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Quantum Spectral Stabilization of Riemann Zeros: A Deformed Semiclassical Hamiltonian Model and Variational Quantum Simulation

Santiago Maroto

Published May 17, 2026• 0 citations
Publisher

Abstract

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).

DOI
10.5281/zenodo.20261023

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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).
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\phi^4ϕ4 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 introduce harmonic Hamiltonian relaxation (HHR), a variational spectral framework for locating ground states of operators defined on high-dimensional discrete lattices. The method embeds the discrete configuration space into a continuous Hilbert space equipped with a periodic modulation potential, transforming a combinatorial optimization problem into a smooth variational one. We construct an effective Schro ̈dinger operator whose localized ground state encodes the shortest non-zero lattice vector. Under a polynomial spectral gap hypothesis supported by numerical evidence for BKZ-reduced bases, the method admits an adiabatic optimization path compatible with gradient-based variational sampling. As an application, we show that HHR forms the computational core of the causal-quantum-spectral (CQS) framework and numerically reproduces arithmetic invariants of semistable elliptic curves, in numerical agreement with predictions of the Birch–Swinnerton-Dyer conjecture [Timakov, 2025, doi:10.5281/zenodo.17768557].
This study investigates the Variational Quantum Eigensolver (VQE), a hybrid quantum-classical algorithm that partitions the computational workload between a quantum processing unit and a classical optimiser to determine the ground-state energies of molecular systems. The Qiskit SDK v1.x framework is employed, utilising the PySCFDriver module within the qiskit-nature package to construct second-quantised molecular electronic Hamiltonians for two representative systems: Hydrogen ( ) and Lithium Hydride ( ). For the complete active space is retained without spatial reduction, mapping the full Hamiltonian onto a 4-qubit register. However for the full STO-3G representation demands 12 qubits, rendering direct classical parameter optimisation computationally intractable. This bottleneck is resolved by applying an ActiveSpaceTransformer (num_electrons=2, num_spatial_orbitals=3) to freeze the chemically inactive lithium core electrons, reducing the active register to 6 qubits. Fermionic operators are subsequently converted to Pauli strings via the Jordan-Wigner transformation ( ) enabling the quantum processor to evaluate the molecular Hamiltonian as a circuit while preserving the fermionic anti-commutation relations. Under ideal, noise-free simulation using the StatevectorEstimator primitive paired with the COBYLA optimiser (maxiter=200), all VQE energy points converge precisely onto the Full Configuration Interaction (FCI) classical baseline across the full range of interatomic separations. This confirms that the hardware-efficient TwoLocal ansatz ( rotation layers and entanglement layers with reps=3) possesses sufficient expressive power to capture the molecular ground-state wavefunction. Introducing gate-level noise through AerSimulator backends calibrated from the FakeManilaV2 and FakeNairobiV2 hardware profiles—with optimisation driven by the SPSA algorithm (maxiter=150)—reveals two critical failure modes. A systematic upward energy bias is observed, attributable to depolarising gate errors and measurement readout faults that prevent the optimiser from reaching the true minimum energy eigenvalue. Additionally, severe stochastic oscillations between 1.0 Å and 1.5 Å on the curve induce premature convergence of the SPSA optimiser to spurious local minima. These results demonstrate that unmitigated hardware noise renders VQE incapable of reliably predicting equilibrium bond lengths on current NISQ devices, motivating the future integration of Zero-Noise Extrapolation (ZNE), readout error mitigation, and Virtual Distillation into the computational pipeline. Keywords—Variational Quantum Eigensolver (VQE); Qiskit 1.x; Potential Energy Curve; Molecular Ground-State Energy; Hardware-Efficient Ansatz; Gate-Level Noise Simulation; ActiveSpaceTransformer; NISQ Devices.
We propose a scheme to restore spatial symmetry of Hamiltonian in the variational-quantum-eigensolver (VQE) algorithm for which the quantum circuit structures used usually break the Hamiltonian symmetry. The symmetry-adapted VQE scheme introduced here simply applies the projection operator, which is Hermitian but not unitary, to restore the spatial symmetry in a desired irreducible representation of the spatial group. The entanglement of a quantum state is still represented in a quantum circuit but the nonunitarity of the projection operator is treated classically as postprocessing in the VQE framework. By numerical simulations for a spin-1/21/21/2 Heisenberg model on a one-dimensional ring, we demonstrate that the symmetry-adapted VQE scheme with a shallower quantum circuit can achieve significant improvement in terms of the fidelity of the ground state and has a great advantage in terms of the ground-state energy with decent accuracy, as compared to the non-symmetry-adapted VQE scheme. We also demonstrate that the present scheme can approximate low-lying excited states that can be specified by symmetry sectors, using the same circuit structure for the ground-state calculation.
This project investigates how quantum bound states behave when a standard physical model,the one-dimensional Quantum Harmonic Oscillator (QHO),is exposed to an extreme asymmetric force modeled by an exponential potential. This perturbation breaks the spatial symmetry of the system and serves as an idealized model for studying quantum systems under intense localized external stress fields, such as strong gravitational acceleration or structural stresses in semiconductor nanostructures. The project successfully marries theoretical mathematical derivations with an automated Python simulation environment. Mechanistically, as the asymmetric strength parameter increases, the uniform energy levels characteristic of a standard harmonic oscillator begin to distort and converge. Eventually, the system crosses a precise mathematical threshold where the confining potential well disappears entirely. At this exact branch boundary, the excited states completely evaporate, and the ground-state wavefunction collapses tightly against the domain boundary.
We present a mathematical analysis of the one-dimensional infinitely deep square quantum well (1D IDSQW) within the paradigm of q-deformed quantum mechanics. The central technical contribution is a realization of the naive application of Sturm-Liouville theory (S-L theory): we prove that the eigenvalue parameter of the q-hypergeometric differential equation is a formal spectral parameter that does not directly correspond to physical en- ergies. Physical eigenvalues are extracted only after imposing q-Dirichlet boundary conditions on the Jackson lattice, which forces a truncation of the q-hypergeometric series, leading to q-quantized energy levels Eq,n = π2􏰀2 [n]2. We establish that the spectrum-generating algebra is the non- 2mL2 q compact quantum group SUq(1,1), whose positive discrete series D+(κ) furnishes the complete set of bound states. The theory is illustrated with precise numerical calculations of q-deformed quantum numbers, energy spec- tra, and wavefunctions, demonstrating the breakdown of the classical Sturm- Liouville eigenvalue-readoff technique.