Cloned Units in Quantum & Software Systems

• Cloned units are entities duplicated via explicit copying, feature extraction, or unitary processes in both classical and quantum frameworks.
• In quantum contexts, cloned units include state cloning machines and encrypted protocols that navigate the no-cloning theorem to optimize fidelity and resource scaling.
• In software and machine learning, cloned units refer to replicated code fragments and network modules that improve maintainability, repairability, and performance.

A cloned unit is an entity, operation, or subsystem—classical or quantum—that has been duplicated (exactly or approximately) via explicit copying, feature extraction and replication, or physical/mathematical unitary processes. In the quantum regime, the topic includes quantum state cloning machines and their variants, encrypted quantum cloning mechanisms, process super-replication, as well as the creation and use of cloned code or architectural units in classical and embedded software. The mathematical and practical properties of cloned units, their fidelity, resource scaling, and their role in fundamental constraints such as the no-cloning theorem, define the scope of this concept and its relevance across quantum information theory, quantum engineering, machine learning, and classical/quantum software systems.

1. Mathematical Foundations and No-Cloning Constraints

The canonical no-cloning theorem establishes that, in quantum mechanics, no unitary transformation $T$ exists such that for all unknown pure states $|\psi\rangle$,

$$T(|\psi\rangle \otimes |0\rangle) = |\psi\rangle \otimes |\psi\rangle$$

This principle extends under unitary scaling: even if the second output is allowed to be $\mathcal{U}|\psi\rangle$ for any fixed unitary $\mathcal{U}$, universal deterministic cloning remains impossible for non-orthogonal states. Formally, it is shown that the inner-product preservation under unitarity leads to $$\langle\psi|\phi\rangle = (\langle\psi|\phi\rangle)^2$$, which can only be satisfied for orthogonal or identical states, and hence general cloning is forbidden (Li, 24 Apr 2026).

Yet, approximate and probabilistic forms of cloning, as well as special-purpose constructions (such as for orthogonal pairs or specific input sets), are permitted under quantum mechanics. Furthermore, certain protocols (see Section 2) decouple cloning from the strict no-cloning bound by leveraging encryption, entanglement, or process approximation (Yamaguchi et al., 6 Jan 2025, Dür et al., 2014).

2. Quantum State Cloned Units: Universal, Asymmetric, and Encrypted Cloning

2.1 Universal and Asymmetric Quantum Cloning Machines

Universal (symmetric) cloning machines produce $N$ output (clone) systems from $K$ identical inputs, optimizing the average single-clone fidelity

$F_{K \rightarrow M} = \frac{M K + M + K}{M(K+2)}$

For $K=1$, $|\psi\rangle$0, the optimal fidelity of an individual cloned unit is $|\psi\rangle$1; for instance, $|\psi\rangle$2 (Pelofske et al., 2022). Asymmetric universal cloning machines distribute fidelity non-uniformly among the output clones, subject to tight quadratic ellipsoidal inequalities—see, e.g., the trade-off constraints for a $|\psi\rangle$3 cloner,

$|\psi\rangle$4

where $|\psi\rangle$5, $|\psi\rangle$6, $|\psi\rangle$7 parametrize the expectation values associated with the three outputs (Jiang et al., 2012).

2.2 Encrypted Quantum Cloning: Perfect Clones under Encryption

Encrypted quantum cloning creates arbitrarily many encrypted clones of a qubit via a specific unitary transformation involving entanglement with ancillary "noise qubits," such that each clone by itself is maximally mixed and no information about the input is locally accessible. The decryption of any one encrypted clone consumes the quantum decryption key (the set of "noise qubits"), preventing further decryptions, thus sidestepping the no-cloning theorem. Explicitly, for $|\psi\rangle$8 desired encrypted clones, $|\psi\rangle$9 Bell pairs

$$T(|\psi\rangle \otimes |0\rangle) = |\psi\rangle \otimes |\psi\rangle$$0

are prepared. The unitary

$$T(|\psi\rangle \otimes |0\rangle) = |\psi\rangle \otimes |\psi\rangle$$1

carries out the encryption. Decryption is realized by

$$T(|\psi\rangle \otimes |0\rangle) = |\psi\rangle \otimes |\psi\rangle$$2

where only one clone $$T(|\psi\rangle \otimes |0\rangle) = |\psi\rangle \otimes |\psi\rangle$$3 can be decrypted given all noise qubits $$T(|\psi\rangle \otimes |0\rangle) = |\psi\rangle \otimes |\psi\rangle$$4; afterwards, the remaining clones are no longer decryptable (Yamaguchi et al., 6 Jan 2025). Hardware demonstrations confirm that encrypted cloning is stable with respect to hardware noise up to clone counts $$T(|\psi\rangle \otimes |0\rangle) = |\psi\rangle \otimes |\psi\rangle$$5 ($$T(|\psi\rangle \otimes |0\rangle) = |\psi\rangle \otimes |\psi\rangle$$6 floor), and circuit depth scaling is linear in $$T(|\psi\rangle \otimes |0\rangle) = |\psi\rangle \otimes |\psi\rangle$$7 (Yamaguchi et al., 11 Feb 2026).

2.3 Specialized Cloning Architectures

Pauli cloners provide a systematic method for $$T(|\psi\rangle \otimes |0\rangle) = |\psi\rangle \otimes |\psi\rangle$$8 cloning of $$T(|\psi\rangle \otimes |0\rangle) = |\psi\rangle \otimes |\psi\rangle$$9-qubit registers, implementing the entire family of (possibly asymmetric or noise-tailored) cloners via a fixed architecture and programmable ancillary state $\mathcal{U}|\psi\rangle$0. The output disturbance is precisely a Pauli channel, and the approach enables optimal adaptive attacks in quantum key distribution protocols or tailored fidelity distributions suited to MUB or noise-specific requirements (Kerstan et al., 31 Jan 2026).

3. Telecloning and Super-Replication: Quantum Information Propagation via Cloned Units

Quantum telecloning leverages multipartite entanglement and teleportation to distribute $\mathcal{U}|\psi\rangle$1 optimal approximate clones to physically separate systems. Utilizing a Dicke-type resource state,

$\mathcal{U}|\psi\rangle$2

telecloning protocols scale gracefully, as recent implementations on all-to-all trapped-ion processors demonstrate $\mathcal{U}|\psi\rangle$3 telecloning with measured average output fidelity $\mathcal{U}|\psi\rangle$4 ($\mathcal{U}|\psi\rangle$5 theoretical), and reduced control overhead via circuit optimization (Pelofske et al., 2022).

Super-replication (cloned units as process simulacra) extends the concept to unitary channels, allowing $\mathcal{U}|\psi\rangle$6 uses of a black-box $\mathcal{U}|\psi\rangle$7 to produce $\mathcal{U}|\psi\rangle$8 approximate yet near-perfect process clones. The protocol employs ancillary embedding and basis change unitaries, with the Choi–Jamiołkowski fidelity obeying

$\mathcal{U}|\psi\rangle$9

where $\mathcal{U}$0 is the Gaussian error function, and the overall scaling saturates the Heisenberg limit (Dür et al., 2014).

4. Cloned Units in Classical and Quantum Software Engineering

Cloning also refers to identical or similar code and architectural fragments proliferated in classical or hybrid codebases ("software cloned units"). In quantum programming (e.g., with Qiskit), analysis reveals that Type-2 (renamed identifiers) and Type-3 (near-miss) code clones are prevalent, with 13.6% of repos containing Type-1 and 20.5% containing Type-2/3 clones, and average clone sizes $\mathcal{U}$1 tokens, $\mathcal{U}$2 tokens (Manoku et al., 11 Jan 2025). Such cloned units impact maintainability and scalability and motivate the need for quantum-aware clone-detection and refactoring tools.

IEC 61131-3 programmable controller systems exhibit clone-and-own cloning at both fine-grained (POU-level) and coarse-grained (entire system configuration) scales. Formal type I–IV clone definitions and tree-edit similarity measures support automated detection and subsequent refactoring (e.g., library extraction, parameterization), as validated in practical case studies (Rosiak et al., 2021).

5. Cloned Units in Machine Learning and Repair Automation

In deep learning, cloned network units are leveraged to extract robust, invariant features—e.g., networks trained with many weight-sharing clones on differently noised versions of the same utterance, driving feature-extractor outputs to consensus and yielding stable low-dimensional salient representations. This architecture has been applied in generative speech enhancement, where clone-based training outperforms single-network baselines in MUSHRA-style perceptual tests, with statistically significant robustness to noise (Chinen et al., 2019).

Automated program repair frequently benefits from recognizing cloned edit units: a large-scale study of multi-hunk bug-fix patches finds that 67.7% contain at least one change clone group, and 70.3% are strictly-composed of cloned changes. 88.6% of strictly-cloned patches are made of identical edits, whether or not the context is also identical. This strongly supports repair algorithms that seek and propagate clone edits for multi-location repair (Madeiral et al., 2021).

6. Cloned Units as Quantum Resources and Information-Carrying Media

Cloned units, whether output by universal quantum cloners or specialized protocols, can act as valuable resources in information processing tasks. For example, the two-qutrit cloned output of a Buzek–Hillery-type cloner functions as an NPT-entangled (negativity via partial transpose) state, which—when locally distilled—achieves nontrivial teleportation fidelities and is dense-codable for $\mathcal{U}$3 in a specific range, with $\mathcal{U}$4 bits at maximal entanglement (Roy et al., 2013).

Encrypted clones, in particular, are directly relevant for encrypted quantum multi-cloud storage: each cloud holds an encrypted clone (maximally mixed on its own), while the decryption key (noise qubits) may be kept on-site or distributed. The protocol fulfills the requisite criteria of redundancy, off-site storage, and perfect encryption in a unitary, resource-efficient manner, assuming the key remains secret (Yamaguchi et al., 6 Jan 2025, Yamaguchi et al., 11 Feb 2026).

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In summary, cloned units exemplify a multi-faceted concept spanning quantum information, quantum and classical software systems, machine learning, and information-theoretic protocol design. In quantum settings, fundamental physical constraints (no-cloning, Heisenberg bound) are encountered and sometimes circumvented through encryption, approximation, entanglement, or resource-sharing. In algorithmic and software contexts, cloned units influence robustness, maintainability, multi-location repairability, and system scalability, justifying targeted detection and management strategies.