Verification-Fidelity Scaling Law

• Verification-Fidelity Scaling Law is a quantitative framework that defines how verification error decreases as allocated resources increase, often following power-law or exponential relations.
• It applies across fields like scientific machine learning, quantum information, and inference-time reasoning, providing clear operational strategies for balancing cost and fidelity.
• Empirical studies demonstrate its practical impact by quantifying resource trade-offs and scaling transitions, enabling precise optimization of verification protocols.

A verification-fidelity scaling law describes the quantitative relationship between the resources allocated for verification—such as dataset size, computational cost, or number of measurement rounds—and the attainable verification fidelity or error for a learned model, quantum state, or inference pipeline. Such scaling laws arise across classical scientific machine learning, quantum information, and inference-time reasoning, providing explicit power-law or exponential relations between resource investment and the ability to verify correctness at prescribed precision. These laws provide not only predictive formulas for expected error at fixed resource levels but also operational guidance for optimally allocating cost, computational effort, or data fidelity when verifying complex systems.

1. Precise Definitions and Mathematical Formulation

Verification-fidelity scaling laws formalize how the fidelity (or conversely, the verification error) scales with key resource parameters. The general structure is process-dependent but typically takes a power-law or exponential form. For example, in the context of multi-fidelity neural surrogate datasets for CFD, the verification-fidelity scaling law is

$$E(C, \alpha) \approx A(\alpha) \, C^{-\beta(\alpha)}$$

where $E$ is the verification error (e.g., MSE evaluated on high-fidelity test data), $C$ is the total compute budget (e.g., core-hours), and $\alpha$ parametrizes the fraction of the budget allocated to high-fidelity simulations (Setinek et al., 3 Nov 2025).

In quantum information, verification-fidelity for an N-qubit register prepared at finite temperature scales exponentially: $$\mathcal{F}(N, \beta) = [1 + e^{-\beta \Delta E}]^{-N}$$ where $N$ is the system size, $\beta$ the inverse temperature, and $\Delta E$ is the energy gap (Buffoni et al., 2022).

For sampling-based search with self-verification (as in LLM inference), verification accuracy increases as a power law in both the number of generation samples $N$ and verifier calls $M$: $$\mathrm{Verification@}N = \mathrm{Pass@}N \times F(N, M), \quad F(N, M) \sim 1 - cN^{-\alpha} - dM^{-\beta}$$ with empirical $\alpha \approx 0.2{-}0.3$, $\beta \approx 0.4$ (Zhao et al., 3 Feb 2025).

In quantum state certification, the infidelity $\epsilon$ after $N$ optimal tests scales as $\epsilon \sim N^{-1}$ (Heisenberg scaling), while sub-optimal protocols saturate at the standard quantum limit $\epsilon \sim N^{-1/2}$ (Jiang et al., 2020).

2. Multi-Fidelity Trade-Offs in Scientific Machine Learning

Scientific ML settings involving expensive data-generation naturally motivate verification-fidelity scaling laws parameterized by data fidelity and compute cost. Setinek et al. (Setinek et al., 3 Nov 2025) introduce the explicit cost model: $$C = c_{\mathrm{LF}} n_{\mathrm{LF}} + c_{\mathrm{HF}} n_{\mathrm{HF}}$$ where $n_{\mathrm{LF}}$ (resp. $n_{\mathrm{HF}}$) is the number of low-fidelity (resp. high-fidelity) samples and $c_{\mathrm{LF}}, c_{\mathrm{HF}}$ their per-sample costs. The fidelity mix is

$\alpha = \frac{c_{\mathrm{HF}} n_{\mathrm{HF}}}{C}$

allowing dataset composition to be specified as a coordinate in $[0,1]$.

Empirical investigation reveals that the optimal allocation $\alpha^*(C)$ minimizing verification error $E(C, \alpha)$ transitions from low to high-fidelity dominance as the budget increases. Specifically, at low budgets ($C \lesssim 10^3$ h), best results are achieved with $\alpha^* \ll 1$ (mostly LF), whereas for very high budgets ($C \gg 10^3$ h), $\alpha^* \approx 1$ becomes optimal. This scaling law enables practitioners to deterministically choose data-generation strategies to minimize error given a verification tolerance, balancing the superior initial coverage of LF with the steeper error decay rate $\beta(\alpha)$ available from HF data (Setinek et al., 3 Nov 2025).

Parameter	Symbol	Typical Value
LF cost/sample	$c_{\mathrm{LF}}$	$4.8$ core-hours
HF cost/sample	$c_{\mathrm{HF}}$	$13.4$ core-hours
Optimal $\beta$	$\beta(\alpha)$	Not given (inferred from data)
Error scaling law	$E(C, \alpha)$	$A(\alpha) C^{-\beta(\alpha)}$

3. Verification-Fidelity Scaling in Quantum Systems

Quantum verification tasks, including state certification and process tomography, display a rich variety of scaling laws for fidelity as a function of system size and sampling resources. For multi-qubit initialization at finite temperature, the third law of thermodynamics yields the exponential scaling law

$$\mathcal{F}(N, \beta) = (1 + e^{-\beta\Delta E})^{-N}$$

implying that as the system size $N$ grows, the maximum achievable verification fidelity falls off exponentially unless the effective temperature is reduced exponentially, setting a fundamental limit for quantum computer scaling (Buffoni et al., 2022). Similar exponential decay governs verification for deep circuits with cumulative gate noise, as gate errors can be lumped into an effective decrease in $\beta$.

For quantum state verification (e.g., entangled photon pairs), optimal protocols exhibit Heisenberg scaling: $\epsilon(N) \sim N^{-1}$ contrasting with the standard quantum limit scaling $\epsilon \sim N^{-1/2}$ found in tomographic estimation. These scalings are realized by verification strategies leveraging locally projective or LOCC-adaptive measurements and are confirmed experimentally, with fitted exponents $r = -0.88\pm0.03$ (nonadaptive) and $r = -0.78\pm0.07$ (adaptive) (Jiang et al., 2020). In the case of maximally entangled states, the minimal number of tests needed to guarantee infidelity at most $\epsilon$ with significance $\delta$ scales as

$N \sim \frac{1}{\epsilon} \ln \frac{1}{\delta}$

across both adversarial and honest scenarios, with only modest overheads for measurement-parsimonious LOCC strategies (Zhu et al., 2019).

For continuous-variable bosonic channels, verification protocols via fidelity witnesses require a number of channel uses that scales polynomially in all parameters (number of modes $m$, maximum squeezing $r_{\rm max}$, target error $\epsilon$, confidence level $\delta$), i.e.,

$$N_U = O\bigl( m^3 \|S\|_\infty^4 \max \{ \|d\|^2, 1\} \sigma^2 / (\epsilon^2 \ln(1/\delta)) \bigr)$$

for unitary Gaussian channels, thereby enabling efficient verification for large CV systems without exponential blow-up (Wu et al., 2019).

4. Inference-Time and Sampling-Based Verification Scaling

Verification-fidelity scaling laws govern not only training or data-generation, but also test-time procedures in reasoning and control. In LLM inference via sampling-based search and self-verification (Zhao et al., 3 Feb 2025), increasing the number of candidate samples $N$ and the number of verifier calls per candidate $M$ yields

$$\mathrm{Verification@}N = \mathrm{Pass@}N \times F(N, M),\quad F(N, M) \sim 1 - cN^{-\alpha} - dM^{-\beta}$$

with $\alpha \approx 0.2{-}0.3$ and $\beta \approx 0.4$ for difficult benchmarks, and power-law scaling persisting well beyond the self-consistency regime. This indicates substantial accuracy gains from scaling verification, not merely generation. The effect is attributed to "implicit scaling," where sampling more candidates not only promotes greater solution diversity but also increases the probability of generating a high-quality, verifiable correct answer.

Analogous scaling laws have been demonstrated for vision-language-action (VLA) models in robotic manipulation, where the action error with $k$ sampled candidates and an oracle verifier obeys an exponentiated power law: $e(k) \approx a\,k^{b}$ with $a$ and $b$ in the range $a\sim 0.14{-}0.20$, $b\sim -0.06{-}-0.22$ depending on architecture and sampling method (Kwok et al., 21 Jun 2025). Pairing cheap diversification (e.g., Gaussian perturbation) with a fast learned verifier enables these scaling benefits in real-time robotic systems, with closed-loop task success rates rising logarithmically in the size of preference-comparison datasets used to train the verifier.

Pipeline	Error Scaling Law	Empirical Exponent(s)
LLM inference	$1 - cN^{-\alpha} - dM^{-\beta}$	$\alpha=0.2{-}0.3,\ \beta=0.4$
VLA action error	$a\,k^{b}$	$b=-0.06\ldots-0.22$

5. Quantum Criticality: Fidelity Scaling and Universality

In quantum many-body systems, "quantum fidelity" between ground states at neighboring parameter values can be used to extract critical exponents via scaling theory. For a $d$-dimensional system near a quantum critical point, the scaling law holds: $-\ln \mathscr{F} \sim L^d |\delta|^{\nu d}$ in the thermodynamic limit ($L \to \infty$, fixed $\delta$), where $\nu$ is the correlation length exponent (Adamski et al., 2015, Mukherjee et al., 2011). In 2D models, the presence of direction-dependent correlation lengths $\xi_i \sim \delta^{-\nu_i}$ introduces multiple scaling regimes; only when a single relevant exponent dominates do fidelity scaling methods yield accurate universal critical indices.

This scaling law provides an alternative to two-point correlation function asymptotics for extracting universality data, provided the critical behavior is isotropic and $d \nu < 2$. In the presence of multicriticality or anisotropy (as in certain Kitaev or pairing models), fidelity scaling alone is insufficient to resolve all exponents, and hybrid analysis becomes necessary.

6. Operational Implications and Best-Practice Recommendations

Verification-fidelity scaling laws furnish practitioners with concrete, quantitative strategies for optimizing verification workflows subject to limited resources:

• In scientific ML, allocate as much budget as possible to LF data at low compute budgets to maximize coverage, transitioning to HF once budget allows, following the empirically determined $\alpha^*(C)$ trajectory (Setinek et al., 3 Nov 2025).
• For quantum devices, efficient verification requires protocols achieving Heisenberg scaling (e.g., optimal or adaptive strategies), with requisite number of tests scaling inversely with target infidelity; overheads for LOCC or adversarial settings are bounded and minimal (Zhu et al., 2019, Jiang et al., 2020).
• In reasoning and control, substantial verification accuracy gains are possible by increasing the number of sampled candidates and exploiting efficient, potentially learned, verification mechanisms. Diminishing returns with respect to $M$ or $k$ can be computed directly from empirical exponents; practical regimes for $N, M$ in LLMs or $k$ in VLAs can be selected from scaling curves to achieve given error rates within runtime constraints (Zhao et al., 3 Feb 2025, Kwok et al., 21 Jun 2025).
• In quantum many-body and criticality studies, fidelity scaling enables clean extraction of universality class exponents provided system anisotropy and multicriticality are controlled (Adamski et al., 2015).

7. Limitations, Breakdown Regimes, and Future Outlook

While verification-fidelity scaling laws provide broadly applicable quantitative predictions, several breakdown and limitation regimes warrant attention:

• In multi-fidelity dataset construction, improper balancing at intermediate budgets may yield suboptimal error scaling, especially if downstream models are not able to effectively leverage LF signals (Setinek et al., 3 Nov 2025).
• In verification of quantum states, the scaling advantage may be lost if optimal measurement protocols cannot be implemented due to experimental restrictions or decoherence, and SQL scaling dominates in such cases (Jiang et al., 2020).
• In reasoning or control applications, the empirical exponents governing error scaling may saturate or degrade due to model limitations, distribution shifts, or overfitting in verifier training; further, computational latency imposes hard upper bounds on $N$ or $M$ in real-time settings (Zhao et al., 3 Feb 2025, Kwok et al., 21 Jun 2025).
• In quantum critical models with multiple divergent length scales or direction-dependent exponents, the fidelity scaling approach cannot unambiguously disentangle all relevant universality data (Adamski et al., 2015).

A plausible implication is that continued research into scalability of both verification protocols and underlying data/model architectures is essential to fully realizing the operational advantages predicted by scaling law theory. Extensions to other domains, including licensing of synthetic data, fine-tuning strategies, and control over adversarial verification, are ongoing areas of development.