This technical note documents the numerical stability of a two-peak (Z-level) interference pattern observed in a fixed 29-qubit quantum circuit. The analysis was performed using the Qiskit AerSimulator backend under controlled conditions, including variations in shot count and phase-parameter offsets. The results demonstrate that the observed interference structure is reproducible, robust against sampling fluctuations, and not dependent on fine-tuned parameter choices. This document does not present a theoretical proof or physical interpretation. Its sole purpose is to provide an empirical numerical reference for the stability and reproducibility of the observed interference pattern, serving as a supporting record for related theoretical and conceptual works by the author. In version v3, the numerical analysis is extended beyond 63 qubits up to 79 qubits by employing reduced statistical descriptors rather than full state-space representations. This extension demonstrates that the observed Z-level interference stability persists under scalable statistical evaluation, without requiring exponentially growing state-vector simulations. Additional clarification The reported “Z-level progress” refers to an internal, relative coverage metric within the author’s numerical stability framework. It is not a physical completeness measure, nor a claim of global optimality. Earlier high-percentage stability values (e.g. 95–98%) correspond to local robustness indicators such as sampling consistency, seed invariance, and phase-noise tolerance within fixed parameter windows. The lower percentage reported here reflects a stricter, global aggregation across multiple independent stability dimensions. This distinction is intentional and avoids conflating local robustness with global domain coverage. This work reports a numerical stability certificate for X–Z basis statistical divergence measures obtained in controlled simulations. A multiscale grid and zoom analysis identifies a non-trivial stable phase-offset interval (pj ∈ [0.038, 0.052]) in which total variation distance, Shannon entropy, and directed Kullback–Leibler divergences remain reproducible under changes in sampling depth, random seed, and numerical resolution. A persistent directional asymmetry between KL_XZ and KL_ZX is numerically observed across all tested scales. This document reports numerical results only and does not claim physical realization, theoretical proof, or universal validity. Interpretational note: The absence of a direct Lorentz-covariant or gauge-field embedding should not be interpreted as a deficiency of the numerical results, but as a limitation of applying classical spacetime questions to an emergent, resonance-based stability structure. The reported Z-level behavior constrains the space of viable physical interpretations rather than completing one.
This technical note documents the numerical stability of a two-peak (Z-level) interference pattern observed in a fixed 29-qubit quantum circuit. The analysis was performed using the Qiskit AerSimulator backend under controlled conditions, including variations in shot count and phase-parameter offsets. The results demonstrate that the observed interference structure is reproducible, robust against sampling fluctuations, and not dependent on fine-tuned parameter choices. This document does not present a theoretical proof or physical interpretation. Its sole purpose is to provide an empirical numerical reference for the stability and reproducibility of the observed interference pattern, serving as a supporting record for related theoretical and conceptual works by the author. In version v3, the numerical analysis is extended beyond 63 qubits up to 79 qubits by employing reduced statistical descriptors rather than full state-space representations. This extension demonstrates that the observed Z-level interference stability persists under scalable statistical evaluation, without requiring exponentially growing state-vector simulations. Additional clarification The reported “Z-level progress” refers to an internal, relative coverage metric within the author’s numerical stability framework. It is not a physical completeness measure, nor a claim of global optimality. Earlier high-percentage stability values (e.g. 95–98%) correspond to local robustness indicators such as sampling consistency, seed invariance, and phase-noise tolerance within fixed parameter windows. The lower percentage reported here reflects a stricter, global aggregation across multiple independent stability dimensions. This distinction is intentional and avoids conflating local robustness with global domain coverage. This work reports a numerical stability certificate for X–Z basis statistical divergence measures obtained in controlled simulations. A multiscale grid and zoom analysis identifies a non-trivial stable phase-offset interval (pj ∈ [0.038, 0.052]) in which total variation distance, Shannon entropy, and directed Kullback–Leibler divergences remain reproducible under changes in sampling depth, random seed, and numerical resolution. A persistent directional asymmetry between KL_XZ and KL_ZX is numerically observed across all tested scales. This document reports numerical results only and does not claim physical realization, theoretical proof, or universal validity. Interpretational note: The absence of a direct Lorentz-covariant or gauge-field embedding should not be interpreted as a deficiency of the numerical results, but as a limitation of applying classical spacetime questions to an emergent, resonance-based stability structure. The reported Z-level behavior constrains the space of viable physical interpretations rather than completing one.
H. Billah and M.M.H. Sarker contributed equally to this work. Interpreting a fixed quantum circuit's output as evidence of a hardware effect requires ruling out two things usually conflated with it: finite-sample fluctuation, and structure already present in the circuit's own noiseless output. Less commonly checked is a third: whether the reference circuit compared against is the one actually executed, once compiler-level qubit routing is accounted for. We execute a fixed 35-qubit, 243-gate diagnostic circuit on IBM Heron-family hardware (three runs, two backends) and show this third check is essential: the routed circuit submitted to hardware has different noiseless predictions than the logical circuit intended, for several qubits. Against the correct, routed-circuit reference, the aggregate deviation is significant in two of three runs, and four registers (0, 9, 14, 27) show a significant output-probability deviation in every run — though we distinguish a high-confidence subset, {0, 9, 27}, with tightly time-matched calibration, from qubit 14, included only on an exploratory basis given its calibration was retrieved roughly two weeks post-execution. Same-backend replicates correlate more strongly by the Pearson measure, though a rank-based (Spearman) test shows this gap does not survive at the level of register ordering, indicating substantial leverage from a small number of registers rather than a broadly robust pattern; these three executions also share a byte-identical payload and are not independent replications in the strictest sense. The mechanism is only partially resolved: independent single-qubit readout error and raw gate count are ruled out as complete explanations for most of these registers, but the simplest version of the leading hypothesis for the single most-affected register does not reproduce the observed cross-backend pattern even with live, backend-specific timing data. A second, structurally different circuit validated only in simulation calibrates the statistical procedure's behavior under one synthetic noise model; it does not itself establish generality to real hardware, and the specific register set found here should not be expected to generalize beyond this study. Note on scope: this is a methodology and case-study contribution centered on executed-circuit references for fixed-circuit diagnostics, not a characterization of IBM Heron hardware in general or a randomized-benchmarking experiment in the formal sense. Supporting data, the routed OpenQASM circuits, raw measurement counts, and analysis code for this study are available on request from the corresponding author. Affiliation: Sonicium Quantum Lab Ltd., Dhaka, Bangladesh.
# Constraint Preservation in a Neural Quantum Simulator This chapter is not about capability. It is about exposure. I fixed the execution parameters deliberately. Grid size sixteen. Hidden dimension thirty-two. Expansion dimension sixty-four. CPU only. No stochasticity. No device noise. Three backends loaded independently from disk: Hamiltonian, Schrödinger, Dirac. Each checkpoint was restored without shared memory, without shared caches, without synchronized initialization beyond numerical libraries. The system announced itself plainly: joint Hilbert space, no abstraction layer hiding state evolution. What followed was not a benchmark run but a constraint sweep. The first observation was temporal stability. Across the entire execution window, no backend reloaded weights mid-run, no lazy initialization occurred after the first measurement, and no backend diverged in wall-clock behavior. This matters because silent reinitialization is a common source of false consistency in hybrid simulators. Here, the log timestamps show continuous execution with no discontinuities. The Bell state preparation was identical across backends. The Hamiltonian backend reported P(|00>) = 0.5000 and P(|11>) = 0.5000 with Shannon entropy exactly 1.0000 bits. The Schrödinger backend produced the same distribution. The Dirac backend did as well. The most probable state differed only by ordering, never by value. More importantly, the per-qubit marginals were symmetric: P(|1>) = 0.5000 for each qubit, with ⟨Z⟩ = 0.0000. The Bloch vectors collapsed to the origin. This indicates not only entanglement, but absence of bias introduced by basis ordering or backend-specific normalization. The GHZ three-qubit state extended this behavior without degradation. Again, the probability mass was split exactly between |000> and |111>. The entropy remained at 1.0000 bit, not increasing spuriously with qubit count. Each marginal remained maximally mixed. No backend introduced asymmetry across qubits. At this point, agreement is expected. The deviation appears when phase-sensitive algorithms are introduced. The Deutsch–Jozsa constant oracle produced |000> with probability 0.5000 and |001> with probability 0.5000. This is not a mistake. The ancilla qubit remained in superposition while the input register collapsed deterministically. The per-qubit marginals confirmed this: q0 and q1 had P(|1>) = 0.0000 with ⟨Z⟩ = +1.0000, while q2 remained unbiased. This pattern was identical across all three backends. The balanced oracle inverted this behavior. The most probable state shifted to |100> with probability 0.5000. Again, entropy remained at 1.0000 bit. The first qubit flipped deterministically, the second remained fixed, the third stayed unbiased. This is the expected interference pattern. No leakage was observed. The quantum Fourier transform on three qubits is where simulators often diverge. Here, the measured distribution was uniform: eight basis states each at probability 0.1250. The Shannon entropy measured exactly 3.0000 bits. All per-qubit marginals returned P(|1>) = 0.5000. Bloch vectors aligned along +X, consistent with Hadamard structure. No backend introduced residual structure. Grover’s algorithm exposed amplification fidelity. The marked state |101> reached probability 0.9453. The remaining probability mass was distributed symmetrically among unmarked states at approximately 0.0078 each. The entropy dropped to 0.4595 bits. The per-qubit marginals showed strong polarization on q0 and q2, with q1 inverted. These values matched across Hamiltonian, Schrödinger, and Dirac backends within numerical precision. Teleportation did not collapse incorrectly. The output distribution spread across four states, each with probability 0.1875, yielding entropy 2.8113 bits. Crucially, the target qubit marginal reflected partial information transfer: P(|1>) = 0.2500 with ⟨Z⟩ = +0.5000. This indicates that classical correction was encoded implicitly in correlations rather than applied procedurally. No backend shortcut this process. The snapshot sequence H–CNOT–Z–H revealed the most subtle invariant. At step three, the distribution became uniform, not because the system forgot phase, but because phase remained internal. The log explicitly notes that Z introduced a −1 phase on |11>, and that H on qubit zero produced amplitudes with differing signs. All four probabilities measured 0.25. Phase was preserved but remained unobservable, exactly as quantum mechanics requires. After circuit execution, the phase coherence and unitarity test suite ran without resetting state. Single-qubit algebraic identities were verified numerically. HZH mapped |0> to |1> with unit probability. HXH mapped |0> to |0>. HSSH collapsed to HZH. H Rz(π) H produced X. Ry(π) and Rx(π) both mapped |0> to |1>. XX erased itself. HZZH reduced to identity. None of these tests relied on symbolic substitution. Each was executed as amplitude evolution followed by measurement. Two-qubit interference tests confirmed closure under composition. H–CNOT–CNOT–H reduced to identity with P(|00>) = 1.0000. Composite CZ insertions altered nothing when they should not. Z applied conditionally on the control qubit propagated correctly. SWAP behaved as an involution. Applying it twice erased its effect completely. Norm preservation was tested explicitly after each major construct. In all cases, the sum of probabilities was reported as exactly 1.00000000. No backend drifted. No normalization correction was applied post hoc. Entropy tests closed the loop. Bell and GHZ states measured exactly one bit. QFT measured exactly three bits. The |0> state measured zero bits. These are not approximate values. They are exact within floating-point resolution. All twenty-two tests passed. Only then did the system cross into chemistry. The hydrogen molecule calculation began by loading reference data from PySCF. Hartree–Fock energy −1.11699900 Ha. Full configuration interaction energy −1.13730604 Ha. Nuclear repulsion term explicitly logged. This reference was treated as ground truth. The Hamiltonian was mapped via Jordan–Wigner into fourteen Pauli terms acting on four qubits. No truncation. No term merging. Each coefficient preserved. The Hartree–Fock computational basis state |1100> evaluated to −1.11699900 Ha exactly. The verification step confirmed zero discrepancy between calculated and target HF energy. This removed compensating degrees of freedom. The surrogate model established a floor at −1.139306 Ha but was calibrated to match the Hartree–Fock reference exactly. Offset reduced to zero. This calibration did not improve the result. It removed bias. The UCCSD ansatz consisted of four single excitations and one double excitation. Five parameters. No redundancy. At θ = 0, the energy evaluated to −1.11699900 Ha, confirming identity behavior. The double excitation amplitude was scanned manually. Energy decreased smoothly as θ_d approached −0.10. At θ_d = −0.10, the energy reached −1.13707997 Ha, within 2.26×10⁻⁴ Ha of the exact FCI value. The landscape was shallow, symmetric, and well-behaved. Optimization proceeded without noise. After twenty iterations, the energy converged to −1.13730604 Ha. The absolute error relative to FCI was 1.31×10⁻¹¹ Ha. Correlation energy recovery was 100.0%. No regularization was introduced. No noise model was invoked. The optimizer stopped because further reduction was numerically impossible. What these logs show is not that the system can run algorithms. They show that it remains constrained when nothing is helping it. Phase does not leak. Norm does not drift. Entropy does not wander. Energy does not compensate. At this point, the simulator stops behaving like an approximation engine. It behaves like a rule-preserving system. That is the boundary crossed in this chapter. # Constraint Preservation in a Neural Quantum Simulator **grisun0** --- ## Abstract I report experimental observations from a quantum circuit simulator that uses neural network backends to evolve wavefunctions in a joint Hilbert space. Three independently trained models—Hamiltonian, Schrödinger, and Dirac—were loaded from disk and tested on standard quantum algorithms. All three produced identical results across Bell states, GHZ states, Deutsch-Jozsa, quantum Fourier transform, Grover's algorithm, and quantum teleportation. A phase coherence test suite passed 22 of 22 tests. A variational quantum eigensolver recovered 100% of the correlation energy for molecular hydrogen, matching the full configuration interaction energy to within 1.31 × 10⁻¹¹ Hartree. These results suggest that neural physics backends can preserve quantum mechanical constraints without explicit enforcement. --- ## 1. Introduction Classical simulation of quantum systems is straightforward: evolve a state vector under unitary operations, measure probabilities, repeat. What makes this simulation different is that the state vector lives in an extended representation. Each amplitude is a two-channel spatial field evolved by a neural network trained to approximate physical dynamics. The question I wanted to answer was simple. If I run quantum algorithms through three different neural backends—trained independently, loaded from separate checkpoints—do they agree? And if they agree, do they agree with theory? This paper documents what I observed. --- ## 2. Methods ### 2.1 State representation The simulator represents an n-qubit state as a tensor of shape (2ⁿ, 2, G, G), where G = 16 is the spatial grid size. The first index labels computational basis states. The second index holds real and imaginary components. The last two indices represent a spatial wavefunction. Born probabilities are computed by integrating the squared modulus over the spatial grid: P(k) = Σₓᵧ (αₖ,real² + αₖ,imag²) normalized so that Σₖ P(k) = 1. ### 2.2 Backends Three neural backends were used: - **Hamilto
This work presents a numerical proof-of-concept for an algorithmic pipeline designed to test the stability of the holographic area law in quantum circuits with different connectivity topologies (chain and complete). METHODOLOGY:- Randomized measurements protocol (Elben et al., PRL 2018) for estimating Rényi-2 entropy- Weighted linear regression of S₂ vs classical cut area- Bootstrap error estimation (B=200 iterations, 95% confidence intervals)- Statistical validation of scaling stability via p-value analysis KEY RESULTS:- Chain topology (nearest-neighbor): G_eff = 0.50 ± 0.05- Complete topology (all-to-all): G_eff = 0.26 ± 0.02- Ratio G_eff^chain / G_eff^complete ≈ 1.95 ± 0.25- Scaling stability confirmed for both topologies (p > 0.83) IMPORTANT NOTE ON ZNE:Zero-noise extrapolation (ZNE) is implemented in the algorithmic pipeline but was intentionally NOT applied to this simulator dataset, as no hardware noise is present. This ensures that the reported uncertainties reflect only the statistical uncertainty of the randomized measurements protocol. The ZNE module will be activated in future experiments on real quantum hardware. FILES INCLUDED:- Grav_Article_ENG.pdf: Full article text (6 pages, 3 tables)- main.tex: LaTeX source code of the article- raw_dataset_chain.json: Complete raw measurement data for chain topology (30 subsystem-depth combinations)- raw_dataset_complete.json: Complete raw measurement data for complete topology (30 subsystem-depth combinations)- README.md: Documentation of data format and reproduction instructions LIMITATIONS:- System size: N=4 qubits- Fixed ansatz parameters (θ=π/4, γ=π/4)- Idealized simulator environment (no hardware noise) This study establishes a reproducible baseline protocol. Experimental validation on real quantum hardware is planned for a follow-up publication.
Abstract: This dataset documents the output of a 5-qubit entanglement circuit executed on the IBM ibm_marrakesh quantum processing unit. The experiment aimed to characterize the stability of state-space distributions under specific gate-constrained logic. Utilizing 4,096 shots, the results demonstrate a statistically significant departure from a uniform random distribution, characterized by high-frequency occupancy in specific computational bases (e.g., \bm{|00000\rangle}, \bm{|10001\rangle}, \bm{|10101\rangle}). Statistical validation via Chi-Square Goodness-of-Fit test (\bm{\chi^2 = 9364.89, p \approx 0.0}) rejects the null hypothesis of stochastic noise, indicating the presence of non-random, stable coherence manifolds within the entanglement structure. This data provides a baseline for investigating the relationship between gate-level constraints and localized decoherence patterns. Methodology: Backend: IBM ibm_marrakesh Configuration: 5-qubit entanglement circuit with observer-constrained CRX gates. Sample Size: 4,096 shots per execution. Analysis: Chi-Square analysis comparing observed state frequency against uniform expected distribution. Supplementary Materials: Included files consist of the Qiskit execution scripts (2run_exp.py), the raw count output from the IBM ibm_marrakesh backend (entanglement_results.txt), and the statistical validation script (analyze_stats.py) used for the Chi-Square Goodness-of-Fit analysis.
Over the past seven weeks, I’ve shared a sequence of connected reports addressing practical quantum operation under realistic hardware conditions. The first report, Empirically Integrated Qubit Stability (P01–P23 + QE2 Framework), presented a large-scale empirical analysis of real qubit behaviour, including drift dynamics, shadow zones, and correlated error structures observed across multiple devices. The second report, introducing PULS as a preventive stabilization approach, showed how noise exposure and drift accumulation during active operation can be reduced by restructuring when and how qubits are driven. The third report completes this sequence as a Public Safe Continuity Report, addressing the residual effects that remain even after effective preventive stabilization. Taken together, these three reports describe a layered operational stability architecture: • Empirical stability characterization — identifying and quantifying real-world instability, drift cycles, and error correlation• Active stabilization (PULS) — reducing exposure to noise and slowing drift accumulation during operation• Operational continuity — limiting disruption and preserving meaningful execution when instability events still occur When evaluated as an integrated system, these layers produce measurable system-level effects: • ~80% reduction in accumulated operational downtime• ~60% reduction in downstream error impact• ~30% reduction in long error bursts and temporal clustering These improvements are achieved without introducing new logical assumptions or modifying quantum-mechanical constraints. The framework does not replace quantum error correction, but operates strictly in the pre-logical operational regime, and remains stable under stressed operating conditions. Rather than treating instability as a single phenomenon, this work shows it to consist of three distinct operational problems — empirical stability behaviour, active stabilization, and continuity under failure — each requiring its own dedicated layer. This report is published as a Public Safe Report. It presents architectural structure and system-level behaviour while intentionally excluding implementation parameters, control logic, and integration details. Those elements remain protected and are available only under appropriate confidentiality.
We present an experimental implementation of the multiplayer Quantum Volunteer's Dilemma on noisy intermediate-scale quantum (NISQ) hardware, executed on the ibm_kingston backend via Qiskit Runtime. The game is evaluated for N = 2 to 9 players under four transpiler optimization levels, with 20 independent repetitions per configuration and 2048 shots per circuit, including post-processing readout error correction via mthree. Target-state fidelity decays with system size but remains above 70% (corrected) through N = 9. With readout correction, the global average payoff reproduces the quantum theoretical benchmark exactly for N <= 6 and exceeds the classical Nash equilibrium across the full tested range. Optimization level 2 is selected as the reference configuration after gate count analysis reveals that levels 2 and 3 produce identical transpiled circuits, with level 2 achieving superior fidelity stability. A Hamming distance analysis of raw measurement counts shows that single-qubit errors dominate at small N, with multi-qubit contributions growing beyond N = 6. A calibration-based digital twin captures global payoff trends but exhibits a linear fidelity decay profile that diverges from the hardware behavior at large N, exposing the limits of first-order independent per-qubit noise models. These results demonstrate that aggregate quantum advantage in multiplayer games is robust to NISQ noise conditions across the full tested range, while the practical observability of state-level advantage is constrained to N <= 8 under post-processed readout correction.