we may modify $\tilde{V}_{n+2}$ such that for all $m \notin K_n \cup \ell$ and all $K_n \not\ni \ell$, $V_{K_n, \ell}^{\to m} = 0$ holds – which then ensures the analogous property for the adjoint of the graph. For the associated branch graph, incoming green dashed arrows will generally always vanish if the respective branch is now certain either to happen or not to happen, and analogously for the red dashed arrows. In such a QC-QC, there is always an implementation in which $A_\ell$ never acts in the $(n+1)$th position, which allows to remove the corresponding arrows.

**Example 5.9.** Consider a bipartite process compatible with a fixed order $A \prec B$. Then clearly, there exist implementations of this process where $V_{\emptyset, \emptyset}^B = V_{\emptyset, B}^A = 0$. Hence, for any choice of agent operations, $\mathcal{X}_1^B = \mathcal{X}_2^A = \emptyset$, and there exists an implementation in which $V_{B, A}^F = 0$ as well. Accordingly, we may remove the arrows

$$V_1 \xrightarrow{\mathcal{X}_0^B} B, \quad B \xrightarrow{\mathcal{X}_0^B} V_2, \quad V_2 \xrightarrow{\mathcal{X}_B^A} A \quad \text{and} \quad A \xrightarrow{\mathcal{X}_B^A} V_3, \qquad (5.15)$$

from the graph in Fig. 4.2a, leaving us with the acyclic routed graph

$$\rightarrow V_1 \xrightarrow{\mathcal{X}_0^A} A \xrightarrow{\mathcal{X}_0^A} V_2 \xrightarrow{\mathcal{X}_A^B} B \xrightarrow{\mathcal{X}_A^B} V_3 \rightarrow . \qquad (5.16)$$

Here, the indices always take the values $\mathcal{X}_1^A = \{A\}$, $\mathcal{X}_2^B = \{A, B\}$, and could be omitted from the graph. The resulting branch graph is then almost identical to the routed graph, as for each node, there is only a single branch, which always happens.

$$V_1^\emptyset \xrightarrow{\mathcal{X}_0^A} A^\emptyset \xrightarrow{\mathcal{X}_0^A} V_2^A \xrightarrow{\mathcal{X}_A^B} B^A \xrightarrow{\mathcal{X}_A^B} V_3^{AB}. \qquad (5.17)$$

# 6 Discussion

In providing a representation of arbitrary QC-QCs in terms of routed quantum circuits, we have established an application of the latter framework well beyond the particular examples previously considered, extending it to a wide class of processes. Our construction notably provides a more fine-grained picture of the logical structure of QC-QCs, with the control of the causal order being represented by indices $\mathcal{X}_n^k$ in the routed graph.

The fact that we achieved this by specifying just a single template for the routed graph and a generic fleshing out is testament to the ability of the framework to condense entire classes of processes in terms of their underlying information-processing structure, which is concisely encapsulated in the routed graph. In addition, the more fine-grained branch graph it brings forth may be closely related to the underlying causal structure of the respective process, as has been tentatively demonstrated in [51].

## 6.1 Contrasting routed circuit decompositions

In Section 4 of [39], explicit constructions were given for two instances of QC-QCs: the quantum switch [21] and the Grenoble process [34]. These constructions differ from our generic routed graph $G_{QC-QC(N)}$, with fewer nodes: internal nodes corresponding to the agents $A_k$, as well as nodes for the past and future $P$ and $F$, which play roles analogous to our internal nodes $V_1$ and $V_{N+1}$. By featuring additional internal nodes $V_n$, our construction appears to yield a significantly more verbose routed graph representation. In Section 5.3, we discussed how one can, under certain conditions, nonetheless recover the structure of their routed graph from ours by merging nodes. However, for $N > 3$, a representation without further internal (i.e., non-agent) nodes can no longer generally be found, as we exemplify in Appendix E. The generalisation of the routed graphs given in [39], without additional internal nodes, is thus unable to capture the full expressivity of QC-QCs.

Even in the $N = 2$ and $N = 3$ cases, the more compact routed graph representation of [39] comes at the cost of a more complex fleshing out for the agent nodes $A_k$, which must not only act as an adapter for allowing the local operations of the agents to be plugged in, but also includes the, in general non-trivial, (internal) routing of the process.

However, with our generic routed graph (modified to represent the ancillary systems explicitly in $G_{QC-QC(N)}$), we can mostly forgo the necessity of a fleshing out to describe the class of QC-QCs.
Indeed, for the V-nodes the routes actually restrict the associated slots precisely to valid QC-QC isometries, and any routed isometries following the routes constitutes a valid fleshing out. For the A-nodes, as observed in Section 5.1, allowing the agents to perform any routed operation<sup>50</sup> following the routes essentially gives them just the power to perform routed operations dependent on the causal order – which in fact mirrors many physical implementations suggested so far more closely, while still conveying a form of ICO [51, 53]. One can moreover show that each such routed process may still be simulated as a conventional (non-routed) QC-QC, potentially decreasing the relevance of this distinction.

Indeed, consider a process where a coherent copy of the control system $\bar{C}^{(k)}$ is passed to each agent $A_k$. Then, they may perform their operation controlled on the set of agents $K_n \setminus k$ who acted previously, as in a routed process, or perform a measurement on the causal order, destroying the coherence of the quantum control in the QC-QC. Any $N$-slot QC-QC process may be amended to pass a copy of the control to the agents in such a fashion, increasing the dimensions of $\mathcal{H}_k^{A_k/N}$ accordingly. After their operation, the coherent copy may then be undone – presuming, the agents did not modify the copy – and be traced out without harming the coherence of the control. If the agents modify their copy of the control however, they are transcending the set of operations which would be allowed in a routed switch, and may collapse the action of the process to a QC-CC (with classical control of causal order).

Finally, we briefly comment on the index structure we utilised. While in [39], all indices used in practice only take values $\{0, 1\}$, we use the index value to encode subsets $K_n \subseteq \mathcal{N}$ (or $\emptyset$) – with the notable exception of Eq. (5.5). Thereby, we demonstrate how a more complex set of possible index values may be helpful to obtain an adequate routed graph.

## 6.2 Characterising classes of processes

Several previous works have proposed subclasses of QC-QCs with specific properties of interest. These include a restriction to classical control (QC-CC) [34], to superpositions of fixed orders (Sup [54, 55]), or the more general QC-supFO class [56]), and to non-influenceable causal orders (QC-NIQC), capturing a weak kind of dynamicality of causal order [56].

Typically, the restrictions on these subclasses can be characterised as restrictions on the internal operations of a QC-QC. Our routed quantum circuit construction provides a new perspective from which to study these classes, as these restrictions may correspond to additional constraints on the routed graph. For instance, modifying the index structure to capture the full order $(k_1, \dots, k_N)$ from $V_1$ all the way to $V_{N+1}$ should yield a superposition of fixed orders. Generally, constraining the routed graph will involve introducing additional structure into the indices, effectively fine-graining the sectorisation of the systems to store additional information on the causal order, and imposing restrictions on the branch structure of the internal nodes $V_{n+1}$. The resulting graphs can then potentially be simplified following techniques analogous to those presented in Section 5.

Conversely, imposing constraints directly on the routed graph may be an interesting avenue to study new types of QC-QCs, a potential example being the subclass of processes which can be represented without involving internal transformations beyond local pre- and post-processing in-between the agents (cf. Appendix E). When assuming the operations associated with each branch of the respective nodes to be localised in spacetime, this property in particular affects the embeddability of the associated circuit into spacetime [11, 57].

In addition to capturing these classes in terms of routed quantum circuits, the properties of these representations may also serve as starting points to generalise the concepts they capture to non-QC-QC processes. For instance, in the language of routed quantum circuits, green dashed arrows in the branch graph may indicate whether a node has an influence on the order of later nodes: beyond QC-QCs, this has been showcased for both the Lugano process [39], and more recently, for a process with “unilateral determination of causal order” [48]. Studying this relation in detail would potentially allow the notion of dynamicality, which has so far been formalised for QC-QCs only [56], to be studied in processes beyond QC-QCs.

<footnote n="50">⁵⁰With the skeletal supermap associated with the routed graph from Section 5.1, that is, to prevent the agents from accessing the ancillary systems $\alpha_n$.</footnote>
## 6.3 Consequences for the routed quantum circuit framework

Last but not least, we consider the impact of our results on the general status of the routed quantum circuit framework. By extending the number of processes with an established routed quantum circuit decomposition from a small handful to the entire class of QC-QCs, we confirm the applicability of the framework to a wide class of physically implementable processes. Moreover, recent results connecting QC-QCs with the causal box framework [33, 41, 58] indicate that QC-QCs appear to be the most general class of processes compatible (and implementable) within a fixed (acyclic) spacetime. Combined with our results, this would imply that skeletal supermaps associated with routed graphs (without weak loops) – or even the subset of routed graphs from our construction – are capable of expressing any quantum process implementable in a fixed spacetime.<fnref n="51" /> This would provide some additional support the first main conjecture of [39], concerning the hypothesis that any unitary process may be obtained from fleshing out some skeletal supermap associated with a routed graph. The results of [52] further indicate that these statements may also be generalisable from fixed acyclic spacetimes to any settings where all agents (independently) record a monotonically increasing time, e.g. capturing gravitational quantum switch scenarios. Further progress in this direction, extending the result beyond processes implementable within a fixed spacetime, would most likely require significant progress in the study of causal decompositions of unitary channels [59, 60].

Considering the fact that QC-QCs do not violate causal inequalities [34], our results are in agreement with the second major conjecture of [39] – that a skeletal supermap violates causal inequalities if and only if its branch graph features weak loops. However, as our procedure only yields a small subset of valid routed graphs without weak loops, we are unable to provide any stronger statement in that regard.

The joint perspective obtained by studying QC-QCs and routed quantum circuits may also be valuable to provide further insight on the composability of processes. Indeed, although the validity conditions for routed graphs provide a condition to affirm the validity of compositions, a systematic study of the resulting family of routed graphs has been lacking so far. Restricting to compositions of routed graphs which represent faithful reconstructions of QC-QCs may help both to render the scope of this study more manageable and to restrict oneself to physically admissible implementations [40, 61–64]. Conversely, the clear graphical representation of connectivity may also allow one to restrict the class of processes attainable by composing a restricted set of “elementary” processes.

# 7 Acknowledgements

We thank Hlér Kristjánsson, Ved Kunte, Raphaël Mothe, Pierre Pocreau, Matthias Salzger, V. Vlasini and Emilien de Bank for helpful discussions.

This research was funded in part by l'Agence Nationale de la Recherche (ANR) projects ANR-15-IDEX-02 and ANR-22-CE47-0012, the PEPR integrated project EPiQ ANR-22-PETQ-0007 as part of Plan France 2030, and the STEP2 grant ANR-22-EXES-0013. For the purpose of open access, the authors have applied a CC-BY public copyright license to any Author Accepted Manuscript (AAM) version arising from this submission.

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# Appendix

## A Derivation of the choice function for $\mathcal{G}_{QC-QC}(N)$

Here we will evaluate explicitly the choice relation $\Lambda_{\mathcal{G}_{QC-QC}(N)}$, given by Eq. (4.13):

$$ \Lambda_{\mathcal{G}_{QC-QC}(N)} = \lambda_{\mathbf{A}_1}^{\text{aug}} * \cdots * \lambda_{\mathbf{A}_N}^{\text{aug}} * \lambda_{\mathbf{V}_1}^{\text{aug}} * \cdots * \lambda_{\mathbf{V}_{N+1}}^{\text{aug}}. \quad (A.1) $$

For this, let us start by explicitly writing the Boolean matrix elements of the different augmented relations in the graph:

* For $\lambda_{\mathbf{A}_k}^{\text{aug}}$, according to Eqs. (4.2) and (4.9), one can write (using the label $\mathcal{K}_{n_k-1}^{[\mathbf{A}_k]}$ rather than $\mathcal{K}_n \setminus k$ for the branch that happens at node $\mathbf{A}_k$, to distinguish it further below from the labels at the other nodes):

$$ (\lambda_{\mathbf{A}_k}^{\text{aug}})^{(\mathcal{X}_n^k)^{N}_{n=1}, \mathcal{K}_{n_k-1}^{[\mathbf{A}_k]}} (\mathcal{X}_n^k)^{N}_{n=1} = \delta_{\mathcal{X}_n^k, \mathcal{K}_{n_k-1}^{[\mathbf{A}_k]}} \cup k \left( \prod_{n \neq n_k} \delta_{\mathcal{X}_n^k, \emptyset} \right). \quad (A.2) $$

* For $\lambda_{\mathbf{V}_{n+1}}^{\text{aug}}$ ($n=1, \dots, N-1$), according to Eqs. (4.4) and (4.10), one can write (here for ease of notations we keep the label $\mathcal{K}_n$ for the branch that happens at node $\mathbf{V}_{n+1}$, rather than introducing $\mathcal{K}_n^{[\mathbf{V}_{n+1}]}$):

$$ \left(\Lambda_{\mathbf{V}_{n+1}}^{\text{aug}}\right)^{(\mathcal{X}_{n+1}^{\ell})^{N}_{\ell=1}, \mathcal{K}_n} (\mathcal{X}_n^k)^{N}_{k=1}, \vec{\mathcal{E}}_n = (\ell^{(\mathcal{K}_n')})_{\mathcal{K}_n'} = \left(\sum_{k \in \mathcal{K}_n} \left[\delta_{\mathcal{X}_n^k, \mathcal{K}_n} \left(\prod_{k' \in \mathcal{N} \setminus k} \delta_{\mathcal{X}_n^{k'}, \emptyset}\right)\right] \left(\delta_{\mathcal{X}_{n+1}^{\ell(\mathcal{K}_n)}, \mathcal{K}_n \cup \ell(\mathcal{K}_n)} \left(\prod_{\ell' \in \mathcal{N} \setminus \ell(\mathcal{K}_n)} \delta_{\mathcal{X}_{n+1}^{\ell'}, \emptyset}\right)\right)\right), \quad (A.3) $$

while for $\lambda_{\mathbf{V}_1}^{\text{aug}}$ and $\lambda_{\mathbf{V}_{N+1}}^{\text{aug}}$, according to Eqs. (4.3), (4.11) and (4.12), one can write (ignoring the trivial inputs and outputs, see Fig. 4.4):

$$ (\Lambda_{\mathbf{V}_1}^{\text{aug}})^{(\mathcal{X}_1^{\ell})^{N}_{\ell=1}} = \delta_{\mathcal{X}_1^{\ell(\emptyset)}, \ell(\emptyset)} \left( \prod_{\ell' \in \mathcal{N} \setminus \ell(\emptyset)} \delta_{\mathcal{X}_1^{\ell'}, \emptyset} \right) \quad (A.4) $$

$$ \text{and} \quad (\Lambda_{\mathbf{V}_{N+1}}^{\text{aug}})^{(\mathcal{X}_N^k)^{N}_{k=1}} = \sum_{k \in \mathcal{N}} \left[ \delta_{\mathcal{X}_N^k, \mathcal{N}} \left( \prod_{k' \in \mathcal{N} \setminus k} \delta_{\mathcal{X}_N^{k'}, \emptyset} \right) \right]. \quad (A.5) $$

Recalling from Section 2.1.1 how the “link product” * in Eq. (A.1) evaluates, we can write the Boolean matrix elements of the choice relation as

$$ \left(\Lambda_{\mathcal{G}_{QC-QC}(N)}\right)^{(\mathcal{K}_{n_k-1}^{[\mathbf{A}_k]})_{k \in \mathcal{N}}, (\mathcal{K}_n)^{N-1}_{n=1}} \vec{\mathcal{E}} = (\ell^{(\mathcal{K}_n')})_{n=0, \dots, N-1}; \mathcal{K}'_n = \sum_{(\mathcal{X}_n^k)_{n;k}} \left( \prod_{k \in \mathcal{N}} (\lambda_{\mathbf{A}_k}^{\text{aug}})^{(\mathcal{X}_n^k)^{N}_{n=1}, \mathcal{K}_{n_k-1}^{[\mathbf{A}_k]}} (\mathcal{X}_n^k)^{N}_{n=1} \right) \left( \Lambda_{\mathbf{V}_1}^{\text{aug}}\right)^{(\mathcal{X}_1^{\ell})^{N}_{\ell=1}} \left( \prod_{n=1}^{N-1} \left( \Lambda_{\mathbf{V}_{n+1}}^{\text{aug}}\right)^{(\mathcal{X}_{n+1}^{\ell})^{N}_{\ell=1}, \mathcal{K}_n} (\mathcal{X}_n^k)^{N}_{k=1}, \vec{\mathcal{E}}_n = (\ell^{(\mathcal{K}_n')})_{\mathcal{K}_n'} \right) \left(\Lambda_{\mathbf{V}_{N+1}}^{\text{aug}}\right)^{(\mathcal{X}_N^k)^{N}_{k=1}}. \quad (A.6) $$

To evaluate this expression, using Eqs. (A.2) to (A.5) above, one may start by noticing that for $m=1, \dots, N$,

$$ \left( \delta_{\mathcal{X}_m^{\ell(\mathcal{K}_{m-1})}, \mathcal{K}_{m-1} \cup \ell(\mathcal{K}_{m-1})} \left( \prod_{\ell' \in \mathcal{N} \setminus \ell(\mathcal{K}_{m-1})} \delta_{\mathcal{X}_m^{\ell'}, \emptyset} \right) \right) \left( \sum_{k \in \mathcal{K}_m} \left[ \delta_{\mathcal{X}_m^k, \mathcal{K}_m} \left( \prod_{k' \in \mathcal{N} \setminus k} \delta_{\mathcal{X}_m^{k'}, \emptyset} \right) \right] \right) = \delta_{\mathcal{K}_m, \mathcal{K}_{m-1} \cup \ell(\mathcal{K}_{m-1})} \delta_{\mathcal{X}_m^{\ell(\mathcal{K}_{m-1})}, \mathcal{K}_{m-1} \cup \ell(\mathcal{K}_{m-1})} \left( \prod_{\ell' \in \mathcal{N} \setminus \ell(\mathcal{K}_{m-1})} \delta_{\mathcal{X}_m^{\ell'}, \emptyset} \right). \quad (A.7) $$
From this one easily obtains, recursively, that

$$ (\Lambda_{\mathbf{V}}^{\text{aug}})_{\ell(\emptyset)}^{(\mathcal{X}_1^{\ell})^N_{\ell=1}} \left( \prod_{n=1}^{m} (\Lambda_{\mathbf{V}_{n+1}}^{\text{aug}})_{(\mathcal{X}_n^k)^N_{k=1}, \vec{\ell}_n = (\ell^{(\mathcal{K}_n')})_{\mathcal{K}_n'}}^{(\mathcal{X}_{n+1}^{\ell})^N_{\ell=1}, \mathcal{K}_n} \right) \\ = \delta_{\mathcal{K}_1, \ell(\emptyset)} \delta_{\mathcal{K}_2, \mathcal{K}_1 \cup \ell^{(\mathcal{K}_1)}} \cdots \delta_{\mathcal{K}_m, \mathcal{K}_{m-1} \cup \ell^{(\mathcal{K}_{m-1})}} \prod_{n=0}^{m} \left( \delta_{\mathcal{X}_{n+1}^{\ell(\mathcal{K}_n)}, \mathcal{K}_n \cup \ell^{(\mathcal{K}_n)}} \left( \prod_{\ell' \in \mathcal{N} \setminus \ell^{(\mathcal{K}_n)}} \delta_{\mathcal{X}_{n+1, \emptyset}^{\ell'}} \right) \right), \quad (A.8) $$

and that the third line in Eq. (A.6) takes the same form as above (for $m = N-1$), so that (recalling also Eq. (A.2))

$$ (\Lambda_{\mathcal{G}_{\text{QC-QC}(N)}})_{\vec{\ell}=(\ell^{(\mathcal{K}_n')})_{n=0,...,N-1}; \mathcal{K}_n'}^{(\mathcal{K}_{n-1}^{[\mathbf{A}_k]})_{k \in \mathcal{N}}, (\mathcal{K}_n)^{N-1}_{n=1}} \\ = \delta_{\mathcal{K}_1, \ell(\emptyset)} \delta_{\mathcal{K}_2, \mathcal{K}_1 \cup \ell^{(\mathcal{K}_1)}} \cdots \delta_{\mathcal{K}_{N-1}, \mathcal{K}_{N-2} \cup \ell^{(\mathcal{K}_{N-2})}} \\ \sum_{(\mathcal{X}_n^k)_{n;k}} \left[ \prod_{k \in \mathcal{N}} \left( \delta_{\mathcal{X}_{n_k}^k, \mathcal{K}_{n-1}^{[\mathbf{A}_k]} \cup k} \left( \prod_{n \neq n_k} \delta_{\mathcal{X}_n^k, \emptyset} \right) \right) \right] \left[ \prod_{n=0}^{N-1} \left( \delta_{\mathcal{X}_{n+1}^{\ell(\mathcal{K}_n)}, \mathcal{K}_n \cup \ell^{(\mathcal{K}_n)}} \left( \prod_{\ell' \in \mathcal{N} \setminus \ell^{(\mathcal{K}_n)}} \delta_{\mathcal{X}_{n+1, \emptyset}^{\ell'}} \right) \right) \right]. \quad (A.9) $$

Recalling the recursive definition of $\mathcal{K}_1^{\vec{\ell}} := \ell(\emptyset)$, $\mathcal{K}_{n+1}^{\vec{\ell}} := \mathcal{K}_n^{\vec{\ell}} \cup \ell^{(\mathcal{K}_n')}$ we introduced in Eq. (4.14), one can see that the above expression can only be nonzero if the $\mathcal{K}_n$'s satisfy the same recursive constraints, i.e. if $\mathcal{K}_n = \mathcal{K}_n^{\vec{\ell}}$ for all $n = 1, \dots, N-1$. The last product term in square brackets above then restricts the Boolean sum to only index values $\mathcal{X}_n^k$ such that, for any $n = 1, \dots, N$, $\mathcal{X}_n^k = \mathcal{K}_{n-1} \cup \ell^{(\mathcal{K}_{n-1})} = \mathcal{K}_n^{\vec{\ell}}$ if $k = \ell^{(\mathcal{K}_{n-1})} = \ell^{(\mathcal{K}_{n-1})}$ and $\mathcal{X}_n^k = \emptyset$ otherwise. For the first product term in square bracket to be nonzero, one then needs these to satisfy $\mathcal{X}_{n_k}^k \neq \emptyset$: this requires that for each $k \in \mathcal{N}$, $n_k$ be such that $k = \ell^{(\mathcal{K}_{n_k-1})}$, i.e., $n_k$ must be equal to $n_k^{\vec{\ell}}$ as we defined in Eq. (4.14). In that case $\mathcal{X}_{n_k}^k = \mathcal{K}_{n_k}^{\vec{\ell}}$, and the first Kronecker delta in the last line of Eq. (A.9) requires $\mathcal{K}_{n_k}^{\vec{\ell}} = \mathcal{K}_{n_k-1}^{[\mathbf{A}_k]} \cup k$ or equivalently, $\mathcal{K}_{n_k-1}^{[\mathbf{A}_k]} = \mathcal{K}_{n_k}^{\vec{\ell}}$ to be nonzero. Reciprocally, satisfying these various conditions indeed gives an overall value of 1 for the expression in Eq. (A.9) above.

All in all, we find that we can write

$$ (\Lambda_{\mathcal{G}_{\text{QC-QC}(N)}})_{\vec{\ell}=(\ell^{(\mathcal{K}_n')})_{n=0,...,N-1}; \mathcal{K}_n'}^{(\mathcal{K}_{n-1}^{[\mathbf{A}_k]})_{k \in \mathcal{N}}, (\mathcal{K}_n)^{N-1}_{n=1}} = \left( \prod_{k \in \mathcal{N}} \delta_{\mathcal{K}_{n_k-1}^{[\mathbf{A}_k]}, \mathcal{K}_n^{\vec{\ell}}_{n_k-1}} \right) \left( \prod_{n=1}^{N-1} \delta_{\mathcal{K}_n, \mathcal{K}_n^{\vec{\ell}}} \right). \quad (A.10) $$

As one sees, for any fixed input value $\vec{\ell} = (\ell^{(\mathcal{K}_n')})_{n=0,...,N-1}; \mathcal{K}_n'$ of the choice relation, there exists a unique output value $((\mathcal{K}_{n_k-1}^{[\mathbf{A}_k]})_{k \in \mathcal{N}}, (\mathcal{K}_n)^{N-1}_{n=1})$ for which all Kronecker deltas above give the value 1, so that the Boolean matrix element is 1. The choice relation thus defines a proper function, as described in Eq. (4.15) of the main text.

## B Obtaining a QC-QC by fleshing out a skeletal supermap

We provide here the detailed calculations that allowed us to obtain

$$ \begin{aligned} & \left(\bigotimes_{n=0}^{N} |\mathbf{V}_{n+1}\rangle\rangle\right) * \left(\bigotimes_{k \in \mathcal{N}} |\mathbf{A}_k\rangle\rangle\right) \\ = & \sum_{(k_1, k_2, \dots, k_N)} |V_{\emptyset, \emptyset}^{\to k_1}\rangle\rangle^{P, A_{k_1}^I \alpha_1} * |V_{\emptyset, k_1}^{\to k_2}\rangle\rangle^{A_{k_1}^O \alpha_1, A_{k_2}^I \alpha_2} * |V_{\{k_1\}, k_2}\rangle\rangle^{A_{k_2}^O \alpha_2, A_{k_3}^I \alpha_3} * \dots \\ & \cdots * |V_{\{k_1, \dots, k_{N-2}\}, k_{N-1}}\rangle\rangle^{A_{k_{N-1}}^O \alpha_{N-1}, A_{k_N}^I \alpha_N} * |V_{\{k_1, \dots, k_{N-1}\}, k_N}\rangle\rangle^{A_{k_N}^O \alpha_N, F} = |w_{\text{QC-QC}}\rangle\rangle, \end{aligned} \quad (B.1) $$

as in Eq. (4.34).
To prove this, we first show recursively that, for any $m = 0, \dots, N-1$,

$$ egin{aligned} & \left(\bigotimes_{n=0}^{m} |\mathbf{V}_{n+1}\rangle\rangle\right) * \left(\bigotimes_{k \in \mathcal{N}} |\mathbf{A}_k\rangle\rangle\right) \\ = & \sum_{(k_1, k_2, \dots, k_m)} |V_{\emptyset, \emptyset}^{\to k_1}\rangle\rangle^{P, A_{k_1}^I \alpha_1} * |V_{\emptyset, k_1}^{\to k_2}\rangle\rangle^{A_{k_1}^O \alpha_1, A_{k_2}^I \alpha_2} * \dots * |V_{\{k_1, \dots, k_{m-2}\}, k_{m-1}}^{\to k_m}\rangle\rangle^{A_{k_{m-1}}^O \alpha_{m-1}, A_{k_m}^I \alpha_m} \\ & \qquad * \bigoplus_{k_{m+1}} |V_{\{k_1, \dots, k_{m-1}\}, k_m}^{\to k_{m+1}}\rangle\rangle^{A_{k_m}^O \alpha_m, A_{k_{m+1}}^I \alpha_{m+1}} * |1\rangle\rangle^{A_{k_{m+1}}^O \alpha_{m+1}, \mathcal{H}_{\{k_1, \dots, k_m\}, k_{m+1}}^{\mathbf{A}_{k_{m+1}} \to \mathbf{V}_{m+2}}} \\ & \qquad * \left( \bigotimes_{k \in \mathcal{N} \setminus \{k_1, \dots, k_{m+1}\}} |\mathbf{A}_k\rangle\rangle\right). \end{aligned} \quad (B.2) $$

Indeed, starting from $m = 0$, one has (changing the dummy index $\ell$ in Eq. (4.26) to $k_1$)

$$ egin{aligned} & |\mathbf{V}_1\rangle\rangle * \left(\bigotimes_{k \in \mathcal{N}} |\mathbf{A}_k\rangle\rangle\right) = \bigoplus_{k_1 \in \mathcal{N}} |V_{\emptyset, k_1}^{\to k_1}\rangle\rangle^{P, \mathcal{H}_{\emptyset, k_1}^{\mathbf{V}_1 \to \mathbf{A}_{k_1}}} * \left( |\mathbf{A}_{k_1}\rangle\rangle \otimes \bigotimes_{k \in \mathcal{N} \setminus \{k_1\}} |\mathbf{A}_k\rangle\rangle \right) \\ = & \bigoplus_{k_1} |V_{\emptyset, \emptyset}^{\to k_1}\rangle\rangle^{P, \mathcal{H}_{\emptyset, k_1}^{\mathbf{V}_1 \to \mathbf{A}_{k_1}}} * \left( \bigoplus_{n=1}^{N} \bigoplus_{\mathcal{K}_n \ni k_1} |1\rangle\rangle^{\mathcal{H}_{\mathcal{K}_n \setminus k_1, k_1}^{\mathbf{V}_n \to \mathbf{A}_{k_1}}, A_{k_1}^I \alpha_n} * |1\rangle\rangle^{A_{k_1}^O \alpha_n, \mathcal{H}_{\mathcal{K}_n \setminus k_1, k_1}^{\mathbf{A}_{k_1} \to \mathbf{V}_{n+1}}} \right) * \left( \bigotimes_{k \in \mathcal{N} \setminus \{k_1\}} |\mathbf{A}_k\rangle\rangle \right) \\ = & \bigoplus_{k_1} |V_{\emptyset, \emptyset}^{\to k_1}\rangle\rangle^{P, A_{k_1}^I \alpha_1} * |1\rangle\rangle^{A_{k_1}^O \alpha_1, \mathcal{H}_{\emptyset, k_1}^{\mathbf{A}_{k_1} \to \mathbf{V}_2}} * \left( \bigotimes_{k \in \mathcal{N} \setminus \{k_1\}} |\mathbf{A}_k\rangle\rangle \right), \end{aligned} \quad (B.3) $$

as in Eq. (B.2) – with the understanding that for $m = 0$, only the direct sum over $k_1$ was left, and that the link product of all $|V_{\cdot, \cdot}^{\to k_1}\rangle\rangle$'s reduced to just $|V_{\emptyset, \emptyset}^{\to k_1}\rangle\rangle^{P, A_{k_1}^I \alpha_1}$.

Supposing that Eq. (B.2) holds for some value $m = 0, \dots, N-2$, one can then write (similarly changing the dummy index $\ell$ in Eq. (4.25) to $k_{m+2}$)

$$ egin{aligned} & \left(\bigotimes_{n=0}^{m+1} |\mathbf{V}_{n+1}\rangle\rangle\right) * \left(\bigotimes_{k \in \mathcal{N}} |\mathbf{A}_k\rangle\rangle\right) = \left(\bigotimes_{n=0}^{m} |\mathbf{V}_{n+1}\rangle\rangle\right) * |\mathbf{V}_{m+2}\rangle\rangle * \left(\bigotimes_{k \in \mathcal{N}} |\mathbf{A}_k\rangle\rangle\right) \\ = & \sum_{(k_1, k_2, \dots, k_m)} |V_{\emptyset, \emptyset}^{\to k_1}\rangle\rangle^{P, A_{k_1}^I \alpha_1} * |V_{\emptyset, k_2}^{\to k_2}\rangle\rangle^{A_{k_1}^O \alpha_1, A_{k_2}^I \alpha_2} * \dots * |V_{\{k_1, \dots, k_{m-2}\}, k_{m-1}}^{\to k_m}\rangle\rangle^{A_{k_{m-1}}^O \alpha_{m-1}, A_{k_m}^I \alpha_m} \\ & \qquad * \bigoplus_{k_{m+1}} |V_{\{k_1, \dots, k_{m-1}\}, k_m}^{\to k_{m+1}}\rangle\rangle^{A_{k_m}^O \alpha_m, A_{k_{m+1}}^I \alpha_{m+1}} * |1\rangle\rangle^{A_{k_{m+1}}^O \alpha_{m+1}, \mathcal{H}_{\{k_1, \dots, k_m\}, k_{m+1}}^{\mathbf{A}_{k_{m+1}} \to \mathbf{V}_{m+2}}} \\ & \qquad * |\mathbf{V}_{m+2}\rangle\rangle * \left( \bigotimes_{k \in \mathcal{N} \setminus \{k_1, \dots, k_{m+1}\}} |\mathbf{A}_k\rangle\rangle \right) \\ = & \sum_{(k_1, k_2, \dots, k_m)} \dots * \bigoplus_{k_{m+1}} |V_{\{k_1, \dots, k_{m-1}\}, k_m}^{\to k_{m+1}}\rangle\rangle^{A_{k_m}^O \alpha_m, A_{k_{m+1}}^I \alpha_{m+1}} * |1\rangle\rangle^{A_{k_{m+1}}^O \alpha_{m+1}, \mathcal{H}_{\{k_1, \dots, k_m\}, k_{m+1}}^{\mathbf{A}_{k_{m+1}} \to \mathbf{V}_{m+2}}} \\ & \qquad * \bigoplus_{\substack{\mathcal{K}_{m+1}, k \in \mathcal{K}_{m+1} \\ k_{m+2} \notin \mathcal{K}_{m+1}}} |V_{\mathcal{K}_{m+1} \setminus k, k}^{\to k_{m+2}}\rangle\rangle^{\mathcal{H}_{\mathcal{K}_{m+1} \setminus k, k}^{\mathbf{A}_{k_{m+1}} \to \mathbf{V}_{m+2}}, \mathcal{H}_{\mathcal{K}_{m+1}, k_{m+2}}^{\mathbf{V}_{m+2} \to \mathbf{A}_{k_{m+2}}}} \\ = & \sum_{(k_1, k_2, \dots, k_m)} \dots * \bigoplus_{k_{m+1}} |V_{\{k_1, \dots, k_{m-1}\}, k_m}^{\to k_{m+1}}\rangle\rangle^{A_{k_m}^O \alpha_m, A_{k_{m+1}}^I \alpha_{m+1}} * |1\rangle\rangle^{A_{k_{m+1}}^O \alpha_{m+1}, \mathcal{H}_{\{k_1, \dots, k_m\}, k_{m+1}}^{\mathbf{A}_{k_{m+1}} \to \mathbf{V}_{m+2}}} \\ & \qquad * \bigoplus_{k_{m+2}} |V_{\{k_1, \dots, k_m\}, k_{m+1}}^{\to k_{m+2}}\rangle\rangle^{\mathcal{H}_{\{k_1, \dots, k_m\}, k_{m+1}}^{\mathbf{A}_{k_{m+1}} \to \mathbf{V}_{m+2}}, \mathcal{H}_{\{k_1, \dots, k_{m+1}\}, k_{m+2}}^{\mathbf{V}_{m+2} \to \mathbf{A}_{k_{m+2}}}} \\ & \qquad * \left( \bigoplus_{n=1}^{N} \bigoplus_{\mathcal{K}_n \ni k_{m+2}} |1\rangle\rangle^{\mathcal{H}_{\mathcal{K}_n \setminus k_{m+2}, k_{m+2}}^{\mathbf{V}_n \to \mathbf{A}_{k_{m+2}}, A_{k_{m+2}}^I \alpha_n} * |1\rangle\rangle^{A_{k_{m+2}}^O \alpha_n, \mathcal{H}_{\mathcal{K}_n \setminus k_{m+2}, k_{m+2}}^{\mathbf{A}_{k_{m+2}} \to \mathbf{V}_{n+1}}} \right) * \left( \bigotimes_{k \in \mathcal{N} \setminus \{k_1, \dots, k_{m+2}\}} |\mathbf{A}_k\rangle\rangle \right) \end{aligned} $$
$$
= \sum_{(k_1, k_2, \dots, k_m)} \cdots \sum_{k_{m+1}} |V_{\{k_1, \dots, k_{m-1}\}, k_m}^{\to k_{m+1}}|^{A_{k_m}^O \alpha_m} A_{k_{m+1}}^I \alpha_{m+1} \\
\qquad * \bigoplus_{k_{m+2}} |V_{\{k_1, \dots, k_m\}, k_{m+1}}^{\to k_{m+2}}|^{A_{k_{m+1}}^O \alpha_{m+1}} A_{k_{m+2}}^I \alpha_{m+2} \\
\qquad * |1\rangle^{A_{k_{m+2}}^O \alpha_{m+2}} \mathcal{H}_{\{k_1, \dots, k_{m+1}\}, k_{m+2}}^{A_{k_{m+2}} \to \mathbf{V}_{m+3}} \\
\qquad * \left( \bigotimes_{k \in \mathcal{N} \setminus \{k_1, \dots, k_{m+2}\}} |\mathbf{A}_k\rangle \right), \quad (B.4)
$$

so that we obtain the same form as Eq. (B.2), for the value of $m+1$ instead of $m$ – which indeed
proves recursively that Eq. (B.2) holds for all $m = 0, \dots, N-1$. For $m=N-1$ more specifically,
we obtain:

$$
\begin{align}
& \left( \bigotimes_{n=0}^{N-1} |\mathbf{V}_{n+1}\rangle \right) * \left( \bigotimes_{k \in \mathcal{N}} |\mathbf{A}_k\rangle \right) \nonumber \\
&= \sum_{(k_1, \dots, k_{N-1})} |V_{\emptyset, \emptyset}^{\to k_1}\rangle^{P, A_{k_1}^I \alpha_1} * |V_{\emptyset, k_2}^{\to k_2}\rangle^{A_{k_1}^O \alpha_1, A_{k_2}^I \alpha_2} * \dots * |V_{\{k_1, \dots, k_{N-3}\}, k_{N-2}}^{\to k_{N-1}}\rangle^{A_{k_{N-2}}^O \alpha_{N-2}, A_{k_{N-1}}^I \alpha_{N-1}} \nonumber \\
&\quad * \bigoplus_{k_N} |V_{\{k_1, \dots, k_{N-2}\}, k_{N-1}}^{\to k_N}\rangle^{A_{k_{N-1}}^O \alpha_{N-1}, A_{k_N}^I \alpha_N} * |1\rangle^{A_{k_N}^O \alpha_N, \mathcal{H}_{\{k_1, \dots, k_{N-1}\}, k_N}^{A_{k_N} \to \mathbf{V}_{N+1}}} \tag{B.5}
\end{align}
$$

With this in place, we can then write

$$
\begin{align*}
& \left( \bigotimes_{n=0}^{N} |\mathbf{V}_{n+1}\rangle \right) * \left( \bigotimes_{k \in \mathcal{N}} |\mathbf{A}_k\rangle \right) = \left( \bigotimes_{n=0}^{N-1} |\mathbf{V}_{n+1}\rangle \right) * \left( \bigotimes_{k \in \mathcal{N}} |\mathbf{A}_k\rangle \right) * |\mathbf{V}_{N+1}\rangle \\
&= \sum_{(k_1, \dots, k_{N-1})} |V_{\emptyset, \emptyset}^{\to k_1}\rangle^{P, A_{k_1}^I \alpha_1} * |V_{\emptyset, k_2}^{\to k_2}\rangle^{A_{k_1}^O \alpha_1, A_{k_2}^I \alpha_2} * \dots * |V_{\{k_1, \dots, k_{N-3}\}, k_{N-2}}^{\to k_{N-1}}\rangle^{A_{k_{N-2}}^O \alpha_{N-2}, A_{k_{N-1}}^I \alpha_{N-1}} \\
&\quad * \bigoplus_{k_N} |V_{\{k_1, \dots, k_{N-2}\}, k_{N-1}}^{\to k_N}\rangle^{A_{k_{N-1}}^O \alpha_{N-1}, A_{k_N}^I \alpha_N} * |1\rangle^{A_{k_N}^O \alpha_N, \mathcal{H}_{\{k_1, \dots, k_{N-1}\}, k_N}^{A_{k_N} \to \mathbf{V}_{N+1}}} \\
&\quad * \bigoplus_{k \in \mathcal{N}} |V_{\mathcal{N} \setminus k, k}^{\to F}\rangle^{\mathcal{H}_{\mathcal{N} \setminus k, k}^{A_k \to \mathbf{V}_{N+1}, F}} \\
&= \sum_{(k_1, \dots, k_{N-1})} |V_{\emptyset, \emptyset}^{\to k_1}\rangle^{P, A_{k_1}^I \alpha_1} * |V_{\emptyset, k_2}^{\to k_2}\rangle^{A_{k_1}^O \alpha_1, A_{k_2}^I \alpha_2} * \dots * |V_{\{k_1, \dots, k_{N-3}\}, k_{N-2}}^{\to k_{N-1}}\rangle^{A_{k_{N-2}}^O \alpha_{N-2}, A_{k_{N-1}}^I \alpha_{N-1}} \\
&\quad * \sum_{k_N} |V_{\{k_1, \dots, k_{N-2}\}, k_{N-1}}^{\to k_N}\rangle^{A_{k_{N-1}}^O \alpha_{N-1}, A_{k_N}^I \alpha_N} * |V_{\{k_1, \dots, k_{N-1}\}, k_N}^{\to F}\rangle^{A_{k_N}^O \alpha_N, F}, \tag{B.6}
\end{align*}
$$

which gives precisely Eq. (4.34).

C Proofs for Section 5

**Lemma 5.2.** Considering the index values $\mathcal{N}$ to be 1-dimensional, the split-node generic routed graph $\mathcal{G}_{\text{QC-QC}(N)}^{\text{split}}$ is a valid routed graph.

*Proof.* The split-node generic routed graph is a modified version of the generic routed graph, featuring multiple nodes $\hat{\mathbf{V}}_{n+1}^{\mathcal{K}_n}$ in place of a single original node $\mathbf{V}_{n+1}$, being endowed with a subset of its arrows. However, all existing branches are preserved under this change.

The modification preserves the property of univocality: For the additional branches $\bar{\mathbf{V}}_{n+1}^{\mathcal{K}_n}$ with their singleton input/output index sets, no non-trivial bifurcation choice $\text{Ind}_{\bar{\mathbf{V}}_{n+1}^{\mathcal{K}_n}}^{\text{out}}$ is made. Therefore, the choice function for $\mathcal{G}_{\text{QC-QC}(N)}$ (cf. Eq. (2.7)) remains practically unchanged: Whenever $\Lambda_{(\Gamma, (\lambda'_N)_N)}$ evaluates to branch $\mathbf{V}_{n+1}^{\mathcal{K}_n}$ for node $\mathbf{V}_{n+1}$, the new choice relation $\Lambda_{(\Gamma', (\lambda'_N)_N)}$ will evaluate to branch $\mathbf{V}_{n+1}^{\mathcal{K}_n}$ for node $\hat{\mathbf{V}}_{n+1}^{\mathcal{K}_n}$, and to branch $\bar{\mathbf{V}}_{n+1}^{\mathcal{K}'_n}$ for any node $\bar{\mathbf{V}}_{n+1}^{\mathcal{K}'_n}$ with $\mathcal{K}'_n \neq \mathcal{K}_n$. An analogous argument applies for the adjoint for the graph, confirming bi-univocality of the resulting graph.
For the branch graph, the existing nodes $V_{n+1}^{\mathcal{K}_n}$ and the arrows connected therewith remain unchanged, while we receive new arrows $\bar{V}_{n+1}^{\mathcal{K}_n}$ endowed with the same incoming green arrows and outgoing red arrows as $V_{n+1}^{\mathcal{K}_n}$. By contrast, the $V_{n+1}^{\mathcal{K}_n}$ do not participate in any strong parent/child relationship, as they feature only a single, one-dimensional index value. Hence, the partial order of arrows, independent of colour, in the branch graph is preserved from the generic routed graph, and the rewriting does not introduce any loops into the new branch graph. $\square$

**Lemma 5.3.** Consider the two routed graphs $\mathcal{G}_{QC-QC(N)}$ and $\mathcal{G}_{QC-QC(N)}^{\text{split}}$. Restricting to the case in which all $V_n$ and $V_{n+1}^{\mathcal{K}_n}$ are fleshed out by routed isometries, there is a canonical mapping between equivalent routed circuit decompositions (as defined in Definition 2.26) for the two routed graphs.

*Proof.* Any routed isometry $V$ fleshing out the slot for a node $V_{n+1}$ needs to follow the branched route $\lambda V_{n+1}$ and therefore be of the form $V = \bigoplus_{\mathcal{K}_n} V^{\mathcal{K}_n}$ with $V^{\mathcal{K}_n} : \mathcal{H}_{\mathcal{K}_n}^{\text{in}(V_{n+1})} \to \mathcal{H}_{\mathcal{K}_n}^{\text{out}(V_{n+1})}$. Therefore, we can take these $V^{\mathcal{K}_n}$ individually to flesh out the slots associated with the branch nodes $\hat{V}_{n+1}^{\mathcal{K}_n}$, with $\hat{V}^{\mathcal{K}_n} := V^{\mathcal{K}_n} \oplus \mathbb{1}_{\bar{V}_{n+1}^{\mathcal{K}_n}} \to \text{out}(\hat{V}_{n+1}^{\mathcal{K}_n})$, following the respective routes. Therefore, any routed circuit decomposition based on $\mathcal{G}$ can equivalently be obtained using $\mathcal{G}^{\text{split}}$, provided that the slot for $V_{n+1}$ is fleshed out with a routed isometry rather than an routed superisometry.

Conversely, consider arbitrary routed circuit compositions for $\mathcal{G}^{\text{split}}$. While the most general fleshing out for the slot associated with $\hat{V}_{n+1}^{\mathcal{K}_n}$ would be

$$ \hat{V}^{\mathcal{K}_n} = V^{\mathcal{K}_n} \oplus e^{i\varphi_{\mathcal{K}_n}} \cdot \mathbb{1}_{\bar{V}_{n+1}^{\mathcal{K}_n}} \to \text{out}(\hat{V}_{n+1}^{\mathcal{K}_n}), \quad (C.1) $$

we can restrict to $\varphi_{\mathcal{K}_n} = 0$, as a given phase $\varphi_{\mathcal{K}_n}$ could always be absorbed into the remaining $V_{n+1}^{\mathcal{K}_n}$. Therefore, any routed circuit decomposition based on $\mathcal{G}_{QC-QC(N)}^{\text{split}}$ can equivalently be obtained using $\mathcal{G}_{QC-QC(N)}$ as well (again, given that the slot for $V_{n+1}$ is fleshed out with a routed isometry rather than an routed superisometry). $\square$

**Lemma 5.4.** Let $\mathcal{G} = (\Gamma, (\lambda_N)_{N \in \text{Nodes}_\Gamma})$ be a valid routed graph without any weak loops, and $\mathbf{A}, \mathbf{V} \in \text{Nodes}_\Gamma$ such that $\mathbf{A}$ is the sole child of $\mathbf{V}$ (cf. Fig. 5.4a), and the associated branch structure is such that

$$ \forall \mathbf{A}^{\beta} \in \text{Bran}(\lambda_{\mathbf{A}}) \quad \exists! \mathbf{V}^{\gamma} \in \text{Bran}(\lambda_{\mathbf{V}}) : \text{LinkVal}(\mathbf{V}^{\gamma}, \mathbf{A}^{\beta}) \neq \emptyset \quad (5.10) $$

Consider another routed graph $\mathcal{G}^{\text{merge}\uparrow}$ obtained by merging $\mathbf{V}$ into $\mathbf{A}$, as follows:

* the two nodes being replaced by a single node $\mathbf{A}'$, with incoming arrows $\text{in}(\mathbf{A}') = \text{in}(\mathbf{V}) \cup (\text{in}(\mathbf{A}) \setminus \text{out}(\mathbf{V}))$ and outgoing arrows $\text{out}(\mathbf{A}') = \text{out}(\mathbf{A})$, as shown in Fig. 5.4c,

* the resulting node $\mathbf{A}'$ being endowed with the route $\lambda_{\mathbf{A}'} = \lambda_{\mathbf{A}} * \lambda_{\mathbf{V}}$.

Then $\mathcal{G}^{\text{merge}\uparrow}$ is a valid routed graph.

*Proof.* By construction, every $k \in \text{Ind}_{\mathbf{V} \to \mathbf{A}}^{\text{prac}} = \bigsqcup_{\mathbf{V}^{\gamma}} \text{Ind}_{\mathbf{V}^{\gamma}}^{\text{out}}$ can be associated with a branch $\mathbf{V}^{\gamma}$. Equation (5.10) additionally ensures that each $k$ is associated with exactly one branch $\mathbf{A}^{\beta}$ and $\bigsqcup_{\mathbf{A}^{\beta}: \text{LinkVal}(\mathbf{V}^{\gamma}, \mathbf{A}^{\beta}) \neq \emptyset} \text{Ind}_{\mathbf{A}^{\beta}}^{\text{in}} = \text{Ind}_{\mathbf{V}^{\gamma}}^{\text{out}}$.

Concatenating the branches $\mathbf{A}^{\beta}$ and $\mathbf{V}^{\gamma}$ (or rather, their respective part of the relation) to a single branch $\mathbf{A}'^{\gamma}$ within a joint node $\mathbf{A}'$ yields a “completely connected” (sub-)relation $\text{Ind}_{\mathbf{V}^{\gamma}}^{\text{in}} \to \bigsqcup_{\mathbf{A}^{\beta}} \text{Ind}_{\mathbf{A}^{\beta}}^{\text{out}}$, which may correspond to a branch (if the entire relation turns out to be branched). Concatenating all branches in this fashion, altogether we indeed confirm the resulting relation $\lambda_{\mathbf{A}'} = \lambda_{\mathbf{V}} * \lambda_{\mathbf{A}}$ to be branched.

We now consider how this changes the choice function $\Lambda_{\mathcal{G}}$: For the arguments, the bifurcation choices $\text{Ind}_{\mathbf{V}^{\gamma}}^{\text{out}}$ and $\text{Ind}_{\mathbf{A}^{\beta}}^{\text{out}}$ are affected, while in its image, multiple branches of $\mathbf{A}^{\beta}$ may be coarse-grained to a single value of the choice function. However, due to the structure of the graph, any choice made at $\mathbf{V}^{\gamma}$ is completely washed out at $\mathbf{A}^{\beta}$ anyway, keeping the arguments of the
choice function practically unchanged for determining the remaining indices. Therefore, the choice function remains a function, and the graph is univocal.

As the scenario is inherently asymmetric, we also consider univocality for the adjoint graph $G^T$ explicitly. In this graph, we have $\mathcal{L}_{A^\beta, \text{LinkVal}(A^\beta, V^\gamma) \neq \emptyset} \text{Ind}_{A^\beta}^{\text{out}} = \text{Ind}_{V^\gamma}^{\text{in}}$, and a branched route $\lambda_{A^\beta}$. However, the arguments of the choice function change in a different manner, with any bifurcation choice made within the branch $V^\gamma$ being preserved through $A^\beta$, and therefore leaving the choice function practically invariant once more.

Finally, if the original graph did not feature any loops (which means that the branch graph is acyclic even ignoring any colours), contracting across an arrow can only introduce loops into the graph if another directed path exists between the respective nodes in the branch graph. As $A$ is the sole child of $V$, this can not be the case here. Therefore, the routed graph is valid. $\square$

**Lemma 5.5.** Let $\mathcal{G} = (\Gamma, (\lambda_N)_{N \in \text{Nodes}_\Gamma})$ be a valid routed graph without any weak loops, and $A, V \in \text{Nodes}_\Gamma$ such that $A$ is the sole parent of $V$ (cf. Fig. 5.4d) and the associated branch structure is such that

$$ \forall A^\beta \in \text{Bran}(\lambda_A) \quad \exists! V^\gamma \in \text{Bran}(\lambda_V) : \text{LinkVal}(A^\beta, V^\gamma) \neq \emptyset \quad (5.11) $$

Consider another routed graph $\mathcal{G}^{\text{merge}\downarrow}$ obtained by merging $V$ into $A$, similarly to Lemma 5.4. Then $\mathcal{G}^{\text{merge}\downarrow}$ is a valid routed graph.

*Proof.* This is the adjoint scenario for Lemma 5.4, so the proof follows by analogy. $\square$

**Proposition 5.6.** Any process with a routed circuit decomposition (as defined in Definition 2.26) according to the routed graph $\mathcal{G}$, where $V$ is fleshed out with a routed isometry (an “internal” node), can also be expressed with a routed circuit decomposition according to $\mathcal{G}^{\text{merge}\uparrow} / \mathcal{G}^{\text{merge}\downarrow}$, related to $\mathcal{G}$ via Lemma 5.4 / Lemma 5.5, and vice versa.

*Proof.* The backward direction of the proof is obvious: For any fleshing out of a mergeable node pair $A, V$ in $\mathcal{G}$, we obtain an associated fleshing out for $\mathcal{G}^{\text{merge}\uparrow/\downarrow}$ by fleshing out the resulting node with $|A\rangle * |V\rangle$, composing the respective routed (super)maps along the composition of the respective routes.

For the forward direction, it remains to show that splitting the merged node into two, “undoing” the merge, does not restrict the set of possible operations due to an insufficient dimension for (some) sectors of $\mathcal{H}^{V \to A} / \mathcal{H}^{A \to V}$, which could lead to a more restricted set of processes. To do so, we remind of the fact that the practical (and therefore operationally relevant) Hilbert space $\mathcal{H}_{\text{prac}}^{\text{in/out}(N)}$ of a node $V$ is given by the direct sum of all individual branch spaces $\mathcal{H}_{V^\gamma}^{\text{in/out}(N)}$.

**I. Proof for $\mathcal{G}^{\text{merge}\uparrow}$:** For any fleshing out of $V$ (with a routed isometry), we have $\forall V^\gamma \in \text{Bran}(\lambda_V) : \dim(\mathcal{H}_{V^\gamma}^{\text{out}(V)}) \ge \dim(\mathcal{H}_{V^\gamma}^{\text{in}(V)})$. Therefore, no valid dimension assignment for $\mathcal{H}^{V \to A}$ (with $\dim(\mathcal{H}_{V^\gamma}^{\text{out}(V)}) = \dim(\bigoplus_{k \in \text{Ind}_{V^\gamma}^{\text{in}(V)}} \mathcal{H}_{k \to A}^{V \to A})$) can restrict the set of possible operations of form $\mathcal{H}^{\text{in}(V)} \xrightarrow{\lambda_{A^*V}} \mathcal{H}^{\text{out}(A)}$.

**II. Proof for $\mathcal{G}^{\text{merge}\downarrow}$:**

First, consider the case where the skeletal supermap specifies that $V$ must be fleshed out by a routed unitary, which entails that the dimension assignment for the skeletal supermap is such that $\forall V^\gamma \in \text{Bran}(\lambda_V) : \dim(\mathcal{H}_{V^\gamma}^{\text{out}(V)}) = \dim(\mathcal{H}_{V^\gamma}^{\text{in}(V)})$. In this case, the dimension for $\dim(\mathcal{H}_{V^\gamma}^{\text{in}(V)}) = \dim(\bigoplus_{k \in \text{Ind}_{V^\gamma}^{\text{in}(V)}} \mathcal{H}_{k \to A}^{V \to A})$ is clearly sufficient to not restrict the set of possible operations $\mathcal{H}_{\text{in}(A)} \xrightarrow{\lambda_{A^*V}} \mathcal{H}^{\text{out}(V)}$. However, for the routed circuit decomposition as a whole, the restriction to a routed unitary is actually without loss of generality: Specifying a monopartite routed superisometry to flesh out $A$, we may choose it so that it prepares some independent state and sends it to $V$, allowing to recover any (routed) isometry by composition of this state with the (routed) unitary. The specifics of this argument are analogous to the proof of Corollary 2.25. $\square$
## D Reconstruction of the routed circuit decomposition of the Grenoble process in [39]

In Figure 25 of [39], a routed quantum circuit decomposition of the Grenoble process is displayed. While we outlined the general procedure to obtain this representation from the $G_{QC-QC(N)}$ in Example 5.8, we did not specify a detailed mapping between the two representations.

Here, we provide the detailed translations required to relate both representations. For the index values, we have:

$$ 
\begin{array}{l}
l = 1 \rightarrow \chi_{\emptyset}^{A} = \{A\}, \quad m = 1 \rightarrow \chi_{\emptyset}^{B} = \{B\}, \quad n = 1 \rightarrow \chi_{\emptyset}^{C} = \{C\}, \\
l_1 = 1 \rightarrow \chi_{A}^{B} = \{A, B\}, \quad m_1 = 1 \rightarrow \chi_{B}^{C} = \{B, C\}, \quad n_1 = 1 \rightarrow \chi_{C}^{A} = \{C, A\}, \\
l_2 = 1 \rightarrow \chi_{A}^{C} = \{A, C\}, \quad m_2 = 1 \rightarrow \chi_{B}^{A} = \{A, B\}, \quad n_2 = 1 \rightarrow \chi_{C}^{B} = \{B, C\}, \\
f = 1 \rightarrow \chi_{BC}^{A} = \{A, B, C\}, \quad g = 1 \rightarrow \chi_{AC}^{B} = \{A, B, C\}, \quad h = 1 \rightarrow \chi_{AB}^{C} = \{A, B, C\}. 
\end{array} 
\quad (D.1) $$

For all of their indices, if they take the value 0, we set the associated index $\chi_{K_n}^k = \emptyset$.

Slightly modifying the circuit by routing $E^g$ trivially through $W_B$, for the Hilbert spaces we have:

$$ \mathcal{H}^{P_C} \otimes \mathcal{H}^{P_T} \sim \mathcal{H}^P, \quad (D.2) $$

$$ \mathcal{H}^{F_T} \sim \mathcal{H}^F, \quad (D.3) $$

$$ \mathcal{H}^{F_A} \otimes \mathcal{H}^{F_C} \sim \mathcal{H}^{\alpha_F}, \quad (D.4) $$

$$ \mathcal{H}^{E^g} \otimes \mathcal{H}^{Q^{qml_1 n_2}} \sim \mathcal{H}^{\bar{\alpha}} \otimes \mathcal{H}^{\bar{C}^{(k)}}, \quad (D.5) $$

$$ \mathcal{H}^{E^g} \otimes \mathcal{H}^{S^g} \sim \mathcal{H}^{C \to V_4}, \quad (D.6) $$

$$ \mathcal{H}^{E^{g=1}} \otimes \mathcal{H}^{S^{g=1}} \sim \mathcal{H}^{\alpha_3} \otimes \mathcal{H}^{C_3^{(B)}}, \quad (D.7) $$

$$ \mathcal{H}^{X^{ml_1}} \sim \mathcal{H}^{B \to C}, \quad (D.8) $$

$$ \mathcal{H}^{Y^{m_2 n_2}} \sim \mathcal{H}^{B \to A} \quad (D.9) $$

etc. Similarly, we identify

$$ U_P \sim |V_1\rangle\rangle, \quad \tilde{E}_F \sim |V_4\rangle\rangle, \quad (D.10) $$

$$ V_B \sim |J_{V_3^{AC}}^{in}\rangle\rangle * |\tilde{V}_3^{AC}\rangle\rangle * |J_{V_3^{AC}}^{out}\rangle\rangle * |J_{B}^{in}\rangle\rangle, \quad (D.11) $$

$$ W_B \sim |J_{B}^{out}\rangle\rangle * |J_{V_2^{B}}^{in}\rangle\rangle * |\tilde{V}_2^{B}\rangle\rangle * |J_{V_2}^{out}\rangle\rangle \quad (D.12) $$

etc. Indeed we see that system $E^g$ being withheld from the agent B ensures that its output dimension does not change, even though $2 = \dim(\mathcal{H}^{\alpha_3}) > \dim(\mathcal{H}^{\alpha_2}) = 1$, while the agent's system dimensions are constant. While our notations may differ significantly, we reproduce essentially the same representation for the process.
<figure><img src="image_31.png" /></figure>

$$ \exists! (k_1, k_2, k_3, k_4) : \begin{array}{l} \mathcal{X}_1^{k_1} = \{k_1\}, \\ \mathcal{X}_2^{k_2} = \{k_1, k_2\} (= \{A, B\}), \quad \mathcal{X}_3^{k_3} = \{A, B, k_3\}, \\ \mathcal{X}_4^{k_4} = \{A, B, k_3, k_4\} (= \{A, B, C, D\}), \\ \text{while all other } \mathcal{X}_n^k = \emptyset \end{array} $$

(a) $\mathcal{G}_{QC-QC(4)}$, restricted to $AB \prec CD$

$$ \exists! (k_1, k_2, k_3, k_4) : \begin{array}{l} \mathcal{X}_1^{k_1} = \{k_1\}, \\ \mathcal{X}_2^{k_2} = \{k_1, k_2\} (= \{A, B\}), \quad \mathcal{X}_3^{k_3} = \{A, B, k_3\}, \\ \mathcal{X}_4^{k_4} = \{A, B, k_3, k_4\} (= \{A, B, C, D\}), \\ \text{while all other } \mathcal{X}_n^k = \emptyset \end{array} $$

(b) Restricting to $V_3 = V_{A,B} \oplus V_{B,A}$

**Figure E.1:** Two routed graphs for 4-partite processes with $AB \prec CD$. While (a) can represent any QC-QC with this order, representation (b) is only applicable if $V_3$ admits a more fine-grained branch structure into two branches $V_{A,B}$ and $V_{B,A}$. Example E.1 can not be represented with the latter graph.

## E Restricting to agent-localised processing

In Section 5.3, we have demonstrated how to obtain an alternative routed circuit decomposition with fewer internal nodes by modifying our generic routed graph $\mathcal{G}_{QC-QC(N)}$, eliminating $V_2$ and $V_N$. Based on this, we consider an example which demonstrates how a removal of further internal nodes (between $V_2$ and $V_N$) generally fails, and under which conditions we may recover the possibility to get rid of further internal nodes.

**Example E.1** (Zurich process [41, 52]). The Zurich process, introduced in [41] as double quantum switch (and later independently proposed in Appendix B of [52]), is given as follows:

$$ V_{\emptyset, \emptyset}^{\to k_1} = \frac{1}{\sqrt{2}} |\psi\rangle^{A_{k_1}^I} \qquad (E.1) $$

$$ V_{\emptyset, k_1}^{\to k_2} = \mathbb{1}^{A_{k_1}^O \to A_{k_1}^I} \qquad (E.2) $$

$$ V_{\{k_1\}, k_2}^{\to k_3} = \begin{cases} -\frac{1}{\sqrt{2}} \mathbb{1}^{A_{k_2}^O \to A_{k_3}^I}, & k_2 = B \text{ and } k_3 = C \\ \frac{1}{\sqrt{2}} \mathbb{1}^{A_{k_2}^O \to A_{k_3}^I}, & \text{else} \end{cases} \qquad (E.3) $$

$$ V_{\{k_1, k_2\}, k_3}^{\to k_4} = \mathbb{1}^{A_{k_3}^O \to A_{k_4}^I} \qquad (E.4) $$

$$ V_{\{k_1, k_2, k_3\}, k_4}^{\to F} = \mathbb{1}^{A_{k_4}^O \to F} \otimes |k_4 \text{ mod } 2\rangle^{\alpha_F} \qquad (E.5) $$

where it is assumed that $k_1, k_2 \in \{A, B\}$ and $k_3, k_4 \in \{C, D\}$, while all other internal operations are assumed to vanish. Hence, it can be represented using the routed graph depicted in Fig. E.1a. Intuitively, we can see the process as two subsequent quantum switches, with the operation performed
<figure><img src="image_32.png" /></figure>

$$ \exists! (k_1, k_2, k_3, k_4) : \begin{array}{l} \mathcal{X}_1^{k_1} = \{k_1\}, \\ \mathcal{X}_2^{k_2} = \{k_1, k_2\} (= \{A, B\}), \quad \mathcal{X}_{3,k_2}^{k_3} = \{A, B, k_3\}, \\ \mathcal{X}_4^{k_4} = \{A, B, k_3, k_4\} (= \{A, B, C, D\}), \\ \text{while all other } \mathcal{X}_n^k, \mathcal{X}_{3,k_2}^{k_3} = \emptyset \end{array} $$

**Figure E.2:** A routed graph which may represent all 4-partite processes which feature order $AB \prec CD$ and support only agent-localised transformations. Any process supported by this routed graph can be understood as a wiring of independent pre- and post-processing (with memory) localised to the individual agents. This representation is obtained starting from Fig. E.1b, by additional assuming a more fine-grained branch structure imposing $V_3 = V_{A,B} \oplus V_{B,A}$ for the internal node $V_3$. We split the respective node into two nodes $\hat{V}_{B,A}$ and $\hat{V}_{A,B}$ according to Section 5.2, and merge the resulting nodes into their respective parents $A'$ and $B'$ afterwards to obtain $A''$ and $B''$. One would obtain the same final result for the routed graph by assuming a branch structure imposing $V_3 = V^{\rightarrow C} \oplus V^{\rightarrow D}$ instead, albeit with C and D featuring the $''$ instead.
at $\tilde{V}_3$ acting as a basis change of the control system from the computational basis to $\{|+\rangle, |-\rangle\}$.<fnref n="52" />

Having only a single branch, $\tilde{V}_3$ does not admit a direct sum decomposition $\tilde{V}_3 = V_{A,B} \oplus V_{B,A}$ or $\tilde{V}_3 = V \xrightarrow{C} V \xrightarrow{D}$ into independent isometries. Indeed, the respective maps $V_{k_1, k_2} \xrightarrow{k_3}$ do not constitute isometries on the respective subspaces when considered in isolation.

This raises the question whether we can capture relevant subclasses of processes, allowing for such a decomposition, by adjusting the index constraints to produce a more fine-granular branch structure. For instance, the structure $V_{A,B} \oplus V_{B,A}$ suggests that at the third agent C or D, the second agent is encoded in $\mathcal{H}^{\text{in}(C/D)}$<fnref n="53" />. This is not the case for the Zurich process, but would allow for many simpler 4-partite processes, including a concatenation of two quantum switches controlled by the same basis or by two separate, uncorrelated qubits.

Encoding this branch structure using additional index values and constraints upon them, we obtain Fig. E.1b from Fig. E.1a. The changed branch structure then allows to perform further node splitting (according to Lemma 5.3) and node merging (according to Proposition 5.6), getting rid of $V_3$ in the routed graph similarly as for $V_2$. Remarkably, we would have received the same end result for the routed graph if we had chosen a branch structure imposing $V \xrightarrow{C} V \xrightarrow{D}$ instead, ultimately performing a merge of $\hat{V} \xrightarrow{C} C'$ and $\hat{V} \xrightarrow{D} D'$. This reflects the fact that after merging, the routed graph recovers the symmetry of the original routed graph Fig. E.1a under flipping all arrows.

This suggests a more general class of processes with an arbitrary number of agents, which can be represented without any need of internal nodes other than $V_1$ and $V_{N+1}$. The action of the internal isometries is then restricted to local pre- and post-processing<fnref n="54" /> at the respective agents: At each node, the next agent is independently chosen, without interference from any other agents. Altogether, this amounts of a process which decomposes into a set of quantum 1-combs (and two routed isometries) wired together in a routed circuit. We specify the generic routed graph $\mathcal{G}_{\text{QC-QC}(N)}^{\text{local}}$ capturing implementations of this form as follows:

**Definition E.2** ($\mathcal{G}_{\text{QC-QC}(N)}^{\text{local}}$). The routed graph $\mathcal{G}_{\text{QC-QC}(N)}^{\text{local}}$, representing agent-localised processes with minimal internal nodes, is defined as follows:

* There are 2 types of nodes: “A-nodes” $A_k$, for $k = 1, \dots, N$, and “V-nodes” $V_1$ and $V_{N+1}$.

* The indexed graph involves the following indexed arrows:

$$ \begin{array}{l} \text{– open-ended arrows} \quad \rightarrow \quad V_1 \text{ and } V_{N+1} \rightarrow , \\ \text{– } \forall k = 1, \dots, N, \end{array} $$

$$ V_1 \xrightarrow{\chi_1^k} A_k \quad \text{and} \quad A_k \xrightarrow{\chi_N^k} V_{N+1}, \qquad (\text{E.6}) $$

with the indices $\chi_1^k$ taking one of two values each: either $\chi_1^k = k$ or $\chi_N^k = N \setminus k$ as its value, or $\chi_n^k = \emptyset$. The latter value will later be considered 1-dimensional.

$$ \begin{array}{l} \text{– } \forall n = 2, \dots, N-1, \quad \forall k = 1, \dots, N, \quad \forall \ell = 1, \dots, N, \\ \text{– } \forall n = 2, \dots, N-1, \quad \forall k = 1, \dots, N, \quad \forall \ell = 1, \dots, N, \end{array} $$

$$ A_k \xrightarrow{\chi_{n,k}^\ell} A_\ell, \qquad (\text{E.7}) $$

with a given index $\chi_{n,k}^\ell$ taking either $\chi_{n,k}^\ell = K_n$ with $k, \ell \in K_n$ as its value, or $\chi_n^k = \emptyset$. The latter value will later be considered 1-dimensional.

* The routes are specified such that

<footnote n="52">⁵²In particular, this implies that even dependent on isometries A, B, C, D chosen by the agents, no POVM $\mathcal{M}(A, B, C, D)$ on $\mathcal{H}^F \otimes \mathcal{H}^{\alpha_F}$ can return the entire causal order in the process, while this is possible for the quantum switch or the Grenoble process.</footnote>

<footnote n="53">⁵³In absence of an ancillary system $\alpha_3$, it will be encoded in $\mathcal{H}^{C^I/D^I}$. If the respective information is passed through an ancillary, it will be passed further along through the process instead.</footnote>

<footnote n="54">⁵⁴with a side-channel $\bar{\alpha}$</footnote>
- for each node $A_k$, the lists of all input and output index values $(\mathcal{X}_{n,j}^k)_{n,j} \in \text{Ind}_{A_k}^{\text{in}}$ and $(\mathcal{X}_{n,k}^\ell)_{n,k} \in \text{Ind}_{A_k}^{\text{out}}$ must satisfy that there is exactly one $1 \le n \le N-1$ such that

$$ \exists! j, \mathcal{X}_{n,j}^k \neq \emptyset, \quad \exists! \ell, \mathcal{X}_{n+1,k}^\ell \neq \emptyset; \quad \text{and} \quad \mathcal{X}_{n+1,k}^\ell = \mathcal{X}_{n,j}^k \cup \ell. \qquad (\text{E.8}) $$

For all other values of $n$, all indices $\mathcal{X}_{n,j}^k$ and $\mathcal{X}_{n,k}^\ell$ are $\emptyset$.

- for $V_1$, $\mathcal{X}_0^\ell \in \text{Ind}_{V_1}^{\text{out}}$ must satisfy $\exists! \ell, \mathcal{X}_0^\ell \neq \emptyset$.

- for $V_N$, $\mathcal{X}_N^k \in \text{Ind}_{V_{N+1}}^{\text{in}}$ must satisfy $\exists! k, \mathcal{X}_{N \setminus k}^k \neq \emptyset$.

They condense to the global index constraints that $\exists! (k_1, k_2, \dots, k_N)$:

$$ \mathcal{X}_1^{k_1} = \{k_1\}, \quad \mathcal{X}_{2,k_1}^{k_2} = \{k_1, k_2\}, \dots, \quad \mathcal{X}_{N-1,k_{N-2}}^{k_{N-1}} = \{k_1, k_2, \dots, k_{N-1}\}, \quad \mathcal{X}_N^k = \mathcal{N} $$

while all other $\mathcal{X}_{n,k}^\ell = \emptyset$. This captures the understanding that, along each global path, all operations $A_k$ are applied once and only once, in a given order.

For $N=4$, this graph is shown in Fig. E.2.

Actually, one may simplify $\mathcal{G}_{\text{QC-QC}(N)}^{\text{local}}$ by removing the label $n$ for each index $\mathcal{X}_{n,k}^\ell$, as generally $|\mathcal{X}_{n,k}^\ell| = n$, further estranging $\mathcal{G}_{\text{QC-QC}(N)}^{\text{local}}$ from the original generic routed graph $\mathcal{G}_{\text{QC-QC}(N)}$. As expected, we confirm that up to relabelling, $\mathcal{G}_{\text{QC-QC}(2)}^{\text{local}}$ and $\mathcal{G}_{\text{QC-QC}(3)}^{\text{local}}$ reproduce the generic routed graphs we received after node merging in Section 5.3.

We conjecture that there is an implementation-agnostic SDP characterisation of the subclass of QC-QCs which may be implemented in this fashion, and leave proving this conjecture for future work.