Tatopenn/Dense-Evolution-Ising-Tests

🔬 Quantum Phase Transitions, Variational Gradients, and Error Mitigation

This repository contains a rigorous empirical study, raw datasets, and quantum error mitigation protocols executed on Dense Evolution—a high-performance Statevector quantum simulator. Utilizing 64-bit double precision (complex128) and hardware-accelerated static compilation via the JAX XLA engine, this project maps the non-linear physics of the Transverse Field Ising Model (TFIM) and Tight-Binding Fermionic dynamics.

📊 Repository Architecture & Ecosystem

• —`scan_ising.py`: Automated data pipeline responsible for high-resolution parameter sweeps and graphical rendering of the ideal ferromagnetic phase transition using a true variational ansatz.
• —`plot_ising.py`: Computes the first-order numerical derivative (quantum susceptibility) to locate the exact critical phase boundary.
• —`zne_mitigation.py`: Mathematical implementation of a stochastic Richardson Zero-Noise Extrapolation (ZNE) protocol over discrete Pauli-Z phase dephasing channels.
• —`vqe_gradient.py`: Exact numerical finite-difference gradient tracker mapping the variational energy landscape and locating stationary points.
• —`vqe_jax_grad.py`: Advanced VQE gradient execution computing the exact non-fictitious Parameter-Shift Rule over a massively parallel 10,500-track JAX batch array.
• —`quantum_defect_scanner.py`: Isotropic resilience topology mapper evaluating node-by-node quantum coherence under localized parameter-driven Kraus noise.
• —`next_gen_silicon.py`: Solid-state bandstructure designer tracking continuous dispersion shifts induced by mechanical lattice straining.
• —`test_manufacturing_formula.py`: Lattice thermodynamics simulator modeling electron-phonon scattering and decoherence via Bose-Einstein statistical distributions.
• —`vqe_silicon_molecular.py`: Variational Quantum Eigensolver tracking self-consistent Potential Energy Curves (PEC) and Born-Oppenheimer molecular dissociation limits.
• —`transizione_fase_ising.csv`: Raw tabular dataset capturing exact computational basis probabilities extracted directly from JAX memory slices.

🔬 Scientific Discoveries & Empirical Evidence

1. Quantum Phase Transition & Order Parameters

We present a rigorous physical validation of the longitudinal spin-correlation order parameter $\langle H{zz} \rangle$ governed by the 1D Transverse Field Ising Model Hamiltonian: $$H = -\sum{i} Zi Z{i+1} - g\sum{i} Xi$$ As the transverse field coupling strength $g$ sweeps from $0.0$ to $2.5$ over 3,500 high-resolution steps, the structural expectation value smoothly decays from an absolute ferromagnetic alignment of $+1.0000$ down to $+0.0050$. This continuous trajectory maps the exact critical boundaries where quantum fluctuations dismantle long-range magnetic ordering, steering the system toward a disordered paramagnetic regime. The critical phase transition boundary is resolved via quantum susceptibility metrics.

<p align="center"> <img src="transizionefaseising.png" alt="Quantum Ising Phase Scan and Susceptibility" width="85%"> </p>

2. Quantum Error Mitigation via Real Stochastic Richardson Extrapolation (ZNE)

To circumvent non-unitary noise without physical hardware overhead, a classical-quantum hybrid mitigation protocol was deployed under a realistic stochastic Pauli-Z dephasing Kraus channel. By scaling the noise density via stretching coefficients ($\lambda1 = 1.0, \lambda2 = 2.0$) over $2,000$ discrete hardware shots, a linear Richardson extrapolation was computed: $$E(0) = 2E(\lambda1) - E(\lambda2)$$ The ZNE protocol successfully reconstructed the unperturbed, zero-noise ideal target trajectory, respecting the fundamental physical bounds of the Hamiltonian energy operator without introducing non-linear artifacts.

<p align="center"> <img src="transizioneisingmitigata.png" alt="Stochastic Zero-Noise Extrapolation Results" width="85%"> </p>

3. Exact Multi-Particle Variational Optimization (VQE)

Utilizing a mathematically sound hardware-efficient excitation-preserving ansatz based on continuous Givens rotations, we tracked the accurate convergence profile of a single-electron state inside the crystal lattice. By maintaining strict Fock space conservation throughout the parameter optimization loop, the classical-hybrid optimizer successfully isolated the exact analytic minimum bound of the kinetic field: $$E{ground} = -2 \cdot t{hopping}$$

4. Parallel Quantum Defect Mapping via JAX Parallel Batching

Using the native run_parametric_batch_jit() engine, we mapped the isotropic resilience of an entangled state against localized dephasing noise. By altering the noise parameter along the matrix diagonal, JAX XLA compiled $12$ concurrent execution tracks in a single hardware cycle. The evaluation maps the systematic loss of $\langle X \rangle$ single-qubit coherence, capturing the directed noise-propagation properties across deep entangling layers.

<p align="center"> <img src="mappadifettisilicio.png" alt="True Quantum Defect Mapping Graph" width="85%"> </p>

5. Rigorous 1D Crystalline Lattice Dispersion

We resolved the exact 1-electron fermionic Bloch state dispersion relation mapped via Jordan-Wigner transformations. By evaluating the pure exchange interactions ($\langle Xi X{i+1} + Yi Y{i+1} \rangle$) and applying strict periodic boundary conditions (PBC), the engine resolves the full, continuous single-band cosine energy spectrum: $$E(k) = -2t \cos(k)$$ This eliminates artificial scaling factors and rigid offsets, delivering an honest statevector simulation of tight-binding quantum dynamics.

<p align="center"> <img src="bandesilicioibrido.png" alt="Rigorous Quantum Tight-Binding Dispersion" width="85%"> </p>

6. Analytical Gradients via Parallel Parameter-Shift Rule

To evaluate the variational optimization landscape with absolute machine-epsilon stability, we successfully deployed an analytical Parameter-Shift Rule framework mapped across parallel virtual execution tracks: $$\frac{\partial E}{\partial \theta} = \frac{1}{2} \left[ E\left(\theta + \frac{\pi}{2}\right) - E\left(\theta - \frac{\pi}{2}\right) \right]$$ By packing shifted parameters concurrently into run_parametric_batch_jit(), JAX XLA processed 10,500 continuous configurations in a macro-batch execution cycle of 0.86 seconds. The exact quantum derivatives successfully map continuous trajectories, verifying the total absence of vanishing gradient dead-zones or artificial plateaus under compact excitation-conserving ansatze.

<p align="center"> <img src="vqejaxgradient.png" alt="Exact Parameter-Shift Rule Gradients" width="85%"> </p>

7. Strained Silicon Bandstructure Engineering (3,500-Point Sweep)

We modeled a continuous dispersion profile mapping a high-mobility Strained Silicon configuration under a $5\%$ tensile strain ($\varepsilon = 0.05$). By perturbing the atomic equilibrium distances, the physical Hamiltonian undergoes an exponential inter-orbital hopping decay dictated by Harrison's law: $$t(\varepsilon) = t_0 \cdot \frac{1}{(1 + \varepsilon)^2}$$ The high-resolution 3,500-point k-space parameter sweep executed via JAX maps the physical contraction of the modal hopping energy from the standard $\pm 4.2200\text{ eV}$ limits down to the accurate engineered boundary of $\pm 3.8277\text{ eV}$ across the Brillouin zone.

<p align="center"> <img src="confrontonuovosilicio.png" alt="Strained Silicon Next-Gen Bandstructure" width="85%"> </p>

8. Molecular VQE and Potential Energy Dissociation Curves

We mapped the exact Born-Oppenheimer Potential Energy Curve (PEC) for a silicon dimer system via a classical-quantum hybrid variational loop. The effective Hamiltonian tracks electronic hopping integrals $t(R)$ alongside nuclear Coulomb repulsion fields $V{rep}(R)$ decaying over the interatomic coordinate: $$t(R) = t0 e^{-\beta(R - R0)}, \quad V{rep}(R) = V0 e^{-\gamma(R - R0)}$$ The 3,500-point variational sweep cleanly resolves the stable binding landscape, isolating the exact molecular equilibrium coordinates at $R \approx 3.557\text{ \AA}$ with a resolved bound state energy of $-0.273498\text{ eV}$ before steering continuously into the asymptotic free-atom dissociation limit.

<p align="center"> <img src="curvapotenzialesilicio.png" alt="Silicon Dimer Dissociation Curve" width="85%"> </p>

9. Quantum Lattice Thermodynamics & Debye Phonon Simulation

We evaluated the impact of lattice temperature $T$ on electronic conductivity by modeling acoustic phonon population metrics governed by the Bose-Einstein distribution function: $$nB(\omega) = \frac{1}{e^{\hbar\omega / kB T} - 1}$$ The high-resolution 3,500-point sweep from $10\text{ K}$ up to $400\text{ K}$ tracks the non-linear degradation of coherent inter-orbital hopping energy caused by scattering effects. This establishes an honest open-system quantum baseline mapping thermal resistance propagation directly onto active memory states.

<p align="center"> <img src="validazione_fabbricazione.png" alt="Quantum Lattice Thermodynamics Graph" width="85%"> </p>

🔍 Additional Investigation: Hunting Quantum Many-Body Scars

quantum_scar_investigation/ contains a self-contained, honestly-reported investigation into whether a "quantum many-body scar" (the non-thermalizing phenomenon first observed in 2017 Rydberg-atom experiments) shows up in Dense Evolution's frustrated Ising simulations. Full writeup: `report_indagine_scar.md` (Italian).

Short version: an initial-looking scar signature on a 4x4 frustrated TFIM grid did not survive rigorous verification (entanglement entropy, Trotter convergence, and a systematic 25-combination parameter scan) — it turned out to be the wrong observable (energy instead of entanglement entropy) plus a gauge-equivalence coincidence between sign patterns. The verification pipeline was then validated against the PXP model (Rydberg blockade), where scars are known to genuinely exist — confirmed via fidelity revivals and the characteristic "tower" of low-entanglement eigenstates in the exact spectrum. Using Dense Evolution's own NoiseModel.apply_to_sv (real stochastic Kraus channel, averaged over 30 quantum trajectories), the PXP scars turned out to be extremely fragile: a 0.5-1% per-site depolarizing error rate destroys almost the entire revival signal. Projecting the noisy state back onto the exact 13-state scar tower recovers ~31x of the lost revival amplitude — an idealized theoretical bound (not a realizable hardware protocol as-is) showing the protection target exists.

Open for anyone who wants to pick it up: translating the PXP dynamics into an actual circuit and testing revival + a physically realizable protection protocol (e.g. constraint-postselection instead of exact-eigenstate projection) on real quantum hardware.

⚙️ System Specifications & Reproducibility

• —Software Stack: Python 3.9+ | JAX (XLA Hardware Engine) | NumPy | Pandas | Matplotlib | SciPy
• —Memory Efficiency: Active Zero-Reshape memory architecture preserves absolute execution tracking under complex128 float layouts without memory leaks.