The Hilbert-Pólya conjecture proposes that the non-trivial zeros of the Riemann zeta function correspond to the eigenenergy spectrum of a quantum system. Although experiments based on analog quantum probes have determined some low-order zeros, the classical computational bandwidth bottleneck—caused by synthesizing complex macroscopic driving waveforms—has severely hindered the extension of this approach to high-frequency spectra. Here, we propose a native digital quantum emulation framework that maps a continuous $\phi^4$ topological potential well into a Pauli tensor network based on first principles, physically deploying it on a real superconducting quantum processor (Rigetti Ankaa-3). Using a 3-qubit array, we accurately extracted the first non-trivial zero (14.134725). Addressing the "semiclassical aliasing" spectrum divergence triggered by limited local phase space capacity and depleted coherence time (~100 µs) when climbing to higher-order zeros, we introduce a classical-quantum hybrid Quantum Subspace Expansion (QSE) protocol. By injecting an adiabatic geometric phase into an expanded 8-qubit quantum manifold, the system effectively corrected the truncation divergence, accurately restoring the Weyl logarithmic density law (Mean Absolute Error MAE = 0.569), and physically quantified the decoherence leakage rate (~10%) and shot thermal drift on the actual hardware. This native digital method bypasses the computational paradox of classical pre-computation, predicting that in the future, only 14 fault-tolerant logical qubits will be needed to natively resolve the first 10,000 high-frequency zeros. This study provides a viable path for natively emulating quantum chaos on universal quantum architectures.
The Hilbert-Pólya conjecture proposes that the non-trivial zeros of the Riemann zeta function correspond to the eigenenergy spectrum of a quantum system. Although experiments based on analog quantum probes have determined some low-order zeros, the classical computational bandwidth bottleneck—caused by synthesizing complex macroscopic driving waveforms—has severely hindered the extension of this approach to high-frequency spectra. Here, we propose a native digital quantum emulation framework that maps a continuous ϕ4 topological potential well into a Pauli tensor network based on first principles, physically deploying it on a real superconducting quantum processor (Rigetti Ankaa-3). Using a 3-qubit array, we accurately extracted the first non-trivial zero (14.134725). Addressing the "semiclassical aliasing" spectrum divergence triggered by limited local phase space capacity and depleted coherence time (~100 µs) when climbing to higher-order zeros, we introduce a classical-quantum hybrid Quantum Subspace Expansion (QSE) protocol. By injecting an adiabatic geometric phase into an expanded 8-qubit quantum manifold, the system effectively corrected the truncation divergence, accurately restoring the Weyl logarithmic density law (Mean Absolute Error MAE = 0.569), and physically quantified the decoherence leakage rate (~10%) and shot thermal drift on the actual hardware. This native digital method bypasses the computational paradox of classical pre-computation, predicting that in the future, only 14 fault-tolerant logical qubits will be needed to natively resolve the first 10,000 high-frequency zeros. This study provides a viable path for natively emulating quantum chaos on universal quantum architectures.
We propose a novel, physically motivated computational approach to the Hilbert-Pólya conjecture by constructing and numerically simulating a deformed one-dimensional quantum Hamiltonian. While classical formulations of the Berry-Keating Hamiltonian suffer from unbound trajectories, we demonstrate that a spatial confinement based on a log-polynomial deformed potential of the form V(x) = c0 ln(x) + c1 x + c2 x^2 + c3/x yields a discrete, stable, and purely real energy spectrum. The eigenvalues of this stabilized system are successfully mapped to a 2-qubit virtual register, with ground state convergence verified via a Variational Quantum Eigensolver (VQE) and excited states resolved using Quantum Subspace Expansion (QSE).
Quantum computers promise an exciting path to the scalable simulation of quantum matter. However, in their nascent state, the accessible gate depth of quantum computation on available hardware remains highly constrained. In this thesis, I present a suite of hybrid quantum-classical algorithms that aid Hamiltonian simulation across both the Noisy Intermediate-Scale Quantum era and future fault-tolerant architectures. By treating classical computation as an active resource to refine quantum algorithms, I develop automated, variational methods to learn hardware-efficient circuits. This work aims to overcome traditional trainability barriers, such as barren plateaus, by embedding physical symmetries directly into the circuit design. Central to the thesis is the synergistic use of tensor networks to classically pre-compute weakly entangled stages of a quantum simulation. This enables the compression of deep circuits, drastically reducing both the accumulation of gate errors on near-term physical qubits and the prohibitive overhead of operations like magic state distillation on error-corrected hardware, reserving the quantum processor strictly for the most challenging, highly entangled dynamics. These methodologies are ultimately directed toward the ab-initio simulation of subatomic physics, providing a scalable approach for the first-principles study of strongly correlated systems, including nuclear structure and lattice gauge theories.
Dynamical quantum phase transitions (DQPT) provide a non-equilibrium framework for probing quantum criticality, where the non-analytic kinks in the Loschmidt echo rate function λ(t) signal the crossing of Fisher zeros across the real-time axis. In disordered many-body localized (MBL) systems, spatial inhomogeneity typically induces multi-frequency dephasing that melts these non-analytic signatures into diffuse crossovers. Here, we demonstrate that deterministic on-site potential landscapes derived from the unfolded non-trivial zeros of the Riemann zeta function prevent the thermalized melting of DQPT signatures through Gaussian Unitary Ensemble (GUE) level repulsion. While strong unengineered Riemann potentials (W = 2.0J) induce multi-frequency spectral beating resulting in dense multi-kink profiles, we formulate an augmented Lagrangian variational framework that directly targets the trade-off between isochronous periodicity and singular non-analyticity. By minimizing a multi-objective functional encompassing Fisher zero proximity, high-frequency beating penalties, and GUE level-repulsion kernels, the optimization converges to a dual-champion configuration h*. Under quantum quench dynamics of a one-dimensional transverse-field Ising chain (L=10, 1024-dimensional Hilbert space), this engineered landscape drives the Loschmidt echo down to the numerical machine-precision floor (L_min = 2.03e-13), generating an ultra-sharp super-kink with peak amplitude λ_max = 2.9219, nearly three times the baseline transition height. These results reveal a direct route for engineering singular non-equilibrium criticality in digital quantum processors.
Building upon the universal flow equations and infrared attractor dynamics established in the PROMETEO-PRIME framework, this paper addresses the role of high-frequency residual noise—traditionally filtered out by finite bond dimension truncation (χ=16) in Multiscale Entanglement Renormalization Ansatz (MERA) networks. By constructing a coupled differential system that reintegrates the orthogonal fluctuation sector with the macroscopic Gaussian Unitary Ensemble (GUE) bulk, we formulate an expanded block-matrix effective Hamiltonian. We regularize the extended Hilbert space via a non-separable Hermitian cross-coupling term, forcing the Schmidt spectrum to decay as an anomalous power law governed by a non-ergodic extended (NEE) fractal dimension D2≈0.8132, shielding the system from chaotic thermalization while preserving strict self-adjointness (Htotal=Htotal†). Empirical validation shows that the adjacent gap ratio stabilizes at the universal fixed point ⟨r⟩≈0.4678, verifying a topologically protected quantum phase that underpins the Hilbert-Pólya program and advances the spectral resolution of the Riemann Hypothesis.
The relentless pursuit of modeling quantum many-body systems has exposed a profound epistemic and technological schism in modern physics. As the dimension of the Hilbert space expands exponentially with each additional particle, direct computational modeling via classical state vectors becomes a physical impossibility. Concurrently, the physical realization of fault-tolerant Quantum Processing Units (QPUs) is stymied by persistent decoherence, fragile topological phases, and exorbitant error correction overheads. In response to this paradigm, a disparate array of mathematical techniques and classical computational heuristics have emerged, often mis-categorized under the broad, imprecise umbrella of "quantum computing" or "quantum simulation." This exhaustive report serves as the formal proposal and rigorous justification for the establishment of a new, distinct scientific discipline: Quantum Mathematics and Quantum Emulation. By synthesizing avant-garde mathematical constructs—including non-commutative geometry, von Neumann operator algebras, and the Connes-Kreimer Hopf algebra of renormalization—with High-Performance Computing (HPC) techniques such as the Density Matrix Renormalization Group (DMRG) and Projected Entangled Pair States (PEPS), this field systematically truncates quantum state spaces for deterministic emulation. Architected by Syed Abraham Ahmed, Founder and CEO of Lifeline AI, this document serves as the synergetic collection of prior art and modern innovation to create the field of Quantum Mathematics and Quantum Emulation. This document formally synthesizes these isolated mathematical and computational breakthroughs into a unified discipline by acknowledging the foundational prior art, in the noble pursuit of scientific integrity, accuracy, and logical justification. It is fundamentally necessary to correctly re-categorize these algorithmic efforts away from quantum hardware simulation and unify rooting them in rigorous mathematical compression—ensuring continued scientific merit and progress for distinguishing the accurate categorization of research funding mechanisms. Furthermore, the development of the novel field of Quantum Mathematics provides an actionable and definitive pathway to empower the future of scientific discovery, utilizing existing supercomputing constructs to bypass the limitations of physical quantum hardware, and champion the synergistic fusion of these existing elements to unlock entirely novel computational paradigms. Part 1 establishes the historical and taxonomic necessity of this classification, while Part 2 details the rigorous sub-Planckian algebraic architectures required for its execution. Ultimately, by rooting the existing siloed unification of Quantum Mathematics away from the physical Quantum hardware simulation, this structural re-categorization is a fundamental necessity for both scientific integrity and provides the computational pathway to bypass hardware limitations and compute the quantum realm using existing supercomputing powerhouse constructs.
The advancements in classical computational components such as CPUs and GPUs have had a reverberating impact on the democratization of computational power available, with the field of computational chemistry reaping significant benefits from the propulsion of the limits of systems studied. The accurate prediction of chemical properties and reaction dynamics through computational simulation is now a centerpiece of modern chemical research, significantly accelerating discoveries research areas of materials science and drug development, as well as industrially relevant processes such as the emission of aldehydes from soft foams, flame retardants and pyrolysis and the carbonylation of epoxides. These achievements of computational chemistry still are overshadowed in the face of fundamental bottlenecks where classical methods remain constrained by the exponential scaling when treating large and/or strongly correlated quantum many-body systems. This innate computational complexity often constitutes chemically accurate calculations based on first principles impractical, or plain impossible, for systems beyond modest sizes. Quantum computing offers a potential pathway around these limitations, originally formally formulated by David Deutsch as a tool for simulating quantum systems, with the first quantum algorithms indicating speed-ups compared to classical counterparts arising from Peter Shor and Lov Grover for the prime factorization of an integer number and unstructured search respectively. Relying on fundamental theorems by Claude Shannon, it has been demonstrated that through the use of error-correcting codes fault-tolerant quantum computers are able to safeguard information against noise and decoherence, strongly indicating towards exponential speedups in solving central computational chemistry problems, such as determining ground state energies or reaction pathways of chemical systems. Substantial strides in experimental quantum computing over the past decade, including proof-of-principle demonstrations of quantum error correction, mean to bring full-scale fault-tolerant quantum computing within the grasp of humanity. The breakthroughs needed to attain scalable, error-corrected quantum computation with capabilities exceeding those of classical methods in real-world scenarios is likely still some years away. While striving to address the engineering challenges standing in the way of achieving fault tolerant quantum computing, substantial effort has been focused toward developing and benchmarking hybrid algorithms tailored for the Noisy Intermediate- Scale Quantum (NISQ) devices, available now to researchers. These hybrid quantum-classical algorithms, with the most notable being the Variational Quantum Eigensolver (VQE), allow the probing of quantum computing’s utility in computational chemistry even in the face of significant hardware noise and limited coherence times. The work presented in this thesis introduces several key additions the VQE framework, geared especially towards to NISQ hardware, where the mindful use of available resources is imperative. First, the implementation of the Conjugate Model Search optimizer is described, an optimization method that integrates simple models that are surrogate to the original function, combined with conjugate gradient optimization to navigate effectively parameter landscapes typical of variational algorithms, prone to noise. Second, this thesis contains the proposal and extensive study of a hyperparameter tuning procedure employing genetic algorithms, allowing for comprehensive simulation benchmarks that augment the robustness and efficiency of VQE implementations with respect to convergence time through real wall-time models of quantum processors. Continuing, a bridge is presented between quantum simulation and quantum chemistry, through qubit-to-fermion mappings of existing quantum gates, further enabling the direct and accurate modeling of chemical systems on trapped atom quantum hardware with the added arsenal provided by techniques designed for qubit architectures. Additionally, experimental data contained in this thesis include detailed benchmarking data collected from state-of-the-art trapped ion quantum processors available at the Johannes Gutenberg Universität Mainz, demonstrating the actual viability and performance of various methodologies for extracting results. Lastly, the state of VQE is examined, as it served as a springboard for a plethora of advancements in the development of quantum computing hardware, as well as the direction the field of simulation of computational quantum chemistry via quantum devices is headed towards in the coming years.