--- ## Parametric Adiabatic Coherent Optimizer (PACO) **A Critical Review and Technical Analysis of a Novel Computing Paradigm for Combinatorial Optimization** **Version:** 1.0 **Date**: August 19, 2025 [Rowan Brad Quni](mailto:[email protected]), [QNFO](https://qnfo.org/) ORCID: [0009-0002-4317-5604](https://orcid.org/0009-0002-4317-5604) DOI: [10.5281/zenodo.16899179](http://doi.org/10.5281/zenodo.16899179) *Related Works:* - *Quantum Resonance Computing (QRC): The Path Forward for Quantum Computing (DOI: [10.5281/zenodo.16732364](http://doi.org/10.5281/zenodo.16732364))* - *Harmonic Resonance Computing: Harnessing the Fundamental Frequencies of Reality for a Novel Computational Paradigm* (*DOI: [10.5281/zenodo.15833815](http://doi.org/10.5281/zenodo.15833815))* --- **Abstract:** The Parametric Adiabatic Coherent Optimizer (PACO) is a novel computing paradigm addressing computationally intractable combinatorial optimization problems, particularly NP-hard instances formulated as Quadratic Unconstrained Binary Optimization (QUBO) problems. Operating as a hybrid quantum-classical architecture, PACO leverages the collective self-organization and phase-locking of an array of coupled, parametrically driven nonlinear oscillators. Through adiabatic pumping, these oscillators are guided to a global minimum energy configuration, which directly corresponds to the problem's solution. PACO's versatility is demonstrated by its diverse physical implementations, including superconducting, spintronic, photonic, NEMS, magnonic, and exciton-polariton systems. This approach offers a robust alternative to classical heuristics and addresses key limitations of current quantum computing, positioning PACO as a promising technology for next-generation high-performance computing. --- ### 1. The Computational Frontier: Intractability in Combinatorial Optimization The relentless advance of science and industry is increasingly defined by the ability to solve problems of immense complexity. Central to this challenge is the field of combinatorial optimization, which seeks to find the best possible solution from a finite, but often astronomically large, set of discrete alternatives. These problems are not abstract mathematical curiosities; they are the computational bedrock of modern society, dictating the efficiency of global supply chains, the profitability of financial markets, the design of life-saving drugs, and the architecture of communication networks. However, a great number of these vital problems share a daunting characteristic: they are computationally intractable for conventional computers. #### 1.1. The Ubiquity and Complexity of NP-Hard Problems Many of the most important combinatorial optimization problems belong to a complexity class known as NP-hard. This classification signifies that for all known algorithms, the computational resources—specifically time—required to find an exact, guaranteed optimal solution grow superpolynomially, and often exponentially, with the size of the problem instance. For a problem with $N$ variables, the number of possible solutions can scale as $2^N$ or $N!$, a combinatorial explosion that quickly overwhelms even the most powerful supercomputers. This inherent difficulty means that finding an exact solution for large, real-world instances is not merely a matter of building faster processors; it is a fundamental barrier imposed by the nature of the problem itself. The practical implications of this intractability are profound. For example: - **Logistics and Transportation:** The Traveling Salesman Problem, which seeks the shortest possible route to visit a set of cities, is a canonical NP-hard problem. Logistics companies face this challenge daily when optimizing delivery routes, and even a small improvement in efficiency can translate into millions of dollars in savings. - **Finance:** Portfolio optimization, which aims to select a basket of assets to maximize returns for a given level of risk, can be formulated as a variant of the knapsack problem, another NP-hard challenge. Financial institutions constantly seek better solutions to manage trillions of dollars in assets. - **Manufacturing and Scheduling:** Optimizing production schedules in a factory to minimize processing time or resource usage is a complex scheduling problem that directly impacts industrial output and efficiency. - **Network Design:** Designing communication or energy networks to maximize capacity and resilience while minimizing cost is a critical infrastructure challenge that falls into this category of hard optimization problems. The sheer economic and scientific value locked within these problems has driven a decades-long search for more powerful computational paradigms capable of taming their complexity. #### 1.2. A Critical Assessment of Classical Heuristics and Exact Solvers In the face of NP-hardness, practitioners have developed a sophisticated toolkit of classical algorithms. These approaches can be broadly divided into exact solvers and heuristic methods. Exact solvers, such as branch-and-bound or dynamic programming, are guaranteed to find the global optimum but suffer from the exponential scaling that makes them infeasible for large problems. Consequently, much of the focus has been on heuristic and meta-heuristic algorithms. These methods, which include widely used techniques like simulated annealing, genetic algorithms, and ant colony optimization, abandon the guarantee of optimality in favor of finding high-quality, “good-enough” solutions in a practical amount of time. Simulated annealing, for instance, mimics the metallurgical process of slowly cooling a metal to reach a low-energy crystalline state. It introduces thermal-like fluctuations to allow the search process to escape from local energy minima and explore a wider portion of the solution space. However, these classical approaches have significant limitations. A primary failing of many techniques, particularly those based on gradient descent, is their propensity to become trapped in locally optimal solutions. The “energy landscape” of a complex optimization problem is often rugged, featuring countless valleys, with only one representing the true global minimum. An algorithm that only ever moves “downhill” will inevitably get stuck in the first valley it finds, which is unlikely to be the deepest one. While meta-heuristics like simulated annealing are designed to overcome this, their performance is highly sensitive to the problem’s structure and the careful tuning of algorithmic parameters, such as the “cooling schedule,” which can be a difficult art. Furthermore, these limitations persist even when classical optimizers are used as components within larger hybrid systems. In variational quantum algorithms like the Quantum Approximate Optimization Algorithm (QAOA), a classical optimizer is used to tune the parameters of a quantum circuit. The difficulty of navigating the noisy and often barren optimization landscapes generated by these quantum circuits can become the primary bottleneck for the entire hybrid algorithm, limiting its effectiveness. #### 1.3. The Quantum Approach: Promise and Practical Hurdles Quantum computing offers a fundamentally different approach to computation, leveraging principles like superposition and entanglement to explore vast computational spaces. For optimization, the most relevant paradigms have been Adiabatic Quantum Computation (AQC) and its practical, though less idealized, counterpart, Quantum Annealing (QA). ##### 1.3.1. Adiabatic Quantum Computation and Quantum Annealing AQC and QA are designed to find the ground state (lowest energy configuration) of a physical system, a task that is naturally analogous to solving an optimization problem. The process involves encoding the objective function of the optimization problem into a final “problem” Hamiltonian, $H_P$, whose ground state corresponds to the optimal solution. The system of quantum bits (qubits) is initialized in the simple and easy-to-prepare ground state of an initial Hamiltonian, $H_0$. The total Hamiltonian of the system is then slowly evolved over time, for instance as $H(t)=(1-t/T)H_0+(t/T)H_P$, from $H_0$ to $H_P$. According to the adiabatic theorem, if this evolution is performed slowly enough, the system will remain in its instantaneous ground state throughout the process, ultimately arriving at the ground state of $H_P$ and thus solving the problem. Quantum tunneling, a uniquely quantum phenomenon, allows the system to pass through energy barriers that would trap a classical system, offering a potential pathway to avoid local minima. Despite this elegant theoretical promise, practical implementations of AQC and QA face severe physical hurdles that limit their performance and scalability: - **The Energy Gap Problem:** The “slowness” required by the adiabatic theorem is not arbitrary; it is dictated by the minimum energy gap, $\Delta_{min}$, between the ground state and the first excited state of the system’s Hamiltonian during the evolution. The required evolution time, $T$, scales as $T \propto 1/\Delta_{min}^2$. For many NP-hard problems, it is known or suspected that this minimum gap closes exponentially with the problem size ($N$). This implies that the required computation time would also grow exponentially, thereby erasing any potential quantum advantage over classical algorithms. This is arguably the most fundamental obstacle to the power of adiabatic quantum optimization. - **Coherence and Noise:** The theoretical model of AQC assumes a perfectly closed quantum system evolving at zero temperature. Real-world quantum annealers, however, are open systems that are inevitably coupled to their environment. This coupling leads to two detrimental effects: decoherence, which destroys the delicate quantum superpositions needed for the computation, and thermal excitations, which can provide enough energy to knock the system out of its ground state and into a higher-energy, suboptimal state. Maintaining quantum coherence for the long evolution times required by small energy gaps is an immense experimental challenge. - **Mapping Overhead:** Physical quantum annealing hardware, such as the devices built by D-Wave Systems, has a fixed and often sparse connectivity graph between its qubits. Most real-world optimization problems, however, have complex and dense interaction graphs. To run such a problem on the hardware, a process called “minor-embedding” is required, where a single logical variable from the problem is represented by a chain of multiple, strongly coupled physical qubits. This embedding process consumes a significant number of physical qubits, reducing the effective size of the problem that can be solved, and introduces additional control challenges, creating a substantial performance overhead. ##### 1.3.2. Universal Gate-Model Quantum Computers An alternative quantum paradigm is the universal gate-model quantum computer, which is analogous to a classical digital computer. It operates by applying a sequence of discrete quantum logic gates to a register of qubits to perform arbitrary computations. While theoretically capable of universal computation and running algorithms like Shor’s algorithm for factoring, these machines are exceptionally fragile. Their profound sensitivity to environmental noise and operational errors means that any non-trivial computation requires extensive quantum error correction. The resource overhead for this error correction is enormous, requiring thousands or even millions of physical qubits to create a single, stable “logical” qubit. Consequently, today’s Noisy Intermediate-Scale Quantum (NISQ) devices have far too few qubits and are far too noisy to tackle large-scale optimization problems that are classically intractable. This landscape reveals a clear progression of computational challenges. The intrinsic mathematical difficulty of NP-hard problems led to the failure of classical algorithms, which are often defeated by complex energy landscapes and exponential scaling. This, in turn, motivated the development of quantum annealers, which use quantum tunneling to overcome these landscapes. However, the physical realities of implementing quantum systems introduced a new set of formidable obstacles, namely the energy gap problem and the destructive effects of environmental noise. The PACO paradigm can be understood as a direct and logical response to this second layer of challenges. It seeks a computational mechanism that is physically robust and powerful enough to navigate complex optimization problems but does not rely on the most fragile aspects of quantum mechanics, thereby aiming to operate effectively in the pragmatic gap between purely classical and fully fault-tolerant quantum computation. ### 2. The PACO Paradigm: Theoretical Foundations and Architecture The Parametric Adiabatic Coherent Optimizer (PACO) is a novel computing paradigm that synthesizes concepts from early digital computing, modern nonlinear optics, and quantum adiabatic theory into a cohesive and powerful framework for optimization. It is not a purely digital or purely quantum machine but a hybrid architecture that leverages the natural, collective dynamics of a physical system to find the ground state of a computational problem. #### 2.1. Conceptual Origins: From the Classical Parametron to Coherent Ising Machines The intellectual lineage of PACO can be traced back to the mid-20th century and the invention of the parametron by Eiichi Goto in 1954. The parametron was a foundational logic element used in early Japanese computers. Its operation was based on the principle of parametric excitation. A simple resonant circuit, composed of an inductor and a capacitor (an LC circuit), is “pumped” by modulating one of its parameters (e.g., the inductance) at a frequency that is approximately twice its natural resonant frequency, $2f$. This parametric drive excites an oscillation in the circuit at the resonant frequency, $f$. Crucially, this subharmonic oscillation is bistable; it can settle into one of two stable phase states, separated by $\pi$ radians (180 degrees). These two distinct phases provide a robust physical representation of a binary digit, 0 or 1. The parametron demonstrated the power of using the phase of a nonlinear oscillator as a reliable information carrier. This fundamental concept has been revitalized and advanced in recent years with the development of Coherent Ising Machines (CIMs). A CIM is a physical system designed to find the ground state of an Ising Hamiltonian, a mathematical model that is computationally equivalent to the QUBO format. Modern CIMs are often implemented using a network of coupled degenerate optical parametric oscillators (DOPOs). Each DOPO, when pumped above its oscillation threshold, exhibits the same bistable phase behavior as the original parametron, allowing it to represent an Ising spin. By introducing programmable couplings between these DOPOs, the entire network can be made to physically emulate an Ising system. The key insight is that the collective state of the coupled oscillators will naturally evolve towards a configuration that minimizes a global loss function. By designing the system such that this loss function is mathematically equivalent to the energy of the target Ising Hamiltonian, the CIM physically finds the optimal solution simply by settling into its most stable mode of operation. The PACO paradigm generalizes this principle, applying a specific operational methodology—adiabatic parametric pumping—to a broad class of physical oscillator systems beyond just optics. #### 2.2. Core Principle: Collective Self-Organization via Adiabatic Parametric Pumping The computational heart of the PACO system is a carefully orchestrated physical process that guides an array of coupled oscillators to the global minimum of a complex energy landscape. This process unfolds in three key stages: parametric pumping, bifurcation, and adiabatic evolution. - **Parametric Pumping:** The process begins with the oscillator array in a quiescent, non-oscillating state. A global AC excitation signal, the “pump,” is then applied uniformly to every oscillator in the array. This pump signal has a frequency, $\omega_p$, that is approximately twice the natural resonant frequency, $\omega_0$, of the individual oscillators ($\omega_p \approx 2\omega_0$). - **Bifurcation:** Initially, the pump signal’s amplitude is zero or very low. In this state, the potential energy landscape of each oscillator has a single stable minimum, corresponding to a state of no oscillation. As the amplitude of the pump is slowly and continuously increased, it reaches a critical threshold. At this point, the system undergoes a bifurcation: the potential energy landscape of each oscillator transforms from a single-well potential into a double-well potential. This transformation is a fundamental feature of parametrically driven nonlinear systems, including Kerr-nonlinear parametric oscillators (KPOs), which serve as a theoretical model for this behavior. The emergence of the double-well potential forces each oscillator to make a “choice” and settle into one of the two new stable minima, which correspond to the two distinct phase states (0 or $\pi$). - **Adiabatic Evolution:** The rate at which the pump amplitude is increased is the most critical parameter of the entire process. The ramp-up is performed according to an “adiabatic schedule,” meaning it is done slowly enough for the entire coupled system to continuously adapt and remain in its instantaneous ground state as the collective energy landscape evolves. Because the oscillators are coupled, they do not choose their phases independently. Instead, the entire array engages in a process of coherent, collective self-organization. The interactions between oscillators guide this process, and the adiabatic nature of the evolution ensures that the system is gently steered towards the phase configuration that corresponds to the global minimum of the total system energy, effectively avoiding the numerous local minima that would trap a conventional optimization algorithm. This operational principle reveals that the “slowness” of the adiabatic ramp is not merely a constraint but a powerful, tunable hyperparameter. It governs the fundamental trade-off between the speed of computation and the probability of finding the true optimal solution. A rapid ramp might yield a result quickly but increases the risk of non-adiabatic transitions, where the system is “excited” out of the ground state into a suboptimal local minimum. A very slow ramp increases the likelihood of finding the global minimum but extends the computation time. This suggests a more sophisticated control strategy is possible. Rather than a simple linear ramp, the classical host computer could employ techniques from optimal control theory to design a problem-specific, nonlinear ramp profile. Such a “shortcut-to-adiabaticity” protocol would allow the system to evolve rapidly when the energy gap is large and only slow down near the critical points where the gap is small, potentially achieving a dramatic improvement in the overall time-to-solution without sacrificing accuracy. This transforms the PACO from a static solver into an adaptive, intelligent computational system. #### 2.3. The Hybrid Quantum-Classical Architecture: Orchestration and Execution The PACO system is not a standalone analog device but a tightly integrated hybrid quantum-classical architecture. This approach, where powerful classical computers are used to control and interpret the results from a specialized quantum or analog co-processor, has become the dominant paradigm for near-term quantum applications, leveraging the strengths of both worlds. The classical computer provides the programmability, control, and logic, while the co-processor provides the raw computational power for a specific task. ##### 2.3.1. The Role of the Classical Host Computer (CHC) The CHC acts as the master controller and orchestrator of the entire optimization process. Its responsibilities are multifaceted and critical for the system’s operation: - **Problem Formulation and Compilation:** The CHC receives a high-level description of a combinatorial optimization problem. Its first task is to translate this problem into the standardized mathematical form of a QUBO. This step is analogous to a software compiler translating high-level source code into a standardized intermediate representation. - **Parameter Mapping:** Once the problem is in QUBO format, the CHC extracts the matrix of coefficients ($h_i$ and $Q_{ij}$). It then translates these numerical values into a set of specific, physical control signals—such as voltages, currents, magnetic fields, or laser intensities—that are required to program the analog co-processor. This is the equivalent of compiling the intermediate representation into the specific machine code for the target hardware. - **Operational Control:** The CHC manages the entire time-dependent operational sequence. It initiates the system, transmits the programming signals to the co-processor, and, most importantly, precisely controls the timing and shape of the adiabatic pump ramp that drives the computation. - **Result Interpretation:** After the co-processor has settled into its final state, the CHC receives the raw analog measurement data (e.g., the measured phases of the oscillators). It digitizes this data and converts the physical states back into a classical binary string, which represents the solution to the QUBO problem. Finally, it translates this binary string back into the context of the original high-level problem. - **Problem Decomposition:** For problems that are too large to fit on the physical co-processor at once, the CHC can employ decomposition strategies. It can break the large problem into a series of smaller, interconnected subproblems, send each one to the co-processor for solution, and then classically stitch the partial solutions together to form a solution for the original large problem. This workflow reveals the sophistication of the PACO architecture. The CHC functions as a “generalized physical compiler.” It abstracts the complex physics of the co-processor, allowing a user to interact with the system at a high level of mathematical description. The modularity of this approach is a profound strategic advantage: the same CHC software and QUBO formulation could, in principle, be used to target vastly different physical co-processors—be they superconducting, photonic, or spintronic—much as the same C++ code can be compiled to run on different classical processor architectures. ##### 2.3.2. The PACO Co-processor (PCP) The PCP is the dedicated hardware accelerator, the physical embodiment of the optimization problem. While the CHC provides the intelligence and control, the PCP provides the raw, physics-based computational power. Its functions are to: - Receive the analog “machine code” from the CHC to configure its internal state. - Physically realize the problem’s energy landscape through its network of coupled oscillators. - Evolve its state according to the laws of physics under the influence of the global parametric pump, naturally relaxing into its global minimum energy configuration. - Allow its final state to be measured and read out by the CHC. The tight integration of the CHC and PCP creates a symbiotic system where the limitations of one are offset by the strengths of the other, forming a complete and powerful computational platform. #### 2.4. Mapping and Information Encoding: Translating QUBO into Physical Dynamics The bridge between the abstract mathematical problem and the physical hardware is the QUBO formulation. The QUBO model seeks to minimize an objective function of the form $E(x)=\sum_i h_i x_i + \sum_{i<j} Q_{ij} x_i x_j$, where $x$ is a vector of binary variables $x_i \in \{0,1\}$. This formulation is remarkably versatile and has been shown to be a universal representation for a wide range of NP-hard problems. The PACO system leverages a direct and intuitive mapping between the QUBO parameters and the physical parameters of the oscillator array: - **Binary Variables ($x_i$):** Each binary variable $x_i$ in the problem is assigned to a unique physical oscillator unit in the array. The two possible values of the variable, 0 and 1, are encoded in the two stable phase states of the oscillator’s subharmonic oscillation, 0 and $\pi$. - **Linear Coefficients ($h_i$):** The linear, or “bias,” terms of the QUBO correspond to local fields applied to individual oscillators. A non-zero $h_i$ asymmetrically perturbs the oscillator’s double-well potential, making one phase state energetically more favorable than the other. This acts as a “nudge,” encouraging the corresponding variable to take on a specific value in the final solution. - **Quadratic Coefficients ($Q_{ij}$):** The quadratic, or “coupling,” terms represent the interactions between pairs of variables. These map directly to the physical coupling strength and sign between the corresponding pair of oscillators. A positive coupling ($Q_{ij}>0$) implements a “ferromagnetic” interaction, which energetically favors the two connected oscillators settling into the same phase. A negative coupling ($Q_{ij}<0$) implements an “antiferromagnetic” interaction, favoring opposite phases. The magnitude of $Q_{ij}$ determines the strength of this energetic preference. Through this mapping, the abstract QUBO objective function is physically instantiated as the total potential energy of the coupled oscillator network. The problem of finding the binary vector $x$ that minimizes $E(x)$ becomes entirely equivalent to the physical process of the oscillator array settling into its lowest-energy phase configuration. ### 3. A Deep Dive into Physical Implementations The PACO paradigm is not tied to a single technology but represents a unifying computational framework that can be realized across a diverse spectrum of physical systems. This versatility is one of its greatest strengths, allowing for implementations that can be optimized for different environments, performance requirements, and levels of technological maturity. Each implementation leverages a distinct physical phenomenon to create the core bistable oscillator and the necessary couplings, spanning a range from well-understood classical dynamics to the frontiers of macroscopic quantum phenomena. #### 3.1. Superconducting Circuit Implementation (SCI) This implementation leverages the mature technology of superconducting electronics, which operate at cryogenic temperatures to exploit quantum mechanical effects. - **Oscillator Unit (OU):** The core element is the Adiabatic Quantum-Flux-Parametron (AQFP). An AQFP is a sophisticated superconducting circuit based on a radio-frequency Superconducting Quantum Interference Device (rf-SQUID), in which a primary Josephson junction is replaced by a DC-SQUID. This modification allows the effective critical current of the device to be modulated with high sensitivity. The binary state is encoded in the two stable magnetic flux states (e.g., clockwise or counter-clockwise circulating supercurrents) that can exist within the superconducting loop. - **Operating Principle:** The parametric pumping is achieved by applying an AC magnetic flux or excitation current ($I_x$) to the AQFP. This modulation of the magnetic flux effectively changes the critical current of the DC-SQUID, which in turn alters the normalized inductance parameter, $\beta_L$, of the rf-SQUID loop. This change in $\beta_L$ drives the transformation of the system’s potential energy landscape from a single-well potential to a double-well potential, enabling the process of adiabatic switching. This is a direct hardware realization of the bifurcation principle central to PACO. - **Components:** The local bias ($h_i$) for each AQFP is applied by a dedicated control line that threads a small, static magnetic flux through the SQUID loop, tilting the double-well potential to favor one flux state over the other. The couplings ($Q_{ij}$) between AQFPs are implemented physically using shared inductive loops or tunable capacitive elements, which mediate the interaction between the magnetic fields of adjacent oscillators. The readout of the final state is performed by highly sensitive flux-to-voltage converters, such as coupled DC-SQUIDs, which can precisely measure the quantized magnetic flux in each AQFP loop. - **Analysis:** The SCI is arguably the most “quantum-native” of the PACO implementations. Operating at temperatures of a few Kelvin, it minimizes thermal noise and maximizes quantum coherence. This allows the system to genuinely leverage quantum tunneling to navigate the energy landscape, providing a potential speedup over more classical implementations, especially for problems with tall, thin energy barriers. The underlying technology of superconducting circuits is well-established within the quantum computing research community, benefiting from decades of development in fabrication and control. The primary challenges are the necessity of complex and costly cryogenic infrastructure and scaling the intricate on-chip wiring required for control and coupling in very large arrays. #### 3.2. Spintronic Oscillator Implementation (SOI) This implementation moves from superconductivity to the field of spintronics, which utilizes the spin of the electron, in addition to its charge, to process information. This approach offers the significant advantage of room-temperature operation. - **Oscillator Unit (OU):** The fundamental element is the Spin-Torque Nano-Oscillator (STNO). An STNO is a nanoscale multilayered magnetic device, typically a magnetic tunnel junction (MTJ) or a spin valve, composed of ferromagnetic layers separated by a non-magnetic spacer. When a spin-polarized DC current is passed through the device, it exerts a spin-transfer torque on the magnetization of one of the ferromagnetic layers (the “free layer”), causing it to enter a state of steady-state precession at microwave frequencies. - **Operating Principle:** To function as a PACO element, the STNO is driven by an additional AC current at a frequency approximately twice its natural precession frequency ($2f$). This parametric drive excites the STNO, causing its precession to lock into one of two stable phase states, 0 or $\pi$, relative to the pump signal. These two phase states encode the binary variable. - **Components:** Local biases are applied by generating small, localized magnetic fields or by injecting localized DC currents, which alter the magnetic anisotropy and favor one precessional phase. Couplings between STNOs can be achieved through several mechanisms, including the magnetic dipole fields that emanate from each precessing nanomagnet, or more robustly through the propagation of spin waves (magnons) within a shared magnetic substrate or waveguide. The final phase state is read out by measuring the phase of the microwave voltage generated by the STNO’s oscillating magnetoresistance. - **Analysis:** The primary advantages of the SOI are its potential for extremely high integration density, its inherent compatibility with standard CMOS fabrication processes, and its ability to operate at room temperature. This makes it a compelling candidate for practical, large-scale systems. The main challenges lie in engineering strong, controllable, and scalable coupling mechanisms. Dipole coupling is relatively weak and falls off quickly with distance, while spin-wave coupling requires sophisticated magnonic engineering. Furthermore, ensuring phase stability and overcoming thermal noise in a room-temperature nanoscale magnetic system is a significant hurdle. #### 3.3. Photonic and Optoelectronic Implementation (POI) This implementation uses light as the information carrier, leveraging the mature ecosystem of integrated photonics to build a high-speed, low-noise optimizer. - **Oscillator Unit (OU):** The oscillator is a nanophotonic Optical Parametric Oscillator (OPO). An OPO is created by placing a material with a strong optical nonlinearity (typically a $\chi^{(2)}$ or $\chi^{(3)}$ nonlinearity) inside a high-quality-factor optical resonant cavity, such as a silicon nitride microring resonator. - **Operating Principle:** The OPO is pumped by a strong, coherent laser beam at a frequency $\omega_p$. Through the nonlinear process of parametric down-conversion, the pump photons are converted into pairs of “signal” and “idler” photons at lower frequencies. In a degenerate OPO (DOPO), the signal and idler frequencies are identical and equal to half the pump frequency ($\omega_s = \omega_i = \omega_p/2$). Above a certain pump power threshold, the OPO begins to “lase,” generating a coherent signal field at $\omega_p/2$ that has a bistable phase, either 0 or $\pi$, relative to the pump laser. This is the direct optical analog of the parametron and forms the basis of the Coherent Ising Machine. - **Components:** Local biases are implemented by injecting a very weak “seed” laser signal into an individual OPO cavity. The phase of this seed laser is precisely controlled to energetically favor one of the two possible output phases of the OPO. Couplings are realized through the evanescent field overlap between adjacent optical waveguides or resonators, allowing light to leak from one OPO to another. The strength and sign (phase) of this coupling can be controlled using tunable optical couplers and integrated phase shifters. The final solution is read out using on-chip interferometric techniques, such as homodyne detection, which directly measure the phase of the light emitted from each OPO. - **Analysis:** The POI offers the prospect of extremely high-speed computation, with operational clock rates potentially in the tens of GHz, limited only by the photon lifetime in the optical cavities. As an optical system, it is also largely immune to thermal noise. The main scalability challenges are the fabrication of large, dense arrays of nearly identical micro-resonators with high precision, and the physical implementation of the complex optical feedback network required for programmable, all-to-all coupling, which can become a significant challenge in terms of footprint and optical loss. #### 3.4. Nano-Electro-Mechanical Systems (NEMS) Implementation (NEMSI) This implementation utilizes the physical vibration of microscopic mechanical structures to perform computation. - **Oscillator Unit (OU):** The OU is a NEMS resonator, a microor nano-scale mechanical structure such as a doubly-clamped beam, cantilever, or membrane, often fabricated from silicon or graphene. These devices can be made to vibrate at well-defined resonant frequencies, often in the MHz to GHz range. - **Operating Principle:** Parametric oscillation is induced not by a direct driving force at the resonant frequency, but by modulating a physical parameter of the resonator—such as its spring constant or effective mass—at twice the resonant frequency ($2f$). For example, an oscillating electric field can be used to modulate the tension in a beam, thereby modulating its stiffness. When the modulation amplitude exceeds a threshold, the resonator is excited into large-amplitude vibration in one of two stable phase states (e.g., in-phase or out-of-phase with the pump signal), which encode the binary variable. - **Components:** Local biases are applied by adding a weak, continuous electrostatic or piezoelectric force that gently pushes the resonator, making it energetically favorable to settle into one of the two vibrational phases. Couplings are implemented using electrodes placed between adjacent resonators, which exert programmable electrostatic or piezoelectric forces that link their motions. The strength and sign (attractive or repulsive) of these forces can be controlled by the applied voltages. The final vibrational phase of each resonator is read out using sensitive techniques like capacitive sensing or optical interferometry. - **Analysis:** NEMS-based systems offer the potential for extremely low power consumption and high integration densities. However, they face several significant challenges. The operational speed is fundamentally limited by the mechanical resonant frequencies, which are typically lower than their electronic or photonic counterparts. Fabricating large arrays of NEMS resonators with high uniformity is a major manufacturing challenge. Finally, mitigating unwanted mechanical crosstalk and acoustic energy leakage between oscillators in a dense array is critical for reliable operation. #### 3.5. Magnonic Implementation (MI) This implementation enters the domain of magnonics, which uses spin waves—collective excitations of the magnetic order in a material, also known as magnons—as the information carrier. - **Oscillator Unit (OU):** The OU is a magnonic nano-resonator. This is typically a small, patterned region within a magnetic thin film with low magnetic damping, such as yttrium iron garnet (YIG) or permalloy. The geometry of this region is designed to confine and sustain spin waves as standing modes at specific resonant frequencies. - **Operating Principle:** A globally applied microwave magnetic field, oscillating at twice the spin-wave resonant frequency ($2f$), acts as a parametric pump. This pump drives the spin-wave mode above a threshold, causing it to grow in amplitude and settle into one of two stable phase states (0 or $\pi$) relative to the pump phase. - **Components:** Local biases can be applied by locally injecting a weak, phase-coherent spin-wave signal or by using a spin-polarized current from a nano-contact to exert a spin-transfer torque, which influences the oscillation phase. Couplings are mediated by the propagation of spin waves through the shared magnetic film or through dedicated “magnonic waveguides” that connect the resonators. The phase of the spin wave in each resonator can be read out electrically, by converting the spin current into a charge current via the inverse spin Hall effect, or optically, using Brillouin Light Scattering (BLS) microscopy. - **Analysis:** The MI is a compelling solid-state, room-temperature platform that leverages the rich physics of wave-based computing. The use of propagating spin waves for coupling opens up possibilities for designing complex interference-based logic and reconfigurable circuits. This area is closely related to the active development of magnonic Ising machines, which share the same underlying principles. The primary challenges are in the nanofabrication of complex magnonic circuits and in achieving precise, dynamically programmable control over spin-wave propagation, coupling, and phase. #### 3.6. Exciton-Polariton Condensate Implementation (EPCI) This implementation is the most fundamentally quantum-mechanical, utilizing a macroscopic quantum fluid of light and matter as its computational medium. - **Oscillator Unit (OU):** The OU is a polariton condensate. Polaritons are hybrid quasiparticles that form in a semiconductor microcavity when excitons (electron-hole pairs in a quantum well) couple strongly with cavity photons. Under non-resonant optical pumping, these polaritons can undergo a phase transition into a Bose-Einstein condensate, a macroscopic coherent quantum state with a well-defined global phase. - **Operating Principle:** While the non-resonant pump laser creates and sustains the condensate, a separate, weak resonant laser can be used to “inject” a phase reference. The condensate will then lock its phase to the resonant laser, but with two stable possibilities: in-phase (0) or out-of-phase ($\pi$). These two states of the macroscopic quantum wavefunction serve to encode the binary variable. - **Components:** The local bias is effectively applied by the phase of the resonant driving laser. Coupling between adjacent condensates can be engineered through the spatial overlap of their wavefunctions or by creating an array of optical potential traps and allowing polaritons to tunnel between them. The final phase of each condensate is read out optically by interfering the light emitted from the microcavity with a coherent reference laser beam and analyzing the resulting interference pattern. - **Analysis:** The EPCI operates on a true macroscopic quantum state, opening up fascinating possibilities for exploring the role of quantum effects in computation and the boundary between the quantum and classical worlds. However, it is by far the least technologically mature of the proposed implementations. It typically requires cryogenic temperatures to achieve stable condensation and involves complex, free-space optical setups for pumping and readout. The primary challenges are creating stable, uniform, and scalable arrays of condensates and engineering precise and programmable tunneling or coupling links between them. The diversity of these physical implementations reveals a crucial aspect of the PACO paradigm. The various systems exist on a spectrum of “quantumness.” At one end, implementations like NEMSI and SOI can be described almost entirely by classical nonlinear dynamics. In the middle, SCI and POI operate in a quasi-quantum regime where quantum noise and tunneling effects become significant and can be harnessed. At the other extreme, the EPCI is a fundamentally quantum system. This means PACO is not a single, fixed type of computer but a flexible framework. The choice of implementation allows a designer to consciously trade the robustness and technological maturity of classical systems for the potential computational speedups offered by quantum phenomena, tailoring the machine’s characteristics to the specific class of problems it is intended to solve. This adaptability also unifies previously disparate research fields; a breakthrough in coupling techniques in magnonics, for example, could inspire new approaches in spintronics or NEMS, fostering a cross-pollination of ideas that could accelerate progress across the entire field of unconventional computing. | Feature | Superconducting (SCI) | Spintronic (SOI) | Photonic (POI) | NEMS (NEMSI) | Magnonic (MI) | Exciton-Polariton (EPCI) | | :---------------------- | :-------------------------------------------------- | :---------------------------------------------- | :---------------------------------------------- | :---------------------------------------------- | :---------------------------------------------- | :-------------------------------------------------- | | **Core Oscillator** | Adiabatic Quantum-Flux-Parametron (AQFP) | Spin-Torque Nano-Oscillator (STNO) | Optical Parametric Oscillator (OPO) | NEMS Resonator | Magnonic Nano-Resonator | Polariton Condensate | | **Physical State** | Quantized Magnetic Flux | Magnetization Precession Phase | Optical Field Phase | Mechanical Vibration Phase | Spin Wave Phase | Condensate Wavefunction Phase | | **Operating Temp.** | Cryogenic ($\sim$4 K) | Room Temperature | Room Temperature | Room Temperature | Room Temperature | Cryogenic ($\sim$4 K) | | **Potential Speed** | High (GHz) | High (GHz) | Very High (10s of GHz) | Low-Medium (MHz-GHz) | High (GHz) | Medium (GHz) | | **Coupling** | Inductive / Capacitive | Dipolar / Spin Wave | Evanescent / Optical | Electrostatic / Piezoelectric | Spin Wave Propagation | Wavefunction Overlap / Tunneling | | **Fabrication** | Medium-High | High (CMOS compatible) | Medium | Medium | Medium | Low | | **Key Advantages** | High coherence, quantum tunneling, low noise | High density, CMOS compatible, room temp. | Ultra-high speed, low thermal noise | Very low power, high density | Room temp., wave-based computing potential | Macroscopic quantum state, high nonlinearity | | **Challenges** | Cryogenics, complex wiring, scalability | Coupling strength, phase stability, thermal noise | Fabrication uniformity, feedback complexity | Low speed, mechanical crosstalk, uniformity | Nanofabrication, precise control of spin waves | Cryogenics, stability, complex optics, immaturity | ### 4. Comparative Analysis and Performance Landscape To fully appreciate the potential of the Parametric Adiabatic Coherent Optimizer, it must be situated within the broader landscape of high-performance computing. PACO is not proposed in a vacuum; it is a direct response to the specific limitations of existing paradigms. Its strategic value lies in the unique set of trade-offs it makes, sacrificing the universality of some approaches to gain significant advantages in robustness, specialization, and near-term feasibility. #### 4.1. PACO vs. Quantum Annealing At first glance, PACO and Quantum Annealing (QA) appear similar. Both are analog, special-purpose machines that solve optimization problems by finding the ground state of a Hamiltonian, and both use an adiabatic evolution schedule. However, their underlying physical mechanisms and operational constraints are critically different. The primary performance bottleneck for QA is the minimum energy gap problem. The requirement for the evolution time to be inversely proportional to the square of this gap means that for hard problems with exponentially small gaps, QA requires an exponentially long time to guarantee a high probability of success. While PACO’s adiabatic pump schedule is also sensitive to the system’s energy spectrum, its computational power does not derive solely from quantum tunneling in the same way as an ideal AQC system. Instead, it relies on the collective, coherent, nonlinear dynamics of the oscillator network. In its more classical implementations (e.g., NEMS, SOI), the system navigates the energy landscape through classical bifurcation dynamics, which may provide alternative pathways to find the ground state that are less sensitive to the specific quantum energy gap structure. Perhaps the most significant practical distinction is in the coherence requirements. QA, in its ideal form, requires the system to maintain quantum coherence throughout the entire, often lengthy, annealing process. Any premature decoherence or thermal excitation can corrupt the computation. PACO, by contrast, has significantly relaxed coherence demands. While its quantum implementations (SCI, EPCI) benefit from coherence, the classical and quasi-quantum versions are designed to be inherently more robust to environmental noise. The information is encoded in the stable, classical phase states of the oscillators, which are macroscopic properties less susceptible to the quantum-level noise that plagues individual qubits. This increased robustness could allow PACO systems to operate for longer effective computation times or in less pristine environments than their QA counterparts. #### 4.2. PACO vs. Gate-Model Quantum Computers The comparison between PACO and universal gate-model quantum computers is a clear case of specialization versus universality. A gate-model quantum computer is, in principle, a universal Turing machine capable of running any quantum algorithm, including those for which there is a proven exponential speedup, such as Shor’s algorithm. However, this universality comes at the steep price of extreme fragility. The gate-based model requires the precise, sequential manipulation of delicate quantum states, making it exquisitely sensitive to errors and environmental noise. To overcome this, large-scale quantum error correction is required, a technological feat that is likely decades away from practical realization. PACO makes the opposite trade-off. It is a non-universal, special-purpose machine; it is an analog co-processor designed to do one thing exceptionally well: solve QUBO problems. It cannot factor numbers or search unstructured databases. However, by focusing on this specific, high-value problem class, it can employ a computational mechanism that is far more robust. Instead of manipulating individual qubit states with a long sequence of gates, PACO lets the entire physical system evolve in parallel, guided by a single global control parameter (the pump amplitude). This inherent robustness and specialization could allow a PACO machine with thousands of oscillators to solve meaningful optimization problems in the near term, a task that remains far beyond the reach of today’s noisy, small-scale universal quantum processors. #### 4.3. PACO as an Advanced Analog Computer PACO can be understood as a renaissance of analog computing, augmented with modern control techniques and quasi-quantum capabilities. Historically, analog computers were highly efficient for solving specific problems, such as systems of differential equations, by creating a physical system whose dynamics were directly analogous to the mathematical problem. They fell out of favor due to their lack of programmability and susceptibility to noise and drift. PACO revitalizes this concept in a highly controllable and scalable form. The hybrid architecture makes it fully programmable at the QUBO level via the classical host computer. Its reliance on bistable phase states provides a degree of inherent noise resilience. Most importantly, PACO bridges the gap between purely classical analog computing and quantum computing. A purely classical analog optimizer, like its digital counterparts, can still get trapped in local energy minima. The “quasi-quantum” implementations of PACO, such as the superconducting SCI or the polaritonic EPCI, can explicitly leverage quantum effects like tunneling and superposition to explore the energy landscape more effectively. This gives them a distinct performance advantage over any purely classical analog machine, allowing them to find global optima that would otherwise be inaccessible. This positions the PACO framework in a unique strategic space. Its performance characteristics are tunable based on the choice of physical implementation. For a problem whose energy landscape is characterized by many tall, thin barriers, a highly quantum implementation like the SCI, which excels at tunneling, might be the optimal choice. For a problem with a smoother landscape where speed and robustness are paramount, a more classical implementation like the photonic POI might be superior. This means PACO is not a single, monolithic competitor to other computing paradigms but a flexible class of machines whose strengths can be adapted to the problem at hand. This adaptability also redefines the concept of a “solver.” In PACO, the hardware and the algorithm are inextricably linked; the physical evolution of the system is the algorithm. Optimizing a PACO system is therefore a holistic co-design problem, requiring the simultaneous optimization of the physical hardware substrate and the classical control protocol—a new and deeply interdisciplinary challenge for computer engineering. #### 4.4. Potential for Quantum Advantage: A Problem-Class-Specific Evaluation Quantum advantage is the milestone at which a quantum or quasi-quantum device can solve a problem of practical interest significantly faster, more accurately, or more efficiently than the best available classical computer running the best known classical algorithm. For PACO, the potential for achieving this advantage will likely depend not just on the size of the problem, but on its structure. A critical factor is the connectivity of the problem graph. Many classical algorithms excel at problems with sparse or structured connectivity. Similarly, quantum annealers with limited hardware connectivity require a costly minor-embedding overhead for densely connected problems. This suggests a potential “sweet spot” for PACO. Implementations like the photonic POI or the superconducting SCI, which can support all-to-all or very dense connectivity through feedback loops or shared coupling busses, may hold a decisive advantage on problems where every variable interacts with every other variable. Therefore, the most promising target applications for demonstrating a PACO’s superiority are not just any QUBO problem, but specifically those characterized by dense interaction matrices, which represent some of the most challenging instances for competing technologies. | Feature | PACO | Classical Heuristics (e.g., SA) | Quantum Annealing (QA) | Gate-Model QC | | :---------------------- | :------------------------------------------ | :------------------------------------------ | :------------------------------------------ | :------------------------------------------ | | **Core Mechanism** | Collective Bifurcation & Self-Organization | Thermal Fluctuation & Probabilistic Search | Quantum Tunneling & Adiabatic Evolution | Sequential Unitary Gate Operations | | **Information Carrier** | Oscillator Phase (analog, bistable) | Digital Bits (classical) | Qubit State (quantum) | Qubit State (quantum) | | **Noise Resilience** | High to Medium (depending on implementation) | High (by design) | Low to Medium | Very Low | | **Coherence Requirement** | Low to None (for classical versions) | None | High | Very High | | **Universality** | No (Specialized for QUBO/Ising) | No (Specialized heuristics) | No (Specialized for QUBO/Ising) | Yes (Universal) | | **Target Problem Class** | QUBO / Ising | General Optimization | QUBO / Ising | Universal Computation | | **Key Bottleneck** | Scalable, Programmable Coupling | Getting Trapped in Local Minima | Small Energy Gaps, Decoherence | Error Correction Overhead, Qubit Count | | **Near-Term Viability** | High | High (Existing Technology) | Medium | Low (for large problems) | ### 5. Applications and Target Domains The applicability of the PACO paradigm is defined by its native problem format: the Quadratic Unconstrained Binary Optimization (QUBO) model. The remarkable versatility of the QUBO formulation means that a wide array of computationally challenging problems from diverse and high-impact fields can be mapped onto the PACO architecture for potential acceleration. The ideal target problems for PACO are those that not only map well to QUBO but also exhibit characteristics, such as dense connectivity, that are particularly challenging for competing solvers. #### 5.1. Logistics, Finance, and Network Design These domains are rich with NP-hard problems that have natural or well-established QUBO formulations, making them prime candidates for PACO. - **Logistics and Supply Chain:** Problems such as the Vehicle Routing Problem (an extension of the Traveling Salesman Problem) and facility location problems can be directly formulated as QUBOs. Optimizing these systems can lead to massive reductions in fuel consumption, delivery times, and operational costs for global logistics networks. - **Financial Modeling:** The Markowitz portfolio optimization problem, which seeks to find the ideal allocation of assets to maximize returns for a given level of risk, is a cornerstone of modern finance. This problem can be cast as a QUBO, where binary variables represent the inclusion or exclusion of an asset in the portfolio. An efficient QUBO solver like PACO could enable more complex and accurate risk models, potentially leading to better investment strategies. - **Network Optimization:** Designing optimal network topologies for communications, energy grids, or social networks often involves solving the Max-Cut problem, which seeks to partition the nodes of a graph into two sets to maximize the number of edges between them. The Max-Cut problem is canonically equivalent to a QUBO problem on the graph’s adjacency matrix. PACO could be used to design more resilient and efficient network infrastructures. #### 5.2. Materials Science and Pharmaceutical Research Many fundamental problems in the physical and life sciences can be framed as finding the minimum energy configuration of a system of interacting particles, a natural fit for a ground-state solver like PACO. - **Drug Discovery and Protein Folding:** Determining the three-dimensional structure of a protein is critical for understanding its function and for designing drugs that can interact with it. The protein folding problem can be modeled as finding the minimum energy conformation of a chain of amino acids. While this is an extremely complex problem, simplified lattice models can be formulated as QUBOs, where PACO could rapidly explore the vast conformational space to identify stable, low-energy structures. - **Materials Design:** Discovering new materials with desired properties (e.g., high-temperature superconductors, novel catalysts) often involves searching through a vast combinatorial space of possible atomic arrangements or electronic configurations. Ising-like models are frequently used to describe these systems, and a PACO machine could serve as a powerful simulation engine to find the ground-state properties of these candidate materials, accelerating the discovery process. #### 5.3. Machine Learning and Artificial Intelligence Acceleration Several tasks in machine learning and AI are fundamentally optimization problems and can be mapped to the QUBO format. - **Training and Sampling:** Certain types of neural networks, such as Boltzmann machines, have an energy function that is identical in form to an Ising Hamiltonian. Training these models involves finding low-energy states, a task for which PACO is ideally suited. It could also be used as a rapid sampler to generate data from the probability distribution defined by the model. - **Clustering and Feature Selection:** Unsupervised learning tasks like clustering, which involves grouping similar data points together, can be formulated as a QUBO problem where the objective is to minimize the distance between points within a cluster and maximize the distance between clusters. Similarly, the problem of selecting the most informative subset of features for a predictive model can be cast as a QUBO, which PACO could solve to improve model accuracy and reduce complexity. A common thread across these applications is the structure of the underlying problem graph. The performance of many classical solvers and quantum annealers degrades on problems with dense, all-to-all connectivity, as the number of interactions scales quadratically with the number of variables. This is precisely where PACO may have a distinct advantage. Physical implementations of PACO, particularly those using time-multiplexed feedback like photonic CIMs or those with long-range coupling mechanisms, can naturally support dense or even fully connected problem graphs with minimal overhead. This suggests that the ideal applications for demonstrating PACO’s superiority are not just any QUBO problem, but specifically those characterized by dense interaction matrices, which represent some of the most challenging instances for competing technologies. ### 6. Challenges, Scalability, and Future Directions While the PACO paradigm presents a compelling theoretical framework, its transition from concept to a practical, large-scale computational tool depends on surmounting significant engineering, control, and validation challenges. The future development of PACO will require a concerted, interdisciplinary effort focused on hardware fabrication, control algorithm design, and rigorous benchmarking. #### 6.1. Engineering Challenges The viability of any PACO implementation hinges on the ability to fabricate and control large arrays of coupled oscillators with high precision. Two challenges are paramount: - **Inter-Oscillator Coupling:** The single greatest engineering hurdle is the implementation of a scalable, dense, and programmable coupling network. Achieving strong, precise, and reconfigurable interactions between potentially thousands or millions of individual oscillators is non-trivial. In spatially distributed arrays (e.g., NEMS or STNOs), minimizing unwanted crosstalk while ensuring strong desired couplings is a delicate balancing act. In time-multiplexed systems (e.g., photonic CIMs), the challenge shifts to the speed and complexity of the classical feedback electronics needed to calculate and apply the coupling terms for each oscillator in each cycle. - **Parameter Control and Uniformity:** The PACO model assumes that all oscillators in the array are largely identical, with the same natural resonant frequency and response to control signals. In reality, fabrication imperfections will inevitably lead to variations across the array. A lack of uniformity in resonant frequencies can make it difficult for the global pump to effectively drive all oscillators and can complicate the mapping of the QUBO problem. Developing techniques for post-fabrication tuning and calibration, or designing systems that are inherently robust to such variations, will be critical for building large-scale, reliable machines. #### 6.2. The Path to Scalability Scaling PACO to solve commercially relevant problems (which may involve tens of thousands to millions of variables) requires a clear architectural roadmap. The two primary approaches are spatial multiplexing and time multiplexing. - **Spatial Multiplexing:** This involves creating a physical 2D or 3D array of individual oscillator units on a chip. This approach is conceptually straightforward but faces the direct physical challenges of fabrication uniformity, wiring density for control and coupling, and managing crosstalk and heat dissipation at large scales. - **Time Multiplexing:** This approach, pioneered by photonic CIMs, uses a much smaller number of physical oscillators (or even a single one) that are reused in time. A train of pulses circulates in a long delay loop, with each pulse in the train representing a different binary variable. In each round trip, the state of every pulse is measured, a classical feedback processor (e.g., an FPGA) calculates the coupling interactions based on the QUBO matrix, and these interactions are applied to the pulses before the next round of amplification. This architecture elegantly solves the all-to-all connectivity problem but shifts the scalability bottleneck to the classical feedback hardware, which must perform a massive matrix-vector multiplication in real-time. The optimal path forward may involve hybrid architectures that combine spatial and time multiplexing to balance the trade-offs between physical device count, connectivity, and classical processing requirements. #### 6.3. Adiabaticity Control and Noise Resilience The performance of a PACO machine is fundamentally tied to the quality of its adiabatic evolution. Future research must move beyond simple linear pump ramps and focus on developing sophisticated, adaptive control protocols. The classical host computer should play a dynamic role, implementing “shortcuts to adiabaticity” where the pump schedule is tailored to the specific problem being solved. By analyzing the problem structure or even by observing the system’s behavior in real-time, the CHC could learn to speed up the evolution when the system is far from a critical point (i.e., where the energy gap is large) and slow down precisely when needed to navigate a small gap, thus minimizing the total time-to-solution while maintaining a high probability of finding the global optimum. This sophisticated control layer is also key to noise resilience. As PACO systems scale, the complexity of the classical control system will grow dramatically. The CHC will be responsible for managing potentially millions of individual bias and coupling parameters, executing real-time optimal control algorithms for the pump schedule, and running verification and error-checking routines on the solutions. This implies that the future of PACO is not simply a victory of analog over digital, but a deep, symbiotic integration of both. The ultimate bottleneck for a million-oscillator PACO machine might not be the physics of the analog co-processor, but rather the classical I/O bandwidth and real-time processing capability of its digital host. #### 6.4. A Roadmap for Experimental Validation and Benchmarking To establish its credibility and guide development, the PACO paradigm must be subjected to rigorous and standardized experimental validation. It is imperative to benchmark prototype PACO systems against both state-of-the-art classical solvers (e.g., Gurobi, CPLEX) and existing quantum and quasi-quantum devices (e.g., D-Wave’s quantum annealers) on well-defined, publicly available problem sets. This benchmarking process presents a new challenge. Simple metrics like “qubit count” are meaningless for comparing the diverse PACO implementations. A 1,000-oscillator NEMS-based PACO will have entirely different performance characteristics from a 1,000-oscillator superconducting PACO. Therefore, a new frontier in benchmarking is required, one that utilizes multi-dimensional metrics that capture not only problem size but also problem connectivity (e.g., graph density), solution quality (approximation ratio), total time-to-solution (including programming and readout overhead), and, crucially, the total energy-to-solution. Only through such a nuanced and comprehensive benchmarking framework can the true advantages and disadvantages of different PACO implementations be understood and the most promising paths for future development be identified. ### 7. Conclusion: PACO’s Position in the Future of Computing The Parametric Adiabatic Coherent Optimizer represents a significant and compelling conceptual advance in the ongoing quest to overcome the limitations of conventional computing for intractable combinatorial optimization problems. By synthesizing principles from nonlinear dynamics, adiabatic evolution, and hybrid system architecture, PACO charts a pragmatic and powerful course between the well-understood but often inadequate domain of classical computing and the potentially revolutionary but profoundly challenging realm of fault-tolerant quantum computing. The core strength of the PACO paradigm lies in its unique approach to computation. It circumvents the primary failure mode of many classical optimizers—getting trapped in local minima—by using a global, adiabatic evolution to guide a physical system to its true ground state. Simultaneously, it avoids the most daunting obstacles facing quantum computers—the extreme sensitivity to noise and the requirement for massive error correction—by encoding information in robust, macroscopic phase states and relying on collective dynamics rather than fragile, individual quantum superpositions. This positions PACO as a potentially crucial “bridge” technology, capable of delivering tangible computational advantages on high-value problems in the near-to-mid-term, without waiting for the decades of development that may be required for universal quantum computers to mature. Furthermore, the paradigm’s versatility is a key strategic asset. The ability to implement the same fundamental computational principle across a vast spectrum of physical substrates—from room-temperature spintronic and photonic devices to cryogenic superconducting circuits and macroscopic quantum condensates—demonstrates the robustness of the underlying concept. This diversity not only provides multiple parallel paths for technological development but also creates a unique research platform for exploring the complex interplay between classical and quantum dynamics in computation. To be sure, the path from the current conceptual framework to a large-scale, commercially viable PACO machine is fraught with significant engineering challenges. The realization of scalable, dense, and precisely programmable coupling networks remains the most critical hurdle across all proposed physical implementations. Nevertheless, the theoretical coherence of the PACO paradigm, its clear architectural advantages, and its deep connections to multiple active and rapidly advancing fields of physics and engineering make it a high-priority and exceptionally promising direction for future research and investment. In an era where the continued scaling of conventional digital computing is facing fundamental physical limits, paradigms like PACO that harness the rich dynamics of physical systems directly offer a vital and exciting vision for the future of high-performance computation. --- ### Appendix: References 1. “The Power of NP-Hardness in Combinatorial Optimization,” Number Analytics. [https://www.numberanalytics.com/blog/power-np-hardness-combinatorial-optimization](https://www.numberanalytics.com/blog/power-np-hardness-combinatorial-optimization) 2. “Combinatorial Optimisation,” University of Warwick. [https://warwick.ac.uk/fac/soc/wbs/subjects/orms/research/areas/combinatorial_optimization/](https://warwick.ac.uk/fac/soc/wbs/subjects/orms/research/areas/combinatorial_optimization/) 3. “Mastering NP-Hardness in Combinatorial Optimization,” Number Analytics. [https://www.numberanalytics.com/blog/ultimate-guide-np-hardness-combinatorial-optimization](https://www.numberanalytics.com/blog/ultimate-guide-np-hardness-combinatorial-optimization) 4. “Introduction to Combinatorial Optimization,” The University of Texas at Dallas. [https://www.utdallas.edu/~dxd056000/cs6363/LectureNotes.pdf](https://www.utdallas.edu/~dxd056000/cs6363/LectureNotes.pdf) 5. “Revolutionizing Financial Decision-Making: Overcoming the Limitations of Traditional Optimization Methods,” quantumrock | Asset Management. [https://www.quantumrock.group/overcoming-the-limitations-of-traditional-optimization-methods/](https://www.quantumrock.group/overcoming-the-limitations-of-traditional-optimization-methods/) 6. “CLASSICAL OPTIMIZATION TECHNIQUES,” ResearchGate. [https://www.researchgate.net/publication/378635305_CLASSICAL_OPTIMIZATION_TECHNIQUES](https://www.researchgate.net/publication/378635305_CLASSICAL_OPTIMIZATION_TECHNIQUES) 7. “Limitations of classical optimization techniques across key performance dimensions,” ResearchGate. [https://www.researchgate.net/figure/Limitations-of-classical-optimization-techniques-across-key-performance-dimensions_fig3_393742033](https://www.researchgate.net/figure/Limitations-of-classical-optimization-techniques-across-key-performance-dimensions_fig3_393742033) 8. “Can classical optimizers undermine quantum advantage in hybrid algorithms?,” Reddit. [https://www.reddit.com/r/QuantumComputing/comments/1g3ezbd/can_classical_optimizers_undermine_quantum/](https://www.reddit.com/r/QuantumComputing/comments/1g3ezbd/can_classical_optimizers_undermine_quantum/) 9. “Adiabatic quantum computation,” Wikipedia. [https://en.wikipedia.org/wiki/Adiabatic_quantum_computation](https://en.wikipedia.org/wiki/Adiabatic_quantum_computation) 10. “Quantum annealing and adiabatic computation,” Fiveable. [https://library.fiveable.me/quantum-computing/unit-12/quantum-annealing-adiabatic-quantum-computation/study-guide/7Cd7y59tXCBXE8h4](https://library.fiveable.me/quantum-computing/unit-12/quantum-annealing-adiabatic-quantum-computation/study-guide/7Cd7y59tXCBXE8h4) 11. “Adiabatic vs Quantum Computation? : r/AskPhysics,” Reddit. [https://www.reddit.com/r/AskPhysics/comments/a5uspf/adiabatic_vs_quantum_computation/](https://www.reddit.com/r/AskPhysics/comments/a5uspf/adiabatic_vs_quantum_computation/) 12. “Quantum Annealing (QA ...),” Quantum Computing Modalities. [https://postquantum.com/quantum-modalities/annealing-adiabatic/](https://postquantum.com/quantum-modalities/annealing-adiabatic/) 13. “What is Adiabatic Quantum Optimization,” QuEra Computing. [https://www.quera.com/glossary/adiabatic-quantum-optimization](https://www.quera.com/glossary/adiabatic-quantum-optimization) 14. “Quantum Adiabatic Optimization with Rydberg Arrays: Localization Phenomena and Encoding Strategies,” *Physical Review X*. [https://link.aps.org/doi/10.1103/PRXQuantum.6.020306](https://link.aps.org/doi/10.1103/PRXQuantum.6.020306) 15. “Unstructured Adiabatic Quantum Optimization: Optimality with Limitations,” *Quantum*. [https://quantum-journal.org/papers/q-2025-07-11-1790/](https://quantum-journal.org/papers/q-2025-07-11-1790/) 16. “Quantum logic gate,” Wikipedia. [https://en.wikipedia.org/wiki/Quantum_logic_gate](https://en.wikipedia.org/wiki/Quantum_logic_gate) 17. “Gate based QC,” Quantum Flagship. [https://qt.eu/quantum-principles/computing/gate-based-qc](https://qt.eu/quantum-principles/computing/gate-based-qc) 18. “Demystifying Quantum Computing: Gate-Based vs. Quantum Annealers (QML-2),” Medium. [https://medium.com/@shreyasatsangi27/demystifying-quantum-computing-gate-based-vs-quantum-annealers-qml-2-c4e21e1ccc94](https://medium.com/@shreyasatsangi27/demystifying-quantum-computing-gate-based-vs-quantum-annealers-qml-2-c4e21e1ccc94) 19. “Quantum Algorithm Outpaces Classical Solvers in Optimization Tasks, Study Indicates,” The Quantum Insider. [https://thequantuminsider.com/2025/05/17/quantum-algorithm-outpaces-classical-solvers-in-optimization-tasks-study-indicates/](https://thequantuminsider.com/2025/05/17/quantum-algorithm-outpaces-classical-solvers-in-optimization-tasks-study-indicates/) 20. “Quantum Annealing in 2025: Practical Quantum Computing,” Research AIMultiple. [https://research.aimultiple.com/quantum-annealing/](https://research.aimultiple.com/quantum-annealing/) 21. “Parametron,” Wikipedia. [https://en.wikipedia.org/wiki/Parametron](https://en.wikipedia.org/wiki/Parametron) 22. “Milestone-Proposal:Parametron, 1954,” IEEE Milestones. [https://ieeemilestones.ethw.org/Milestone-Proposal:Parametron,_1954](https://ieeemilestones.ethw.org/Milestone-Proposal:Parametron,_1954) 23. “Parametron-Computer Museum,” IPSJ Computer Museum. [https://museum.ipsj.or.jp/en/computer/dawn/0007.html](https://museum.ipsj.or.jp/en/computer/dawn/0007.html) 24. “Parametron Computer,” IEICE. [https://www.ieice.org/eng_r/assets/pdf/publication/milestone/d30.pdf](https://www.ieice.org/eng_r/assets/pdf/publication/milestone/d30.pdf) 25. “The Initial Input Routine of the Parametron Computer PC-1,” IEEE Milestones. [https://ieeemilestones.ethw.org/w/images/7/75/Wada_PC-1.pdf](https://ieeemilestones.ethw.org/w/images/7/75/Wada_PC-1.pdf) 26. “PC-1 Parametron Computer,” University of Tokyo / IPSJ Computer Museum. [http://museum.ipsj.or.jp/en/computer/dawn/0016.html](http://museum.ipsj.or.jp/en/computer/dawn/0016.html) 27. “Computational Principle and Performance Evaluation of Coherent Ising Machine Based on Degenerate Optical Parametric Oscillator Network,” *MDPI*. [https://www.mdpi.com/1099-4300/18/4/151](https://www.mdpi.com/1099-4300/18/4/151) 28. “Boltzmann Sampling by Degenerate Optical Parametric Oscillator Network for Structure-Based Virtual Screening,” *MDPI*. [https://www.mdpi.com/1099-4300/18/10/365](https://www.mdpi.com/1099-4300/18/10/365) 29. “Combinatorial optimization using networks of optical parametric oscillators,” *Optics Express*. [https://opg.optica.org/abstract.cfm?uri=nlo-2017-NM2B.2](https://opg.optica.org/abstract.cfm?uri=nlo-2017-NM2B.2) 30. “Non-von Neumann computing using networks of optical parametric oscillators Abstract: Combinatorial optimization problems,” Berkeley Microlab. [https://microlab.berkeley.edu/text/seminars/prevsemsched/abstracts_2017/08-24-17.pdf](https://microlab.berkeley.edu/text/seminars/prevsemsched/abstracts_2017/08-24-17.pdf) 31. “Accuracy-enhanced coherent Ising machine using the quantum ...,” *Optics Express*. [https://opg.optica.org/oe/abstract.cfm?uri=oe-29-12-18530](https://opg.optica.org/oe/abstract.cfm?uri=oe-29-12-18530) 32. “Coherent Ising machine based on degenerate optical parametric ...,” *Physical Review A*. [https://link.aps.org/doi/10.1103/PhysRevA.88.063853](https://link.aps.org/doi/10.1103/PhysRevA.88.063853) 33. “Combinatorial optimization by simulating adiabatic bifurcations in nonlinear Hamiltonian systems,” *PNAS*. [https://pmc.ncbi.nlm.nih.gov/articles/PMC6474767/](https://pmc.ncbi.nlm.nih.gov/articles/PMC6474767/) 34. “Combinatorial optimization by simulating adiabatic bifurcations in nonlinear Hamiltonian systems,” ResearchGate. [https://www.researchgate.net/publication/332535366_Combinatorial_optimization_by_simulating_adiabatic_bifurcations_in_nonlinear_Hamiltonian_systems](https://www.researchgate.net/publication/332535366_Combinatorial_optimization_by_simulating_adiabatic_bifurcations_in_nonlinear_Hamiltonian_systems) 35. “Bifurcation-based adiabatic quantum computation with a nonlinear oscillator network: Toward quantum soft computing,” ResearchGate. [https://www.researchgate.net/publication/282790373_Bifurcation-based_adiabatic_quantum_computation_by_a_nonlinear_oscillator_network_Toward_quantum_soft_computing](https://www.researchgate.net/publication/282790373_Bifurcation-based_adiabatic_quantum_computation_by_a_nonlinear_oscillator_network_Toward_quantum_soft_computing) 36. “Quantum Computation Based on Quantum Adiabatic Bifurcations of Kerr-Nonlinear Parametric Oscillators,” arXiv. [https://arxiv.org/abs/1808.10609](https://arxiv.org/abs/1808.10609) 37. “Geometrical optimization of pumping power under adiabatic parameter driving,” arXiv. [https://arxiv.org/abs/1901.01669](https://arxiv.org/abs/1901.01669) 38. “Geometrical Optimization of Pumping Power under Adiabatic Parameter Driving,” *JPS Journals*. [https://journals.jps.jp/doi/pdf/10.7566/JPSJ.89.064706](https://journals.jps.jp/doi/pdf/10.7566/JPSJ.89.064706) 39. “Pulse optimization in adiabatic quantum computation and control,” arXiv. [https://arxiv.org/abs/2507.09770](https://arxiv.org/abs/2507.09770) 40. “Comparison between optimal control and shortcut to adiabaticity protocols in a linear control system,” *Physical Review A*. [https://link.aps.org/doi/10.1103/PhysRevA.101.013423](https://link.aps.org/doi/10.1103/PhysRevA.101.013423) 41. “Optimal shortcut-to-adiabaticity quantum control,” ResearchGate. [https://www.researchgate.net/publication/390213413_Optimal_shortcut-to-adiabaticity_quantum_control](https://www.researchgate.net/publication/390213413_Optimal_shortcut-to-adiabaticity_quantum_control) 42. “Pulse optimization in adiabatic quantum computation and ...,” arXiv. [https://arxiv.org/pdf/2507.09770](https://arxiv.org/pdf/2507.09770) 43. “Hybrid Quantum-Classical Approach to Correlated Materials,” *Physical Review X*. [https://link.aps.org/doi/10.1103/PhysRevX.6.031045](https://link.aps.org/doi/10.1103/PhysRevX.6.031045) 44. “A Quantum Optimization Toolbox,” arXiv. [https://arxiv.org/html/2507.07649v1](https://arxiv.org/html/2507.07649v1) 45. “Classification of Hybrid Quantum-Classical Computing,” arXiv. [https://arxiv.org/pdf/2210.15314](https://arxiv.org/pdf/2210.15314) 46. “Improving the Solving of Optimization Problems: A Comprehensive Review of Quantum Approaches,” *MDPI*. [https://www.mdpi.com/2624-960X/7/1/3](https://www.mdpi.com/2624-960X/7/1/3) 47. “Optimizing QUBO on a quantum computer by mimicking imaginary time evolution,” arXiv. [https://arxiv.org/html/2505.22924v2](https://arxiv.org/html/2505.22924v2) 48. “QUBO Formulations of Combinatorial ...,” Lehigh University. [https://engineering.lehigh.edu/sites/engineering.lehigh.edu/files/_DEPARTMENTS/ise/pdf/tech-papers/23/23T_016.pdf](https://engineering.lehigh.edu/sites/engineering.lehigh.edu/files/_DEPARTMENTS/ise/pdf/tech-papers/23/23T_016.pdf) 49. “Quantum Bridge Analytics I: A Tutorial on Formulating and Using QUBO Models,” Wigner Research Centre for Physics. [https://wigner.hu/~koniorczykmatyas/qubo/literature/1811.11538.pdf](https://wigner.hu/~koniorczykmatyas/qubo/literature/1811.11538.pdf) 50. “Mapping NP-Complete Problems to Physics-Based QUBO Solvers: Quantitative Comparison and Understanding,” JuSER. [https://juser.fz-juelich.de/record/1021223/files/Hizzani_NNPC2023.pdf](https://juser.fz-juelich.de/record/1021223/files/Hizzani_NNPC2023.pdf) 51. “Adiabatic Quantum-Flux-Parametron: A Tutorial Review,” ResearchGate. [https://www.researchgate.net/publication/357919726_Adiabatic_Quantum-Flux-Parametron_A_Tutorial_Review](https://www.researchgate.net/publication/357919726_Adiabatic_Quantum-Flux-Parametron_A_Tutorial_Review) 52. “Adiabatic Quantum-Flux-Parametron: A Tutorial Review,” IEICE. [https://search.ieice.org/bin/pdf_advpub.php?category=C&lang=E&fname=2021SEP0003&abst=](https://search.ieice.org/bin/pdf_advpub.php?category=C&lang=E&fname=2021SEP0003&abst=) 53. “Adiabatic Quantum-Flux-Parametron: A Tutorial Review,” IEICE Digital Library. [https://globals.ieice.org/en_transactions/electronics/10.1587/transele.2021SEP0003/_pdf](https://globals.ieice.org/en_transactions/electronics/10.1587/transele.2021SEP0003/_pdf) 54. “Low-latency adiabatic quantum-flux-parametron circuit integrated with a hybrid serializer/deserializer,” arXiv. [https://arxiv.org/abs/2210.03446](https://arxiv.org/abs/2210.03446) 55. “Adiabatic Quantum-Flux-Parametron: Towards Building Extremely Energy-Efficient Circuits and Systems,” *PubMed*. [https://pubmed.ncbi.nlm.nih.gov/31324832/](https://pubmed.ncbi.nlm.nih.gov/31324832/) 56. “Spin transfer nano-oscillators,” arXiv. [https://arxiv.org/pdf/1302.6467](https://arxiv.org/pdf/1302.6467) 57. “Spin torque nano-oscillators with a perpendicular spin polarizer,” *Chinese Physics B*. [https://cpb.iphy.ac.cn/article/2019/1976/cpb_28_3_037503.html](https://cpb.iphy.ac.cn/article/2019/1976/cpb_28_3_037503.html) 58. “Magnetic field detection using spin-torque nano-oscillator combined ...,” *AIP Advances*. [https://pubs.aip.org/aip/adv/article/13/3/035228/2881287/Magnetic-field-detection-using-spin-torque-nano](https://pubs.aip.org/aip/adv/article/13/3/035228/2881287/Magnetic-field-detection-using-spin-torque-nano) 59. “Spin-Torque and Spin-Hall Nano-Oscillators,” Semantic Scholar. [https://www.semanticscholar.org/paper/Spin-Torque-and-Spin-Hall-Nano-Oscillators-Chen-Dumas/c191220e9ee7d739e97cd9ab87d40fcff5ee735e](https://www.semanticscholar.org/paper/Spin-Torque-and-Spin-Hall-Nano-Oscillators-Chen-Dumas/c191220e9ee7d739e97cd9ab87d40fcff5ee735e) 60. “Spin-Torque and Spin-Hall Nano-Oscillators,” DiVA portal. [http://www.diva-portal.org/smash/get/diva2:868735/FULLTEXT01.pdf](http://www.diva-portal.org/smash/get/diva2:868735/FULLTEXT01.pdf) 61. “Synchronization of spin torque nano-oscillators,” *Physical Review B*. [https://link.aps.org/doi/10.1103/PhysRevB.95.144412](https://link.aps.org/doi/10.1103/PhysRevB.95.144412) 62. “Optimization of the threshold for degenerate optical parametric oscillations in a bichromatically pumped microresonator,” *Physical Review Applied*. [https://link.aps.org/doi/10.1103/xphr-cml2](https://link.aps.org/doi/10.1103/xphr-cml2) 63. “Demonstration of chip-based coupled degenerate optical parametric oscillators for realizing a nanophotonic spin-glass,” *PubMed*. [https://pubmed.ncbi.nlm.nih.gov/32807796/](https://pubmed.ncbi.nlm.nih.gov/32807796/) 64. “Combinatorial optimization solving by coherent Ising machines based on spiking neural networks,” arXiv. [https://arxiv.org/abs/2208.07502](https://arxiv.org/abs/2208.07502) 65. “Nanomechanical Resonators: Toward Atomic Scale,” *ACS Nano*. [https://pubs.acs.org/doi/10.1021/acsnano.2c01673](https://pubs.acs.org/doi/10.1021/acsnano.2c01673) 66. “Symmetry breaking of large-amplitude parametric oscillations in few ...,” *Physical Review Applied*. [https://link.aps.org/doi/10.1103/PhysRevApplied.23.054036](https://link.aps.org/doi/10.1103/PhysRevApplied.23.054036) 67. “A Nanoscale Parametric Feedback Oscillator,” *Nano Letters*. [https://pubs.acs.org/doi/10.1021/nl2031162](https://pubs.acs.org/doi/10.1021/nl2031162) 68. “Nonlinear and Parametric NEMS Resonators,” ResearchGate. [https://www.researchgate.net/publication/304195659_Nonlinear_and_Parametric_NEMS_Resonators](https://www.researchgate.net/publication/304195659_Nonlinear_and_Parametric_NEMS_Resonators) 69. “Dynamics-Enabled Nanoelectromechanical Systems (NEMS) Oscillators,” DTIC. [https://apps.dtic.mil/sti/tr/pdf/ADA604923.pdf](https://apps.dtic.mil/sti/tr/pdf/ADA604923.pdf) 70. “Inverse-design magnonic devices,” *Nature Communications*. [https://pmc.ncbi.nlm.nih.gov/articles/PMC8113576/](https://pmc.ncbi.nlm.nih.gov/articles/PMC8113576/) 71. “Nanoscale magnonic Fabry-Pérot resonator for low-loss spin-wave manipulation,” Aalto University Research Portal. [https://research.aalto.fi/en/publications/nanoscale-magnonic-fabry-p%C3%A9rot-resonator-for-low-loss-spin-wave-m](https://research.aalto.fi/en/publications/nanoscale-magnonic-fabry-p%C3%A9rot-resonator-for-low-loss-spin-wave-m) 72. “Perspective on nonvolatile magnon-signal storage and in-memory computation for low-power consuming magnonics,” *Applied Physics Letters*. [https://pubs.aip.org/aip/apl/article/126/16/160502/3344944/Perspective-on-nonvolatile-magnon-signal-storage](https://pubs.aip.org/aip/apl/article/126/16/160502/3344944/Perspective-on-nonvolatile-magnon-signal-storage) 73. “Nanoscale magnonic networks,” *Physical Review Applied*. [https://link.aps.org/doi/10.1103/PhysRevApplied.21.040503](https://link.aps.org/doi/10.1103/PhysRevApplied.21.040503) 74. “3D Magnonic Conduits by Direct Write Nanofabrication,” *MDPI*. [https://www.mdpi.com/2079-4991/13/13/1926](https://www.mdpi.com/2079-4991/13/13/1926) 75. “Realization of inverse-design magnonic logic gates,” arXiv. [https://arxiv.org/html/2411.17546v1](https://arxiv.org/html/2411.17546v1) 76. “Evaluating Spintronics-Compatible Implementations of Ising Machines,” *Physical Review Applied*. [https://link.aps.org/doi/10.1103/PhysRevApplied.20.024005](https://link.aps.org/doi/10.1103/PhysRevApplied.20.024005) 77. “The 2024 magnonics roadmap,” TU Delft Research Portal. [https://research.tudelft.nl/en/publications/the-2024-magnonics-roadmap](https://research.tudelft.nl/en/publications/the-2024-magnonics-roadmap) 78. “Global biasing using a hardware-based artificial Zeeman term in spinwave Ising machines,” *Applied Physics Letters*. [https://pubs.aip.org/aip/apl/article/124/9/092409/3267695/Global-biasing-using-a-hardware-based-artificial](https://pubs.aip.org/aip/apl/article/124/9/092409/3267695/Global-biasing-using-a-hardware-based-artificial) 79. “A spinwave Ising machine,” ResearchGate. [https://www.researchgate.net/publication/373412529_A_spinwave_Ising_machine](https://www.researchgate.net/publication/373412529_A_spinwave_Ising_machine) 80. “Materials and Cavity Design Principles for Exciton-Polariton Condensates,” *ACS Nano*. [https://pubs.acs.org/doi/10.1021/acsnano.4c15929](https://pubs.acs.org/doi/10.1021/acsnano.4c15929) 81. “Optimization results for the polaritonic lens. a) The excitonpolariton...,” ResearchGate. [https://www.researchgate.net/figure/Optimization-results-for-the-polaritonic-lens-a-The-excitonpolariton-condensate_fig2_382526722](https://www.researchgate.net/figure/Optimization-results-for-the-polaritonic-lens-a-The-excitonpolariton-condensate_fig2_382526722) 82. “Scalable continuous-variable entanglement of light beams produced by optical parametric oscillators,” *Physical Review A*. [https://link.aps.org/doi/10.1103/PhysRevA.77.022311](https://link.aps.org/doi/10.1103/PhysRevA.77.022311) 83. “Josephson parametric oscillator based Ising machine,” *Physical Review B*. [https://link.aps.org/doi/10.1103/PhysRevB.109.014511](https://link.aps.org/doi/10.1103/PhysRevB.109.014511) 84. “Dynamically tuned arrays of polariton parametric oscillators,” *Optica*. [https://opg.optica.org/abstract.cfm?uri=optica-7-12-1673](https://opg.optica.org/abstract.cfm?uri=optica-7-12-1673) 85. “Power scaling of a non-resonant optical parametric oscillator based on periodically poled LiNbO3 with spectral narrowing,” ResearchGate. [https://www.researchgate.net/publication/388253283_Power_scaling_of_a_non-resonant_optical_parametric_oscillator_based_on_periodically_poled_LiNbO3_with_spectral_narrowing](https://www.researchgate.net/publication/388253283_Power_scaling_of_a_non-resonant_optical_parametric_oscillator_based_on_periodically_poled_LiNbO3_with_spectral_narrowing) 86. “Dynamically Tuned Arrays of Polariton Parametric Oscillators,” arXiv. [https://arxiv.org/abs/2003.01386](https://arxiv.org/abs/2003.01386) 87. “Optimal control in nearly adiabatic two-level quantum systems via time-dependent resonance,” *Physical Review A*. [https://link.aps.org/doi/10.1103/PhysRevA.111.L050201](https://link.aps.org/doi/10.1103/PhysRevA.111.L050201) 88. “AnalogCoder-Pro: Unifying Analog Circuit Generation and Optimization via Multi-modal LLMs,” arXiv. [https://arxiv.org/html/2508.02518v1](https://arxiv.org/html/2508.02518v1) 89. “Machine Learning Techniques and Optimization Approaches for Analog Validation and Testing,” ITESO. [https://rei.iteso.mx/items/49a75ce2-ad1a-4758-b36c-404c7ff442a0](https://rei.iteso.mx/items/49a75ce2-ad1a-4758-b36c-404c7ff442a0) 90. “Gaussian-Process-Based Surrogate for Optimization-Aided and Process-Variations-Aware Analog Circuit Design,” *MDPI*. [https://www.mdpi.com/2079-9292/9/4/685](https://www.mdpi.com/2079-9292/9/4/685) 91. “Experimental Evaluation of an Analog Gain Optimization Algorithm in Optical Camera Communications,” ResearchGate. [https://www.researchgate.net/publication/346315719_Experimental_Evaluation_of_an_Analog_Gain_Optimization_Algorithm_in_Optical_Camera_Communications](https://www.researchgate.net/publication/346315719_Experimental_Evaluation_of_an_Analog_Gain_Optimization_Algorithm_in_Optical_Camera_Communications) 92. “Analog FeedForward Equalizer Optimization in the Context of 50G-PON,” Zenodo. [https://zenodo.org/records/15119416/files/%5BJOCN_2025%5D%20Analog%20Equalization%2050G-PON.pdf?download=1](https://zenodo.org/records/15119416/files/%5BJOCN_2025%5D%20Analog%20Equalization%2050G-PON.pdf?download=1) 93. “Modeling, Simulation, and Experimental Validation of a Novel MPPT for Hybrid Renewable Sources Integrated with UPQC: An Application of Jellyfish Search Optimizer,” *MDPI*. [https://www.mdpi.com/2071-1050/15/6/5209](https://www.mdpi.com/2071-1050/15/6/5209)