Even-Majorana Ising exchanges normalize logical Paulis

Reordering the $2m$ Majorana factors gives $({\gamma }_1\cdots {\gamma }_{2m}{)}^{\ast }=(-1{)}^m{\gamma }_1\cdots {\gamma }_{2m}$ and $({\gamma }_1\cdots {\gamma }_{2m}{)}^2=(-1{)}^{m}I$. Therefore ${\mathrm{\Pi }}^{\ast }=\mathrm{\Pi }$ and ${\mathrm{\Pi }}^2=I$. Also $\mathrm{\Pi }{\gamma }_j=-{\gamma }_j\mathrm{\Pi }$ for each $j$, so ${\gamma }_1$ bijects the $+1$ and $-1$ eigenspaces; each has dimension $2^{m-1}$. Since each product ${\gamma }_j{\gamma }_{j+1}$ commutes with $\mathrm{\Pi }$, every $U_j$, being a function of that product, preserves both parity sectors.

Put $A_j=i{\gamma }_j{\gamma }_{j+1}$. Then $A_j^{\ast }=A_j$ and $A_j^2=I$, so $Q_1=(I-A_j)\mathrm{/}2$ and $Q_{\psi }=(I+A_j)\mathrm{/}2$ are orthogonal complementary projections. The operator $U_j=2^{-1\mathrm{/}2}(I-{\gamma }_j{\gamma }_{j+1})$ has eigenvalues $e^{-i\pi \mathrm{/}4}$ on $Q_1$ and $e^{i\pi \mathrm{/}4}$ on $Q_{\psi }$, whose ratio is $i$. By [F3], the channel exchange is therefore $(R_1{e}^{i\pi \mathrm{/}4})U_j$; its common scalar has no effect on conjugation.

Let $c=s=2^{-1\mathrm{/}2}$ and $A={\gamma }_j{\gamma }_{j+1}$, so $U_j=cI-sA$ and $U_j^{\ast }=cI+sA$. Multiplication using ${\gamma }_{j}A={\gamma }_{j+1}$, $A{\gamma }_j=-{\gamma }_{j+1}$, ${\gamma }_{j+1}A=-{\gamma }_j$, and $A{\gamma }_{j+1}={\gamma }_j$ gives $U_j{\gamma }_j{U}_j^{\ast }={\gamma }_{j+1}$ and $U_j{\gamma }_{j+1}{U}_j^{\ast }=-{\gamma }_j$. Every other ${\gamma }_k$ commutes with $A$, so $U_j{\gamma }_k{U}_j^{\ast }={\gamma }_k$.

For $A={\gamma }_j{\gamma }_{j+1}$ and $B={\gamma }_{j+1}{\gamma }_{j+2}$, the Majorana relations give $A^2=B^2=-I$ and $AB=-BA$. Expanding with these identities gives $U_j{U}_{j+1}{U}_j=-(A+B)\mathrm{/}\sqrt{2}=U_{j+1}{U}_j{U}_{j+1}\mathrm{.}$ If $\mathrm{\mid }j-k\mathrm{\mid }\ge 2$, then the two products ${\gamma }_j{\gamma }_{j+1}$ and ${\gamma }_k{\gamma }_{k+1}$ commute because moving their four distinct factors past each other contributes sign $(-1{)}^4=1$; hence $U_j{U}_k=U_k{U}_j$. Thus the $U_j$ satisfy the Artin braid relations. The common scalar in step 1.2 preserves these relations, since both sides of each relation have the same number of generators.

For $1\le k<m$ define the even Majorana monomials ${\bar{Z}}_k=i{\gamma }_{2k-1}{\gamma }_{2k}$ and ${\bar{X}}_k={\prod }_{\mathrm{\ell }=k}^{m-1}(i{\gamma }_{2\mathrm{\ell }}{\gamma }_{2\mathrm{\ell }+1})$. Each bilinear is a Hermitian involution; the bilinears in a fixed ${\bar{X}}_k$ have disjoint supports and commute, so every ${\bar{X}}_k$ is also a Hermitian involution. All these even monomials commute with $\mathrm{\Pi }$ and preserve ${\mathcal{H}}_{\eta }$. Their supports show that the ${\bar{Z}}_j$ commute with one another, the ${\bar{X}}_k$ commute with one another, and ${\bar{Z}}_j$ anticommutes with ${\bar{X}}_k$ exactly when $j=k$ (their support intersection has odd size exactly in that case). The commuting involutions ${\bar{Z}}_1,\dots ,{\bar{Z}}_{m-1}$ have joint sign projections ${\prod }_k(I+{ϵ}_k{\bar{Z}}_k)\mathrm{/}2$ whose sum is identity, so at least one joint eigenspace is nonzero. Applying products of the ${\bar{X}}_k$ flips each sign independently; hence all $2^{m-1}$ joint sign eigenspaces are nonzero and orthogonal. Their number equals $\dim {\mathcal{H}}_{\eta }=2^{m-1}$ from step 1.1, so each is one-dimensional. In this basis the ${\bar{Z}}_k,{\bar{X}}_k$ are the standard logical Pauli generators. To see they exhaust the restrictions of all even Majorana monomials, label ${\gamma }_j$ by the $j$th standard vector of ${\mathbb{F}}_2^{2m}$. Even monomial labels form the $2m-1$ dimensional even-sum subspace, and the all-ones parity label lies in it. The labels of the $2m-2$ operators ${\bar{Z}}_k,{\bar{X}}_k$ are independent modulo the parity label: no combination uses coordinate $2m$, while the successive coordinates $1,2,3,4,\dots ,2m-1$ force in turn the coefficients of ${\bar{Z}}_1,{\bar{X}}_1,{\bar{Z}}_2,{\bar{X}}_2,\dots ,{\bar{Z}}_{m-1},{\bar{X}}_{m-1}$ to vanish. They therefore form a basis of the quotient of even labels by the parity label. Reordering and cancelling Majorana factors shows that monomials with the same label differ only by a scalar; on ${\mathcal{H}}_{\eta }$ the parity monomial is scalar. Thus every even Majorana monomial restricts to an element of the logical phase-aware Pauli group generated by the ${\bar{Z}}_k,{\bar{X}}_k$. By step 2.1 each braid sends every Majorana to a signed Majorana, so it maps this finite Pauli group into itself; invertibility and finiteness make the image equal to the group. Hence each exchange normalizes the logical Pauli group.

Conjugation gives a homomorphism from the projective Clifford group to the automorphism group of the phase-aware Pauli group. That Pauli group is finite, so its automorphism group is finite. The kernel consists of unitaries commuting with every logical Pauli; commuting with all logical $Z_k$ makes a unitary diagonal in the computational basis, and commuting with each logical $X_k$ forces every diagonal entry to agree. Thus the kernel is scalar and the projective Clifford group is finite. For $m\ge 2$, $PU(2^{m-1})$ is an infinite Hausdorff metric space by [F2]: the classes of $\mathrm{diag}(1,e^{it},1,\dots ,1)$ are distinct as $t$ varies modulo $2\pi$. A finite subgroup is closed in a metric space, so this braid subgroup is not dense.