slug: wierichs-kottmann-killoran-2025-critical-values
bibkey: wierichs2025brickwall
doi: 10.48550/arXiv.2511.16736
url: https://arxiv.org/abs/2511.16736v1
triage: theorem
motivation_gids:
  - D5/S3/Quantum/Information/PauliRotationCriticalValues.result

Critical values of parametrized quantum circuits

Problem

D. Wierichs, K. Kottmann and N. Killoran, Unitary synthesis with optimal brick wall circuits, arXiv:2511.16736v1, §V (page 9):

Conjecture 3. The critical values F(C) of a PQC that has full rank almost everywhere in T^d form an at most (d−2)-dimensional subset of G.

A PQC (Appendix F, Definition 1) consists of Clifford gates and $d$ rotations from $\{R_X,R_Y,R_Z\}$, each input feeding one rotation without preprocessing, viewed as $F:T^d\to G\subset {\mathbb{C}}^{2^n\times 2^n}$ with $d=\dim G$; $C$ is the set of inputs at which the Jacobian has rank below $d$. The verbatim texts are in the literature note. Issue #15435 preregisters the two-qubit, $G=SU(4)$, $d=15$ specialization with Hausdorff dimension and the refutation.

Motivation

The authors reduce universality of their optimal brick wall circuits to Conjecture 3: if the critical values had codimension at least two they could not separate the image from full-dimensional regions of $G$, and full Jacobian rank almost everywhere would imply surjectivity. With it, they “know [their] circuit families to be universal for n≤5”.

Gap

The paper proves that a dense image is the whole group and that full rank at one point gives full rank almost everywhere, and supports Conjecture 3 by numerical experiments and lower-dimensional examples. The literature check in #15435 found no settlement; not-found-in-searched-scope.

Route

Take the two-qubit Pauli words, in this order, $P=(YY,IY,XX,ZY,ZX,IZ,XY,YZ,IX,XZ,ZI,YI,YX,XI,ZZ)$, the rotations $R_P(a)=\cos \frac{a}{2}I-i\sin \frac{a}{2}P$ (each a Clifford conjugate of a single-qubit $R_Z$) and $F(\alpha )=R_{P_1}({\alpha }_1)\cdots R_{P_{15}}({\alpha }_{15})$.

1. ${\partial }_{j}F=-\frac{i}{2}{H}_{j}F$ with $H_j$ the $j$-th prefix conjugate of $P_j$, so the rank of $DF(\alpha )$ is the rank of the real $15\times 15$ matrix $J(\alpha )$ of Pauli coefficients of the $H_j$.
2. $\det J$ is real-analytic and not identically zero, so $F$ has full rank almost everywhere.
3. At ${\alpha }^{\ast }$ with ${\alpha }_6^{\ast }={\alpha }_7^{\ast }={\alpha }_8^{\ast }={\alpha }_{11}^{\ast }=\pi /4$ and all other inputs $0$, rational changes of coordinates reduce $J({\alpha }^{\ast })$ to a diagonal matrix with fourteen ones and one zero; along the kernel direction $v$ the normalized determinant $g$ has derivative $-1$.
4. The map $\alpha \mapsto (Q(F(\alpha )),g(\alpha ))$, with $Q$ a fixed linear projection onto ${\mathbb{R}}^{14}$, has a surjective derivative at ${\alpha }^{\ast }$; its image therefore contains a neighborhood, and $Q(F(C))$ contains an open subset of ${\mathbb{R}}^{14}$. Hence ${\dim }_{H}F(C)\ge 14>13$.

Falsifier

A reading of the class, the dimension notion or the rotation convention narrower than the source’s; or an error in the certificate at ${\alpha }^{\ast }$. Control in #15435: for all 720 orders of the six two-qubit Pauli words spanning $\mathfrak{so}(4)$, every fold point found numerically has critical-value rank $d-2=4$, the bound the conjecture predicts.

Evidence

D5/S3/Quantum/Information/PauliRotationCriticalValues.lean defines the class of circuits of fifteen nonidentity two-qubit Pauli rotations, the rank of fderiv, the critical set and claim (full rank almost everywhere implies dimH (circuit P '' critical P) ≤ 13, Hausdorff dimension in the Frobenius metric), and proves result : ¬ claim. The axiom closure of result is exactly propext, Classical.choice and Quot.sound; there is no sorry or native_decide. The step from a nonzero analytic determinant to full rank almost everywhere is the zero-set theorem of Mityagin, proved in the module by induction on the dimension with Fubini.

Triage

Tier 1: a named conjecture of a November 2025 preprint, preregistered in #15435 before any Lean. theorem; resolution refuted. result is bind-only in the repository’s sense (instantiation of pinned Mathlib calculus, rank, analyticity and Hausdorff-dimension facts with an exact finite certificate); admission basis open-problem-resolution; utility kind=certified-instance; basis=refutes. There is no digestion atom.

What the refutation shows

• Proved by result: a two-qubit circuit of fifteen Pauli rotations with full rank almost everywhere whose critical values have Hausdorff dimension at least $14=d-1$ in $SU(4)$.
• Mechanism: the critical point is a fold. $J({\alpha }^{\ast })$ has corank one, and the second derivative contracted with its kernel and cokernel vectors is nonzero. The critical set is then a hypersurface transverse to the kernel of $DF$, and $F$ maps it onto a codimension-one wall in $G$. This is the generic singularity of a map between manifolds of equal dimension. In Euler-angle circuits on $SU(2)$ the critical points are gimbal locks, where two rotations merge and the image loses two dimensions; in the $\mathfrak{so}(4)$ control the derivative restricted to the critical set has rank $d-2$ at every fold point found (computed, mechanism not analysed).
• Consequence for the source: the reduction “full rank almost everywhere and Conjecture 3 imply universality” no longer applies. Universality of the authors’ brick wall circuits for $n\le 5$ is not decided by this result and remains open.
• Reported, not verified here (#15435): tensoring the witness with the identity and appending every other Pauli word gives a $(d-1)$-dimensional critical-value patch for every $n\ge 2$.
• Open: whether codimension-one critical walls actually separate the image of a circuit from part of $G$, i.e. whether full rank almost everywhere still implies surjectivity by another route.

ASSUMED-UNVERIFIED

The literature check in #15435 is bounded and does not establish worldwide novelty or priority.