Cross-Family Verification

• Cross-family verification is a set of methodologies comparing system behaviors across diverse system families to ensure correctness, security, and scalability.
• It uses protocols like LOCC and Bell-Basis Measurement in quantum systems and cross-model checks in LLMs to enhance efficiency and reduce bias.
• The framework also addresses pseudorandom sequence validation and formal verification in software, offering scalable benchmarks for heterogeneous architectures.

Cross-family verification refers to a suite of methodological frameworks for assessing correctness, consistency, or security by comparing behaviors, outputs, or properties across different system families, architectures, or modules. The term encompasses diverse domains, including modular quantum devices, pseudorandom sequence families, LLMs, independent quantum architectures, and variational software systems. Cross-family verification enables rigorous and scalable benchmarking, uncovering errors or failures that may be masked within a single class or platform, and is increasingly essential for modular, heterogeneous, or compositional system architectures.

1. Conceptual Foundations and Scope

Cross-family verification arises when independently implemented systems (families) or modules are evaluated for mutual consistency, correctness, or shared property satisfaction. The need for such techniques is especially acute in:

• Modular quantum information processing, where system integrity and scalability require reliable verification within and between heterogeneous modules (Dalton et al., 21 Jul 2025).
• Machine learning, notably LLMs, where independent solution verifiers are drawn from distinct model families to reduce bias and maximize filtering of incorrect outputs (Lu et al., 2 Dec 2025).
• Pseudorandom sequence pseudorandomness, where cross-family correlation measures ensure the absence of structural weaknesses between different code or sequence ensembles (Doğan et al., 2020, Winterhof et al., 2014).
• Quantum device benchmarking, where the output distributions over related computational tasks are compared across different physical implementations for trustworthy certification (Greganti et al., 2019).
• Family-based formal verification, where model checking or property verification is conducted across all variants/configurations of a variational system, often leveraging abstractions and projections to preserve correctness (Dimovski, 2018).

The overarching goal is to design protocols or metrics that remain robust to structural diversity and to detect any loss of performance, security, or correctness that may arise from inter-family differences or implementation biases.

2. Cross-Family Verification in Modular Quantum Systems

Resource-efficient cross-platform verification in modular quantum computing targets the certification of global quantum state preparation and entanglement between distinct modules (e.g., flip-chip superconducting architectures). The verification task is to estimate state overlap or fidelity,

$\mathcal F(\rho_A,\rho_B)=\frac{\tr(\rho_A\rho_B)}{\max\{\tr\rho_A^2,\,\tr\rho_B^2\}},$

between modules $A$ and $B$, each preparing, in principle, the same $n$-qubit state (Dalton et al., 21 Jul 2025).

Protocols and Complexity

• LOCC Protocol: Relies strictly on local operations and classical communication. This class includes quantum state tomography (QST) and randomized measurement protocols. Both scale exponentially in $n$, with empirical sample complexity $N_{\rm QST}, N_{\rm RM} \sim O(4^n), O(2^{bn})$ respectively. For $n=3$ qubits, $\gtrsim 1{,}200$–$1{,}500$ repetitions are needed for high-precision overlap estimation.
• Bell-Basis Measurement (BBM) Protocol: Utilizes an inter-module two-qubit gate per qubit pair, enabling a SWAP test or Bell-basis measurement. In the noiseless case, this achieves $O(1)$ scaling with $n$; under moderate noise, a mild quadratic increase emerges, but remains vastly superior to exponential scaling of LOCC. For $n=3$, BBM requires only $\approx 375$ shots, providing a fourfold reduction in resource requirements over LOCC.
• Scalability: As system size and fidelity improve, the BBM protocol's nearly constant sample complexity makes it the only viable route for large-scale benchmarking.

The critical insight is that inter-module quantum communication fundamentally transforms the resource requirements for end-to-end system verification, enabling modular scalability unattainable with classical-only protocols (Dalton et al., 21 Jul 2025).

3. Cross-Family Verification for LLMs

In contemporary LLM workflows, cross-family verification leverages verifier models from one family to judge the outputs of solvers from another, aiming to overcome self-enhancement bias and limited internal diversity (Lu et al., 2 Dec 2025).

Verification Settings and Metrics

• Self-Verification: Solver and verifier are the same model; gains are minimal for strong, post-trained networks due to overconfidence and sharply peaked output distributions.
• Intra-Family Verification: Verifier and solver are from the same family but differ in size or version; moderate gains, especially for weaker base models.
• Cross-Family Verification: Verifier and solver are from completely different model families (e.g., Llama3 vs Qwen3); yields the largest accuracy gains (2–5 percentage points, up to 7 for highly checkable tasks).

The key quantitative metric is verifier gain,

$$\mathrm{Gain}(S,V;\mathcal{D}) = \mathrm{Precision}(S,V;\mathcal{D}) - \mathrm{SolverAcc}(S;\mathcal{D}),$$

which captures the end-to-end accuracy improvement obtainable through verifier-based rejection sampling. Empirical studies show that cross-family verification consistently leads to the most reliable accuracy boosts and is especially effective on tasks with high inherent verifiability (e.g., puzzles, mathematics, logic).

Self-enhancement bias, whereby verifiers are more tolerant toward solutions similar to their own generative style, is substantially reduced by cross-family pairing, resulting in lower false positive rates and higher discrimination power (Lu et al., 2 Dec 2025).

4. Pseudorandomness and Cross-Correlation in Sequence Families

In the theory of pseudorandom sequence generation, cross-family verification measures the independence and unpredictability of multiple families with respect to both intrafamily and interfamily correlations.

Measures and Inequalities

• Cross-correlation measure of order $\ell$ for family $\mathcal{F}$:

$$\Phi_\ell(\mathcal{F}) = \max_{M, D, I} \left|\sum_{n=1}^M \prod_{j=1}^\ell e_{i_j,n+d_j}\right|,$$

with indices $I$ (sequence choices) and $D$ (shifts).

• $f$-complexity $C(\mathcal{F})$: The maximal integer $j$ such that all patterns of length $j$ occur in some sequence in the family.
• Canonical Inequality (Winterhof–Yayla):

$$C(\mathcal{F}) \ge \lceil \log_2 |\mathcal{F}| - \log_2\left(\max_{1 \le j \le \log_2 |\mathcal{F}|} \Phi_j(\mathcal{F}^\perp)\right) \rceil - 1.$$

Cross-family verification in this context demands that mixed-index autocorrelation sums between two sequence ensembles remain below theoretical bounds, which is crucial for spread-spectrum communications, multiuser cryptography, and other cryptographic primitives (Doğan et al., 2020, Winterhof et al., 2014).

Concrete constructions (e.g., shifted-polynomial and Legendre-symbol families) have been shown to attain low cross-correlation and large complexity simultaneously, both for binary and $k$-ary alphabets, ensuring robust pseudorandomness against both intra- and inter-family attacks.

5. Cross-Verification Across Heterogeneous Quantum Devices

Cross-family verification in quantum computation extends to the rigorous comparison of output distributions from structurally diverse platforms (e.g., NMR, superconducting circuits, trapped ions, photonic cluster states). The method leverages the flexibility of measurement-based quantum computation to generate related tasks whose output statistics are linked by exact theoretical relations (Greganti et al., 2019).

Protocol Overview

• Use a single entangled resource (graph state) and define several MBQC flows, each mapping to a different logical circuit (varying in size or structure).
• Run each circuit on its respective device (same or different physical platform); acquire output strings.
• Compare output distributions via sample-efficient collision statistics and the squared $\ell^2$-distance $\Delta_{12}$.
• Statistical proximity below threshold signifies physical-level consistency, independently of classical simulatability.

Practically, cross-family verification enables robustness checks between distinct technology stacks and mitigates vendor- or architecture-specific anomalies, providing a scalable alternative to classical simulation for benchmarking quantum processors (Greganti et al., 2019).

6. Cross-Family Model Checking in Software Product Lines

In formal methods, efficient cross-family verification is central to the quality assurance of variational or family-based systems, where a large set of configuration variants (feature sets) must simultaneously satisfy temporal properties (typically in CTL* logic) (Dimovski, 2018).

Methodological Components

• Featured Transition Systems (FTS) and Modal Featured Transition Systems (MFTS) provide a symbolic representation encompassing all product variants.
• Galois connections and variability abstractions enable conservative (may) and dual (must) abstractions, preserving the universal ($\forall$ CTL*) and existential ($\exists$ CTL*) properties, respectively.
• Divide-and-conquer partitioning permits the configuration space to be split along logical feature boundaries, with abstraction + projection ensuring that local verification suffices for global correctness.

Empirical results show that this family-based abstraction approach yields orders-of-magnitude speed-ups compared to naïve product-by-product analysis, while preserving rigorous property satisfaction guarantees across all system variants (Dimovski, 2018).

7. Synthesis and Practical Implications

Cross-family verification, as developed across quantum computing, machine learning, pseudorandomness, and formal verification, embodies several unifying principles:

Domain	Verification Target	Sample/Compute Scaling
Modular quantum (BBM)	State overlap/fidelity	$O(1)$–$O(n^2)$ (BBM); $O(2^n)$ (LOCC)
LLM pipelines	Solution correctness (accuracy)	$O(1)$ w.r.t. ensemble size; verifier gain predicts improvement
Sequence families	Pseudorandomness/correlations	Algebraic bounds via character sums; polynomial in length, degree
Quantum devices (MBQC)	Output distribution distance	$O(2^{\delta n})$ with small $\delta$, determined by flow overlap
Software product lines	CTL* property satisfaction	Polynomial via abstraction; exponential avoided via partitioning

The principal takeaways are:

• Cross-family verification systematically reduces the risk of systemic bias, hidden errors, or under-detection due to architectural or inductive similarity.
• Incorporating even minimal quantum or inter-family links (e.g., Bell-basis gates, verifier model diversity) yields dramatic improvements in efficiency, robustness, and scalability.
• Quantitative benchmarks (verifier gain, collision-based distance, cross-correlation bounds) provide rigorous metrics for certification and optimization of verification protocols.

These frameworks are now foundational for scalable benchmarking, certification, and quality assurance in large-scale, modular, or highly variable computational platforms.