Normalized transport schedule for a J/CZ circuit

Let $C$ be a finite $n$-wire circuit over the $J(\alpha )$ and $CZ$ gates, with chronological layers $L_1,\dots ,L_d$ and source wire labels $0,\dots ,n-1$ as in Finite circuit syntax and ordered wire labels. Schedule index $j\in \{1,\dots ,n\}$ denotes source wire $j-1$. For each source layer $L_r$, let $M_r$ be the matching of its $CZ$ occurrences after reindexing a source pair $\{u,v\}$ as $\{u+1,v+1\}$, and let $J_r\subseteq \{1,\dots ,n\}$ be the indices carrying one-qubit gates $J({\alpha }_{rj})$. The source layer has pairwise disjoint gate supports by Finite circuit syntax and ordered wire labels, so no $J$ gate in $J_r$ meets an edge of $M_r$. By Gates on disjoint wires commute, its denotation may be written with the CZ matching first and then at most one $J({\alpha }_{rj})$ per schedule index $j$. The normalized transport schedule $\mathcal{T}(C)$ is the finite chronological list of rounds $(M_s,({\alpha }_{s1},\dots ,{\alpha }_{sn}))$ and the scalar ${\sigma }_C$ defined as follows. A round applies its matching first and then one $J$ gate on each wire. Define ${\sigma }_C={\prod }_{r:0<\mathrm{\mid }{J}_r\mathrm{\mid }<n}{e}^{i\pi (n-\mathrm{\mid }{J}_r\mathrm{\mid })\mathrm{/}4}$. The round list is the concatenation, in source-layer order, of the blocks specified by the first matching case below; the empty product is $1$. This order makes the zero-wire empty-layer case unambiguous:

1. If $J_r=\mathrm{\varnothing }$ and $M_r=\mathrm{\varnothing }$, the block is empty.
2. If $n\ge 1$ and $J_r=\{1,\dots ,n\}$, the block consists of one round with matching $M_r$ and the original angles ${\alpha }_{rj}$.
3. If $J_r=\mathrm{\varnothing }$ and $M_r\mathrm{\ne }\mathrm{\varnothing }$, the block consists of two rounds: the first has matching $M_r$ and angle $0$ on every wire; the second has no matching and angle $0$ on every wire.
4. If $0<\mathrm{\mid }{J}_r\mathrm{\mid }<n$, the block consists of three rounds. The first has matching $M_r$, angle ${\alpha }_{rj}$ on active wires $j\in J_r$, and angle $\pi \mathrm{/}2$ on idle wires. The next two have no matching, angle $0$ on active wires, and angle $\pi \mathrm{/}2$ on idle wires. The factor for this mixed layer is included in the displayed product defining ${\sigma }_C$.

The rounds are numbered $s=1,\dots ,R$ in concatenation order. Their schedule indices continue to denote the corresponding source wires $j-1$. Angles at indices without an original $J$ gate are padding angles. For a mixed layer, the active-wire product is $J(0)J(0)J(\alpha )=J(\alpha )$ and each idle wire contributes $J(\pi \mathrm{/}2{)}^3=e^{i\pi \mathrm{/}4}I$; the scalar is therefore recorded in ${\sigma }_C$ rather than suppressed. The no-$J$ matching case has two $J(0)=H$ transports, whose product is $I$. No claim about the transport graph's simplicity is part of this definition.