Approximate Clones in Diverse Domains

• Approximate clones are near-duplicate entities that retain sufficient structural similarity to an original while allowing controlled deviations defined by domain-specific criteria.
• They are quantified through diverse metrics such as quantum fidelity, total variation distance, rank span, and token similarity, each addressing the challenges of exact duplication.
• Cross-disciplinary studies emphasize that controlled approximation over exact cloning enables robust analysis in quantum information, statistical inference, social choice, RLHF, software mining, and biology.

Searching arXiv for recent and foundational papers on approximate clones across domains. “Approximate clones” is a cross-disciplinary term for objects that preserve enough of the structure of an original to be useful, while relaxing exact duplication, exact adjacency, or exact identity. In quantum information, approximate clones are imperfect copies of an unknown state, quantified by fidelity and constrained by the no-cloning theorem; in asymptotic classical statistics, they are synthetic outputs approximating additional i.i.d. samples in total variation; in social choice, they are candidates ranked close together rather than perfectly adjacently; in RLHF, they are near-duplicate alternatives in context space whose addition should not substantially perturb learned rewards; in software analysis, they are near-miss code fragments detected by token similarity; and in biology, they arise either as inferred latent subpopulations recovered only approximately from bulk data or as effective clonal structures under recombination and sampling limitations (Pelofske et al., 2022, Matsumoto, 2011, Faliszewski et al., 14 Sep 2025, Procaccia et al., 16 Jan 2025, Källén et al., 2020, Marass et al., 2016).

1. Domain-specific meanings and formal criteria

Across the cited literatures, the term does not denote a single universal object. Rather, each field fixes a space of objects, a notion of exact cloning, and a metric or constraint under which “approximate” becomes meaningful. This suggests that the unifying feature is not duplication per se, but controlled deviation from an exact clone notion.

Domain	Object being cloned	Approximation criterion
Quantum information	Unknown qubit state	Single-clone fidelity
Classical statistics	Product distribution $P_\theta^{\otimes rn}$	$L^1$/total variation distance
Ordinal elections	Candidate set or candidate pair	Rank span, $\alpha$-deletion, $\beta$-swap
RLHF	Alternative in context space	$\|x-x'\|_2 \le \epsilon$ and reward stability
Jupyter notebooks	Code snippet	SourcererCC similarity at 80%
Clonal biology	Latent or effective subpopulation	Posterior approximation or concentration functional

In the election literature of Faliszewski et al., a subset $X\subseteq C$ is a $(q,b)$-approximate clone if $|X|\le b$ and every voter ranks the members of $X$ within an interval of at most $b+q$ positions, equivalently allowing at most $L^1$0 non-members inside the span (Faliszewski et al., 14 Sep 2025). In the later pair-based treatment of Delemazure et al., approximate clone proximity is quantified either by the fraction of voters for whom two candidates fail adjacency, $L^1$1, or by the average number of adjacent swaps needed to make them adjacent, $L^1$2 (Delemazure, 28 Jan 2026).

In RLHF, approximate clones are induced geometrically: an added alternative $L^1$3 is an approximate clone of $L^1$4 when $L^1$5, under the assumption that annotator reward functions are Lipschitz in context space (Procaccia et al., 16 Jan 2025). In large-scale notebook mining, approximate clones are operationalized as near-miss snippet clones detected by SourcererCC at 80% token similarity after removing comments and blank lines (Källén et al., 2020).

Biological usage is less uniform. In Cloe, the recovered clones are explicitly described as a Bayesian approximation to an underlying clonal structure inferred from noisy bulk VAFs, rather than direct observations of true cell populations (Marass et al., 2016). In facultatively sexual populations, the relevant phenomenon is the emergence of large exact or effectively clonal genotypes despite recombination; the paper’s exposition treats genetically close descendants of a successful ancestor as “approximate clones” in a practical sense, especially at the haplotype or segment level (Neher et al., 2012). In clonality estimation, the emphasis shifts from recovering every clone to estimating the functional $L^1$6, the probability that two sampled individuals belong to the same clone (Liu et al., 2014).

2. Approximate cloning in information theory

In quantum information, approximate cloning begins with the no-cloning theorem: there is no unitary $L^1$7 and fixed blank state $L^1$8 such that $L^1$9 for all unknown pure states $\alpha$0 (Pelofske et al., 2022). Approximation enters by replacing exact output copies with reduced states that are close, but not identical, to the input.

For universal, symmetric, optimal qubit cloning from $\alpha$1 inputs to $\alpha$2 outputs, the single-clone fidelity bound is

$\alpha$3

For the experimentally realized $\alpha$4 case, this gives

$\alpha$5

The telecloning construction combines optimal cloning with teleportation: a multipartite entangled resource state is prepared on ancilla, port, and clone qubits; the sender performs a Bell measurement on the message and port qubits; and classical communication triggers local Pauli corrections on the clone registers (Pelofske et al., 2022).

The 2022 implementation realized a $\alpha$6 universal, symmetric, optimal telecloning protocol on the Quantinuum H1-1 device. The $\alpha$7 circuit used 19 qubits, had depth 182, and, after optimization, used 211 CNOT gates rather than 330 in the previous construction, a reduction of about 33% while keeping the same circuit depth under all-to-all connectivity. For four tetrahedral input states, the overall mean clone fidelity across all 9 clones and 4 states was 0.59, above the trivial 0.5 benchmark for a maximally mixed qubit but below the ideal $\alpha$8 bound, as expected under noise (Pelofske et al., 2022).

The classical asymptotic theory is structurally parallel but mathematically different. Given $\alpha$9 i.i.d. samples from an unknown member $\beta$0 of a regular parametric family, an $\beta$1-clone is a Markov kernel that outputs a law approximating $\beta$2. The error is measured in $\beta$3, equivalently total variation up to a factor of two. Under local asymptotic normality, the asymptotic local minimax loss is universal: $\beta$4 Hence the asymptotic difficulty depends only on the cloning factor $\beta$5, not on the specific regular family $\beta$6 (Matsumoto, 2011).

Taken together, these two literatures show two distinct senses of approximate cloning. In the quantum case, approximation compensates for a fundamental impossibility of perfect copying of unknown states. In the classical asymptotic case, perfect synthetic regeneration is limited by finite information about $\beta$7, and the residual loss is characterized by a Gaussian comparison limit.

3. Approximate clones in elections and preference learning

In ordinal elections, perfect clones are candidates that every voter ranks consecutively. Approximate variants relax consecutiveness. Faliszewski et al. define a subset $\beta$8 to be a $\beta$9-approximate clone if $\|x-x'\|_2 \le \epsilon$0 and every voter ranks the members of $\|x-x'\|_2 \le \epsilon$1 within a span of at most $\|x-x'\|_2 \le \epsilon$2 positions: $\|x-x'\|_2 \le \epsilon$3 This allows up to $\|x-x'\|_2 \le \epsilon$4 “intruding” candidates inside the block (Faliszewski et al., 14 Sep 2025).

On the algorithmic side, the problems are sharply stratified. Deciding whether there exists a $\|x-x'\|_2 \le \epsilon$5-approximate clone of size exactly $\|x-x'\|_2 \le \epsilon$6 is NP-complete, but fixed-parameter tractable in $\|x-x'\|_2 \le \epsilon$7. Partitioning candidates into approximate clones is NP-complete even when $\|x-x'\|_2 \le \epsilon$8 or when the number of parts is $\|x-x'\|_2 \le \epsilon$9; it is also NP-hard for a constant number of voters, specifically $X\subseteq C$0. Positive results exist for $X\subseteq C$1 and for parameterization by the imperfection level $X\subseteq C$2 (Faliszewski et al., 14 Sep 2025).

Delemazure et al. replace set-based span constraints by pairwise proximity measures. Two candidates are $X\subseteq C$3-deletion clones if deleting at most an $X\subseteq C$4 fraction of voters makes them perfect clones; they are $X\subseteq C$5-swap clones if the average number of adjacent swaps per voter needed to make them adjacent is at most $X\subseteq C$6. For elections with at least four candidates, none of IRV, Ranked Pairs, or Schulze are independent of approximate clones in the general case, despite their known independence of perfect clones. The paper obtains more positive results for the case of three candidates and empirically finds that approximate clones are common in some real datasets, while the closer two candidates are to being perfect clones, the less likely their removal is to change the outcome (Delemazure, 28 Jan 2026).

The RLHF literature imports the clone idea from social choice into preference aggregation over alternative responses. Procaccia et al. define robustness to approximate clones as follows: for every $X\subseteq C$7, there must exist an $X\subseteq C$8 such that adding an alternative $X\subseteq C$9 within $(q,b)$0 of some existing $(q,b)$1 changes the learned rewards on all original alternatives by at most $(q,b)$2, and also ensures $(q,b)$3. Under representative data and Lipschitz annotator rewards, the standard regularized MLE objective fails this property, while a weighted MLE based on Voronoi-style weights $(q,b)$4 guarantees robustness to approximate clones (Procaccia et al., 16 Jan 2025).

The weighted MLE modifies both the likelihood and regularization terms by weights derived from how much of the ambient context space is closest to each observed alternative. Intuitively, near-duplicate alternatives split the influence of a local region instead of multiplying it. The paper also proves a weighted Borda-like characterization and reports a synthetic “Describe Paris” study in which adding “restaurants” responses, treated as near-duplicates of “food,” changed the topic ranking under standard MLE but left the weighted MLE ranking stable (Procaccia et al., 16 Jan 2025).

A common misconception is that clone independence for exact clones automatically extends to approximate clones. The election and RLHF papers jointly show that this is false: exact clone-independence may survive at the axiomatic level, while small deviations from exactness can still alter winners or learned rewards.

4. Approximate clones in software repositories

In notebook mining, approximate clones are near-miss code clones rather than exact duplicates. The large-scale GitHub study treats each code cell in a Jupyter notebook as a snippet and, for Python notebooks, detects approximate clones with SourcererCC at 80% similarity after removing comments and blank lines (Källén et al., 2020).

The scale of the corpus is unusually large: 2.7 million Jupyter notebooks, 37 million snippets, and 227 million lines of code overall; for Python specifically, 2,592,892 notebooks and 35,450,312 snippets were analyzed. Of the Python snippets, 3,673,406 had 0 source lines of code after removing comments and blank lines and were excluded from approximate-clone analysis. Among the remaining snippets, 25,339,340 had at least one approximate clone, corresponding to 79.74% of analyzed Python snippets (Källén et al., 2020).

The notebook-level distribution is similarly concentrated. Among Python notebooks, 744,548 notebooks, or 28.71%, contained at least one approximate clone from themselves; 1,258,545 notebooks, or 48.54%, consisted solely of near-miss clones; and 140,215 notebooks, or 5.41%, contained only unique snippets. Approximate clones were usually short: the median size was 2 lines, the mean was 5.53 lines, and the 90th percentile was 13 lines. Clone frequency had a weak but statistically significant positive correlation with notebook size, with Spearman $(q,b)$5 and $(q,b)$6 (Källén et al., 2020).

The study also separates intra-repository from cross-repository propagation. Aggregate clone links between different repositories were more common than within the same repository, but the single repository with which a notebook shared the most approximate clones was typically its own repository (Källén et al., 2020). Qualitatively, common approximate clones included import blocks, dataframe inspection commands, print-based exploratory cells, and copied-and-patched tutorial or assignment fragments.

Here the approximation criterion is entirely operational. Unlike the election or RLHF settings, where the notion is tied to invariance or robustness, notebook approximate clones are detector-defined similarity classes. This suggests a different use of the concept: not a normative property to be preserved, but a descriptive regularity to be measured.

5. Approximate clonal structure in biology and statistical inference

In clonal deconvolution, approximate clones are often posterior summaries rather than directly observed entities. Cloe models tumor clones as latent binary features arranged on a latent phylogenetic tree, with mutation acquisition, loss, and penalized convergent evolution. The observation model links clone genotypes and sample frequencies to bulk sequencing counts through a Beta-Binomial likelihood, and inference is performed with Metropolis-coupled MCMC over the tree $(q,b)$7, genotype matrix $(q,b)$8, and frequencies $(q,b)$9 (Marass et al., 2016).

The paper is explicit that the inferred clones are approximations in several senses: real tumor cell states are approximated by a finite set of discrete clones; posterior multimodality means several clonal reconstructions may fit the data nearly equally well; and assumptions such as diploidy, copy-number neutrality, and a fixed error model can turn inferred clones into “effective” genotypes rather than literal ones. Empirically, on synthetic data with 3–5 clones and 50×–1000× depth, the correct number of clones was recovered in 26/27 runs, with $|X|\le b$0 and $|X|\le b$1. On a controlled 5-clone cell-line mixture, high-depth data yielded $|X|\le b$2, $|X|\le b$3, and $|X|\le b$4 (Marass et al., 2016).

In population genetics, the emphasis shifts from inference to dynamical emergence. Neher et al. study facultatively sexual populations in which recombination competes with selection. The order parameter

$|X|\le b$5

is the probability that two randomly chosen individuals are genetically identical. Clonal condensation corresponds to $|X|\le b$6, meaning that a few large clones carry a substantial share of the population. In the purely epistatic case, the critical recombination rate is

$|X|\le b$7

and below this threshold large clones emerge despite recombination (Neher et al., 2012). The paper further interprets large high-frequency haplotypes and closely related descendants as effectively clonal structures on the relevant timescale.

A related statistical perspective appears in clonality estimation from replicate sequencing libraries. Matsen et al. define

$|X|\le b$8

the probability that two randomly sampled individuals belong to the same clone. Their central claim is that one may estimate this functional accurately even when the full clonal frequency vector $|X|\le b$9 is poorly estimated. The exactly unbiased cross-replicate estimator

$X$0

is then combined with jackknife and covariance-regularized estimators to reduce MSE; in the reported simulations, the final estimator reduced mean squared error by more than 98% compared to $X$1 alone (Liu et al., 2014).

These biological literatures use “approximate clones” in a broader sense than the formal state-cloning or code-cloning literatures. Sometimes the approximation concerns the latent object being reconstructed; sometimes it concerns the effective persistence of clonal identity under recombination; and sometimes it concerns estimating a clonal concentration functional without identifying all clones individually.

6. Common structure, limits, and recurrent misconceptions

A recurrent pattern is that exact cloning is either impossible, rare, computationally intractable, or statistically unidentifiable. In quantum mechanics, perfect copying of an unknown state is forbidden, but optimal universal symmetric cloning attains a precise fidelity bound (Pelofske et al., 2022). In asymptotic classical statistics, the unattainable ideal $X$2 is replaced by a Markov kernel whose irreducible loss is the Gaussian $X$3 distance $X$4 (Matsumoto, 2011). In elections, perfect clone axioms do not extend cleanly to realistic near-clone situations (Delemazure, 28 Jan 2026). In RLHF, a standard aggregation rule can be destabilized by near-duplicate alternatives unless clone-aware weighting is introduced (Procaccia et al., 16 Jan 2025).

Another recurrent pattern is metric dependence. Fidelity, total variation, rank span, adjacent-swap distance, Euclidean context distance, token similarity, genotype error, frequency error, $X$5, and $X$6 are not interchangeable. This suggests that “approximate clone” is always relative to a chosen observational geometry. Changing the metric changes both the mathematical object and the associated invariance or optimality claim.

A further misconception is that approximate clones are always just noisy versions of exact clones. The surveyed papers show several non-equivalent possibilities. In notebook mining they are similarity classes found by a detector (Källén et al., 2020). In Cloe they are latent features selected by posterior inference under a phylogenetic prior (Marass et al., 2016). In RLHF they are alternatives whose addition should leave a learned reward function approximately invariant (Procaccia et al., 16 Jan 2025). In social choice they are candidate pairs or sets that are close under rank-based proximity measures, yet can still trigger outcome changes even for rules that are independent of perfect clones (Faliszewski et al., 14 Sep 2025, Delemazure, 28 Jan 2026).

Taken together, these literatures support a broad encyclopedia-level characterization: approximate clones are domain-specific surrogates for exact duplicates, defined by preserving enough structure under a chosen metric to make copying, grouping, reconstruction, or robustness analysis meaningful. The concept is therefore unified less by a shared ontology than by a shared methodological move: replace an unattainable or overly brittle exact clone notion with a quantified approximation that can be optimized, bounded, inferred, or tested.