum computer 102 to perform step 203. The quantum state may be used in step 205 as an input quantum state by a quantum machine learning model for providing by the quantum machine learning model an output quantum state, wherein the measurement of the output quantum state represents a proof step for the defined task. The output quantum state may be measured e.g., in step 205. For example, the classical computer 101 may control the quantum computer 102 to perform step 205. The proof step may be provided in step 207 to the proof assistant (e.g., 107). In response to providing the proof step, a next proof state may be received in step 209 from the proof assistant 107. It may be determined in step 211 whether the received proof state indicates that the proof is completed. In response to determining that the proof is not completed, the received proof state may be used as the current proof state for repeating the method steps 201 to 211 . In response to determining that the proof is completed, the method may end.

[0083] Fig. 3 is a block diagram of an exemplary computer system for implementing at least part of the present method in accordance with an example of the present subject matter.

[0084] The components of the computer system 702 may include, but are not limited to, one or more processors or processing units 703, a storage system 711 , a memory unit 705, and a bus 707 that couples various system components including memory unit 705 to processor 703. The storage system 711 may include for example a hard disk drive (HDD). The memory unit 705 may include computer system readable media in the form of volatile memory, such as random access memory (RAM) and / or cache memory.

[0085] The computer system 702 may also communicate with one or more external devices such as a keyboard, a pointing device, a display 713, etc.; one or more devices that enable a user to interact with computer system 702; and / or any devices (e.g., network card, modem, etc.) that enable the computer system 702 to communicate with one or more other computing devices. Such communication can occur via I / O interface(s) 719. Still yet, the computer system 702 can communicate with one or more networks such as a local area network (LAN), a general wide area network (WAN), and / or a public network (e.g., the Internet) via a network adapter 709. As depicted, the network adapter 709 communicates with the other components of the computer system 702 via bus 707.

[0086] The memory unit 705 is configured to store applications that are executable on the processor 703. For example, the memory unit 705 may comprise an operating system as well as one or more application programs. The application programs comprise instructions that when executed enable to perform the method described with reference to Fig. 2.

[0087] As will be appreciated by one skilled in the art, aspects of the present invention may be embodied as an apparatus, method, computer program or computer program product. Accordingly, aspects of the present invention may take the form of an entirely hardware embodiment, an entirely software embodiment (including firmware, resident software, micro-code, etc.) or an embodiment combining software and hardware aspects that may all generally be referred to herein as a “circuit,” “module” or “system.” Furthermore, aspects of the present invention may take the form of a computer program product embodied in one or more computer readable medium(s) having computer executable code embodied thereon. A computer program comprises the computer executable code or "program instructions".

[0088] Any combination of one or more computer readable medium(s) may be utilized. The computer readable medium may be a computer readable storage medium. A ‘computer-readable storage medium’ as used herein encompasses any tangible storage medium which may store instructions which are executable by a processor of a computing device. The computer-readable storage medium may be referred to as a computer-readable non-transitory storage medium. The computer-readable storage medium may also be referred to as a tangible computer readable medium. In some embodiments, a computer-readable storage medium may also be able to store data which is able to be accessed by the processor of the computing device.

[0089] ‘Computer memory’ or ‘memory’ is an example of a computer-readable storage medium. Computer memory is any memory which is directly accessible to a processor. ‘Computer storage’ or ‘storage’ is a further example of a computer-readable storage medium. Computer storage is any non-volatile computer-readable storage medium. In some embodiments computer storage may also be computer memory or vice versa.

[0090] A ‘processor’ as used herein encompasses an electronic component which is able to execute a program or machine executable instruction or computer executable code. References to the computing device comprising “a processor” should be interpreted as possibly containing more than one processor or processing core. The processor may for instance be a multi-core processor. A processor may also refer to a collection of processors within a single computer system or distributed amongst multiple computer systems. The term computing device should also be interpreted to possibly refer to a collection or network of computing devices each comprising a processor or processors. The computer executable code may be executed by multiple processors that may be within the same computing device or which may even be distributed across multiple computing devices.

[0091] Computer executable code may comprise machine executable instructions or a program which causes a processor to perform an aspect of the present invention. Computer executable code for carrying out operations for aspects of the present invention may be written in any combination of one or more programming languages, including an object oriented programming language such as Java, Smalltalk, C++ or the like and conventional procedural programming languages, such as the "C" programming language or similar programming languages and compiled into machine executable instructions. In some instances the computer executable code may be in the form of a high level language or in a pre-compiled form and be used in conjunction with an interpreter which generates the machine executable instructions on the fly.

[0092] Generally, the program instructions can be executed on one processor or on several processors. In the case of multiple processors, they can be distributed over several different entities. Each processor could execute a portion of the instructions intended for that entity. Thus, when referring to a system or process involving multiple entities, the computer program or program instructions are understood to be adapted to be executed by a processor associated or related to the respective entity.

Claims

CLAIMS1 . A method for proving a statement descriptive of a functionality of an electronic circuit using a quantum computer (102) and a proof assistant (107), the method comprising: encoding (201 ) a current proof state into a vector of real numbers of a fixed length, the current proof state defining a task for proving at least part of the statement; encoding (203) the vector into a quantum state of a quantum system of the quantum computer; using (205) the quantum state as an input quantum state by a quantum machine learning model for providing by the quantum machine learning model an output quantum state whose measurement represents a proof step for the defined task; measuring the output quantum state, thereby obtaining the proof step for the defined task; providing (207) the proof step to the proof assistant; in response to providing the proof step, receiving (209) a next proof state from the proof assistant.

2. The method of claim 1 , the electronic circuit being a chip card of an identity document, wherein the functionality is an authentication functionality and / or an encryption functionality.

3. The method of any of the preceding claims, wherein the functionality represents at least one of:a desired architecture of the electronic circuit that provides a target performance value of the electronic circuit; a functional correctness of one or more components of the electronic circuit; or an expected behaviour of the electronic circuit.

4. The method of any of the preceding claims, further comprising receiving a description of the electronic circuit as a hardware description language code; using the description for determining the statement descriptive of the functionality of the electronic circuit.

5. The method of any of the preceding claims, further comprising: in response to determining that the proof is not completed and failed to fulfill the functionality; adapting the electronic circuit accordingly; and repeating the method for proving the statement of the adapted circuit.

6. The method of claim 5, wherein adapting the electronic circuit comprises at least one of: changing the architecture of the components of the electronic circuit; or adapting configuration parameters that are used to perform the functionality.

7. The method of any of the preceding claims, further comprising determining whether the received proof state indicates that the proof is completed; in response to determining that the proof is not completed, using the received proof state as the current proof state and repeating the method.

8. The method of any of the preceding claims, wherein encoding the current proof state into the vector comprises: providing a recurrent neural network that is trained using proof states as sequential data in order to encode the proof state into a fixed-size vector;inputting the current proof state into the recurrent neural network for obtaining the vector.

9. The method of any of the preceding claims, wherein encoding the vector into the quantum state is performed using amplitude encoding or angle encoding.

10. The method of any of the preceding claims, being automatically performed.

11. The method of any of the preceding claims, wherein the proof assistant is configured to be executed on a classical computer, wherein the method is implemented by a hybrid quantum-classical algorithm whose execution on the classical computer causes the classical computer to perform the method comprising controlling the quantum computer to perform the encoding of the vector, the provision of the output quantum state and the measurement of the output quantum state.

12. The method of any of the preceding claims, wherein the quantum system is defined by a number of qubits which is fixed based on the number of elements of the vector.

13. The method of any of the preceding claims, wherein the size of the vector is fixed to a value dependent on available number of qubits in the quantum computer.

14. The method of any of the preceding claims, comprising before the proving of the statement, training the quantum machine learning model, the training comprising: providing a training dataset, the training dataset comprising entries, wherein each entry comprises a proof state and corresponding tactic to build a proof step; or each entry comprises a proof and corresponding statement; training the quantum machine learning model for proving statements using the training dataset.

15. The method of claim 14, wherein the training dataset is obtained using the proof assistant and / or one or more other proof assistants.

16. The method of claim 14 or 15, further comprising generating at least part of the training dataset using a machine learning model to generate true statements and proofs.

17. The method of any of the preceding claims 1 to 13, the quantum machine learning model being a quantum reinforcement learning model, wherein before the proving of the statement of the computer program, the method comprises training the quantum reinforcement learning model using the proof assistant or another proof assistant as an environment.

18. The method of any of the preceding claims 14 to 17, wherein the encoding of the current proof state into the vector is performed using a recurrent neural network, wherein the training of the quantum machine learning model comprises training the quantum machine learning model and the recurrent neural network in unison.

19. The method of any of the preceding claims, the quantum machine learning model being a quantum neural network a Quantum Support Vector Machine (QSVM), a quantum large language model or a quantum reinforcement learning model.

20. A computer system (100, 702) for proving a statement descriptive of a functionality of an electronic circuit using a quantum computer (102) and a proof assistant (107), the computer system comprising the proof assistant, the computer system (100, 702) being configured for: encoding a current proof state into a vector of real numbers of a fixed length, the current proof state defining a task for proving at least part of the statement;controlling the quantum computer for encoding the vector into a quantum state of a quantum system of the quantum computer; controlling the quantum computer for using the quantum state as an input quantum state by a quantum machine learning model for providing by the quantum machine learning model an output quantum state whose measurement represents a proof step for the defined task; controlling the quantum computer for measuring the output quantum state for obtaining the proof step for the defined task; providing the proof step to the proof assistant; in response to providing the proof step, receiving a next proof state from the proof assistant.21 . The computer system of claim 20, comprising the quantum computer.

22. A computer program product comprising instructions for causing a computer system for performing at least the following: encoding a current proof state into a vector of real numbers of a fixed length, the current proof state defining a task for proving at least part of a statement descriptive of a functionality of an electronic circuit; controlling the quantum computer for encoding the vector into a quantum state of a quantum system of the quantum computer; controlling the quantum computer for using the quantum state as an input quantum state by a quantum machine learning model for providing by the quantum machine learning model an output quantum state whose measurement represents a proof step for the defined task; controlling the quantum computer for measuring the output quantum state for obtaining the proof step for the defined task; providing the proof step to the proof assistant; in response to providing the proof step, receiving a next proof state from the proof assistant.

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