Cloning is as Hard as Learning for Stabilizer States

Abstract: The impossibility of simultaneously cloning non-orthogonal states lies at the foundations of quantum theory. Even when allowing for approximation errors, cloning an arbitrary unknown pure state requires as many initial copies as needed to fully learn the state. Rather than arbitrary unknown states, modern quantum learning theory often considers structured classes of states and exploits such structure to develop learning algorithms that outperform general-state tomography. This raises the question: How do the sample complexities of learning and cloning relate for such structured classes? We answer this question for an important class of states. Namely, for $n$-qubit stabilizer states, we show that the optimal sample complexity of cloning is $Θ(n)$. Thus, also for this structured class of states, cloning is as hard as learning. To prove these results, we use representation-theoretic tools in the recently proposed Abelian State Hidden Subgroup framework and a new structured version of the recently introduced random purification channel to relate stabilizer state cloning to a variant of the sample amplification problem for probability distributions that was recently introduced in classical learning theory. This allows us to obtain our cloning lower bounds by proving new sample amplification lower bounds for classes of distributions with an underlying linear structure. Our results provide a more fine-grained perspective on No-Cloning theorems, opening up connections from foundations to quantum learning theory and quantum cryptography.

• The paper demonstrates that both cloning and learning stabilizer states require Θ(n) samples, showing no structural advantage in cloning over learning.
• It employs advanced representation theory and coding-theoretic techniques, including character measurements and random purification, to derive tight lower bounds.
• The findings have practical implications for quantum cryptography and error correction by reinforcing the security and limits of cloning structured quantum states.

Cloning is as Hard as Learning for Stabilizer States: An Authoritative Synthesis

Introduction and Problem Context

The fundamental No-Cloning theorem asserts that non-orthogonal quantum states cannot be cloned, and even approximate cloning of arbitrary pure states demands resources commensurate with full quantum state tomography. However, much of modern quantum learning theory, both for classical and quantum tasks, focuses not on arbitrary unknown states, but rather on structured families—such as stabilizer states—where computational or sample advantages over generic tomography can sometimes be exploited. This motivates a central theoretical question: For structured quantum states, can approximate cloning ever be significantly easier (in terms of sample complexity) than learning?

This work resolves the question emphatically for $n$-qubit stabilizer states: cloning is as hard as learning, up to constant factors. The sample complexity for either task is $\Theta(n)$. This result generalizes the classical finding that for certain structured distributions, sample amplification is as hard as learning, and integrates this perspective into the quantum domain via heavy use of modern representation theory and connections to hidden subgroup problems.

Technical Frameworks

The study is grounded in two parallel frameworks:

Structured Sample Amplification (Classical)

Sample amplification asks: given $t$ random samples from an unknown member of a class of probability distributions, can one efficiently generate additional samples that are indistinguishable from fresh samples from the same distribution? For general distributions over a $k$-element support, amplification can be quadratically easier than learning ([axelrod2019sampleamplificationincreasingdataset]).

Here, the authors focus on structured function classes (e.g., parities, Reed-Muller codes). They formalize amplification for distribution classes of the form $\{(U_n, f)\}_{f\in \mathcal{F}}$, where $U_n$ is the uniform distribution over $n$ bits and $f$ is drawn from a function class $\mathcal{F}$.

Structured Quantum Cloning

Quantum cloning is formulated as: given $t$ i.i.d. copies of a quantum state from a structured set (such as stabilizer states), can one generate $\Theta(n)$0 copies approximating true i.i.d. samples, with total trace distance at most $\Theta(n)$1? The main structural focus is on stabilizer states, but the results extend to representations associated with Abelian State Hidden Subgroup Problems (Abelian StateHSPs).

Main Results

1. Sample Amplification Lower Bounds

For any Boolean function class $\Theta(n)$2, let $\Theta(n)$3 be its teaching dimension. If $\Theta(n)$4, then structured sample amplification necessarily incurs error proportional to the probability that $\Theta(n)$5 samples uniquely specify $\Theta(n)$6. For $\Theta(n)$7-bit parity functions, sample amplification from $\Theta(n)$8 samples incurs at least $\Theta(n)$9 total variation distance—nearly the same as the sample complexity for learning parities, which is $t$0. Thus, amplification is no easier than learning for parities.

Succinctly:

Function Class	Learning Complexity	Amplification Error Lower Bound
$t$1-bit parities	$t$2	$t$3 if $t$4
$t$5-dim linear subspace	$t$6	In terms of code-theoretic erasure probability

The result is generalized to any linear subspace of Boolean functions, using properties of the related linear code and its dual, specifically the behavior under random erasures.

2. Quantum Cloning Lower Bounds

Stabilizer States

For the class of $t$7-qubit stabilizer states, the optimal sample complexity to clone with small constant error ($t$8) is $t$9. The result is established by considering an Abelian StateHSP associated with stabilizer states: $k$0-independent generators are required to identify the hidden stabilizer group, and character measurements (Bell sampling) are shown to be optimal for extracting subgroup information. Thus, any cloning map capable of producing fresh copies from $k$1 input states would violate fundamental linear independence and information-theoretic bounds.

General Abelian StateHSPs

A general framework for cloning lower bounds is established: for an Abelian group $k$2 and a hidden subgroup $k$3, one cannot clone the hiding states better than one can learn—sample complexity lower bounds are linked to the teaching dimension or generator rank of the dual group $k$4 and the code-theoretic erasure resilience.

3. Technical Machinery

• Representation Theory: The proofs leverage block-diagonalization under the Clifford group (for stabilizer states) and the use of character POVMs (optimal projective measurements corresponding to irreducible representations).
• Random Purification Channel: To extend from mixed hiding states to pure stabilizer states, a structured random purification channel is introduced, generalizing recent developments ([tang2025conjugatequerieshelp], [walter2025randompurificationchannelarbitrary]).
• Coding Theory: The linear independence thresholds for sample amplification directly relate to the code distance and random erasure recovery probabilities for the associated linear function code of $k$5.

Implications and Connections

Theoretical Implications

• No Structural Separation: Even for highly structured and efficiently learnable quantum states, like stabilizers, cloning cannot be performed with fewer samples than learning. There is no sample-complexity separation available via structure for these prominent classes, contrasting sharply with the situation for unstructured distributions or quantum states.
• Coding-Theoretic Reduction: The analysis unifies quantum learning, classical sample amplification, and code theory. This provides new directions for both quantum tomography and classical learning theory, especially regarding combinatorial dimensions (teaching dimension, code distance) and their operational consequences.
• Optimality of Measurements: Character measurements (e.g., Bell sampling for stabilizer states) are shown to be information-theoretically optimal for extracting latent subgroup symmetry information, not just sufficient.

Practical Implications

• Limits of Efficient Cloning: This result systematically limits any approach to machine-learning-inspired quantum data amplification for structured classes relevant to fault-tolerant quantum computation and error correction (stabilizer codes).
• Cryptographic Security: The findings reinforce the security assumptions underlying schemes like quantum money or quantum authentication protocols based on stabilizers and similar codes—difficulty of cloning persists even under symmetry constraints.

Contrasting Amplification and Learning for Unstructured Classes

While structured classes like parities offer no advantage, for the completely unstructured class of all Boolean functions, sample amplification is quadratically easier than learning, matching results for arbitrary classical distributions ([axelrod2019sampleamplificationincreasingdataset]). This contrast is sharp, with structure imposing an informational bottleneck.

Open Questions and Future Directions

Several avenues remain for extending these results:

• Beyond Stabilizers: Does cloning remain as hard as learning for other classes (e.g., hypergraph states, tensor network states, or outputs of shallow circuits)? For instance, for degree-$k$6 polynomials (Reed-Muller codes), does the erasure threshold similarly dictate cloning hardness?
• Restricted Distinguishers: What changes if distinguishers are computationally bounded? This has relevance for cryptography and pseudorandomness.
• Combinatorial Dimensions: Can one characterize sample amplification and quantum cloning complexity for arbitrary function/state classes via suitable combinatorial dimensions, analogous to VC or teaching dimension?
• Average-Case Hardness: The results here are worst-case; average-case bounds remain to be systematically explored.
• Amplification with Access Restrictions: How does the sample complexity change if ancilla or entangling operations are restricted?

Conclusion

This paper closes a central question in quantum learning theory for a broad and important class of structured quantum states, demonstrating that no sample-complexity separation exists between learning and cloning for stabilizer states. The connection between classical sample amplification, quantum cloning, and linear/coding-theoretic structures is rigorously established, and a suite of technical tools—representation theory, random purification, and coding theory—are marshaled to provide tight, nonasymptotic lower bounds. The implications are substantial for both foundational quantum information and for practical tasks in quantum cryptography and tomography, and they outline a template for further study of quantum machine learning under symmetry and structure.