YZ-Insertion in MBQC

• YZ-Insertion is a graph-rewriting operation in MBQC that adds YZ-plane measured qubits while preserving deterministic computation flows.
• It extends the traditional causal flow framework by incorporating non-Pauli planar measurements under explicit combinatorial and algebraic conditions.
• This operation enhances MBQC by enabling optimization, obfuscation, and routing while supporting universal computation with YZ-plane measurement patterns.

Searching arXiv for papers on YZ-insertion in MBQC and related YZ-plane measurement-based quantum computation. YZ-insertion is a graph-rewriting operation in measurement-based quantum computation (MBQC) that adds a new vertex measured in the $YZ$ plane to a labelled open graph while preserving, under explicit combinatorial and algebraic conditions, the flow structure required for deterministic computation. In the formulation developed for MBQC rewriting, a labelled open graph is written as $\Gamma=(G,I,O,\lambda)$, where $G=(V,E)$ is a simple graph, $I$ and $O$ are input and output sets, and $\lambda$ assigns measurement labels from $$\{\mathrm{X},\mathrm{Y},\mathrm{Z},\mathrm{XY},\mathrm{XZ},\mathrm{YZ}\}$$ to vertices. The central problem addressed by YZ-insertion is whether one may enlarge an MBQC pattern by introducing a non-Pauli, planar-measured qubit without destroying causal flow, gflow, or Pauli flow, which are the structural conditions underpinning deterministic execution and efficient circuit extraction (Backens et al., 20 Aug 2025).

1. Definition and formal setting

YZ-insertion is defined for a labelled open graph $\Gamma=(G,I,O,\lambda)$ by adjoining a new vertex $z$ with a prescribed neighbourhood $S\subseteq V$, and assigning the new vertex the measurement label $\Gamma=(G,I,O,\lambda)$0. The resulting graph is

$\Gamma=(G,I,O,\lambda)$1

with

$\Gamma=(G,I,O,\lambda)$2

$\Gamma=(G,I,O,\lambda)$3, and $\Gamma=(G,I,O,\lambda)$4 for all $\Gamma=(G,I,O,\lambda)$5 (Backens et al., 20 Aug 2025).

In MBQC, such insertions are not merely graph-theoretic augmentations. They alter the correction dependencies induced by measurement outcomes and therefore interact directly with the existence of flow. The significance of the operation is that planar measurements in the $\Gamma=(G,I,O,\lambda)$6 plane are necessary for universal MBQC in settings that go beyond purely Pauli or $\Gamma=(G,I,O,\lambda)$7-plane rewrites (Backens et al., 20 Aug 2025). This places YZ-insertion within a broader program of flow-preserving rewriting, optimization, obfuscation, and routing of MBQC patterns.

The underlying context is the one-way model of quantum computation, where computation proceeds by single-qubit measurements on a resource state. In that setting, previous rewrite techniques had mainly focused on deleting qubits or inserting Pauli-measured qubits. YZ-insertion extends the rewrite repertoire to arbitrary planar measurements in a principal plane of the Bloch sphere, a class of measurements that is required for universal computation (Backens et al., 20 Aug 2025).

2. Extended causal flow and the basic preservation criterion

A principal contribution of the YZ-insertion framework is the extension of causal flow, previously restricted to $\Gamma=(G,I,O,\lambda)$8-measurements, to also permit $\Gamma=(G,I,O,\lambda)$9-measurements. For labelled open graphs with $G=(V,E)$0, an extended causal flow is a pair $G=(V,E)$1 consisting of a correction map and a strict partial order satisfying the following conditions for all $G=(V,E)$2: $G=(V,E)$3 These clauses explicitly distinguish $G=(V,E)$4-measured vertices from $G=(V,E)$5-measured ones by forcing a $G=(V,E)$6 vertex to correct to itself (Backens et al., 20 Aug 2025).

For extended causal flow, the preservation condition under YZ-insertion is particularly sharp. If $G=(V,E)$7 admits an extended causal flow $G=(V,E)$8, then the YZ-inserted graph $G=(V,E)$9 admits extended causal flow if and only if no vertex of the insertion neighbourhood $I$0 is equal to or precedes any vertex in $I$1 under $I$2. Formally,

$I$3

where

$I$4

This criterion characterizes exactly when the new $I$5 vertex can be integrated without violating the correction dependencies already encoded in $I$6 (Backens et al., 20 Aug 2025).

Two immediate structural consequences are emphasized. First, two $I$7-measured vertices cannot be adjacent. Second, the number of neighbours of the new $I$8 vertex cannot exceed the number of outputs (Backens et al., 20 Aug 2025). These are not merely incidental restrictions; they indicate that $I$9 measurements impose nontrivial topological constraints on the admissible graph states.

3. YZ-insertion for gflow and Pauli flow

The more general theory of YZ-insertion concerns gflow and Pauli flow. In this setting, the relevant question is not only whether a local correction order survives, but whether the inserted pattern continues to admit a correction structure strong enough to support deterministic MBQC and circuit extraction.

For a YZ-insertion producing $O$0, the criterion is expressed in terms of the existence of a subset

$O$1

The graph $O$2 has Pauli flow, and therefore in particular may preserve gflow in the appropriate special case, if and only if there exists such a subset $O$3 satisfying three conditions (Backens et al., 20 Aug 2025):

1. Focusing: $O$4 is focused over $O$5, and this set is the largest set over which it is focused.
2. Parity constraint: $O$6 is even, i.e. $O$7.
3. Order condition: with

$O$8

and

$O$9

where $\lambda$0 denotes planar-measured vertices and $\lambda$1 is symmetric difference, one requires that for all $\lambda$2 and $\lambda$3,

$\lambda$4

When these conditions hold, the correction function can be extended by defining

$\lambda$5

with a corresponding adjustment of the partial order (Backens et al., 20 Aug 2025).

This formulation makes clear that YZ-insertion is algebraically more delicate than the insertion of a Pauli-measured qubit. The inserted vertex does not simply inherit a local correction rule; instead, its admissibility depends on a compatibility relation between the insertion neighbourhood, parity constraints, and the focused correction sets already present in the pattern.

4. Relation to XZ-insertion and planar-measurement rewriting

The YZ-insertion results have a closely parallel counterpart for $\lambda$6-insertion. For $\lambda$7-measured vertex insertion, the same focusing and order requirements apply, but the parity constraint changes from even to odd: $\lambda$8 This formal symmetry shows that the distinction between $\lambda$9 and $$\{\mathrm{X},\mathrm{Y},\mathrm{Z},\mathrm{XY},\mathrm{XZ},\mathrm{YZ}\}$$0 insertions is encoded at the level of parity, while the correction-set geometry remains otherwise analogous (Backens et al., 20 Aug 2025).

The broader significance of this observation is that flow-preserving insertion of planar-measured qubits can be developed uniformly across the principal planes once the appropriate flow formalism is generalized. The paper introducing YZ-insertion emphasizes that previous work had concentrated on rewrites that either reduced the number of qubits or inserted Pauli-measured vertices; by contrast, YZ- and XZ-insertions preserve a purely planar setting, which is important because planar measurements are necessary for universal MBQC (Backens et al., 20 Aug 2025).

This places YZ-insertion within a wider classification of one-plane MBQC. Universal $$\{\mathrm{X},\mathrm{Y},\mathrm{Z},\mathrm{XY},\mathrm{XZ},\mathrm{YZ}\}$$1-plane-only measurement patterns were later established in a separate study, which proved that universal MBQC can be achieved using only $$\{\mathrm{X},\mathrm{Y},\mathrm{Z},\mathrm{XY},\mathrm{XZ},\mathrm{YZ}\}$$2-plane measurements when correction strategies are allowed to go beyond uniform determinism and use Pauli flow (Kysela et al., 31 Mar 2026). That work also connected $$\{\mathrm{X},\mathrm{Y},\mathrm{Z},\mathrm{XY},\mathrm{XZ},\mathrm{YZ}\}$$3-plane-only and $$\{\mathrm{X},\mathrm{Y},\mathrm{Z},\mathrm{XY},\mathrm{XZ},\mathrm{YZ}\}$$4-plane-only patterns, reinforcing the conceptual role of YZ-insertion as a rewrite primitive compatible with a universal, one-plane measurement regime (Kysela et al., 31 Mar 2026).

5. Vertex splitting, neighbour unfusion, and pivoting

A notable structural result is that the previously known “vertex splitting” or “neighbour unfusion” rule can be derived from YZ-insertion together with pivoting. In the construction described in the MBQC rewriting framework, one first inserts a $$\{\mathrm{X},\mathrm{Y},\mathrm{Z},\mathrm{XY},\mathrm{XZ},\mathrm{YZ}\}$$5-measured vertex $$\{\mathrm{X},\mathrm{Y},\mathrm{Z},\mathrm{XY},\mathrm{XZ},\mathrm{YZ}\}$$6 linked to an existing vertex $$\{\mathrm{X},\mathrm{Y},\mathrm{Z},\mathrm{XY},\mathrm{XZ},\mathrm{YZ}\}$$7, then inserts a second $$\{\mathrm{X},\mathrm{Y},\mathrm{Z},\mathrm{XY},\mathrm{XZ},\mathrm{YZ}\}$$8-measured vertex $$\{\mathrm{X},\mathrm{Y},\mathrm{Z},\mathrm{XY},\mathrm{XZ},\mathrm{YZ}\}$$9 linked to $\Gamma=(G,I,O,\lambda)$0, and finally pivots on the edge $\Gamma=(G,I,O,\lambda)$1 (Backens et al., 20 Aug 2025).

This composition yields the effect of splitting a vertex while staying inside the planar-measurement framework. The result is important because vertex splitting and neighbour unfusion had already been used in the literature as rewrite rules, but the YZ-insertion analysis shows that they can be derived rather than postulated. The same source states that the resulting pattern preserves Pauli flow or gflow if and only if the flow-preservation criteria of the corresponding theorem are satisfied, with a specialized corollary for the neighbour-unfusion case when $\Gamma=(G,I,O,\lambda)$2 is a singleton (Backens et al., 20 Aug 2025).

A plausible implication is that YZ-insertion serves as a more primitive rewriting generator from which other local graph transformations can be synthesized. In that sense, it is not only a standalone operation but also a foundational component in a larger rewrite calculus for MBQC patterns.

6. Graph-structural context: YZ-plane MBQC and bipartite or register-logic graphs

The study of YZ-insertion interacts closely with the structural theory of YZ-plane MBQC. One earlier result established that universal parity quantum computing with a constant-depth decoding procedure is equivalent to MBQC on a bipartite graph using only YZ-plane measurements, and further showed that any unitary MBQC using only YZ-plane measurements must occur on a bipartite graph (Smith et al., 2024). In that setting, the bipartite structure is not an implementation detail but a necessity for deterministic unitary computation.

A later study refined the graph-theoretic characterization by defining register-logic graphs and proving that any uniformly deterministic MBQC where inputs coincide with outputs must be driven on such graphs by measurements in the $\Gamma=(G,I,O,\lambda)$3 plane (Kysela et al., 31 Mar 2026). In the same work, it was shown that for gflow with $\Gamma=(G,I,O,\lambda)$4, one must have $\Gamma=(G,I,O,\lambda)$5, and that any open graph of register-logic form with $\Gamma=(G,I,O,\lambda)$6 and all non-outputs measured in the $\Gamma=(G,I,O,\lambda)$7 plane admits a gflow (Kysela et al., 31 Mar 2026).

These results supply the ambient theory within which YZ-insertion operates. They indicate that YZ-plane measurement patterns are tightly constrained by graph structure, whether described as bipartite graphs in parity MBQC or as register-logic graphs in the gflow-based characterization. YZ-insertion is therefore best understood as a local rewrite that must respect a global architecture of determinism.

This also explains why the insertion conditions are combinatorially restrictive. If YZ-plane-only MBQC with unitary semantics already forces strong graph-structural properties, then the insertion of a new $\Gamma=(G,I,O,\lambda)$8-measured vertex cannot be arbitrary; it must preserve the architecture that makes deterministic correction possible.

7. Role in optimization, obfuscation, routing, and the ZX-calculus

The practical motivation for YZ-insertion is the preservation of flow under pattern rewriting. In MBQC, flow conditions are required for deterministic execution in a suitable sense and for efficient translation into quantum circuits (Backens et al., 20 Aug 2025). Consequently, any rewrite used for optimization or hardware adaptation must preserve the existence of flow rather than merely preserve graph-state equivalence.

The MBQC rewriting study identifies several application domains. In optimization, YZ-insertion enlarges the set of sound pattern transformations beyond qubit deletion and Pauli insertions. In obfuscation, it supports the insertion of planar-measured vertices while retaining determinism, which is relevant to hiding computational structure. In routing, it permits modification of resource-state graphs to adapt to connectivity constraints without breaking the correction structure (Backens et al., 20 Aug 2025).

The same work also situates YZ-insertion in relation to the ZX-calculus. Flow-preserving rewrites, including YZ-insertions and pivots, correspond to equivalence-preserving graphical moves in ZX-based diagrammatic reasoning, circuit extraction, and optimization (Backens et al., 20 Aug 2025). This suggests that YZ-insertion is not only an MBQC-specific device but also part of a broader graphical infrastructure for reasoning about quantum computations.

A plausible implication is that YZ-insertion contributes to a unification of MBQC rewrite theory and ZX-calculus rewrite theory by providing a local operation that is simultaneously sensitive to measurement planes, correction dependencies, and graph transformations.

8. Significance and limitations

YZ-insertion broadens the range of admissible deterministic rewrites in MBQC by allowing the insertion of non-Pauli planar-measured qubits. This is significant because planar measurements are necessary for universal MBQC, and because restricting to Pauli insertions can force a transition out of the gflow setting into more general Pauli flow constructions (Backens et al., 20 Aug 2025). By contrast, YZ-insertion preserves a planar formalism and can remain compatible with deterministic flow when its combinatorial criteria are satisfied.

At the same time, the operation is intrinsically constrained. Under extended causal flow, the insertion neighbourhood must avoid violating the order relation against the preimage $\Gamma=(G,I,O,\lambda)$9 (Backens et al., 20 Aug 2025). Under gflow and Pauli flow, admissibility depends on focusing, parity, and order conditions formulated through the subset $z$0 and the interaction of correction sets with the neighbourhood $z$1 (Backens et al., 20 Aug 2025). These constraints rule out arbitrary local insertions and make clear that YZ-insertion is a controlled rewrite rather than a generic graph augmentation.

Within the larger landscape of one-plane MBQC, YZ-insertion is part of a sequence of results establishing the expressivity of $z$2-plane computation. Bipartite YZ-plane MBQC was shown equivalent to parity quantum computing (Smith et al., 2024), universal YZ-plane-only patterns were later constructed using Pauli flow (Kysela et al., 31 Mar 2026), and YZ-insertion supplies a local mechanism for transforming patterns while preserving the flow structures on which those models depend (Backens et al., 20 Aug 2025).

In that sense, YZ-insertion occupies a distinct place in MBQC theory: it is the rewrite operation that brings planar-measured qubit insertion into the same rigorous, flow-preserving framework that earlier work had developed for deletion, Pauli insertion, and related transformations.