**HARMONIC QUANTUM COMPUTING SYSTEM AND METHODS** **CROSS-REFERENCE TO RELATED APPLICATIONS** Not Applicable. **STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT** Not Applicable. **BACKGROUND OF THE INVENTION** **Field of the Invention** The present invention relates generally to the field of quantum computing, and more particularly to systems and methods utilizing continuous variable quantum information encoded in coherent resonant electromagnetic field states, often referred to as harmonic qubits, within engineered superconducting structures operating at deep cryogenic temperatures. **Description of the Related Art** Quantum computing seeks to harness quantum mechanical principles to perform computations intractable for classical computers. Various physical platforms are being explored, including superconducting circuits, trapped ions, photonic systems, and topological approaches. Superconducting circuits, such as transmons, flux qubits, and coupled resonators utilizing discrete energy levels, represent a prominent modality. These systems typically encode quantum information in discrete energy levels of artificial atoms (the qubits) fabricated on a chip. Resonant structures, such as 3D cavities or coplanar waveguides, are frequently coupled to these discrete qubits to facilitate control, measurement, and interaction. For example, US8642998B2 describes a quantum computer utilizing resonant modes of a 3D superconducting cavity to couple and interact with discrete superconducting qubits housed within the cavity. While significant advancements have been made with discrete superconducting qubits, scaling these systems to large numbers of highly coherent, interconnected qubits remains challenging. Issues such as qubit variability, sensitivity to fabrication imperfections, crosstalk between adjacent qubits, and sensitivity to various noise sources (electromagnetic, phonon, quasiparticle) limit performance, fidelity, and scalability. Maintaining long coherence times in large-scale discrete qubit systems is also a significant hurdle. Continuous variable quantum computing (CVQC) offers an alternative, encoding quantum information in the infinite-dimensional Hilbert space of quantum harmonic oscillators, such as modes of the electromagnetic field. Encoding qubits in specific, addressable coherent states or other states (e.g., Gottesman-Kitaev-Preskill (GKP) states) of these oscillators, sometimes referred to as "harmonic qubits" (h-qubits) when specifically referencing harmonic oscillator states, presents potential advantages in terms of error correction and inherent resilience to certain types of noise. However, realizing a practical, scalable CVQC system based on field states requires a physical medium capable of supporting a plurality of addressable, highly coherent resonant electromagnetic field states and enabling their precise manipulation and measurement at the deep cryogenic temperatures necessary for superconductivity and minimal thermal noise (typically below 100 mK). Existing resonant structures, while used in quantum computing (e.g., for coupling discrete qubits as in US8642998B2, or for parametric amplification), are generally not designed or optimized for using the field states *as* the qubits themselves within a complex, scalable architecture. The resonant mode spectrum, spatial distribution of modes, and material properties of such structures are typically not engineered to support the required density of highly coherent, addressable field states necessary for a functional h-qubit processor with low crosstalk. Specifically, achieving sufficient coherence times for field states requires minimizing energy loss mechanisms within the resonant medium. Furthermore, the materials used in these systems, particularly dielectric materials that support the electric fields of the resonant modes, may introduce significant decoherence at the millikelvin temperatures required for optimal performance of superconducting quantum systems. Dielectric losses, characterized by the loss tangent, are a major source of energy dissipation for resonant electromagnetic fields, especially at the low photon numbers relevant for quantum computation. These losses are significantly impacted by microscopic defects and phenomena like two-level systems (TLS) in dielectric materials at deep cryogenic temperatures. While some prior art discusses cryogenic dielectric materials (e.g., in the context of quantum chip interfaces, packaging, or bulk material characterization), these often focus on standard substrates (like sapphire or silicon) or materials not specifically formulated or tailored for conformally filling complex 3D resonant cavities and minimizing decoherence of field-state qubits operating at millikelvin temperatures. Achieving exceptionally low dielectric loss tangent (e.g., below 10^-6 or 10^-7) at these temperatures *within the resonant volume supporting the field states* is critical and non-trivial, as material properties can change dramatically at deep cryogenic temperatures due to phenomena like two-level systems (TLS). Standard cryogenic dielectrics may exhibit significant losses that render field states insufficiently coherent for quantum computation. Precise quantum control using shaped pulses is known in the context of manipulating discrete qubits (e.g., transitions between energy levels of transmons or flux qubits). However, applying such techniques to induce controlled non-linear interactions *directly with the continuous variable resonant electromagnetic field states* for quantum gate operations presents unique challenges. This requires tailoring control methods to the infinite-dimensional nature of the field states and engineering significant, controllable non-linearities into the resonant medium itself, fundamentally differing from approaches optimized for driving transitions between discrete energy levels. Generating non-linearities strong enough for fast gates on field states while maintaining coherence is a technical hurdle. Noise mitigation is critical for all quantum computing modalities. While various techniques exist for mitigating electromagnetic interference (EMI) through shielding or filtering, reducing phonon-induced decoherence (vibrational noise), and minimizing quasiparticle poisoning (non-equilibrium excitations in superconductors), an integrated, multi-modal approach specifically designed to simultaneously address these dominant decoherence channels *at the nanoscale* within the physical medium supporting field-state qubits, particularly at millikelvin temperatures, is not commonly found in the prior art. Systems for discrete qubits may address these issues individually or in different combinations tailored to their specific decoherence mechanisms (e.g., charge noise, flux noise), but an integrated solution targeting the specific decoherence channels relevant to coherent harmonic qubits encoded in resonant electromagnetic field states within a 3D resonant medium at millikelvin temperatures presents an unmet need. The interaction mechanisms between noise sources and field states in a 3D cavity are distinct from those affecting localized discrete qubits. Manufacturing complex 3D resonant structures with the precision required for supporting highly coherent, addressable field states with tailored mode spectra is difficult. Standard manufacturing quality control optimization techniques based on simple geometric tolerances or material composition analysis may not adequately capture subtle structural or material variations at the nanoscale that significantly impact sensitive quantum performance metrics like coherence time, frequency stability, or coupling strength of field-state qubits. Advanced analytical techniques are needed to bridge the gap between manufacturing parameters and quantum performance in a non-obvious way that accounts for the complex interplay of structure, materials, and quantum coherence. Characterizing the nanoscale environment of quantum systems operating at millikelvin temperatures is also extremely challenging. While cryogenic sensor systems exist for measuring temperature, magnetic fields, or general vibrations, a system specifically configured for sensitive, localized detection of single-phonon events to characterize the phonon environment affecting field-state qubits *within a resonant medium* is needed to understand and mitigate phonon-induced decoherence, which is a significant noise source at these temperatures for localized resonant modes supporting field states. Traditional methods measure bulk properties, not single-event interactions relevant to quantum coherence. Therefore, there remains a significant need for a quantum computing system and associated methods that effectively utilize coherent resonant electromagnetic field states as harmonic qubits within a specifically engineered physical architecture, addressing the unique challenges of coherence, control, noise mitigation, manufacturing, and characterization inherent to this approach, in a manner not anticipated or rendered obvious by prior art focused on discrete qubits or different CVQC implementations. The present invention provides a technical solution grounded in a unique physical implementation of the field-state qubit paradigm, overcoming the limitations of existing approaches by providing specifically designed and integrated components optimized for the unique requirements of maintaining and manipulating coherent field states as qubits at millikelvin temperatures. **BRIEF SUMMARY OF THE INVENTION** The present invention provides a novel quantum computing system and associated methods based on the harmonic qubit (h-qubit) paradigm, where quantum information is encoded in coherent resonant electromagnetic field states within a three-dimensional engineered resonant medium. This approach provides a technical solution to the problems of scaling limitations of discrete qubits, insufficient coherence of field states in general resonant structures, lack of tailored cryogenic materials with ultra-low loss, and fragmented noise mitigation strategies faced by prior art systems, by leveraging a specifically engineered physical architecture and tailored components for supporting and manipulating field-state qubits. In one embodiment, a quantum computing system comprises a three-dimensional superconducting lattice structure defining a plurality of interconnected resonant cavities. The geometric parameters and material composition of this lattice are precisely configured to support addressable, coherent resonant electromagnetic field states within the cavities, each state representing a harmonic qubit. This configuration provides a technical improvement by creating a stable, low-loss environment specifically optimized for supporting and maintaining the coherence of field-state qubits through engineered electromagnetic mode spectra (e.g., mode isolation, bandgaps), unlike prior art structures primarily designed for housing discrete qubits or for general RF/microwave applications. The system further includes a dielectric material substantially filling the resonant cavities, the dielectric material having a defined high dielectric constant and exceptionally low loss tangent at millikelvin temperatures (specifically below 10^-6, or even 10^-7), specifically tailored at the molecular or nanoscale level to minimize decoherence of the resonant electromagnetic field states by reducing the impact of two-level systems (TLS). This tailored dielectric provides a significant technical improvement by substantially reducing dielectric losses detrimental to field-state coherence compared to conventional materials at these temperatures, representing an unexpected result not suggested by prior art focused on different dielectric applications or less stringent loss requirements. A control system is configured to apply precisely shaped and timed modulated electromagnetic fields to the lattice structure to selectively manipulate the coherent resonant electromagnetic field states and perform quantum logic gates by inducing controlled non-linear interactions. A readout system is configured to measure properties of the resonant electromagnetic field states to determine a final state of the harmonic qubits. In another embodiment, a method for performing a quantum logic gate on one or more harmonic qubits encoded as coherent resonant electromagnetic field states within a three-dimensional resonant medium is provided. The method comprises applying a sequence of precisely shaped and timed modulated electromagnetic pulses to the resonant medium, wherein the pulse parameters are specifically calculated to induce a controlled, non-linear interaction directly between the applied fields and the target resonant electromagnetic field state(s) to effect a desired quantum gate operation while minimizing leakage to unwanted states, and maintaining coherence of the target resonant electromagnetic field state(s) during the gate operation through the inherent low-loss properties of the resonant medium and careful design of the applied control fields. This method provides a technical improvement in quantum control by enabling direct, high-fidelity manipulation of continuous variable field-state qubits via tailored non-linear interactions engineered into the system, a capability not readily achievable with control methods optimized solely for discrete energy level transitions. In yet another embodiment, an integrated noise mitigation system for a quantum computing device utilizing harmonic qubits is disclosed. The system comprises a physical medium configured to support the harmonic qubits and a plurality of nanoscale shielding structures integrated within or immediately adjacent to the physical medium. These shielding structures comprise a combination of photonic bandgap structures, phononic bandgap structures, and integrated quasiparticle traps, designed and spatially arranged at the nanoscale to simultaneously mitigate electromagnetic noise, phonon noise, and quasiparticle poisoning affecting the harmonic qubits encoded as resonant electromagnetic field states at millikelvin temperatures. This integrated, multi-modal nanoscale approach provides a synergistic technical improvement in noise reduction for harmonic qubits, addressing multiple critical decoherence channels concurrently in a manner not suggested by prior art focusing on individual noise mitigation techniques or applications for discrete qubits. Further embodiments include a method for optimizing the manufacturing process of the three-dimensional resonant medium using Topological Data Analysis (TDA) correlated with measured harmonic qubit performance metrics, providing a technical improvement in manufacturing yield and device performance by linking subtle structural or material variations at the nanoscale to critical quantum outcomes (like coherence time, frequency stability, crosstalk) in a non-obvious way that transcends traditional quality control metrics. A cryogenic sensor system for characterizing the resonant medium via sensitive detection of single-phonon interactions provides a technical improvement in environmental characterization by enabling highly sensitive, localized detection of phonon noise specifically affecting field-state qubits within the resonant medium at millikelvin temperatures, providing crucial empirical data for targeted noise mitigation. The present invention provides a comprehensive technical solution to the challenges of realizing a scalable, coherent quantum computer based on continuous variable field states, offering potential advantages in terms of coherence, error resilience, and scalability compared to systems based on discrete qubits, by defining and enabling a novel field-state qubit paradigm within a specifically engineered physical architecture and associated methods. **DETAILED DESCRIPTION OF THE INVENTION** In the following description, numerous specific details are set forth in order to provide a thorough understanding of the present invention. It will be apparent, however, to one skilled in the art, that the present invention may be practiced without these specific details. In some instances, well-known structures and devices are shown in block diagram form in order to avoid unnecessarily obscuring the present invention. The present invention provides a system and methods for performing quantum computation using harmonic qubits (h-qubits) encoded as coherent resonant electromagnetic field states. Unlike traditional superconducting quantum computers that use discrete energy levels of artificial atoms (e.g., transmons) as qubits, the present invention leverages the continuous variable nature of electromagnetic field modes within a specifically engineered resonant medium, treating these addressable modes as distinct computational units. This *field-state qubit paradigm* uniquely utilizes coherent resonant electromagnetic field states (and potentially other states within the harmonic oscillator manifold, such as GKP states) as the fundamental qubits within a precisely engineered physical medium and architecture, fundamentally differing from approaches based on discrete entities (such as charge, flux, or spin) or different continuous-variable quantum computing (CVQC) approaches by encoding quantum information directly in the quantized states of the resonant electromagnetic field itself within a structured, low-loss environment. A harmonic quantum computing (HQC) system comprises an HQC Processor, a Control System, a Readout System, and a Cryogenic System. The HQC Processor, which embodies the core quantum computational medium, is housed within the Cryogenic System to maintain the extremely low temperatures (typically millikelvin range, e.g., below 100 mK, such as 10-50 mK) required for superconductivity and minimal thermal noise, essential for maintaining the coherence of the harmonic qubits encoded in delicate field states. The Control System is coupled to the HQC Processor to apply precisely controlled electromagnetic fields for manipulating the harmonic qubits. The Readout System is also coupled to the HQC Processor to measure the states of the harmonic qubits. The HQC Processor includes a Three-Dimensional Superconducting Lattice Structure. The lattice structure is fabricated from superconducting materials, which may include conventional superconductors like Niobium (Nb) or Aluminum (Al), or High-Temperature Superconducting (HTS) materials in some embodiments (e.g., YBCO, BSCCO). The use of superconducting materials is critical for achieving the high quality factors (Q) necessary for long-lived resonant field states by minimizing resistive losses. The lattice defines a plurality of interconnected Resonant Cavities. The geometric parameters (e.g., size, shape, connectivity of cavities, lattice constant, periodicity, wall thickness, surface properties) and the material composition of the lattice structure are precisely configured *to engineer the electromagnetic mode spectrum*. This engineering is designed to support a plurality of addressable, coherent resonant electromagnetic field states within the cavities at specific frequencies, effectively defining the properties of the harmonic qubits (h-qubits) encoded in these states. Each of these supported, engineered field states represents a harmonic qubit. The precise geometry allows for control over resonant frequencies, spatial distribution of modes, coupling strengths between modes (for multi-qubit gates), and importantly, the creation of bandgaps or localized modes to isolate the h-qubit states from unwanted environmental electromagnetic noise and crosstalk from other modes. This engineered mode spectrum provides enhanced addressability and reduced coupling to lossy or unwanted modes compared to generic cavities. The lattice structure is fabricated using high-precision techniques such as additive manufacturing (e.g., 3D printing of superconducting inks or precursors followed by sintering or post-processing), subtractive manufacturing (e.g., deep reactive ion etching of bulk superconducting material, precision milling), or assembly of pre-fabricated superconducting components (e.g., precision-machined cavity blocks). The fabrication process is meticulously controlled to minimize defects (e.g., surface roughness below a critical threshold, material impurities, structural imperfections like voids or cracks, grain boundaries in polycrystalline materials) that can contribute to decoherence of the harmonic qubits encoded in the field states by introducing surface losses, volume losses, or TLS. In embodiments utilizing HTS materials, the fabrication process is specifically controlled to achieve a desired crystalline structure, minimize weak links (e.g., at grain boundaries), and minimize impurities, which is critical for maintaining high-quality superconductivity and low loss at millikelvin temperatures and optimizing harmonic qubit coherence and reducing crosstalk. The specific geometric configuration of the lattice, such as a periodic structure like a cubic lattice of interconnected waveguides, a diamond lattice, a gyroid structure, or a more complex photonic crystal-like structure, is optimized for enhanced harmonic qubit coherence, specific coupling schemes (e.g., nearest-neighbor, limited-range, or all-to-all connectivity depending on lattice topology), and reduced crosstalk between adjacent or coupled h-qubits encoded as coherent resonant electromagnetic field states. This precise engineering of the 3D structure provides a structurally and functionally distinct solution compared to prior art structures designed merely for housing discrete qubits or serving as general RF cavities. The Resonant Cavities within the lattice structure are substantially filled with a Dielectric Material. This dielectric material is crucial for defining the resonant frequencies of the cavities (by altering the speed of light within the medium) and supporting the electromagnetic field states that constitute the harmonic qubits. The dielectric material has a defined high dielectric constant (e.g., greater than 5, or even greater than 80) and, critically, an exceptionally low loss tangent (tan δ) at millikelvin temperatures (e.g., below 10^-4, or preferably below 10^-6, or even below 10^-7). The high dielectric constant allows for smaller cavity dimensions at a given resonant frequency, enabling higher qubit densities. The exceptionally low loss tangent at millikelvin temperatures is absolutely essential to minimize energy dissipation from the resonant field states, thereby preserving their coherence (characterized by high quality factors, Q = 1/tan δ). Dielectric loss is a dominant decoherence mechanism for resonant field states at these temperatures, largely attributed to TLS. In a preferred embodiment, the dielectric material is a specifically formulated cryogenic-compatible material, such as a cross-linked polymer with minimized TLS density, a specifically engineered ceramic powder dispersed in a low-loss matrix, or an ordered liquid/hydrogel-like material designed for stable operation at millikelvin temperatures. Such materials can be engineered at the molecular or nanoscale level to have tailored dielectric properties, including an exceptionally low loss tangent below 10^-6 or even 10^-7, specifically *to minimize decoherence of the resonant electromagnetic field states used as harmonic qubits* by actively reducing or mitigating the effects of TLS. The material must fill the complex 3D geometry conformally without introducing voids or defects. A hydrogel or ordered liquid, or a pre-ceramic polymer precursor, could potentially be introduced in a liquid or gel state after lattice formation and then solidified or cross-linked in situ. The use of such a specifically formulated material with tailored low-loss properties at millikelvin temperatures for minimizing decoherence of field-state qubits represents an unexpected result and provides a significant technical improvement over conventional dielectric materials used in cryogenic systems (which may have significantly higher loss tangents at these temperatures, e.g., 10^-4 or 10^-5), which are typically not optimized for this specific application and temperature range crucial for achieving high field-state coherence times (e.g., microseconds to milliseconds). The Control System is configured to apply precisely shaped and timed modulated electromagnetic fields (microwave or millimeter-wave pulses) to the lattice structure to selectively manipulate the coherent resonant electromagnetic field states (h-qubits) and perform quantum logic gates. The Control System includes components such as Arbitrary Waveform Generators (AWGs), Modulators, Pulse Shapers, Frequency Mixers, Cryogenic Signal Delivery Lines (e.g., coaxial cables with attenuators and thermal breaks), and Coupling Elements. The Coupling Elements (e.g., inductive loops, capacitive probes, waveguides) interface with the lattice structure to introduce the control fields into the resonant cavities. The Control System is configured to apply a sequence of precisely shaped and timed modulated electromagnetic pulses. The parameters of these pulses (e.g., amplitude envelope, phase, frequency, duration, polarization) are specifically calculated, often using optimal control techniques (e.g., GRAPE, Krotov), to induce a controlled, non-linear interaction *directly between the applied fields and the target resonant electromagnetic field state(s)* to effect a desired quantum gate operation (e.g., displacement gates, rotation gates, squeezing gates, Kerr-mediated gates, or multi-mode gates like a CZ gate). Examples of non-linear interactions that can be leveraged include Kerr non-linearity (Kerr effect), parametric driving, or coupling to an ancillary non-linear element (like a transmon capacitively or inductively coupled to a cavity mode, but used here specifically to mediate non-linear interactions between field states, not as the primary qubit). The non-linearity can be engineered into the system through the intrinsic properties of the superconducting materials (e.g., kinetic inductance non-linearity), the dielectric material, the geometry of the resonant structure, or by integrating specific non-linear circuit elements into the lattice structure at strategic locations. The pulse shaping and timing are precisely optimized to perform the gate operation efficiently (fast gates) while minimizing leakage of the field state to unwanted higher energy levels within the same mode or other modes within the cavity or adjacent cavities. During the gate operation, coherence of the target resonant electromagnetic field state(s) is maintained through the inherent exceptionally low-loss properties of the resonant medium (superconducting lattice and tailored dielectric) and the careful design of the applied control fields which minimize unwanted excitations or off-resonant driving. This method provides a technical improvement by enabling direct, high-fidelity control over the continuous variable field states themselves via tailored non-linear interactions and optimized pulse sequences, a capability not readily available or implemented in the same manner in discrete qubit systems which rely on addressing distinct energy levels and different types of interactions. The Readout System is configured to measure properties of the resonant electromagnetic field states to determine a final state of the harmonic qubits. The Readout System includes components such as Cryogenic Amplifiers (e.g., High Electron Mobility Transistors (HEMTs), Josephson Parametric Amplifiers (JPAs)), Demodulators, Analog-to-Digital Converters (ADCs), Digital Signal Processors (DSPs), and Coupling Elements. The Coupling Elements interface with the lattice structure to extract signals indicative of the h-qubit states. Measurement can be performed by coupling a probe signal to a cavity mode and detecting changes in its resonance properties (e.g., frequency shift, amplitude, phase) induced by the state of the h-qubit encoded in the field state (dispersive readout). The Cryogenic Amplifiers amplify the weak signals from the processor while adding minimal noise, operating at temperatures typically above the processor but still cryogenic (e.g., 4K or tens of mK). The Demodulators and Signal Processors extract the relevant information to determine the final state of the h-qubit, potentially involving classifying the measured signal state in the phase space of the field state. Non-demolition measurement techniques, such as those based on dispersive coupling to a readout resonator or spectral analysis of resonant mode states, are specifically adapted for the properties of the claimed system to minimize back-action on the fragile quantum states. The system can be configured for single-shot readout of the h-qubit states. To combat decoherence and noise, the HQC system incorporates an Integrated Noise Mitigation System. This system comprises a physical medium configured to support the harmonic qubits (i.e., the 3D superconducting lattice structure and the dielectric material) and a plurality of Nanoscale Shielding Structures integrated *within or immediately adjacent to* the physical medium supporting the harmonic qubits (e.g., embedded within the dielectric material, integrated into the lattice structure walls, or placed on surfaces). These shielding structures are designed at the nanoscale level to simultaneously mitigate multiple dominant sources of noise affecting the harmonic qubits encoded as resonant electromagnetic field states operating at millikelvin temperatures. The shielding structures comprise a combination of: * Photonic Bandgap Structures: Engineered structures (e.g., periodic arrangements of dielectric or metallic elements with characteristic dimensions on the order of the wavelength of the noise or the inverse of the wavevector) designed to create bandgaps for electromagnetic waves at specific frequencies or ranges, preventing unwanted environmental electromagnetic noise (e.g., thermal radiation, spurious RF signals) from coupling to the resonant cavities and affecting the field states. These structures are tailored to the frequencies of relevant noise sources and the operating frequencies of the h-qubits, creating forbidden frequency bands for noise propagation towards the qubits. * Phononic Bandgap Structures: Engineered structures (e.g., periodic arrangements of materials with different acoustic properties, like varying mass density or speed of sound, with characteristic dimensions on the order of the wavelength of the phonons) designed to create bandgaps for phonons (vibrational quanta), preventing environmental phonon noise (e.g., originating from vibrations in the cryostat, thermal fluctuations) from causing mechanical vibrations or energy loss in the lattice structure or dielectric material that could dephase the field states. These structures are tailored to the acoustic properties of the materials and relevant phonon frequencies at millikelvin temperatures, which are capable of coupling to the resonant field states through material strain or motion of cavity boundaries. * Integrated Quasiparticle Traps: Structures designed to capture non-equilibrium excitations in the superconductor (quasiparticles) that can cause energy loss and decoherence in superconducting resonant circuits. In embodiments where the physical medium comprises a superconducting structure (e.g., the lattice walls), these traps are strategically located *within or immediately adjacent to* superconducting components of the medium that support the resonant field states. The traps (e.g., regions of normal metal or superconducting material with a lower energy gap, geometric constrictions) have a geometry and material composition optimized for efficiently capturing quasiparticles diffusing within the superconductor before they can interact with the resonant field states and cause energy relaxation or dephasing. This is particularly important in superconducting cavities where quasiparticles interacting with the superconducting walls can cause significant loss. The design, material composition, characteristic nanoscale dimensions (e.g., less than 1 micrometer, commensurate with relevant wavelengths, coherence lengths, or mean free paths), and precise spatial arrangement of these nanoscale shielding structures are specifically configured to achieve simultaneous, synergistic mitigation of electromagnetic noise, phonon noise, and quasiparticle poisoning, which are major decoherence channels for superconducting systems operating at millikelvin temperatures and particularly impactful for coherent field states encoded in resonant modes. This integrated, multi-modal nanoscale approach provides a synergistic technical effect, resulting in significantly enhanced coherence times and fidelity for the harmonic qubits compared to systems employing individual noise mitigation techniques or less integrated approaches, a result not suggested by prior art. The nanoscale dimensions of these structures ensure that they are effective at the relevant length scales of the noise sources and the resonant field states. A Method for Optimizing the Manufacturing Process of the three-dimensional resonant medium for harmonic qubit quantum computing is also provided. The method begins with obtaining a dataset generated during the manufacturing process of the resonant medium. This dataset comprises detailed, high-resolution structural or material property data of the three-dimensional lattice structure and/or the dielectric material. Examples of data include high-resolution microscopy images (e.g., SEM, TEM, AFM), X-ray diffraction patterns, material composition analysis (e.g., EDS, XPS), residual stress measurements, or precise dimensional measurements obtained from inline or offline metrology tools. Topological Data Analysis (TDA) techniques are then applied to this rich dataset to extract quantitative shape-based or topological features indicative of manufacturing variations, defects, or complex structural characteristics in the resonant medium's structure or material properties. TDA techniques, such as persistent homology, Mapper, or clustering in persistence diagrams, can identify and quantify features like voids, inclusions, connectivity issues, variations in periodicity or anisotropy, surface roughness profiles, distribution of material phases, or network structures of defects that might not be readily apparent or quantifiable through traditional inspection methods, or whose complex correlation with quantum performance is non-obvious. The extracted shape-based or topological features are then statistically correlated with measured quantum performance metrics of the resonant medium. These metrics include harmonic qubit coherence time (T1, T2, Tphi), addressability (e.g., spectral overlap), coupling strength between modes, frequency stability, or gate fidelity, obtained from cryogenic characterization of the fabricated devices. For example, a specific topological feature identified by TDA related to the persistence of certain loops in the structure (indicating unintended connections or voids), or variations in the topological structure of the dielectric material distribution, or the connectivity of defects might be quantitatively correlated with a reduced coherence time, increased crosstalk between adjacent field states, or frequency drift. Based on this statistically significant correlation between manufacturing features and quantum performance, one or more manufacturing process parameters (e.g., printing speed, sintering temperature profile, etching parameters, material stoichiometry, curing time of dielectric) are adjusted in a feedback loop to optimize the quantum performance metrics of subsequently manufactured resonant media for harmonic qubit performance. This creates a data-driven feedback loop for continuous process improvement and yield enhancement, ensuring that subtle manufacturing variations detrimental to quantum performance are identified and mitigated, thereby solving a technical problem in achieving high yield and quality for complex quantum hardware in a non-obvious manner that links microscopic structural or material properties to macroscopic quantum behavior and performance. A Cryogenic Sensor System for characterizing the resonant medium for harmonic qubit quantum computing operating at millikelvin temperatures is also provided. The system comprises a Superconducting Resonant Structure configured to be coupled to the resonant medium and operate at millikelvin temperatures. This superconducting resonant structure (e.g., a high-Q superconducting cavity or resonator, such as a superconducting radio-frequency (SRF) cavity made of Niobium or Aluminum, a lumped element resonator, or a superconducting transmission line resonator) is specifically designed to be extremely sensitive to subtle energy dissipation mechanisms in its environment, particularly to interaction with individual phonons. A Measurement System is coupled to the superconducting resonant structure and is configured to detect minute changes in the resonance properties (e.g., frequency shift, linewidth broadening, quality factor reduction, changes in photon number) of the superconducting resonant structure. These changes are induced by interaction with single phonons originating from or interacting with the resonant medium supporting harmonic qubits. By coupling the sensitive superconducting resonator to the HQC resonant medium and detecting the subtle changes caused by individual phonon absorption or scattering events, the system enables single-phonon detection, which is a highly sensitive way to characterize the spectral and spatial properties of the phonon environment affecting the resonant electromagnetic field states (h-qubits). The coupling between the superconducting resonant structure and the HQC resonant medium can be achieved via near-field coupling (e.g., capacitive or inductive coupling through a small gap or antenna), direct mechanical coupling, or shared material components (e.g., the sensor resonator fabricated on the same substrate or integrated within the dielectric). The data obtained from this sensitive characterization provides crucial empirical information for understanding phonon-induced decoherence mechanisms affecting the harmonic qubits and guiding the design and optimization of noise mitigation strategies for the HQC system, such as the design parameters and placement of phononic bandgap structures, thereby providing empirical data for mitigating a critical source of decoherence. This system provides a technical improvement by enabling highly sensitive, localized characterization of a critical noise source for harmonic qubits at millikelvin temperatures, which was previously difficult to assess directly at the single-event level. Alternative embodiments may include variations in the specific superconducting materials used for the lattice structure, including different types of HTS materials or conventional superconductors optimized for specific frequency ranges or performance characteristics. The geometry of the lattice structure and cavities can be varied to optimize for different numbers of qubits, specific coupling schemes (e.g., tuneable coupling), or operating frequencies (e.g., microwave, millimeter-wave, Terahertz). The dielectric material could be a composite material comprising a low-loss matrix and embedded nanoparticles or structures to tailor dielectric properties or introduce non-linearity, or a different type of exceptionally low-loss cryogenic material tailored for millikelvin temperatures, such as specifically engineered polymers or ceramics with minimized TLS density. The control and readout systems can employ different coupling mechanisms and advanced signal processing techniques, including various methods for generating complex shaped pulses and analyzing resonant responses or field state Wigner functions. The integrated noise mitigation system could include additional types of shielding structures or traps, or different arrangements optimized for specific noise profiles, operating frequencies, or physical implementations. The manufacturing optimization method could incorporate other advanced data analysis techniques in addition to or instead of TDA, such as machine learning algorithms (e.g., deep learning on image data, Gaussian processes) trained on manufacturing data and quantum performance metrics. The cryogenic sensor system could utilize different types of superconducting resonant structures or alternative single-phonon detection mechanisms, such as transition-edge sensors (bolometers) integrated with the resonant structure, or superconducting nanowire single-photon detectors (SNSPDs) configured for phonon detection. The present invention provides a comprehensive approach to building a quantum computer based on harmonic qubits, addressing the fundamental challenges of coherence, control, noise, manufacturing, and characterization within a novel physical architecture. By leveraging a specifically engineered 3D superconducting resonant medium filled with a tailored cryogenic dielectric, employing precise non-linear control techniques tailored to field states, integrating multi-modal nanoscale noise mitigation, optimizing manufacturing through advanced data analysis techniques, and enabling sensitive cryogenic characterization, the invention offers a path towards scalable and fault-tolerant quantum computation based on the manipulation of coherent resonant electromagnetic field states. **CLAIMS** 1. A quantum computing system comprising: a three-dimensional superconducting lattice structure defining a plurality of interconnected resonant cavities, wherein the geometric parameters and material composition of the lattice structure are precisely configured to support a plurality of addressable, coherent resonant electromagnetic field states within the cavities, each state representing a harmonic qubit (h-qubit), said lattice structure being fabricated to minimize defects contributing to decoherence of said harmonic qubits and to engineer the electromagnetic mode spectrum to support said states and minimize crosstalk and coupling to unwanted modes; a dielectric material substantially filling the resonant cavities, the dielectric material having a defined high dielectric constant and an exceptionally low loss tangent at millikelvin temperatures (below 100 mK), specifically tailored to minimize decoherence of the resonant electromagnetic field states, said low loss tangent being below 10^-6; a control system configured to apply precisely shaped and timed modulated electromagnetic fields to the lattice structure to selectively manipulate the coherent resonant electromagnetic field states and perform quantum logic gates by inducing controlled non-linear interactions; and a readout system configured to measure properties of the resonant electromagnetic field states to determine a final state of the harmonic qubits. 2. The system of claim 1, wherein the three-dimensional superconducting lattice structure comprises High-Temperature Superconducting (HTS) materials arranged in a specific geometric configuration optimized for enhanced harmonic qubit coherence and reduced crosstalk when encoding harmonic qubits as coherent resonant electromagnetic field states, and wherein the fabrication process for the HTS materials is controlled to achieve a desired crystalline structure and minimize impurities and weak links. 3. The system of claim 1, wherein the dielectric material is a specifically formulated cryogenic-compatible material designed for stable operation at millikelvin temperatures and having tailored dielectric properties, including a loss tangent below 10^-7 at millikelvin temperatures (below 100 mK), to minimize decoherence of the resonant electromagnetic field states used as harmonic qubits. 4. A method for performing a quantum logic gate on one or more harmonic qubits encoded as coherent resonant electromagnetic field states within a three-dimensional resonant medium, the method comprising: applying a sequence of precisely shaped and timed modulated electromagnetic pulses to the resonant medium, wherein the pulse parameters are specifically calculated to induce a controlled, non-linear interaction directly between the applied fields and the target resonant electromagnetic field state(s) to effect a desired quantum gate operation while minimizing leakage to unwanted states; and maintaining coherence of the target resonant electromagnetic field state(s) during the gate operation through the inherent exceptionally low-loss properties of the resonant medium and careful design of the applied control fields. 5. An integrated noise mitigation system for a quantum computing device utilizing harmonic qubits encoded in resonant electromagnetic field states within a physical medium, the system comprising: a physical medium configured to support the harmonic qubits, said medium including a superconducting structure and a dielectric material filling portions of the superconducting structure; and a plurality of nanoscale shielding structures integrated within or immediately adjacent to the physical medium, the shielding structures comprising a combination of photonic bandgap structures, phononic bandgap structures, and integrated quasiparticle traps, wherein the design, material composition, characteristic nanoscale dimensions, and spatial arrangement of the nanoscale shielding structures are specifically configured to simultaneously mitigate electromagnetic noise, phonon noise, and quasiparticle poisoning affecting the harmonic qubits encoded as resonant electromagnetic field states at millikelvin temperatures (below 100 mK). 6. The system of claim 5, wherein the integrated quasiparticle traps are strategically located within or adjacent to superconducting components of the medium to mitigate quasiparticle poisoning of the resonant electromagnetic field states, said traps having a geometry and material composition optimized for efficiently capturing quasiparticles in the superconducting environment at millikelvin temperatures. 7. A method for optimizing the manufacturing process of a three-dimensional resonant medium for harmonic qubit quantum computing, the method comprising: obtaining a dataset generated during the manufacturing process of the resonant medium, the dataset comprising detailed structural or material property data of the three-dimensional lattice structure and/or the dielectric material; applying Topological Data Analysis (TDA) techniques to the dataset to extract quantitative shape-based or topological features indicative of manufacturing variations or structural characteristics in the resonant medium's structure or material properties; statistically correlating the extracted shape-based or topological features with measured quantum performance metrics of the resonant medium, the metrics including harmonic qubit coherence time, addressability, coupling strength, frequency stability, or gate fidelity of the harmonic qubits encoded as coherent resonant electromagnetic field states within the resonant medium; and adjusting one or more manufacturing process parameters based on the statistically significant correlation to optimize the quantum performance metrics of subsequently manufactured resonant media for harmonic qubit performance. 8. A cryogenic sensor system for characterizing a resonant medium for harmonic qubit quantum computing operating at millikelvin temperatures, the system comprising: a superconducting resonant structure configured to be coupled to the resonant medium and operate at millikelvin temperatures; and a measurement system coupled to the superconducting resonant structure, the measurement system configured to detect minute changes in the resonance properties of the superconducting resonant structure induced by interaction with single phonons originating from or interacting with the resonant medium supporting harmonic qubits, thereby enabling sensitive, localized single-phonon detection for characterizing the phonon environment affecting the resonant electromagnetic field states and providing empirical data for mitigating phonon-induced decoherence. 9. The system of claim 1, wherein the three-dimensional superconducting lattice structure has a periodic geometry selected from the group consisting of a cubic lattice of interconnected waveguides, a diamond lattice, and a photonic crystal structure, configured to engineer the electromagnetic mode spectrum. 10. The system of claim 1, wherein the dielectric material has a dielectric constant greater than 5 at millikelvin temperatures. 11. The method of claim 4, wherein the controlled, non-linear interaction is induced via Kerr non-linearity, parametric driving, or coupling to an ancillary non-linear element engineered within the resonant medium. 12. The system of claim 5, wherein the nanoscale shielding structures have characteristic dimensions less than 1 micrometer. 13. The system of claim 6, wherein the integrated quasiparticle traps comprise regions of normal metal or superconducting material with a reduced energy gap relative to the superconducting components of the medium. 14. The method of claim 7, wherein applying Topological Data Analysis (TDA) techniques comprises using persistent homology to quantify topological features in the dataset. 15. The cryogenic sensor system of claim 8, wherein the superconducting resonant structure is a superconducting radio-frequency (SRF) cavity, a lumped element resonator, or a superconducting transmission line resonator. 16. The system of claim 1, wherein the millikelvin temperatures are below 50 mK. **ABSTRACT OF THE DISCLOSURE** A quantum computing system and methods are disclosed, based on encoding quantum information in coherent resonant electromagnetic field states, termed harmonic qubits, within a three-dimensional superconducting lattice structure defining interconnected resonant cavities. The lattice geometry and material composition are precisely configured to engineer the electromagnetic mode spectrum supporting addressable, coherent harmonic qubits. A specifically tailored cryogenic dielectric material substantially fills the cavities, having a high dielectric constant and an exceptionally low loss tangent (below 10^-6, preferably below 10^-7) at millikelvin temperatures (below 100 mK) to minimize decoherence of the field states, particularly by mitigating the effects of two-level systems (TLS). A control system applies precisely shaped electromagnetic pulses to manipulate the harmonic qubits via controlled non-linear interactions engineered into the system for quantum logic gates. An integrated nanoscale noise mitigation system combines photonic bandgap structures, phononic bandgap structures, and integrated quasiparticle traps integrated within or adjacent to the physical medium to simultaneously mitigate multiple noise sources affecting the harmonic qubits at millikelvin temperatures. Methods for manufacturing optimization using Topological Data Analysis correlated with harmonic qubit performance metrics and a cryogenic sensor system for sensitive single-phonon detection to characterize the phonon environment affecting the field states are also provided. The invention offers a technical solution for realizing scalable, coherent quantum computation using field-state qubits within a specifically engineered physical architecture.