Method and system for large-scale computation of quantum algorithms using neuromorphic quantum computing

Abstract

Current quantum computing technology, while promising, faces significant challenges that limit its scalability and practical application. These challenges primarily stem from the limited number of qubits available, the high susceptibility of qubits to errors, and the complexity of error correction protocols necessary to maintain coherence in quantum states. As quantum systems grow in size, the need for sophisticated error correction becomes increasingly critical, further complicating the development of reliable quantum computers. The invention described enables efficient, large-scale computation of quantum algorithms and circuits on traditional hardware, without sacrificing the fidelity or capabilities of quantum mechanics-based systems, thus bridging the gap between classical and quantum computing paradigms.

Description

Description METHOD AND SYSTEM FOR LARGE-SCALE COMPUTATION OF QUANTUM ALGORITHMS USING NEUROMORPHIC QUANTUM COMPUTING BACKGROUND Field of the Invention

[0001] The embodiments of this invention pertain to the domain of quantum computing, specifically addressing methods, apparatus, and systems for large- scale computation of quantum algorithms and quantum circuits. These embodiments leverage an innovative neuromorphic quantum computing architecture, designed to enhance computational efficiency and scalability in simulating and executing complex quantum processes. Description of the Related Art

[0002] Quantum computing is a field of research focused on computational systems that utilise quantum mechanical phenomena to manipulate data. These phenomena, such as superposition—where a quantum state can exist in multiple configurations simultaneously—and entanglement—where quantum states remain correlated regardless of distance—have no direct analogs in classical computing and cannot be replicated using traditional devices. While Schrödinger's equations theoretically allow the simulation of quantum phenomena through partial differential equations, solving these equations on classical hardware would require exponentially increasing resources, rendering efficient simulation impractical. BRIEF DESCRIPTION OF THE DRAWINGS

[0003] FIG 1. illustrates one embodiment of a neuromorphic quantum computing system.

[0004] FIG 2. illustrates an example of the conversation of a simple quantum algorithm into Hamiltonians.

[0005] FIG 3. illustrates an example transformation of a hamiltonian into Binary Quadratic Model (BQM) representation.

[0006] FIG 4. illustrates the conversion from BQM to Weighted Max-SAT representation.Description

[0007] FIG.5 illustrates the implementation in a physical neuromorphic quantum circuit using memristors.

[0008] FIG 6. illustrates an ideal memristor.

[0009] FIG 7. illustrates the expression of the physical circuit as Ordinary Differential Equations (ODEs).

[0010] FIG 8. illustrates the process from solving the system of Ordinary Differential Equations (ODEs) and the mapping to the initial quantum algorithm. DETAILED DESCRIPTION

[0011] In the following description, numerous specific details are provided to facilitate a comprehensive understanding of the embodiments of the invention. However, it should be recognised by those skilled in the art that these embodiments can be implemented without certain specific details. In some cases, well-known structures and devices are depicted in block diagram form to avoid obscuring the fundamental principles of the invention. This approach ensures clarity while maintaining the focus on the novel aspects of the embodiments discussed. Introduction

[0012] A traditional quantum computer leverages quantum-mechanical phenomena like superposition and entanglement to perform computations. Unlike digital computers, which store data as definite states (0 or 1), quantum computation uses quantum bits (qubits), capable of existing in a superposition of states. Qubits can be implemented using distinguishable quantum states of elementary particles such as electrons or photons. For instance, a photon's polarisation can represent qubits, with vertical and horizontal polarisation as states, or an electron's spin, with "up" and "down" spins representing the two states.

[0013] Qubit states are typically represented using the bra-ket notation as |0^ and | 1^. In traditional computing, a bit is confined to one state, either ‘0’ or ‘1’. However, in quantum computing, qubits can exist in a superposition of both states simultaneously, a property that is central to the power and uniqueness of quantum computation. This ability to be in multiple states at once allows quantum computers to perform parallel computations, significantly enhancing their processing capabilities compared to classical systems.Description

[0014] Quantum computing systems execute algorithms composed of quantum logic operations performed on qubits. These operations are compiled into a predetermined schedule, with qubits addressed through an indexing scheme. The algorithm is then executed multiple times to ensure the accuracy of the results. The process continues until the confidence interval for the computed outcome exceeds a specified threshold (e.g., 95+%). Reaching this threshold indicates that the desired result of the algorithm has been reliably achieved, ensuring the correctness of the quantum computation.

[0015] Qubits have been implemented using various advanced technologies capable of manipulating and reading quantum states. These technologies include, but are not limited to, quantum dot devices (both spin-based and spatial-based), trapped-ion devices, superconducting quantum computers, optical lattices, nuclear magnetic resonance (NMR) computers, solid-state NMR Kane quantum devices, electrons-on-helium quantum computers, cavity quantum electrodynamics (CQED) devices, molecular magnet computers, and fullerene-based ESR quantum computers. Each of these approaches leverages different physical principles to create and control qubits, contributing to the diverse landscape of quantum computing technologies.

[0016] Current quantum computing technology, while promising, faces significant challenges that limit its scalability and practical application. These challenges primarily stem from the limited number of qubits available, the high susceptibility of qubits to errors, and the complexity of error correction protocols necessary to maintain coherence in quantum states. As quantum systems grow in size, the need for sophisticated error correction becomes increasingly critical, further complicating the development of reliable quantum computers.

[0017] Given these constraints, there is a growing interest in simulating quantum computers on classical hardware. One of the most promising methods involves reformulating quantum algorithms into their corresponding partial differential equations (PDEs), governed by Schrödinger's equations. Schrödinger's equations, which describe the quantum state of a system, provide a theoretical framework for simulating quantum dynamics on classical systems. However, this approach presents its own set of challenges: the computational resources required to solve these PDEs scale exponentially with the size of the quantum system being simulated.

[0018] This exponential growth in resource requirements stems from the inherent complexity of quantum systems, where the state space increases exponentially with the number of qubits. Consequently, while the theoretical formulation offers a pathway to simulate quantum algorithms, the practical limitations of classicalDescription computational resources pose significant barriers. Overcoming these barriers is crucial for advancing the field of quantum computing and for enabling more accurate and scalable simulations that can bridge the gap between current quantum capabilities and their potential applications. Method and System for Large-Scale Computation of Quantum Algorithms using Neuromorphic Quantum Computing

[0019] Over the past five decades, classical computing has greatly benefited from continuous technological advancements, including the evolution of Moore’s Law at the device and transistor level. These developments, alongside significant progress in algorithms and hardware technologies, have consistently pushed the boundaries of classical computing. The invention described enables efficient, large-scale computation of quantum algorithms and circuits on traditional hardware, without sacrificing the fidelity or capabilities of quantum mechanics- based systems, thus bridging the gap between classical and quantum computing paradigms.

[0020] One embodiment of the invention enables the reformulation of quantum algorithms into a system of ordinary differential equations (ODEs), which require computational resources that scale linearly with the size of the quantum system being simulated. This approach contrasts with the current state-of-the-art, where quantum algorithms are typically reformulated into partial differential equations (PDEs) governed by Schrödinger's equations. In such cases, the computational resources necessary to solve these PDEs scale exponentially with the system's size, presenting significant challenges for large-scale simulations.

[0021] This reformulation of quantum algorithms into ordinary differential equations (ODEs) is crucial because it significantly reduces the computational resources required for quantum simulations. Unlike the traditional approach, where resources scale exponentially, the ODE-based method scales linearly with the size of the quantum system. This improvement allows for the simulation of much larger quantum systems on classical hardware, making it more feasible to explore and develop quantum algorithms. Additionally, it enhances accessibility, enabling broader research and application in quantum computing by reducing the computational overhead.

[0022] One embodiment of the invention facilitates the efficient integration of ordinary differential equations (ODEs) derived from quantum algorithms, enabling simulated time to reach real-time execution under ideal conditions. This is achieved through an optimised numerical integration approach that leverages advanced computational techniques to minimise latency and computationalDescription overhead. By aligning simulated time closely with real-time, this embodiment allows for more accurate and practical simulations of quantum systems, making it feasible to model complex quantum behaviours in a timely manner. This real-time capability is crucial for applications requiring immediate feedback, such as quantum control systems and real-time quantum data processing.

[0023] FIG 1. illustrates one embodiment of a neuromorphic quantum computing system, allowing to perform large-scale efficient computations of quantum mechanical phenomena based algorithms, based on the following six steps: Step 1: Quantum Algorithm Conversion to Hamiltonians

[0024] In one embodiment of the invention, a quantum algorithm or quantum circuit is initially converted into its corresponding Hamiltonians. The Hamiltonians represent the energy states and interactions within the quantum system, providing a foundational mathematical framework that describes the quantum dynamics necessary for further processing. This conversion is a critical step in translating the quantum computational problem into a form that can be addressed through classical computational methods.

[0025] FIG 2. illustrates an example of the conversation of a simple quantum algorithm into Hamiltonians. The corresponding Hamiltonians for each quantum gate operation are defined as followed:

[0026] The Pauli-X gate, often referred to as the quantum equivalent of the classical NOT gate, flips the state of a qubit from ∣0^ to ∣1^ or from ∣1^ to ∣0^. It is one of the fundamental quantum gates and plays a crucial role in quantum algorithms, particularly in operations requiring bit-flip actions. Its matrix representation is:One embodiment of the invention is the corresponding Hamiltonian for the Pauli- X gate to be: HX= qi− 2qiqiWhere qirepresents the flipped state of qi.

[0027] The Pauli-Y gate is a fundamental quantum gate that represents a 180- degree rotation around the Y-axis of the Bloch sphere. It is used to manipulate theDescription phase and amplitude of qubits, playing a crucial role in quantum algorithms and operations. The Pauli-Y gate has the following matrix representation in quantum mechanics: 0−iOne embodiment of the invention is the corresponding Hamiltonian for the Pauli- Y gate to be: = −Where qirepresents the flipped state of qi.

[0028] The Controlled-NOT (CNOT) gate is one of the fundamental two-qubit gates in quantum computing. It operates on a pair of qubits, where the first qubit, known as the control qubit, determines whether the second qubit, the target qubit, undergoes a NOT operation (flipping its state from ∣0^ to ∣1^ or vice versa). Specifically, the CNOT gate flips the state of the target qubit if and only if the control qubit is in the ∣1^ state. The CNOT gate is essential for creating entanglement between qubits, a key resource for quantum algorithms. The truth table for the CNOT gate is:One embodiment of the invention is the corresponding Hamiltonian for the CNOT gate to be:

[0029] The Toffoli gate, also known as the Controlled-Controlled-NOT (CCNOT) gate, is a fundamental three-qubit quantum gate used in quantum computing. It operates on three qubits: two control qubits and one target qubit. The Toffoli gate flips the state of the target qubit if and only if both control qubits are in the ∣1^ state. This gate is significant because it is a universal gate for classical reversible computation and plays a key role in constructing more complex quantum circuits, particularly in error correction and quantum algorithms like Grover's search algorithm. The truth table for the Toffoli gate is:Description One embodiment of the invention is the corresponding Hamiltonian for the Toffoli gate to be:

[0030] The Fredkin gate, also known as the SWAP gate, is a three-qubit quantum gate that performs a conditional swap operation. It swaps the states of two qubits, referred to as the target qubits, based on the state of the third qubit, which acts as the control. Specifically, if the control qubit is in the ∣1^ state, the gate swaps the states of the two target qubits; if the control qubit is in the ∣0^ state, no swap occurs. The Fredkin gate is essential in quantum computing for implementing reversible logic and plays a key role in quantum algorithms and error correction protocols. It is also a universal gate for classical computation when combined with other quantum gates. The truth table for the Fredkin gate is:One embodiment of the invention is the corresponding Hamiltonian for the Fredkin gate to be:

[0031] The Hadamard (H) gate is a fundamental single-qubit quantum gate that creates a superposition state from a standard basis state. When applied to a qubit, it transforms the ∣0^ state into an equal superposition of ∣0^ and ∣1^, and the ∣1^Description state into an equal superposition of ∣0^ and −∣1^. Mathematically, the Hadamard gate is represented by a matrix that, when applied, rotates the qubit’s state on the Bloch sphere. This gate is crucial in quantum algorithms like Grover’s search and quantum Fourier transform, as it facilitates the creation of quantum parallelism. The mathematical representation of the Hadamard (H) gate is:One embodiment of the invention is the corresponding Hamiltonian for the Hadamard (H) gate to be:

[0032] The Controlled-Z (CZ) gate is a two-qubit quantum gate that applies a Z gate (Pauli-Z) to the target qubit only when the control qubit is in the ∣1^ state. If the control qubit is in the ∣0^ state, the target qubit remains unchanged. The CZ gate introduces a conditional phase shift, flipping the phase of the target qubit’s ∣1^ state. It plays a crucial role in quantum entanglement and is widely used in quantum algorithms and error correction, often facilitating the creation of complex quantum states. The matrix representation of the Controlled-Z (CZ) gate is:One embodiment of the invention is the corresponding Hamiltonian for the Controlled-Z (CZ) gate to be:

[0033] The Controlled-Rotation-X (CRX) gate is a two-qubit quantum gate that performs a rotation around the X-axis on the target qubit, conditioned on the control qubit being in the ∣1^ state. The rotation angle is specified by a parameter θ, allowing for precise manipulation of the target qubit's state. The CRX gate is crucial in quantum circuits for tasks such as quantum state preparation and entanglement creation. It extends the capabilities of the standard controlled gatesDescription by introducing tunable rotations, enabling more complex quantum operations.The matrix representation of the Controlled-Rotation-X (CRX) gate is: One embodiment of the invention is the corresponding Hamiltonian for the Controlled-Rotation-X (CRX) gate to be:

[0034] The Controlled-Rotation-Y (CRY) gate is a fundamental two-qubit quantum gate that performs a rotation around the Y-axis on the Bloch sphere of a target qubit, conditioned upon the state of a control qubit. Specifically, if the control qubit is in the ∣1^ state, the CRY gate applies a rotation by a specified angle θ to the target qubit; if the control qubit is in the ∣0^ state, the target qubit remains unaffected. This conditional rotation allows for precise and adjustable manipulation of quantum states, facilitating the creation of superposition and entanglement between qubits. The CRY gate is widely used in various quantum algorithms and quantum state preparation protocols, where controlled and continuous transformations of qubit states are required to perform complex computations and operations within a quantum circuit. The matrix representation of the Controlled-Rotation-Y (CRY) gate is:One embodiment of the invention is the corresponding Hamiltonian for the Controlled-Rotation-Y (CRY) gate to be:

[0035] The Controlled-Rotation-Z (CRZ) gate is a two-qubit quantum gate that applies a rotation around the Z-axis of the Bloch sphere to a target qubit, conditional on the control qubit being in the ∣1^ state. The rotation is defined by a specific angle θ, allowing for precise phase adjustments to the target qubit. The CRZ gate is particularly useful in quantum circuits for implementing controlled phase shifts, enabling more complex quantum algorithms and operations thatDescription require conditional and tunable phase transformations. The matrix representationof the The Controlled- gate One embodiment of the invention is the corresponding Hamiltonian for the Controlled-Rotation-Z (CRZ) gate to be:

[0036] The T gate, also known as the π / 8 gate, is a single-qubit quantum gate that applies a phase shift of π / 4 (or 45 degrees) to the qubit’s state. It is a crucial gate in quantum computing because it, along with the Hadamard and CNOT gates, forms a universal set of gates, meaning any quantum circuit can be constructed using these gates. The T gate is particularly important in fault-tolerant quantum computation as it enables more complex quantum operations and algorithms, including the implementation of quantum error correction. The matrix representation of the T gate is:One embodiment of the invention is the corresponding Hamiltonian for the T gate to be:

[0037] The Basis Embedding gate is a quantum operation used to map classical binary data directly onto the computational basis states of qubits in a quantum circuit. This gate allows each qubit to be set to the ∣0^ or ∣1^ state based on the corresponding bit in the input binary string. Basis Embedding is particularly useful in preparing the initial quantum state for quantum algorithms that require classical data as input, ensuring that the quantum system starts in a well-defined configuration that reflects the classical information. A general basis transformation can be represented by a unitary matrix U. For example, let’s consider the transformation to the Fourier basis. The discrete Fourier transform (DFT) matrix for a single qubit is the Hadamard matrix H:Description One embodiment of the invention is the corresponding Hamiltonian for the Basis Embedding gate to be:

[0038] The Controlled Phase Shift gate is a two-qubit quantum gate that applies a phase shift to the target qubit depending on the state of the control qubit. Specifically, if the control qubit is in the ∣1^ state, the gate introduces a phase shift of eiθto the target qubit, where θ is the specified phase angle. This gate is essential in quantum circuits for creating entanglement and manipulating quantum states in a controlled manner, making it a critical component in various quantum algorithms and protocols. The mathematical representation of the Controlled Phase Shift gate is:One embodiment of the invention is the corresponding Hamiltonian for the Controlled Phase Shift gate to be:

[0039] The Controlled Qubit Unitary (CU) gate is a two-qubit quantum gate that applies a specified unitary operation U to the target qubit, conditioned on the control qubit being in the ∣1^ state. The unitary operation U can be any single- qubit gate, such as a rotation or phase shift. This gate generalises controlled operations by allowing the application of a wide range of quantum transformations to the target qubit based on the control qubit's state, making it versatile for complex quantum algorithms and circuit designs. The mathematical representation of the Controlled Qubit Unitary (CU) gate is:One embodiment of the invention is the corresponding Hamiltonian for the Controlled Qubit Unitary (CU) gate to be:Description

[0040] The Quantum Fourier Transform (QFT) is a crucial quantum algorithm that performs a discrete Fourier transform on the amplitudes of a quantum state. It maps a quantum state into its frequency components and is the quantum counterpart of the classical discrete Fourier transform (DFT). The QFT is a fundamental building block in many quantum algorithms, such as Shor’s algorithm for integer factorization. It operates by applying a series of controlled phase shift gates and Hadamard gates, efficiently transforming quantum states to reveal periodic structures. For an n-qubit system, the QFT is defined as:One embodiment of the invention is the corresponding Hamiltonian for the Quantum Fourier Transform (QFT) to be:

[0041] The Adjoint (or Dagger) operation in quantum computing refers to taking the conjugate transpose of a quantum gate or unitary matrix. If U is a unitary matrix, its adjoint U† is defined as the complex conjugate of the transpose of U. The adjoint operation is crucial for reversing quantum operations, as it effectively "undoes" the action of a quantum gate when applied after the original gate. This is essential in many quantum algorithms, particularly in tasks like uncomputing and in the preparation of certain quantum states. One embodiment of the invention is the corresponding Hamiltonian for the adjoint T†to be:One embodiment of the invention is the corresponding Hamiltonian for the adjoint CRX(θ)† to be:Description One embodiment of the invention is the corresponding Hamiltonian for the adjoint CRY (θ)† to be:One embodiment of the invention is the corresponding Hamiltonian for the adjoint CRZ(θ)† to be:One embodiment of the invention is the corresponding Hamiltonian for the adjoint Basis Embedding Gate to be:One embodiment of the invention is the corresponding Hamiltonian for the Quantum Fourier Transform (QFT) and its Adjoint (Inverse QFT) to be:

[0042] The Grover Operator, also known as the Grover diffusion operator, is a key component in Grover's algorithm, a quantum search algorithm. This operator is used to amplify the amplitude of the correct solution(s) in an unsorted database. It consists of two main steps: an inversion about the mean and a reflection through the initial state. The Grover Operator is iteratively applied to increase the probability of measuring the correct solution, making it significantly more efficient than classical search algorithms for certain types of problems. The Grover operator (G) is defined as:Description The Oracle Penalty Function is defined as:The Diffusion Operator Penalty Function is defined as:The Full Grover Operator Penalty Function is defined as:One embodiment of the invention is the corresponding Hamiltonian for the Grover Operator to be:

[0043] Quantum Phase Estimation (QPE) is a fundamental quantum algorithm that estimates the phase (or eigenvalue) associated with an eigenvector of a unitary operator. The algorithm is crucial for various quantum applications, including factoring large integers (as used in Shor's algorithm) and finding the eigenvalues of matrices. QPE works by preparing a quantum state, applying the controlled unitary operations, and then performing an inverse Quantum Fourier Transform (QFT) to extract the phase information. This phase information is then measured, providing a highly accurate estimate of the eigenvalue. The mathematical representation of Quantum Phase Estimation (QPE) is:One embodiment of the invention is the corresponding Hamiltonian for Quantum Phase Estimation (QPE) to be:Description

[0044] A basis state in quantum computing refers to one of the standard quantum states in the computational basis, typically denoted as ∣0^ and ∣1^ for a single qubit. These states represent the fundamental building blocks from which more complex quantum states are constructed. In a multi-qubit system, the basis states are the tensor products of single-qubit basis states, forming a complete set of orthonormal states that span the quantum system's Hilbert space. These states serve as the reference points for measurements and computations in quantum circuits. For an n-qubit system, the mathematical representation of a basis state is:One embodiment of the invention is the corresponding Hamiltonian for a basis state to be:

[0045] The FlipSign operation in quantum computing refers to a technique used to invert the sign of a quantum state or specific qubit based on certain conditions. This operation is often employed within quantum algorithms to modify the amplitude of particular states, effectively "flipping" the sign of the target state’s amplitude while leaving others unchanged. This sign inversion is crucial in algorithms like Grover's search, where it helps differentiate the marked state, allowing it to be amplified through subsequent iterations of the algorithm. The mathematical representation of the FlipSign is: One embodiment of the invention is the corresponding Hamiltonian for FipSign to be:

[0046] The QubitUnitary operation in quantum computing refers to the application of a general unitary transformation to a single qubit or a system of qubits. A unitary operation is a complex matrix that preserves the norm of the quantum state, meaning it is reversible and maintains the probability distributionDescription of the quantum system. The QubitUnitary gate allows for the implementation of any arbitrary quantum operation on qubits, enabling complex quantum state manipulations necessary for executing sophisticated quantum algorithms and circuits. The mathematical representation for a QubitUnitary operation is: One embodiment of the invention is the corresponding Hamiltonian for a QubitUnitary operation to be:Step 2: Transformation of Hamiltonians into Binary Quadratic Model (BQM) Representation

[0047] In another embodiment of the invention, the Hamiltonians derived from the quantum algorithm are expressed in a Binary Quadratic Model (BQM) representation. This transformation allows the quantum problem, typically represented by the Hamiltonians, to be reformulated as a classical optimization problem. The BQM formulation is particularly advantageous because it enables the use of classical optimization techniques to address quantum computational challenges, thereby facilitating more efficient processing.

[0048] FIG 3. illustrates an example transformation of a hamiltonian into Binary Quadratic Model (BQM) Form. The process begins by representing quantum states as binary variables. In a typical quantum system, each qubit can be in a state ∣0^ or ∣1^, which are mapped to binary variables, xi∈ {0,1}. This allows the quantum states described by the Hamiltonian to be represented in a form suitable for classical binary computation.

[0049] The Hamiltonian is then decomposed into terms representing interactions between pairs of qubits. These interactions are represented as quadratic terms in the binary variables, resulting in a quadratic function. The quadratic terms capture the pairwise interactions, while linear terms represent the contributions of individual qubits to the system's energy.

[0050] The Pauli-Z operator σz, which has eigenvalues ±1, is typically mapped to binary variables using a transformation such as σz= 1 - 2x, where x is a binary variable. This transformation enables the Hamiltonian’s terms to be expressed asDescription a quadratic function in the binary variables. For example, a term like σizσizis mapped to a quadratic expression involving xiand xj.

[0051] The Binary Quadratic Model (BQM) consists of both linear and quadratic terms. The linear terms correspond to the contributions from individual qubits (or binary variables), while the quadratic terms represent interactions between pairs of qubits. The Hamiltonian’s energy function is rewritten as a sum of these linear and quadratic terms, preserving the interaction structure of the original quantum system.

[0052] The complete Hamiltonian is then expressed as a Binary Quadratic Model, where the objective is to find the binary variable configuration that minimises the quadratic function. This involves summing all the linear and quadratic terms derived from the Hamiltonian, resulting in a model that captures the essential physics of the quantum system in a classical binary framework. Step 3: Conversion to Weighted Max-SAT Representation

[0053] In a further embodiment, the BQM problem is subsequently converted into a weighted maximum satisfiability (Max-SAT) representation. This step involves translating the optimization problem into a Boolean satisfiability format, wherein the goal is to satisfy the maximum number of clauses weighted by their importance. The conversion to a weighted Max-SAT problem is essential for leveraging classical logic-based methods to solve what was originally a quantum optimization problem.

[0054] FIG 4. illustrates the conversion from BQM to Weighted Max-SAT Representation. The linear terms of the BQM are initially scaled and adjusted using a precision factor to ensure numerical stability. Each linear term viis rounded and then corrected for its sign. Then the linear terms are being reduced aggregating opposite-sign variables (e.g., xiand −xi) and calculating their net effect. If the net effect is non-zero, the term is retained and added to the clause list with its corresponding weight, specifically, if the resulting weight Wi> 0, the mapping is (Wi, Xi), otherwise (-Wi, -Xi).

[0055] The quadratic terms of the BQM are handled by examining pairwise interactions and mapping them into clauses. This function ensures that interactions are encoded such that they respect the Rydberg energy level constraints, which involve interactions between qubits at specific distances. If the resulting weight Wi> 0, the mapping is (Wi, -Xi, -Xj), otherwise (Wi, Xi, Xj),Description (Wi, -Xi, Xj) and (Wi, Xi, -Xj). The result is a set of clauses that efficiently represents the BQM in a format suitable for large-scale quantum-inspired computations. Step 4: Implementation in a Physical Circuit Using Memristors

[0056] In one embodiment of the invention, the weighted Max-SAT representation is implemented in a physical neuromorphic quantum circuit composed of logic gates and memristors. Memristors, which exhibit a memory function due to their hysteresis behaviour, closely mimic the synaptic functions of the human brain. This characteristic positions them as ideal components in what is termed "neuromorphic quantum computing." By leveraging memristors, each gate in the circuit can emulate quantum superposition through multiple simultaneous states, with controllers ensuring an analogy to quantum entanglement. This design enables the entire circuit to function as an interconnected network, where all variables influence each other simultaneously, akin to the processing architecture of entanglement in quantum mechanics.

[0057] FIG.5 illustrates the implementation in a physical neuromorphic quantum circuit using memristors. We utilise analog logic gates that operate with voltages at both the input and output terminals, rather than traditional binary logic gates. These gates are connected to ideal memristors, which allow for the superposition of voltages and currents at the terminals. This means that unlike binary systems where a terminal is either high or low, our analog gates can process a continuous range of values, reflecting the complex quantum-like behaviours that are critical in advanced computational models. The ability to maintain superposition in the circuit enhances the system's capability to handle multiple states simultaneously, making it highly efficient for complex problem-solving tasks.

[0058] FIG 6. illustrates an ideal memristor. An ideal memristor, short for memory resistor, is a fundamental circuit element that not only controls the flow of current but also retains a memory of the amount of charge that has passed through it. This memory is manifested through a property known as hysteresis, where the resistance of the memristor depends on the history of voltage and current applied to it. This pinched hysteresis feature allows the memristor to "memorise" past electrical states, making it invaluable for analog computing applications where the previous states influence current operations. The memristor's capability to remember and adjust its resistance based on past inputs enables the implementation of highly efficient, stateful logic operations in circuits designed for advanced computational tasks, such as quantum-inspired algorithms.Description

[0059] The circuit exhibits a related phenomenon through long-distance connections stemming from a state known as criticality. In this setup, each gate within the circuit is designed to be responsive to distant gates, creating a network where each gate can influence and be influenced by others across the system. As the circuit evolves, the connections between different gates reach a critical state, making the system prone to sudden, large-scale changes or "avalanches" triggered by minor disturbances anywhere in the circuit. These avalanches, driven by the circuit's long-range correlations, enable the system to quickly find the lowest energy state, significantly speeding up computation compared to traditional methods.

[0060] The circuit utilises the behaviour of electrical currents and voltages to expedite the transition to more favourable states. The concept of an instanton, borrowed from the study of dynamical systems, explains how these circuits navigate through energy barriers. This happens as the system encounters "saddle" points within the electrical landscape, which possess characteristics that attract the system initially but then repel it as it gets closer. This interaction results in the system being propelled away from these points at high speed, effectively allowing it to jump from one state to another across the landscape.

[0061] This phenomenon mirrors quantum tunnelling but occurs in the realm of voltages and currents. In neuromorphic quantum computing circuits, this behaviour emerges from the collective action of memristor based gates, which induce widespread fluctuations in voltages, thereby swiftly steering the circuit towards superior configurations. These inherent physical mechanisms enable neuromorphic quantum computing circuits to navigate towards the best possible solution out of a vast array of potential configurations, by effectively mapping the solution into their system.

[0062] Each clause in the weighted Max-2-SAT problem are represented by an analog OR gate, as it needs at least one of its literals to be true. For instance, a clause like (x1∨ ¬x2) can be expressed as an analog OR gate with inputs x1and ¬x2.

[0063] The negation of a variable ¬x2is handled by an analog NOT gate, ensuring that if x2is true, ¬x2will be false, and vice versa.

[0064] The weights in a weighted Max-2-SAT problem are integrated by associating a voltage equally to the weighted sum at the output of each analog OR gate. The gate outputs are multiplied by their respective weights, and the goal is to maximise the total weighted sum.Description Step 5: Expression of the Physical Circuit as Ordinary Differential Equations (ODEs)

[0065] In a further embodiment, the physical neuromorphic quantum circuit realised through logic gates and memristors is mathematically expressed using ordinary differential equations (ODEs). This mathematical formulation replaces the quantum mechanical principles with standard physical components, including memristors, allowing the dynamic behaviour of the circuit to be fully described by ODEs. The ODE representation enables the application of classical numerical methods to simulate the behaviour of the quantum system.

[0066] FIG 7. Illustrates the expression of the physical circuit as Ordinary Differential Equations (ODEs). Linear quantum gates in the neuromorphic quantum circuit are expressed as Ordinary Differential Equations with their following equation of motion: ^With vnexpressing the qubit’s voltage, whereas: QGm = ^ 2.0 Defining the state of each quantum gate, Wmdefining each quantum gate’s weight and the quantum gate polarity is defined as: 1.0 i > 0i<^= 0

[0067] Quadratic quantum gates in the neuromorphic quantum circuit are expressed as Ordinary Differential Equations with their following equation of motion: ^m With vnexpressing the qubit’s voltage, whereas Wmdefines the gate of each quantum gate,1.0 i, j > 0i, j < = 0^ is defining the polarity of the quantum gate, andDescription min((1.0 − pivi), (1.0 − pjvj)) ^is defining the quantum gate state. The mathematical model of the ideal memristors used are using the gradient terms − ^and − ^as well as the rigidity terms ^ ij^ jby defining the auxiliary variables amand bmas^ and bm= α(1.0 +W )(QG − δ ) ^

[0068] Together, they ensure the exact behaviour of the memristor based neuromorphic quantum circuits when represented in their Ordinary DifferentialDescription Equations. Note that optimal values for the applied parameters α, β, γ, δ, ^, ε, ζ may be different for each computational problem class. Step 6: Efficient Solution of ODEs and Reconstruction of Quantum States

[0069] In yet another embodiment, the ordinary differential equations (ODEs) derived from the physical circuit are solved using efficient numerical integration techniques. The solutions to these ODEs provide the necessary data to reconstruct the quantum states of the original system. This final step completes the simulation process, allowing the quantum algorithm to be executed on classical hardware while preserving the integrity and accuracy of the quantum computation.

[0070] FIG 8. illustrates the process from solving the system of Ordinary Differential Equations (ODEs) and the mapping to the initial quantum algorithm. The initial conditions of the variables in the ODEs are being randomly set, they define the starting point for the system. These conditions provide the values of the variables at the initial time, from which the solution will evolve.

[0071] Numerical methods are being used to solve the ODEs. These include Euler’s method, Runge-Kutta methods, and multistep methods. These techniques iteratively approximate the values of the variables at discrete time steps, using the ODEs to calculate how the variables change over each step.

[0072] The core of solving the ODEs numerically involves integration over time. The numerical method approximates the integral of the derivatives to update the variables' values. The choice of time step is critical; smaller steps increase accuracy but require more computation, while larger steps are faster but less accurate. Adaptive step size control are employed to manage potential errors, ensuring that the solution remains stable and accurate. Stability refers to the solution's behaviour over time periods, where it should remain bounded and behave consistently with the expected physical system.

[0073] Once the system has been integrated over the desired time interval, the results are interpreted: This involves the mapping of the resulting voltages of the variables back to the original quantum algorithm’s quantum states to be measured.

[0074] Though the invention has been described in connection with certain preferred embodiments depicted in the various figures, it should be understood that other similar embodiments may be used, and that modifications or additions may be made to the described embodiments for practicing the invention withoutDescription deviating therefrom. The invention, there fore, should not be limited to any single embodiment, but rather should be construed in breadth and scope in accordance with the following claims.

Claims

Claims [1] A neuromorphic quantum computing system, compromising: A quantum algorithm or quantum circuit is initially converted into its corresponding Hamiltonians. The Hamiltonians represent the energy states and interactions within the quantum system, providing a foundational mathematical framework that describes the quantum dynamics necessary for further processing. The Hamiltonians derived from the quantum algorithm are expressed in a Binary Quadratic Model (BQM) representation. This transformation allows the quantum problem, typically represented by the Hamiltonians, to be reformulated as a classical optimization problem. The BQM problem is subsequently converted into its weighted maximum satisfiability (Max-SAT) representation. This step involves translating the optimization problem into a Boolean satisfiability format, wherein the goal is to satisfy the maximum number of clauses weighted by their importance. The conversion to a weighted Max-SAT problem is essential for leveraging classical logic-based methods to solve what was originally a quantum optimization problem. The weighted Max-SAT representation is implemented in a physical neuromorphic quantum circuit composed of logic gates and memristors. By leveraging memristors, each gate in the circuit can emulate quantum superposition through multiple simultaneous states, with controllers ensuring an analogy to quantum entanglement. The physical neuromorphic quantum circuit realised through logic gates and memristors is mathematically expressed using ordinary differential equations (ODEs). This mathematical formulation replaces the quantum mechanical principles with standard physical components, including memristors, allowing the dynamic behaviour of the circuit to be fully described by ODEs. The ODE representation enables the application of classical numerical methods to simulate the behaviour of the quantum system. The ordinary differential equations (ODEs) derived from the physical circuit are solved using efficient numerical integration techniques. The solutions to these ODEs provide the necessary data to reconstruct the quantum states of the original system. This final step completes the simulation process, allowing the quantum algorithm to be executed on classical hardware while preserving the integrity and accuracy of the quantum computation. [2] A quantum algorithm or quantum circuit of claim 1, created with a quantum computing language, a quantum computing simulator or any framework supporting the definition of quantum algorithms or quantum circuits. [3] A conversion of a quantum algorithm or quantum circuit into a Binary Quadratic Model (BQM) of claim 1 instead of Hamiltonians. [4] A conversion of a quantum algorithm or quantum circuit into a Quadratic Unconstrained Binary Optimisation (QUBO) model of claim 1 instead of Hamiltonians. [5] A conversion of a quantum algorithm or quantum circuit into any other constrained or unconstrained optimization model of claim 1 instead of Hamiltonians. Page 1 of 3Claims [6] Expression of the physical neuromorphic quantum circuit into its corresponding Ordinary Differential Equations (ODEs) of claim 1 using alternative ODE formulations to mapping the solution space to the system. [7] The numerical integration of the system of Ordinary Differential Equations (ODEs) of claim 1 by using Euler’s Method. [8] The numerical integration of the system of Ordinary Differential Equations (ODEs) of claim 1 by using Heun’s Method. [9] The numerical integration of the system of Ordinary Differential Equations (ODEs) of claim 1 by using Runge-Kutta’s Method. [10] The numerical integration of the system of Ordinary Differential Equations (ODEs) of claim 1 by using Adams-Bashforth’s Method. [11] The numerical integration of the system of Ordinary Differential Equations (ODEs) of claim 1 by using Adams-Moulton’s Method. [12] The numerical integration of the system of Ordinary Differential Equations (ODEs) of claim 1 by using a Backward Differentiation Formula. [13] The numerical integration of the system of Ordinary Differential Equations (ODEs) of claim 1 by using a Leapfrog Method. [14] The numerical integration of the system of Ordinary Differential Equations (ODEs) of claim 1 by using Verlet Integration. [15] The numerical integration of the system of Ordinary Differential Equations (ODEs) of claim 1 by using Multistep Methods. [16] The numerical integration of the system of Ordinary Differential Equations (ODEs) of claim 1 by using Predictor-Corrector Methods. [17] The numerical integration of the system of Ordinary Differential Equations (ODEs) of claim 1 by using Finite Difference Methods. [18] The numerical integration of the system of Ordinary Differential Equations (ODEs) of claim 1 by using Gear’s Method. [19] The numerical integration of the system of Ordinary Differential Equations (ODEs) of claim 1 by using Implicit Runge-Kutta Methods (e.g., Gauss-Legendre). [20] The numerical integration of the system of Ordinary Differential Equations (ODEs) of claim 1 by using Symplectic Integrators. [21] The numerical integration of the system of Ordinary Differential Equations (ODEs) of claim 1 by using Monte Carlo Methods for Stochastic ODEs. Page 2 of 3Claims [22] A system or apparatus compromised of a physical implementation of the neuromorphic quantum circuits of claim 1 using standard electronic components, which also may include physical realisations of the ideal memristor circuit elements as well as digital, analog or optical components. [22] A system or apparatus compromised of a physical implementation of a configurable system with the same functionality as the Neuromorphic Quantum Circuit of claim 1 using standard electronic components, which also may include physical realisations of the ideal memristor circuit elements as well as digital, analog or optical components. [23] A system or apparatus compromised of a physical implementation representing and solving the the system of Ordinary Differential Equations (ODEs) of claim 1 using standard electronic components. These may include digital, analog or optical components. Page 3 of 3

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