bibkey: kerenidis2026scalablequantumml authors: Iordanis Kerenidis year: 2026 title: “Scalable Quantum Machine Learning: Trainability, Expressivity and Efficiency” doi: 10.48550/arXiv.2607.24014 url: https://arxiv.org/abs/2607.24014v2 claim: “Conjecture 20: the unitary butterfly at uniformly random parameters is a relative-error two-design in the second exterior representation, with error O(1/n) in completely positive order.” strata_touched:

• D5/S3/Quantum/QuantumChannels/ButterflyLambdaTwoRelativeDesignRefutation license: citation-only triage: anchor

The butterfly relative-error exterior-square design conjecture

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Iordanis Kerenidis, Scalable Quantum Machine Learning: Trainability, Expressivity and Efficiency, arXiv:2607.24014v2 [quant-ph], version 2 (20 August 2026). DOI: https://doi.org/10.48550/arXiv.2607.24014. Source: https://arxiv.org/abs/2607.24014v2.

Statement

Conjecture 20, PDF p. 16, is the following verbatim TeX quotation from quantum_v2.tex, \label{conj:lambda2_design} in \label{sec:butterfly_circuit}:

The unitary butterfly W_n at uniformly random parameters is an ε-approximate unitary 2-design on U(n) in the antisymmetric 2-particle representation Λ² ℂⁿ, in the relative (multiplicative) sense, with ε = O(1/n): writing Φ₂^Λ²,W_n[Y] := E_Wn[Λ²(W_n)^{⊗ 2} Y Λ²(W_n)^{†⊗ 2}], for Y ∈ End(Λ²ℂⁿ ⊗ Λ²ℂⁿ), and Φ₂^Λ²,Haar for the corresponding Haar second moment, (1−ε) Φ₂^Λ²,Haar ⪯ Φ₂^Λ²,W_n ⪯ (1+ε) Φ₂^Λ²,Haar as completely positive maps. In particular Φ₂^Λ²,W_n has fixed-point space of dimension 3, matching the Haar decomposition Λ² ⊗ Λ² = V_(2,2) ⊕ V_(2,1,1) ⊕ V_(1,1,1,1) into three irreducible U(n)-representations.

\begin{conjecture}[Butterfly is a relative-error $2$-design on $\Lambda^2$]
\label{conj:lambda2_design}
The unitary butterfly $W_n$ at uniformly random parameters is an
$\varepsilon$-approximate unitary $2$-design on $U(n)$ in the antisymmetric
$2$-particle representation $\Lambda^2 \mathbb{C}^n$, in the \emph{relative}
(multiplicative) sense, with $\varepsilon = O(1/n)$: writing
\begin{equation}\label{eq:lambda2-secondmoment}
\Phi_2^{\Lambda^2,W_n}[Y] := \mathbb{E}_{W_n}\!\left[\Lambda^2(W_n)^{\otimes 2}\,
Y\, \Lambda^2(W_n)^{\dagger\otimes 2}\right],
\end{equation}
for $Y \in \mathrm{End}(\Lambda^2\mathbb{C}^n \otimes \Lambda^2\mathbb{C}^n)$,
and $\Phi_2^{\Lambda^2,\mathrm{Haar}}$ for the corresponding Haar second
moment,
\begin{equation}
\label{eq:relative_design}
  (1-\varepsilon)\,\Phi_2^{\Lambda^2,\mathrm{Haar}}
  \;\preceq\; \Phi_2^{\Lambda^2,W_n}
  \;\preceq\; (1+\varepsilon)\,\Phi_2^{\Lambda^2,\mathrm{Haar}}
\end{equation}
as completely positive maps. In particular $\Phi_2^{\Lambda^2,W_n}$ has
fixed-point space of dimension $3$, matching the Haar decomposition
$\Lambda^2 \otimes \Lambda^2 = V_{(2,2)} \oplus V_{(2,1,1)} \oplus
V_{(1,1,1,1)}$ into three irreducible $U(n)$-representations.
\end{conjecture}

The literal circuit definition, PDF p. 10, section \label{sec:butterfly_circuit}, is:

An $n$-qubit unitary butterfly circuit of depth $K = \log_2 n$ consists of $K$
layers. Layer $\ell \in \{1,\ldots,K\}$ applies a full-width layer of $n$
single-qubit phase gates followed by RBS gates on each of the $n/2$ disjoint
pairs with stride $2^{\ell-1}$:
\begin{align}
  U(\btheta,\bphi) &= U^{(K)}\cdots U^{(1)}, \\
  U^{(\ell)} &= \Bigl[\bigotimes_{j=1}^{n/2}
  \mathrm{RBS}(\theta_\ell^{(j)})\Bigr] \cdot
  \bigotimes_{i=1}^{n} R_z(\phi_\ell^{(i)}).
\end{align}
Equivalently, each pair carries the three-parameter primitive $\calG$ of
Eq.~\eqref{eq:gate_primitive}, one $R_z$ on each of its two qubits.
The $R_z$ gate acts as $R_z(\phi)\ket{0} = e^{i\phi/2}\ket{0}$,
$R_z(\phi)\ket{1} = e^{-i\phi/2}\ket{1}$: a diagonal one-body unitary, hence
passive FLO and particle-number preserving. Within each layer, all $n/2$ gate
pairs act on disjoint qubits, so generators from different pairs commute---the
key structure exploited by the multi-layer parallel parameter-shift rule.

The circuit consists of layers in increasing stride order, with matrix product U^(K) ... U^(1). Layer ell applies a full-width phase layer, then disjoint RBS gates at stride 2^(ell-1). The one-excitation RBS matrix is [[cos(theta), sin(theta)], [-sin(theta), cos(theta)]]. The literal phase convention is R_z(phi)|0> = exp(i phi/2)|0> and R_z(phi)|1> = exp(-i phi/2)|1>; the one-particle phase at mode j is exp(i (sum_i phi_i/2 - phi_j)), including the common vacuum phase.

The encoding uses Fin K -> Fin 2, with bit zero of stride one, identified with Fin (2^K) by finFunctionFinEquiv. There is one angle for every layer/pair and one phase for every layer/mode, all independently uniform on [0,2*pi). Exterior-square coordinates are the two-by-two minors in the ordered basis e_i wedge e_j, i<j. Completely positive order uses Mathlib’s bundled maps and their finite matrix amplifications. The asymptotic statement is read as one constant c and one threshold N0, followed by every power-of-two n above N0 and an error 0 <= epsilon <= c/n.

Theorem 19 (\label{thm:sharp_rate}, pp. 16–17) assumes this conjecture for the literal butterfly. Theorem 16’s additive-error statement uses one copy of the exterior representation. The independent-halves ensemble of Theorem 13 (\label{thm:interior_halves}) is a different ensemble. The refutation concerns the displayed relative two-copy bound; it makes no assertion about those other statements.