Routed Quantum Circuits Overview

• Routed quantum circuits are defined as circuits where wires carry partitioned Hilbert spaces and explicit routing relations enforce sector constraints.
• They enable both coherent quantum-controlled path selection and hardware-compliant circuit transformation by embedding routing directly into the computational model.
• This approach bridges categorical semantics, practical implementations in quantum routers/QRAM, and advanced compilation techniques for sparse architectures.

Routed quantum circuits most specifically denote the extension of standard quantum-circuit semantics introduced through routed linear maps, where wires carry partitioned Hilbert spaces and boxes carry explicit routing relations between sectors, so that direct sums of tensor products and sectorial constraints become part of the circuit language itself (Vanrietvelde et al., 2020). The literature also uses the same expression for two other, partially overlapping ideas: circuits in which the wiring pattern is coherently controlled by quantum states, as in quantum routers and bucket-brigade QRAM, and hardware-feasible circuits obtained by qubit routing on sparse devices (Zhang et al., 20 May 2025, Banerjee et al., 2022). The common thread is that “routing” elevates path selection, sector selection, or physical placement from an external convention to an explicit component of the computational model.

1. Scope, motivation, and terminological landscape

The literature surveyed here suggests that “routed quantum circuits” is an overloaded label rather than a single universally fixed concept. In the semantic sense developed in 2020, routing addresses a deficiency of standard circuit formalisms whenever the relevant Hilbert spaces are not just tensor products but direct sums of tensor products, and whenever physical processes must respect explicit sector decompositions. In operational and architectural work, routing instead refers either to coherent path control of a signal or to the transformation of a logical circuit into a hardware-compliant one.

Usage of the term	Core object	Typical problem
Semantic routed circuits	Partitioned Hilbert spaces and routed maps	Express direct-sum/tensor-product blends and sector constraints
Coherent quantum routing	Quantum-controlled path selection	Route a signal into output paths in superposition
Compilation-oriented routing	Hardware-feasible mapped circuits	Satisfy sparse coupling constraints

The formal motivation for the semantic framework is the observation that standard circuit categories such as $\mathbf{FHilb}$ and $\mathbf{CPM}[\mathbf{FHilb}]$ do not natively encode internal decompositions $$\mathcal{H}_A = \bigoplus_{k\in K_A}\mathcal{H}_A^k$$, nor the fact that only specific sector combinations may be physically populated (Vanrietvelde et al., 2020). The canonical examples are communication in a superposition of trajectories, where only the one-particle subspace of $\mathcal{H}_A \otimes \mathcal{H}_B$ is accessible, and causal decompositions of unitaries, where Lorenz–Barrett “extended circuit diagrams” require sector matching beyond plain tensor-product wiring (Vanrietvelde et al., 2020). In later operational work, a routed quantum circuit is described as one in which “the wiring diagram” of where quantum information flows is itself controlled by quantum states, with QRouters as primitives for bucket-brigade QRAM (Zhang et al., 20 May 2025). In compilation, a routed quantum circuit is a logical circuit transformed into a hardware-feasible circuit by inserting SWAPs or equivalent moves so that all two-qubit gates respect a coupling graph (Banerjee et al., 2022).

2. Partitioned wires, routes, and diagrammatic semantics

In the formal framework, a wire does not merely carry a Hilbert space $\mathcal{H}_A$; it carries a partitioned Hilbert space

$A^k := (\mathcal{H}_A, K_A, (\pi_A^k)_{k\in K_A}),$

where $(\pi_A^k)$ is an orthogonal partition and

$$\mathcal{H}_A = \bigoplus_{k\in K_A}\mathcal{H}_A^k.$$

Tensor product preserves the partition data by using sector labels $K_A \times K_B$, so the wire type remembers which direct-sum sectors exist and how they combine (Vanrietvelde et al., 2020).

A route is a relation $\lambda:K_A\to K_B$, equivalently a Boolean matrix $\mathbf{CPM}[\mathbf{FHilb}]$0. A linear map $\mathbf{CPM}[\mathbf{FHilb}]$1 follows $\mathbf{CPM}[\mathbf{FHilb}]$2 when

$\mathbf{CPM}[\mathbf{FHilb}]$3

or, equivalently, whenever $\mathbf{CPM}[\mathbf{FHilb}]$4 the block $\mathbf{CPM}[\mathbf{FHilb}]$5 vanishes. A routed linear map is then a pair $\mathbf{CPM}[\mathbf{FHilb}]$6. Sequential composition, tensor product, and adjoint are defined componentwise: $\mathbf{CPM}[\mathbf{FHilb}]$7

This structure yields the dagger symmetric monoidal category RoutedFHilb, with partitioned Hilbert spaces as objects and routed linear maps as morphisms. Because dagger symmetric monoidal categories admit a sound graphical calculus, routed maps have a sound representation as circuit diagrams. The diagrammatics use repeated indices and Kronecker-delta contraction to encode identity routes compactly; the Lorenz–Barrett style of repeated sector labels is thereby given a precise semantics rather than an informal one (Vanrietvelde et al., 2020).

A central distinction is between formal space and accessible space at a slice of a circuit. For wires $\mathbf{CPM}[\mathbf{FHilb}]$8 and $\mathbf{CPM}[\mathbf{FHilb}]$9, the formal space is always

$$\mathcal{H}_A = \bigoplus_{k\in K_A}\mathcal{H}_A^k$$0

whereas the accessible space is the subspace actually reachable once the routes above and below the slice are taken into account. In the two-path one-particle example, the formal space is all of $$\mathcal{H}_A = \bigoplus_{k\in K_A}\mathcal{H}_A^k$$1, while the accessible space is only

$$\mathcal{H}_A = \bigoplus_{k\in K_A}\mathcal{H}_A^k$$2

The accessible space is computed by closing the route data under connectivity and summing over all indices except those of the chosen slice; this yields an effective Boolean mask specifying which sector tuples remain physically reachable (Vanrietvelde et al., 2020).

3. Pure and mixed routed theories

In pure theory, routed maps complicate the usual notion of isometry because a routed isometry can delete sectors. The framework therefore introduces practical isometries. Given a route $$\mathcal{H}_A = \bigoplus_{k\in K_A}\mathcal{H}_A^k$$3, the practical input set is

$$\mathcal{H}_A = \bigoplus_{k\in K_A}\mathcal{H}_A^k$$4

with practical input space $$\mathcal{H}_A = \bigoplus_{k\in K_A}\mathcal{H}_A^k$$5. A routed map $$\mathcal{H}_A = \bigoplus_{k\in K_A}\mathcal{H}_A^k$$6 is a practical isometry when $$\mathcal{H}_A = \bigoplus_{k\in K_A}\mathcal{H}_A^k$$7 is an isometry on $$\mathcal{H}_A = \bigoplus_{k\in K_A}\mathcal{H}_A^k$$8, equivalently a partial isometry with that initial domain; it is a practical unitary when both it and its adjoint are practical isometries (Vanrietvelde et al., 2020).

Sequential composition is not automatic. Theorem 3 gives a purely route-theoretic condition ensuring that the composition of practical isometries is again a practical isometry: $$\mathcal{H}_A = \bigoplus_{k\in K_A}\mathcal{H}_A^k$$9 A second symmetric condition ensures that practical unitaries compose as practical unitaries. The terminology “proper” composition refers to route pairs satisfying these constraints. This makes explicit that physicality is not a property of the linear maps alone; it depends on compositional compatibility of their routes (Vanrietvelde et al., 2020).

For mixed theory, routed maps are lifted from Hilbert spaces to operator spaces and from relations $\mathcal{H}_A \otimes \mathcal{H}_B$0 to completely positive relations

$\mathcal{H}_A \otimes \mathcal{H}_B$1

The diagonal $\mathcal{H}_A \otimes \mathcal{H}_B$2 records sector connectivity, while off-diagonal components record which coherences between sector pairs are permitted. The relevant routes are precisely the completely positive relations obtained by a “doubling and tracing out” construction, mirroring the usual CPM construction. This yields the dagger symmetric monoidal category RoutedCPMFHilb (Vanrietvelde et al., 2020).

A routed CPM $\mathcal{H}_A \otimes \mathcal{H}_B$3 is a routed quantum channel when $\mathcal{H}_A \otimes \mathcal{H}_B$4 is practically trace-preserving on density operators supported in the practical input space determined by $\mathcal{H}_A \otimes \mathcal{H}_B$5. The composition criterion depends only on the diagonals: $\mathcal{H}_A \otimes \mathcal{H}_B$6 The framework also characterizes special coherence regimes. In the full-coherence case, $\mathcal{H}_A \otimes \mathcal{H}_B$7 follows $\mathcal{H}_A \otimes \mathcal{H}_B$8 iff each Kraus operator follows the diagonal route $\mathcal{H}_A \otimes \mathcal{H}_B$9. In the fully decohering case, $\mathcal{H}_A$0 follows $\mathcal{H}_A$1 iff there exists a Kraus decomposition whose operators are each supported on a single allowed sector pair. This makes coherence constraints a structural part of circuit syntax rather than a side condition written next to a diagram (Vanrietvelde et al., 2020).

4. Superposition of channels, causal decomposition, and extended diagrams

The two-path “one particle in two trajectories” scenario is the principal application motivating the framework. Each transmission line has vacuum and one-particle sectors,

$\mathcal{H}_A$2

and the physically accessible joint subspace is

$\mathcal{H}_A$3

Alice’s and Bob’s channels satisfy a no-leakage constraint preserving vacuum and one-particle sectors. In standard semantics, the encoding must be artificially extended to all of $\mathcal{H}_A$4, which introduces spurious structure and breaks the direct unitary picture; in routed semantics, the encoder route $\mathcal{H}_A$5 specifies exactly the sector pairs $\mathcal{H}_A$6 and $\mathcal{H}_A$7, so the one-particle constraint is part of the circuit itself (Vanrietvelde et al., 2020).

The same formalism extends to three trajectories and to mixed channels. Routes can distinguish sector-preserving coherent channels,

$\mathcal{H}_A$8

from sector-preserving fully decohering channels,

$\mathcal{H}_A$9

The route

$A^k := (\mathcal{H}_A, K_A, (\pi_A^k)_{k\in K_A}),$0

encodes “exactly one particle across three arms”, and the choice of coherent versus diagonal CP route determines whether coherence between vacuum and one-particle sectors is preserved or erased (Vanrietvelde et al., 2020).

The second major application is causal decomposition. For some no-influence patterns, standard unitary circuits on plain tensor-product wires are not general enough. The paradigmatic example is the “diamond” causal pattern with

$A^k := (\mathcal{H}_A, K_A, (\pi_A^k)_{k\in K_A}),$1

Lorenz and Barrett showed that the intended equivalence between no-influence constraints and circuit realizability can be restored by extended circuit diagrams with repeated sector indices and block-diagonal constraints (Lorenz et al., 2020). The routed-map formalism absorbs these as index-matching routed maps, a special case in which routes are built entirely from Kronecker deltas, and gives them a compositional semantics in terms of routed practical unitaries (Vanrietvelde et al., 2020).

This recasts subsystem structure itself. A routed wire can represent a partitioned space with matched indices, exactly-one-particle constraints, or block-diagonal evolution, and the accessible space of an internal slice becomes a precise notion of the effective subsystem realized by the global circuit. The conclusion of the 2020 work leaves open whether future causal decompositions will require genuinely non-index-matching routes, since index-matching is a proper subframework of full routed circuits (Vanrietvelde et al., 2020).

5. Coherent routers, QRAM, and experimental realizations

In a different operational usage, routed quantum circuits are computations in which the flow of quantum information is itself coherently controlled. Linear-optical work introduced a programmable quantum router that coherently routes a polarization-encoded signal photon into spatial modes while preserving the polarization qubit, with the splitting ratio determined by the states of control photons (Lemr et al., 2012). In spin-network and superconducting settings, a coherent router was proposed in which one or more ancilla qubits control whether an input qubit is transferred to one of several outputs, allowing both directed transport and generation of distributed entanglement (Christensen et al., 2019).

A particularly explicit network-level formulation models the path as a quantum degree of freedom. In a two-route link, a path qubit initialized in $A^k := (\mathcal{H}_A, K_A, (\pi_A^k)_{k\in K_A}),$2 controls whether a message system passes through one channel or another, producing a genuine superposition of channels. For suitable vacuum extensions, coherent routing can sustain nonzero classical capacity over arbitrarily many hops in cases where any definite path has asymptotically zero capacity; more generally, coherent routing through multiple intermediate nodes can increase transmission distance even for classical information (Kristjánsson et al., 2022). Earlier spin-network constructions had already exhibited perfect routing of an unknown qubit state from any node to any other node in time linear in the distance on regular local networks, with entanglement generation as a secondary use case (Pemberton-Ross et al., 2010).

Superconducting implementations have turned this operational notion into concrete hardware primitives. A quantum addressable router, or Q²-router, was realized with four fixed-frequency transmons using two native controlled-iSWAP gates and a large ZZ interaction; the reported estimated average routing fidelity was $A^k := (\mathcal{H}_A, K_A, (\pi_A^k)_{k\in K_A}),$3, with errors arising primarily from decoherence or state preparation and measurement (Miao et al., 6 Mar 2025). For bucket-brigade QRAM, coherent QRouters were demonstrated on a superconducting processor using a transition composite gate scheme and non-adjacent qutrit states $A^k := (\mathcal{H}_A, K_A, (\pi_A^k)_{k\in K_A}),$4 and $A^k := (\mathcal{H}_A, K_A, (\pi_A^k)_{k\in K_A}),$5 for eraser detection, achieving individual QRouter fidelities up to $A^k := (\mathcal{H}_A, K_A, (\pi_A^k)_{k\in K_A}),$6 and a two-layer routing network with average fidelity $A^k := (\mathcal{H}_A, K_A, (\pi_A^k)_{k\in K_A}),$7 (Zhang et al., 20 May 2025). A separate NISQ-era experiment combined quantum routing with channel-adapted quantum error correction on ibmq_jakarta and verified the quantum nature of the routed state by tomography, identifying the correction step as beneficial for routing under the assumed noise model (Shi et al., 2022).

These works use “routing” in the sense of coherent path control rather than sectorial semantics. The relation to routed maps is therefore interpretive rather than identical. A plausible implication is that the semantic framework of partitioned spaces and routes can serve as a natural language for such devices whenever vacuum sectors, single-particle sectors, or branch-dependent coherence constraints must be represented compositionally.

6. Compilation-oriented qubit routing

Compilation literature uses the same phrase for a distinct problem: converting a logical circuit into a hardware-feasible one on a sparse coupling graph. In this sense, a routed quantum circuit is the output of a mapping stage that inserts SWAPs or equivalent moves so that every two-qubit gate acts on adjacent physical qubits. For grid architectures, a locality-aware routing algorithm based on routing via matchings directly constructs parallel SWAP layers and was reported to produce circuits of comparable depth to approximate token swapping, better on random permutations, while being an order of magnitude faster than a typical implementation of approximate token swapping (Banerjee et al., 2022).

For CNOT-only circuits, dynamic remapping changes the formulation. The PermRowCol algorithm synthesizes a parity matrix to a permutation matrix rather than strictly to the identity, allowing logical qubits to finish on different physical registers and thereby reducing CNOT count relative to Steiner-Gauss and RowCol on many benchmarks (Griend et al., 2022). In a more synthesis-oriented direction, lazy synthesis treats routing and architecture-aware resynthesis jointly: subgroup operations such as permutations, linear reversible maps, or Clifford operators are accumulated lazily in a classical data structure and only partially extracted into connectivity-compliant circuits when required by non-subgroup gates, yielding significant reductions in entangling-gate overhead on several standard circuit families (Martiel et al., 2020).

Scalability has become an explicit objective for larger devices. Route-Forcing, a force-directed routing algorithm aimed at monolithic and multi-core architectures, was reported to show an average speedup of $A^k := (\mathcal{H}_A, K_A, (\pi_A^k)_{k\in K_A}),$8 compared to scalable state-of-the-art techniques and to reduce mapped-circuit depth by $A^k := (\mathcal{H}_A, K_A, (\pi_A^k)_{k\in K_A}),$9 at the expense of adding $(\pi_A^k)$0 more SWAP gates (Escofet et al., 2024). In this compilation sense, “routing” concerns logical-to-physical placement and movement under architecture constraints, not coherent path superposition and not the sectorial semantics of partitioned Hilbert spaces. The shared terminology reflects a common concern with explicitly managing where quantum information is allowed to travel.

7. Network coding, routed Bell tests, and broader significance

Quantum networking introduces a further operational layer. In graph-state repeater theory, intermediate nodes of degree greater than two are termed quantum routers and can perform basic measurement-based quantum computations rather than mere entanglement swapping. From a classical linear network code over $(\pi_A^k)$1, one can construct a measurement-based quantum network code that distributes a two-colorable graph state whose adjacency matrix is the global transfer matrix between sources and sinks. The same framework links router-based graph-state distribution to stabilizer error-correction codes and shows that for any stabilizer error-correction code there exists a corresponding quantum network code with similar error-correcting capabilities (Epping et al., 2016).

Routed architectures also enter device-independent cryptography. In routed DIQKD, an active switch sends one wing of an entangled pair either to a nearby test device or to a distant key-generation device. Security can then be analyzed through noncommutative polynomial optimization and the Brown–Fawzi–Fawzi method. For high-quality short-path tests, the routed protocols were found to be significantly more robust to losses, with an improvement of approximately $(\pi_A^k)$2 in detection efficiency compared to nonrouted counterparts, and the routed BB84 protocol achieved a positive key rate with a detection efficiency as low as $(\pi_A^k)$3 for the distant device (Roy-Deloison et al., 2024).

Taken together, these strands suggest that routed quantum circuits sit at an intersection of categorical semantics, quantum-controlled communication, and architecture-aware compilation. In the semantic framework, routing makes sectorial constraints first-class and subsystems contextual; in operational routers, it makes path choice a quantum variable; in compilation, it makes physical movement explicit. The terminological overlap can obscure important differences, but it also highlights a common structural concern: the path by which quantum information propagates is often as important as the local transformation applied to it.