{"id":"fa3a33b7-d31f-4e9b-a6c7-3c166c388b0d","arxiv_id":"2510.18977","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Real and diagonal gates' stabilizer extent can be computed exactly over the real or diagonal Clifford subgroups, enabling optimal decompositions up to seven qubits and exponential speedups for sum-over-Cliffords simulation of QFT and Union Jack MBQC circuits.","lead":"This paper proves that the optimal way to write real, diagonal, or real-diagonal quantum gates as weighted sums of Clifford gates — the \"stabilizer extent\" — can be found by searching only a small subgroup of all Clifford gates. This pushes exact decompositions from two qubits to seven qubits on a laptop and claims large speedups for classically simulating circuits such as the quantum Fourier transform and measurement-based computations.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Appendix C's 'set a=0' claim is false: for C=SHS=(1/√2)[[1,i],[i,1]], the a=0 matrix J/√2 is rank-1 and not Clifford; Lemma 4's proof of Theorem 1 for real unitaries is invalid as written.","rationale":"The reader correctly located the load-bearing point: Lemma 1 is the engine of Theorem 1, and the real case depends on the structural claim about the Choi parametrization in Appendix C. My independent check goes further than 'unproved': the claim is contradicted by an explicit Clifford, C=SHS. This invalidates the proof of Lemma 4 as written, and therefore Theorem 1's real-unitary case is not established. I credit the paper for its verified small examples, honest fSim counterexample, and useful numerical applications; however, a false statement in the main proof cannot be treated as a presentational gap. The diagonal/real-diagonal applications may remain valuable, but the central theorem as stated lacks a valid proof. Hence the appropriate verdict moves from CONDITIONAL to REJECT: the paper would need a substantially different proof of Lemma 1 (or a restricted theorem) before the central claim can be accepted.","tokens_in":34293,"tokens_out":48865,"duration_ms":384166,"concrete_test":"Compute C=SHS=1/√2 [[1,i],[i,1]] and its Choi state C⊗I|φ+⟩. Verify that in the form (C2) one has W=Z_2^2, s=1, a=(1,1), q=b=0 (i-exponent mod 2). Then set a=0 in Eq. (C3) to obtain C'_xy=1/√2 for all x,y. Check whether C' is unitary: its rank is 1 (or C'C'†≠I). If this reproduces the contradiction, the post-Eq. (C3) claim is false and Lemma 4 cannot be repaired by filling a gap.","verdict_should_be":"REJECT","load_bearing_attack":"The proof of Theorem 1 rests on Lemma 1, whose real case (Lemma 4) explicitly relies on the assertion after Eq. (C3): in the Choi parametrization, partial-trace constraints restrict W and q but not a,b, so setting a=0 yields another Clifford unitary. This assertion is not merely unproved; it is false. Take the one-qubit Clifford C=SHS=1/√2 [[1,i],[i,1]]. Its Choi state is 1/2(|00⟩+i|10⟩+i|01⟩+|11⟩), which fits Eq. (C2) with W=Z_2^2, s=1, a=(1,1), q=b=0 (using the paper's convention that the i-exponent is evaluated mod 2). Setting a=0 in Eq. (C3) gives matrix entries all equal to 1/√2, i.e. C'=J/√2, which has rank 1 and is not unitary, hence not a Clifford unitary. Thus the general claim after Eq. (C3) is false. Lemma 4 case 1b/2b explicitly identifies C'=η+κ with this a=0 matrix and concludes it is a real Clifford; that conclusion is unsupported and in general wrong. Consequently Lemma 1 is not proved for G=K_n, and the real-unitary part of Theorem 1 lacks a valid proof as written. The diagonal and real-diagonal cases (Lemmas 3 and 5) may survive, but the theorem as stated is not established.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the stabilizer extent of unitaries and proves a 'strong symmetry reduction': for real, diagonal, and real-diagonal unitaries, the optimal decomposition into Clifford unitaries can be restricted to the corresponding real, diagonal, or real-diagonal Clifford subgroups without increasing the ℓ1-norm. This is formalized as Theorem 1, with the central engine being Lemma 1, whose proof is deferred to Appendix C. The authors combine this with a 'weak symmetry reduction' for additional invariances, enabling numerical computation of stabilizer extents for up to seven qubits, and they apply the results to multi-controlled-phase gates, QFT simulation, hypergraph states, and Union-Jack-lattice MBQC, reporting large exponential improvements in simulation cost. The paper includes an open-source implementation and validation against known extents from Bravyi et al.","tokens_in":34500,"tokens_out":5819,"duration_ms":49888,"significance":"If the theorem is fully established, the work is significant: it substantially extends the size of unitaries for which stabilizer extents can be computed exactly and demonstrates concrete classical-simulation speedups. The diagonal and real-diagonal cases are likely correct and already support most of the numerical and application claims; the code, reproducibility, and benchmarking against known values are clear strengths. However, the real case is currently not proven, so the full statement of Theorem 1 and the associated abstract/introduction claims overreach. The paper's value is contingent on repairing or restricting the real-case proof.","major_comments":[{"comment":"The assertion that 'the matrix C′ obtained by setting a=0 is another valid Clifford unitary' is false. For the one-qubit Clifford C=SHS = 1/√2 [[1,i],[i,1]], the Choi state in the convention of Eq. (C2) is 1/2(|00⟩+i|10⟩+i|01⟩+|11⟩), i.e. W=Z_2^2, s=1, a=(1,1), q=b=0. Setting a=0 gives C′=J/√2, a rank-1 matrix that is not unitary and therefore not Clifford. Lemma 4, case 1b/2b, explicitly identifies C′=η+κ as this a=0 matrix and concludes it is a real Clifford; that conclusion is unsupported. Hence Lemma 1 for G=K_n and the real part of Theorem 1 are not proved as written. The counterexample does not by itself disprove Lemma 1 (ξ_{K_n}(I/√2)=1/2≤1), so a repaired proof or a restriction of the theorem to G∈{Z_n,K_n×Z_n} is needed.","section":"Appendix C, after Eq. (C3)"},{"comment":"The 'phase-convention reduction' asserts without proof that any optimal decomposition can be chosen to contain only Cliffords from Cn, the subset fixed by Eq. (C3). This is used in the proof of Lemmas 3–5. It is a missing step, not a mere formality, because the global-phase ambiguity ω_8^k C changes the decomposition coefficients and the ℓ1 norm; a proof or reference is needed before the lemmas can be accepted.","section":"Appendix C, final paragraph"}],"minor_comments":[{"comment":"The phrase 'the exponent of i is meant to be evaluated modulo two' is confusing: under this convention i^2=1, which is nonstandard and should be stated explicitly with an example. This is also relevant to the counterexample above.","section":"Appendix C, Eq. (C1)–(C3)"},{"comment":"The entries for C^6P(θ_max) and θ_max are missing. If the computation could not be performed, this should be stated in the caption or text.","section":"Table I(c), n=7 row"},{"comment":"The color maps in Fig. 2 should include a colorbar or numeric scale; currently the extent of the ratios is not quantitatively readable.","section":"Fig. 2"}],"recommendation":"major_revision","confidential_remarks":"The concrete false claim in Appendix C is the main obstacle. The paper's application results and most numerical demonstrations rely on the diagonal and real-diagonal cases, which appear unaffected; if the authors repair Lemma 4 or restrict Theorem 1, a publishable contribution likely remains. The referee report should push the authors to address the counterexample head-on rather than only adding a caveat."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The real-unitary case of the main theorem is not proved as written. Appendix C asserts that in the Choi parametrization, setting a=0 in Eq. (C3) yields another Clifford unitary. That is false: take C=SHS=(1/sqrt(2))[[1,i],[i,1]]. Its Choi state fits (C2) with a=(1,1), q(x,y)=xy, b=0. Setting a=0 gives J/sqrt(2), which is rank-1 and not unitary, let alone Clifford. Lemma 4 explicitly uses that construction to claim eta+kappa is a real Clifford, so the proof of Lemma 4 is invalid as written, and Theorem 1 for G=K_n is unproved. The diagonal and real-diagonal cases are separate arguments; I do not see an immediate counterexample there, and they may survive, but they need careful rechecking too.\n\nThat is the headline. The rest of the package is genuinely useful. The strong-reduction idea is new, and the fSim counterexample is a good honest boundary—it shows transposition and permutation reductions genuinely fail, so the result is not a trivial generalization. The numerical work is reproducible: the code matches known extents for T^otimes k, CS, CCZ, and the 3-7 qubit extent values for C^{n-1}Z and C^{n-1}S are the kind of data people in the simulation community will want to use.\n\nThe softer spots are minor but real. The Union Jack exponents (2^{0.4150n}, 2^{1.6601n}, 2^{1.2451n}) are asserted without derivation; I would want to see the per-cell calculation. The theta_max values in Table I(c) come from interpolation with no error estimate. And the abstract's \"~10^74 faster\" comparison is to a specific Clifford+T synthesis from Nam et al., not to a generic prior algorithm; the text says this more carefully, but the abstract overreaches.\n\nNet: this is a serious paper that deserves a full referee pass, not a desk reject. The referee will need to get to the bottom of Lemma 4; the authors should either fix the real case or restate the theorem for diagonal and real-diagonal unitaries. The applications mostly use the diagonal cases, so the practical conclusions may survive. I would send it out and also ask the authors to pin a commit hash for the GitHub repo.","headline":"The real-unitary case of the main theorem is not proved as written—the Appendix C 'set a=0' claim is false—but the diagonal/real-diagonal core and the numerics justify a full referee pass.","tokens_in":709,"tokens_out":1684,"would_cite":true,"duration_ms":75074,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P68"],"pacs":["03.67.-a","03.67.Ac"],"model":"deepseek-v4-flash","headline":"For real, diagonal, and real-diagonal unitaries, the stabilizer extent—the minimal ℓ₁-norm squared of a Clifford decomposition—is exactly the restricted optimization over the corresponding Clifford subgroups, unlocking optimal decomposition","keywords":["stabilizer extent","Clifford group","symmetry reduction","classical simulation","magic resource theory","quantum Fourier transform","measurement-based quantum computation","hypergraph states"],"falsifier":"Compute the true stabilizer extent of a small real or diagonal unitary (e.g., a four-qubit multi-controlled phase gate) by solving the full-Clifford second-order cone program and compare it with the extent restricted to the real or diagonal subgroup; equality must hold for every instance. Alternatively, enumerate all Clifford unitaries on two or three qubits and check whether setting a=0 in their Choi-state parametrization always produces another valid Clifford unitary.","tokens_in":33986,"feed_emoji":"⚛️","tokens_out":5033,"duration_ms":40168,"temperature":0.7,"pith_summary":"The paper proves that if a unitary is real, diagonal, or both, its stabilizer extent can be computed by optimizing only over the real, diagonal, or real-diagonal Clifford subgroups without losing optimality. This collapses a superexponentially large optimization problem into a tractable one, allowing optimal decompositions of unitaries on up to seven qubits on a laptop—previously the ceiling was two qubits. The payoff is dramatic: a 16-qubit quantum Fourier transform block becomes classically simulable about 10⁷⁴ times faster than via the Clifford+T-compiled circuit, and any measurement-based computation on a Union Jack lattice with Pauli measurements gains an exponential runtime improvement. The proof rests on a structural claim about how Clifford unitaries look in their Choi-state parametrization, which the paper states but does not fully prove.","feed_headline":"Symmetry shrinks stabilizer search from 2 to 7 qubits","feed_subtitle":"For real and diagonal gates, optimal Clifford decompositions now run on a laptop and speed up QFT simulation by 10⁷⁴.","key_machinery":"The engine is Lemma 1, which bounds the G-symmetric stabilizer extent of the projection of a single Clifford term. Its proof uses the Choi-state parametrization of Clifford unitaries: every Clifford C is encoded as a stabilizer state C⊗1|φ⁺⟩ with an affine subspace W, a quadratic form q, and linear forms a,b. The paper asserts that restricting to real/diagonal Cliffords truncates W and q but leaves a,b free, so that each projected term becomes a convex combination of subgroup Cliffords. A second tool, weak symmetry reduction (Lemma 2), exploits additional symmetries such as qubit permutations by projecting the search set onto orbits, cutting the number of terms by orders of magnitude (for si","core_discovery":"The central claim is Theorem 1: for G equal to the complex-conjugation group, the diagonal-Pauli group, or their product, any G-invariant unitary U satisfies ξ(U) = ξ_G(U), meaning the stabilizer extent computed over the full Clifford group equals the extent computed over only the G-invariant Cliffords (real, diagonal, or real-diagonal). The proof takes an optimal expansion U = Σ x_C C and shows, via Lemma 1, that the G-projection of each term x_C C can itself be expanded over the subgroup with ℓ₁-norm at most |x_C|. Summing these bounds gives ξ_G(U) ≤ ξ(U), and the reverse inequality is automatic. This yields the first optimal stabilizer-extent decompositions for multi-qubit unitaries up to","pith_inferences":["The paper's link between diagonal unitaries and equatorial stabilizer states suggests that state-based stabilizer-extent solvers, which already handle ~10 qubits, could be adapted to compute unitary extents and push beyond the seven-qubit limit reported here.","The fSim counterexample shows strong symmetry reduction is not a general principle for arbitrary symmetry groups; classifying exactly which subgroups admit it is a concrete open problem that this paper poses.","The headline speedups rely on numerical linear-programming solutions being exact to five decimals; a certification pass or exact rational solver would turn the seven-qubit numbers into rigorous bounds.","Because the method exploits only the final unitary's symmetry, it applies to any symmetric gate block regardless of internal compilation, suggesting it could combine naturally with recompilation and circuit-optimization routines."],"forward_implications":["Optimal stabilizer-extent decompositions become computable for real, diagonal, and real-diagonal unitaries on up to seven qubits on consumer hardware.","Sum-over-Cliffords simulation of QFT blocks speeds up by factors up to ~8.8×10⁷⁴ for a 16-qubit circuit compared with Clifford+T-synthesized versions.","Classical simulation of any measurement-based quantum computation on a Union Jack lattice with Pauli measurements gains an exponential speedup, O(2^{1.2451n}).","T-count lower bounds for synthesizing multi-controlled-phase gates improve: CS needs at least 3 T gates, CCZ at least 4, and C³Z, C³S, C⁴Z, C⁴S at least 6.","Strict submultiplicativity of the stabilizer extent is observed for QFT blocks, so jointly decomposing gate blocks yields far cheaper simulations than gate-by-gate decomposition."],"fun_headline_variants":["Symmetry slashes stabilizer search to 7 qubits","From 2 to 7 qubits: symmetry scales stabilizer optimization","Laptop solves 7-qubit stabilizer extent via symmetry","Symmetry reduction unlocks 7-qubit Clifford decompositions"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof depends on an unproven structural claim in Appendix C: that the Choi-state parametrization of a Clifford unitary can be restricted to real/diagonal subgroups by zeroing out the linear phase terms a and b while still yielding valid Clifford unitaries—if this fails, Lemma 1 and Theorem 1 collapse.","fun_headline_variants_meta":{"raw":{"variants":["Symmetry slashes stabilizer search to 7 qubits","From 2 to 7 qubits: symmetry scales stabilizer optimization","Laptop solves 7-qubit stabilizer extent via symmetry","Symmetry reduction unlocks 7-qubit Clifford decompositions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00053,"raw_usage":{"total_tokens":2420,"prompt_tokens":805,"completion_tokens":1615,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":549,"completion_tokens_details":{"reasoning_tokens":1543}},"tokens_in":549,"tokens_out":1615,"duration_ms":13149,"temperature":1.0,"reasoning_tokens":1543,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T08:47:14.443206+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the true stabilizer extent of a small real or diagonal unitary (e.g., a four-qubit multi-controlled phase gate) by solving the full-Clifford second-order cone program and compare it with the extent restricted to the real or diagonal subgroup; equality must hold for every instance. Alternatively, enumerate all Clifford unitaries on two or three qubits and check whether setting a=0 in their Choi-state parametrization always produces another valid Clifford unitary.","supporting_citations":[],"review_version":1}