This paper presents the application of Relational Field Theory (RFT) to random circuit sampling (RCS), a standard benchmark in quantum computing. Building on the reduced relational law: {\dot{R}}_c=-\lambda R_c+F_{int},\bigm and the operational coherence functional: R_c=\alpha\cdot\beta\cdot\gamma,\bigm the framework is applied to reported RCS data from superconducting qubit experiments. Within this setting: α corresponds to distribution sharpness (e.g., cross-entropy benchmark or peak statistics) β reflects circuit connectivity and gate structure γ captures persistence across circuit depth and repeated runs Using published results, the relational coherence functional is evaluated against observed behaviors in RCS systems, including: Persistence of structured output distributions beyond expected circuit depth Formation of coherence wells, where structure remains stable under noise Attractor-like clustering of high-probability output states Suppression of entropy relative to fully randomized expectations These observations are consistent with the four falsifiable predictions introduced in RFT v2.0: critical slowing near coherence thresholds hysteresis-like behavior under parameter variation persistence in coherence wells emergence of structural anchors This work does not modify quantum mechanics or attempt to reproduce standard RCS results. Instead, it provides a structural interpretation of observed behavior using a coherence-based framework applied to existing experimental data. The results extend Relational Field Theory from classical and toy-model systems into the quantum domain, supporting its potential as a cross-domain framework for understanding persistence, structure, and transition dynamics.
This paper presents the application of Relational Field Theory (RFT) to random circuit sampling (RCS), a standard benchmark in quantum computing. Building on the reduced relational law: {\dot{R}}c=-\lambda R_c+F{int},\bigm and the operational coherence functional: R_c=\alpha\cdot\beta\cdot\gamma,\bigm the framework is applied to reported RCS data from superconducting qubit experiments. Within this setting: α corresponds to distribution sharpness (e.g., cross-entropy benchmark or peak statistics) β reflects circuit connectivity and gate structure γ captures persistence across circuit depth and repeated runs Using published results, the relational coherence functional is evaluated against observed behaviors in RCS systems, including: Persistence of structured output distributions beyond expected circuit depth Formation of coherence wells, where structure remains stable under noise Attractor-like clustering of high-probability output states Suppression of entropy relative to fully randomized expectations These observations are consistent with the four falsifiable predictions introduced in RFT v2.0: critical slowing near coherence thresholds hysteresis-like behavior under parameter variation persistence in coherence wells emergence of structural anchors This work does not modify quantum mechanics or attempt to reproduce standard RCS results. Instead, it provides a structural interpretation of observed behavior using a coherence-based framework applied to existing experimental data. The results extend Relational Field Theory from classical and toy-model systems into the quantum domain, supporting its potential as a cross-domain framework for understanding persistence, structure, and transition dynamics.
Standard Random Circuit Sampling (RCS) predicts full thermalization at sufficient depth, with output distributions converging rapidly toward uniform noise under realistic hardware conditions. Yet several recent experimental results have displayed unexpected persistence of structured output distributions far beyond the expected decoherence horizon. This paper presents a vendor-agnostic benchmark demonstrating that such persistence is not an architectural anomaly, but a direct signature of a deep Rc coherence well as defined in Relational Field Theory (RFT). Rc wells arise when curvature-stiffened regions of the universal scalar field resist chaotic dispersion, enabling stable probability attractors even in circuits where standard quantum dynamics predict complete entropy saturation. We provide a minimal, architecture-independent circuit that any quantum hardware vendor can execute to detect Rc well behavior. The benchmark uses symmetric entanglers, global phase rotation, chaotic scrambling, and convergence layers to probe curvature-stiffened regimes. Four measurable signatures—XEB decay behavior, entropy suppression, HOG persistence, and inverse participation ratio (IPR)—constitute a clear, falsifiable Rc-well fingerprint. This work generalizes beyond any specific architecture and requires no specialized routing, collapse engineering, or deterministic control systems. It offers the first universal test for detecting Rc-governed coherence persistence and forms a reproducible foundation for cross-vendor validation of RFT predictions
This note presents an RFT interpretation of an anomalous coherence persistence effect reported publicly in 2025 RCS benchmark data. Standard quantum-circuit expectations predict full thermalisation on noisy 16-qubit superconducting hardware by depth ~30, driving XEB toward zero and eliminating internal structure. Instead, the publicly reported metrics indicated XEB > 4, HOG > 0.8, and a measurable entropy deficit, revealing structured attractors in a regime where standard models predict complete chaotic dispersion. Relational Field Theory (RFT) provides a natural theoretical explanation for this behavior: the reported anomaly matches the expected signatures of a system entering a deep curvature-stiffened coherence well in the universal scalar field 𝑅 𝑐 R c , allowing attractor structure and causal memory to persist far beyond the expected decoherence horizon. This analysis presents a minimal RFT interpretation based solely on publicly released benchmark metrics. Note: This analysis uses only publicly released circuit metrics. It makes no claims about proprietary architectures, internal mechanisms, trade secrets, or patented methods. No inference is made about the mechanisms or design intentions of QSCE or any other proprietary system.
Random circuit sampling (RCS) on noisy quantum hardware is widely modeled as exhibiting exponential decay of fidelity metrics (e.g., XEB, HOG) with circuit depth, converging toward Haar-random statistics beyond a characteristic noise-limited scale. This work introduces a minimal phenomenological recurrence for a coarse-grained local coherence observable defined on the qubit connectivity graph. The recurrence reproduces standard exponential decay in unstructured (fully scrambled) circuits but predicts delayed thermalization and a nonzero plateau when repeated local motifs are engineered into circuit compilation. The model introduces a single topology-dependent memory parameter α(µ), where µ denotes motif density. A transcritical bifurcation occurs when α(µ) exceeds the effective per-layer decoherence rate γ, producing a calculable plateau height and build-up depth scale. A matched-pair experimental protocol—holding qubit count, two-qubit gate count per layer, and reported hardware error rates fixed while varying only motif density—provides a direct falsification criterion. Absence of statistically significant topology dependence would rule out the model in its minimal form. The recurrence is an effective coarse-grained description at the metric level and does not modify underlying quantum channel structure or violate complete positivity.
This note presents a concrete, falsifiable prediction of Relational Field Theory (RFT) in a regime where standard open-system quantum treatments provide well-defined expectations. Consider a strongly driven open quantum system (e.g., superconducting qubit arrays or cavity-QED platforms) subject to dissipation. Under standard Lindblad or Floquet descriptions, such systems are typically expected to exhibit monotonic coherence decay or steady-state saturation without a revival mechanism of this form. RFT predicts a different outcome. Due to the adaptive domain closure T[R_c], the system’s effective response time increases in the low-coherence regime. This allows accumulated relational structure to persist long enough for the system to spontaneously re-enter a high-coherence state. The result is a non-monotonic coherence revival that arises without fine-tuning or additional parameters. Observable signatures include: Non-monotonic coherence lifetime (decay → plateau → revival) Hysteresis under drive-amplitude cycling Divergence of recovery time near a critical drive strength (critical slowing) A sharp increase in the time-quality factor Q_t=1/D_\theta The key discriminating feature is the combined appearance of these signatures. While individual behaviors may appear in nonlinear or driven systems, their joint occurrence under these conditions is not naturally predicted by standard open-system models. This prediction provides a direct experimental test of RFT. If high-precision measurements in the specified regime show only monotonic decay with no revival or hysteresis, the adaptive-closure mechanism of RFT would be ruled out. This work does not modify quantum mechanics. Instead, it proposes that coherence-constrained dynamics arising from relational persistence may constitute a cross-domain organizing structure that can be tested within existing experimental platforms.
This note clarifies the conceptual role of Relational Field Theory (RFT) and positions it as a cross-domain organizing principle rather than a domain-specific model or a replacement for existing physical theories. RFT introduces a minimal structure: a coherence field R_cgoverned by a persistence law, and an adaptive domain closure that makes memory state-dependent. From this minimal structure, the framework reproduces recurring dynamical signatures across multiple regimes, including: spontaneous threshold crossing, critical slowing, hysteresis, coherence wells, and persistent structured correlations under noise. The purpose of this paper is to show that RFT functions in a role analogous to thermodynamics or information theory: it does not replace the underlying dynamical laws of a system, it constrains the conditions under which coherence and temporal order can emerge within those dynamics, and it remains substrate-independent, with different domains corresponding to different projections of the same underlying structure. To illustrate this organizing role, the paper includes a numerical demonstration of coherence revival in a strongly driven open quantum system using the RFT adaptive kernel. This simulation is not presented as proof of the effect, but as a minimal demonstration that the framework generates the predicted qualitative behavior. A reproducible Python implementation is included in the appendix. This framing helps distinguish RFT from a constructed metric or simulation-specific method. Instead, it supports the interpretation of RFT as a structural constraint on persistence-driven coherence across classical, quantum, and complex systems.
Undesired coupling to the surrounding environment destroys long-range correlations in quantum processors and hinders coherent evolution in the nominally available computational space. This noise is an outstanding challenge when leveraging the computation power of near-term quantum processors1. It has been shown that benchmarking random circuit sampling with cross-entropy benchmarking can provide an estimate of the effective size of the Hilbert space coherently available2–8. Nevertheless, quantum algorithms’ outputs can be trivialized by noise, making them susceptible to classical computation spoofing. Here, by implementing an algorithm for random circuit sampling, we demonstrate experimentally that two phase transitions are observable with cross-entropy benchmarking, which we explain theoretically with a statistical model. The first is a dynamical transition as a function of the number of cycles and is the continuation of the anti-concentration point in the noiseless case. The second is a quantum phase transition controlled by the error per cycle; to identify it analytically and experimentally, we create a weak-link model, which allows us to vary the strength of the noise versus coherent evolution. Furthermore, by presenting a random circuit sampling experiment in the weak-noise phase with 67 qubits at 32 cycles, we demonstrate that the computational cost of our experiment is beyond the capabilities of existing classical supercomputers. Our experimental and theoretical work establishes the existence of transitions to a stable, computationally complex phase that is reachable with current quantum processors. By implementing random circuit sampling, experimental and theoretical results establish the existence of transitions to a stable, computationally complex phase that is reachable with current quantum processors.