On the hardness of cloning and connections to representation theory

Chinmay Nirkhe; Vojt\v{e}ch Havl\'i\v{c}ek

REVIEW 4 major objections 4 minor 13 references

On the hardness of cloning and connections to representation theory

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This note shows that, assuming an unproven conjecture about cloning hidden maximally entangled states, efficient witness cloning would put NP inside BQP.

desk verdict Honest, well-scoped note: the Kronecker-based state-generation hardness is solid, but the headline cloning theorem is conditional on a believable yet unproven conjecture, and there's a likely typo in the phase-estimation formula that needs fixing. read the letter →

arxiv 2411.11805 v1 pith:SYW77YDN submitted 2024-11-18 quant-ph cs.CC

Vojtěch Havlíček , Chinmay Nirkhe This is my paper · ORCID

classification quant-phcs.CC MSC 68Q1220C3081P68

keywords witnesscloningno-cloningtheoremKroneckercoefficientshiddensubspacemaximallyentangledstateweakFouriersamplingBQPvsNPsymmetricgrouprepresentation

topics P versus NP

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This note tries to prove that efficiently cloning a quantum witness—a state accepted by a verification circuit—is as hard as solving NP problems, rather than merely hard for information-theoretic reasons. The authors build a family of verification circuits from representations of the symmetric group, whose unique accepted states are maximally entangled states over hidden subspaces. Deciding whether the relevant Kronecker coefficient is positive is NP-hard, so generating these states is hard unless BQP contains NP. Their main theorem shows that an efficient cloner for this family would also generate states in the hidden subspace, provided Conjecture 2 holds; combined with the Valiant–Vazirani reduction, this would imply BQP ⊇ NP. If correct, the paper reduces the hardness of witness cloning to a single, explicitly stated conjecture about maximally entangled states over hidden subspaces.

What carries the argument

The load-bearing object is the weak Fourier sampling projector $\Xi_\lambda^{(\rho)} = \frac{d_\lambda}{|G|}\sum_{g\in G} \chi_\lambda(g)^* \rho(g)$, which projects onto the $\lambda$-isotypic component of a representation $\rho$. For $\rho = \rho_\mu \otimes \rho_\nu$ of $S_n$, its nonzero-ness coincides with positivity of the Kronecker coefficient $a_{\mu\nu\lambda}$, and the paper adds an 'internal state testing' step that forces the accepted state to be a maximally entangled state across each block. Together these tests produce a verification circuit whose unique accepting state is the hidden maximally entangled state $|\Phi_\Pi\rangle$ whenever $a_{\mu\nu\lambda}=1$. This is the mechanism that converts NP-hardness of Kronecker coefficients into a candidate hard-to-clone family.

What would settle it

Find an efficient quantum circuit that clones the specific hidden maximally entangled states $|\Phi_\Pi\rangle$ constructed in the paper while provably failing to generate any state in $\Pi$; such a circuit would falsify Conjecture 2 and remove the main theorem's premise.

Watch

Extended reading notes

Core claim

The central discovery is a reduction from witness cloning to state generation over hidden subspaces, routed through representation theory. For the symmetric group $S_n$, the weak Fourier sampling projector $\Xi_\lambda$ associated with the representation $\rho_\mu \otimes \rho_\nu$ has dimension $a_{\mu\nu\lambda} d_\lambda$, and it is nonzero exactly when the Kronecker coefficient $a_{\mu\nu\lambda}$ is positive. Because deciding positivity is NP-hard, the subspace is hidden, and the state $|\Phi_\Pi\rangle$ defined as the maximally entangled state over $\Pi$ is the unique state accepted by the constructed $(\mu\otimes\nu,\lambda)$-verification algorithm when $a_{\mu\nu\lambda}=1$. Theorem 8 states that, assuming Conjecture 2 and BQP ⊉ NP, no efficient algorithm clones these states; otherwise an efficient cloner would yield a circuit generating a state in $\Pi$, hence a BQP algorithm for UNIQUE-NP, and by Valiant–Vazirani, BQP ⊇ NP.

Load-bearing premise

The load-bearing premise is Conjecture 2: if an efficient cloner exists for a hidden maximally entangled state uniquely accepted by a verification circuit, then an efficient circuit exists for generating a state in the support of the hidden subspace; the paper gives intuition but no rigorous derivation, and notes that no black-box proof can exist.

Editorial extensions

If this is right

If the main theorem holds, an efficient cloner for the constructed family would put NP inside BQP, so witness cloning is at least as hard as deciding NP under the stated assumptions.
The verification circuits for $a_{\mu\nu\lambda}=1$ have completeness 1 and soundness $\le 8/9$, making them valid verifiers whose unique witnesses are hidden maximally entangled states.
A proof of Conjecture 2 cannot be a black-box reduction; any successful proof must exploit the circuit description of the verifier.
The result reduces the hardness of witness cloning to the hardness of generating states in hidden subspaces, narrowing the problem to a single structural conjecture.
For the family with $a_{\mu\nu\lambda}=1$, state generation is already impossible under BQP ⊉ NP (Corollary 7), and the new result extends this to cloning.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If Conjecture 2 holds, this construction yields the first worst-case complexity-theoretic evidence that witness cloning is hard, complementing existing average-case cryptographic and oracle results.
The template generalizes: any family of projectors whose positivity is NP-hard and whose accepting states are maximally entangled over hidden subspaces would give the same cloning-hardness reduction.
A natural testable extension is to compute small Kronecker coefficient instances and search for explicit cloning circuits for the associated $|\Phi_\Pi\rangle$; a successful cloner that does not reveal a state in $\Pi$ would falsify Conjecture 2.
If average-case hardness of Kronecker coefficients were ever established, the same measurement over $|\Phi_+\rangle$ could be turned into a quantum lightning construction, as the paper notes as a possibility.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick read: the genuinely solid part is the unconditional hardness of state generation for their representation-theoretic verifier. The cloning hardness is a well-designed but explicitly unfinished conditional theorem resting on Conjecture 2, which the authors openly admit they cannot prove.

What's new: they connect NP-hardness of Kronecker coefficients to a family of hidden maximally entangled states, building a verification circuit whose unique accepting states are exactly such states when the Kronecker coefficient is 1. Theorems 6 and 7 follow from known hardness and are clean. That part is a real, citable contribution.

Soft spots, in order of severity. First, Conjecture 2 is the load-bearing step for Theorem 8, and there's no proof, only intuition about disentangling. The authors themselves call the work unfinished and cite NZ24, which rules out any black-box proof, so the conjecture requires a genuinely non-relativizing argument. If it fails, the cloning-to-generation reduction collapses. Second, the claimed soundness of the verification algorithm is asserted as ≤8/9 without a derivation, and the acceptance probability in Eq. (30) has 1/2 + 1/2|⟨B, G(B)⟩|², which does not match standard 1-bit phase estimation; that should involve the real part, not the squared magnitude. This looks repairable and probably doesn't sink the construction, but it needs a fix before the bounds are credible.

Citation pattern is fine. The self-citations to BCG+24 and LH24 are to published work used as ingredients, not as a crutch. Nobody is fitting parameters or assuming the target result.

This paper is for quantum complexity people and quantum money types. It deserves a serious referee: the state-generation result is worth publishing even if Conjecture 2 stays open, and the conjecture itself is a sharp, well-motivated target for future work. A referee should ask for a corrected derivation of the soundness gap and a careful discussion of what evidence beyond intuition supports Conjecture 2. I would send it to peer review rather than desk reject, but the main theorem's status should be presented honestly as conditional.

Referee Report

4 major / 4 minor

Summary. The paper studies the computational hardness of cloning witnesses for quantum verification circuits. It constructs a family of verification algorithms based on weak Fourier sampling for representations of the symmetric group, whose accepting states are maximally entangled states over hidden subspaces associated with Kronecker coefficients. The main result, Theorem 8, shows that, assuming an unproven conjecture (Conjecture 2) about cloning such hidden maximally entangled states and assuming BQP ⊉ NP, no efficient quantum algorithm can clone the witnesses of these verification circuits when the relevant Kronecker coefficient equals 1. The paper also proves conditional hardness of state generation for this family, connecting to NP-hardness of Kronecker coefficient positivity.

Significance. If the technical gaps are repaired, the paper would provide a novel complexity-theoretic reduction: hardness of witness cloning follows from a concrete conjecture about maximally entangled states over hidden subspaces, rather than from cryptographic assumptions or oracles. The connection between quantum state complexity and representation-theoretic multiplicities is interesting, and the verification construction via weak Fourier sampling is elegant. The paper is commendably transparent about the unproven Conjecture 2 and the non-relativizing barrier from [NZ24]. However, the correctness of the verification analysis currently has gaps, and the main theorem remains conditional, so the impact is contingent on both the conjecture and the repair of the technical arguments.

major comments (4)

[§5.2, Eq. (30)] The acceptance probability for the 1-bit phase estimation circuit is stated as 1/2 + 1/2 |⟨ψ|V|ψ⟩|², but the standard formula for the circuit described (ancilla in |0⟩, Hadamard, controlled-V, Hadamard, measure) is 1/2 + 1/2 Re⟨ψ|V|ψ⟩. Since V is unitary and not necessarily Hermitian, the two formulas differ. Lemma 4's proof uses Eq. (30) to translate the acceptance probability into a bound on 1−|⟨B,E(B)⟩|²; with the correct formula the bound would be on 1−Re⟨B,E(B)⟩. Because E is an orthogonal projection under the Hilbert–Schmidt inner product, Re⟨B,E(B)⟩ = ⟨B,E(B)⟩ = ‖E(B)‖² ≥ 0, so the qualitative closeness conclusion can likely be recovered, but the proof as written is incorrect and needs to be revised.
[§6.2, Theorem 8 proof] The proof asserts that the (μ⊗ν,λ)-verification algorithm has completeness 1 and soundness ≤ 8/9, citing Corollary 5. Corollary 5 is a closeness statement saying that states passing with probability 1−ε are close to the accepting subspace; it does not by itself yield the spectral gap required by Definition 1 for soundness 8/9. The authors need to derive an explicit upper bound on the acceptance probability of states orthogonal to the accepting subspace, accounting for the rejection in the weak Fourier sampling step and the internal state test. The constant 8/9 is used in Conjecture 2, so this is load-bearing.
[§6.1, Theorem 6 and §5.2, Corollary 5] The claim that the accepting subspace has dimension exactly a_{μνλ} is inconsistent with the characterization in Lemma 4 and Corollary 5, where the accepting states are of the form |E⟩⊗|Φ+⟩ with |E⟩ ∈ C^{a²}, giving dimension a². For a=1 the two statements agree, so Theorem 8's uniqueness may be unaffected, but Theorem 6's dimension statement is incorrect as written. Please correct the dimension and the definition of D_λ in Eq. (12), which currently spans only the diagonal block states.
[§6.1, Theorem 6 and Corollary 7 proof] The statement that deciding whether a_{μνλ} is 0 or 1 (promised one of the two) is UNIQUE−NP-hard is used for the Valiant–Vazirani reduction in the proofs of Corollary 7 and Theorem 8, but no proof or reference is given. The cited NP-hardness of positivity [IMW17] does not automatically yield hardness of the promise problem; a parsimonious reduction or a separate argument is needed. Please provide a reference or proof.

minor comments (4)

[Abstract and §1] The phrase 'BQP does not equal QMA' is used in the abstract and introduction, but the paper's actual assumption is BQP ⊉ NP; these are not equivalent statements, and the wording should be aligned with the formal results.
[§5.2, Eq. (12)] The definition of D_λ as the span of the block-wise maximally entangled states and its stated dimension a_{ρ→λ} are inconsistent with Lemma 4, which allows arbitrary superposition of blocks. Please align the notation and clarify what subspace is actually characterized.
[Footnote 1] The footnote text appears incomplete in the manuscript ('one.sup'); please provide the full sentence.
[Appendix B, proof of Fact 1] The circuit diagram in the proof of Fact 1 is difficult to parse; please redraw it for clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central hardness claim is a transparent conditional on an explicitly unproven conjecture, and the reductions rest on external NP-hardness results rather than on assumptions equivalent to the conclusion.

full rationale

The paper's main theorem (Theorem 8) is conditional: it assumes Conjecture 2 and BQP not superset NP and derives the absence of an efficient witness cloner. Conjecture 2 is not derived from the target conclusion, nor is it assumed in a form that already contains the conclusion; it is explicitly labeled a conjecture, and Section 7.1 describes the work as 'unfinished', stating 'we were unable to make any of them rigorous' and citing Nehoran and Zhandry [NZ24] to show that no black-box proof is possible. A conditional result with an unproven but clearly stated antecedent is a limitation or open problem, not circularity. The hardness-of-generation step (Theorem 6 and Corollary 7) uses the external NP-hardness of deciding positivity of Kronecker coefficients [IMW17] and the Valiant-Vazirani reduction [VV86]; the internal characterization (Corollary 5) is proven from Schur's lemma and Lemma 4, whose proof is given in Appendix B with no appeal to the no-cloning conclusion. No parameter is fitted, and no quantity called a prediction is fixed from the data it is said to predict. The self-citations ([BCG+24] for the #BQP containment of Kronecker coefficients and [LH24] for an implementation technique) are used as published ingredients, and the relevant weak Fourier sampling measurement is also proven in the appendix; they do not carry the weight of the conditional conclusion. A separate correctness concern is the nonstandard phase-estimation acceptance formula used around Eq. (30), but that would be a repairability issue, not circularity. Accordingly, no circular step is exhibited by the paper's own equations, and the appropriate finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central theorem rests on Conjecture 2 and standard complexity and representation theory facts; no free parameters or invented entities are introduced.

assumptions (6)

domain assumption BQP ⊉ NP
Standard complexity assumption used in Theorems 6 through 8; if BQP contains NP the hardness conclusions are vacuous.
ad hoc to paper Conjecture 2
Load-bearing unproven claim that cloning a hidden maximally entangled state yields a circuit for the hidden subspace; the paper labels it a conjecture and provides only intuition.
standard math NP-hardness of deciding positivity of Kronecker coefficients [IMW17]
Used to construct NP-hard hidden subspaces and to prove no efficient generator exists assuming BQP ⊉ NP.
standard math Efficient quantum Fourier transform over the symmetric group [Bea97]
Required for Fact 2, efficient implementation of weak Fourier sampling for ρ = μ⊗ν.
standard math Valiant-Vazirani randomized reduction from NP to UNIQUE-NP [VV86]
Used in Corollary 7 and Theorem 8 to lift a UNIQUE-NP algorithm to an NP algorithm.
standard math Schur's lemma and Young-Yamanouchi representation matrices [Jam84]
Background for Fact 1, Fact 2, and Lemma 4 in Appendix B.

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Cite this review

Pith. "Pith review of On the hardness of cloning and connections to representation theory." pith.science (2026). https://pith.science/paper/SYW77YDN

@misc{pith2026241111805,
  author       = {Pith},
  title        = {Pith review of: On the hardness of cloning and connections to representation theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SYW77YDN}},
  note         = {Machine review of arXiv:2411.11805}
}

read the original abstract

The states accepted by a quantum circuit are known as the witnesses for the quantum circuit's satisfiability. The assumption BQP does not equal QMA implies that no efficient algorithm exists for constructing a witness for a quantum circuit from the circuit's classical description. However, a similar complexity-theoretic lower bound on the computational hardness of cloning a witness is not known. In this note, we derive a conjecture about cloning algorithms for maximally entangled states over hidden subspaces which would imply that no efficient algorithm exists for cloning witnesses (assuming BQP does not contain NP). The conjecture and result follow from connections between quantum computation and representation theory; specifically, the relationship between quantum state complexity and the complexity of computing Kronecker coefficients.

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Reference graph

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