Parameter Count for a Fixed CNOT Skeleton

A matrix in $SU(2)$ has the form $\left(\begin{array}{cc}a& b\\ -\bar{b}& \bar{a}\end{array}\right)$ with $\mathrm{\mid }a{\mathrm{\mid }}^2+\mathrm{\mid }b{\mathrm{\mid }}^2=1$: its first column is a unit vector, orthogonality and determinant one fix the second column, as in the two-column calculation in the Euler proof. Conversely, direct multiplication verifies that every displayed matrix is unitary with determinant one. Thus $SU(2)$ is $S^3\subset {\mathbb{R}}^4$. On each of the eight open sets where one real coordinate is positive or negative, solve that coordinate as $\pm \sqrt{1-u_1^2-u_2^2-u_3^2}$; these smooth charts show $SU(2)$ is a real 3-manifold. It is closed and bounded, hence compact by [F7]. The tangent Lie algebra is the three-dimensional span in [F6]. Thus $G=SU(2{)}^n$ is a compact smooth real manifold of dimension $3n$. Collecting consecutive one-qubit gates into local layers and moving every scalar factor to the front writes each circuit with the fixed skeleton as $e^{i\varphi }{K}_m{C}_m{K}_{m-1}\cdots C_1{K}_0,$ where $K_j\in G$. The scalar phases commute with all gates and combine into one real parameter $\varphi$; [F1] parameterizes each one-qubit special-unitary factor by three real angles. The chronological matrix convention is [F8], and [F9] fixes the counted gate alphabet.

Direct inspection of the matrices in [F6] shows that every traceless skew-Hermitian $2\times 2$ matrix is a unique real combination of $iX,iY,iZ$: its diagonal entries are $ir,-ir$ and its off-diagonal entries are $z,-\bar{z}$, with $r\in \mathbb{R},z\in \mathbb{C}$. Fix one CNOT $C_j$ with control $c$ and target $t$. Applying its basis action [F2] to the four affected basis vectors gives $C_j{Z}_c{C}_j=Z_c$ and $C_j{X}_t{C}_j=X_t$. By [F10], the CNOT basis permutation is unitary. Its local commuting subgroup $H_j\le G$ therefore consists of arbitrary $SU(2)$ factors on the $n-2$ other wires, rotations $e^{isZ}$ on $c$, and rotations $e^{irX}$ on $t$. This product has real dimension $3(n-2)+1+1=3n-4$. A complementary subspace ${\mathfrak{m}}_j$ of $\mathfrak{s}\mathfrak{u}(2{)}^n$ is spanned by $i{X}_c,i{Y}_c,i{Y}_t,i{Z}_t$ and has dimension four.

Give ${\mathfrak{m}}_j\times H_j$ and $G$ local coordinates from the $S^3$ charts in step 1.1 and angle charts on the two circles in $H_j$. The local product map ${\mathrm{\Phi }}_j:{\mathfrak{m}}_j\times H_j\to G$, $(X,h)\mapsto e^{X}h$, has derivative at $(0,I)$ equal to $(X,Y)\mapsto X+Y$. The direct sum ${\mathfrak{m}}_j\oplus {\mathfrak{h}}_j=\mathfrak{s}\mathfrak{u}(2{)}^n$ makes this derivative an isomorphism between $3n$-dimensional coordinate spaces, so [F3] gives neighborhoods on which every nearby local factor is $e^{X}h$ with four real coordinates in $X$ and $h\in H_j$. The group $G$ is compact: each $SU(2)$ is the closed bounded set of matrices $\left(\begin{array}{cc}a& b\\ -\bar{b}& \bar{a}\end{array}\right)$ with $\mathrm{\mid }a{\mathrm{\mid }}^2+\mathrm{\mid }b{\mathrm{\mid }}^2=1$, and a finite product of compact sets is compact. Translating the neighborhood and using compactness gives a finite cover of $G$ by such local product charts.

Starting with $K_m$ and proceeding downwards, use a finite chart to write the current local factor as $g_{\mathrm{\ell }}{e}^{X_j}{h}_j$, where $g_{\mathrm{\ell }}$ is one of finitely many fixed local translations and $X_j$ has four real coordinates. Since $h_j$ commutes with $C_j$, move it through $C_j$ and multiply it into the next local factor; that product remains in $G$. Repeating this for all $m$ CNOTs leaves $m$ four-parameter factors and one arbitrary local layer $K_0$ with $3n$ parameters. Each chart choice has finitely many possibilities, and the global phase adds one parameter, so the exact image is contained in a finite union of smooth images with dimension $4m+3n+1$.

The exact circuit set is compact, hence Borel: by step 1.1 it is the continuous image of the compact space $S^1\times G^{m+1}$. Each open Euclidean parameter domain has a countable compact exhaustion, so each smooth parameter image is a countable union of compact, hence closed, subsets of $U(2^n)$ and is Borel. If $4m+3n+1<4^n$, every point of each parameter domain is critical for the smooth map into a target chart of $U(2^n)$, since its derivative has rank at most the smaller domain dimension. Apply [F4] on the open preimages of the finitely many target charts from [F5]; each Borel image is Lebesgue-null there, hence volume-null by [F5]. A finite union over parameter-chart choices is still null. Under full AC, [F5] identifies that null set as Haar-null. Thus the fixed skeleton has the claimed null image.