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arXiv:2603.29379

YZ-plane measurement-based quantum computation: Universality and Parity Architecture implementation

Jaroslav Kysela, Katharina Ludwig, Nitica Sakharwade, Anette Messinger, Wolfgang Lechner

quant-phSubmitted Mar 31, 2026• Announced Apr 1, 2026• 0 citations
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Abstract

We define the class of register-logic graphs and prove that any uniformly deterministic measurement-based quantum computation (MBQC) where the inputs coincide with the outputs must be driven on such graphs by measurements in the $YZ$ plane of the Bloch sphere. This observation is revisited in the context that goes beyond uniform determinism, where we present a universal $YZ$-plane-only measurement pattern and establish a connection between $YZ$-plane-only and $XZ$-plane-only patterns. These results conclude the line of research on universal patterns with measurements restricted to one of the principal planes of the Bloch sphere. We further demonstrate, within the framework of the Parity Architecture, that $YZ$-plane patterns with the register-logic graph can be embedded into another graph with purely local interactions, and we extend this case to the scenario of universal quantum computation.

Comments
15 pages, 8 figures

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We define the class of register-logic graphs and prove that any uniformly deterministic measurement-based quantum computation (MBQC) where the inputs coincide with the outputs must be driven on such graphs by measurements in the YZYZYZ plane of the Bloch sphere. This observation is revisited in the context that goes beyond uniform determinism, where we present a universal YZYZYZ-plane-only measurement pattern and establish a connection between YZYZYZ-plane-only and XZXZXZ-plane-only patterns. These results conclude the line of research on universal patterns with measurements restricted to one of the principal planes of the Bloch sphere. We further demonstrate, within the framework of the Parity Architecture, that YZYZYZ-plane patterns with the register-logic graph can be embedded into another graph with purely local interactions, and we extend this case to the scenario of universal quantum computation.
In the one-way model of measurement-based quantum computation (MBQC), computation proceeds via single-qubit measurements on a resource state.Flow conditions ensure that the overall computation is deterministic in a suitable sense, and are required for efficient translation into quantum circuits.Procedures that rewrite MBQC patterns -e.g. for optimisation, or adapting to hardware constraints -thus need to preserve the existence of flow.Most previous work has focused on rewrites that reduce the number of qubits in the computation, or that introduce new Pauli-measured qubits.Here, we consider the insertion of planar-measured qubits into MBQC patterns, i.e. arbitrary measurements in a plane of the Bloch sphere spanned by a pair of Pauli operators; such measurements are necessary for universal MBQC.We extend the definition of causal flow, previously restricted to XY -measurements only, to also permit Y Z-measurements and derive the conditions under which a Y Z-insertion preserves causal flow.Then we derive conditions for Y Z-insertion into patterns with gflow or Pauli flow, in which case the argument straightforwardly extends to XZ-insertions as well.We also show that the 'vertex splitting' or 'neighbour unfusion' rule previously used in the literature can be derived from Y Z-insertion and pivoting.This work contributes to understanding the broad properties of flow-preserving rewriting in MBQC and in the ZX-calculus more broadly, and it will enable more efficient optimisation, obfuscation, or routing.
Abstract Measurement-based quantum computation (MBQC) is a protocol for quantum computation that represents a model distinct from the circuit-based approach. MBQC has been proposed not only for qubits but also for qudits, continuous-variable (CV) qubits, and Bose–Einstein condensates (BECs) qubits. In qubit-based MBQC, arbitrary rotations on the Bloch sphere can be performed by measuring a graph state. This naturally raises the question of whether arbitrary rotations on the Bloch sphere can similarly be achieved through measurements in other types of quantum bits. We have demonstrated that this can indeed be realized for BECs qubits by considering composite graph states involving CV qubits and BECs qubits.
Measurement-based quantum computation (MBQC) is a strong contender for realizing quantum computers. A critical question for MBQC is the identification of resource graph states that can enable universal quantum computation. Any such universal family must have unbounded entanglement width, which is equivalent to the ability to produce any circle graph state from the states in the family using only local Clifford operations, local Pauli measurements, and classical communication. Yet, it was not previously known whether or not circle graph states themselves are a universal resource. We show that, in spite of their expressivity, circle graph states are not efficiently universal for MBQC (i.e., assuming BQP≠BPP\mathsf{BQP} \neq \mathsf{BPP}BQP=BPP). We prove this by articulating a precise graph-theoretic correspondence between circle graph states and a certain subset of fermionic Gaussian states. This is accomplished by synthesizing a variety of techniques that allow us to handle both stabilizer states and fermionic Gaussian states at the same time. As such, we anticipate that our developments may have broader applications beyond the domain of MBQC as well.
We introduce a new family of models for measurement-based quantum computation which are deterministic and approximately universal. The resource states which play the role of graph states are prepared via 2-qubit gates of the form exp⁡(−iπ2nZ⊗Z) . When n=2 , these are equivalent, up to local Clifford unitaries, to graph states. However, when n>2 , their behaviour diverges in two important ways. First, multiple applications of the entangling gate to a single pair of qubits produces non-trivial entanglement, and hence multiple parallel edges between nodes play an important role in these generalised graph states. Second, such a state can be used to realise deterministic, approximately universal computation using only Pauli Z and X measurements and feed-forward. Even though, for n>2 , the relevant resource states are no longer stabiliser states, they admit a straightforward, graphical representation using the ZX-calculus. Using this representation, we are able to provide a simple, graphical proof of universality. We furthermore show that for every n>2 this family is capable of producing all Clifford gates and all diagonal gates in the n -th level of the Clifford hierarchy.
Measurement-based quantum computing (MBQC) is a model of quantum computation where quantum information is coherently processed by means of projective measurements on highly entangled states. Following the introduction of MBQC, cluster states have been studied extensively both from the theoretical and experimental point of view. Indeed, the study of MBQC was catalysed by the realisation that cluster states are universal for MBQC with (X, Y)-plane and Z measurements. Here we examine the question of whether the requirement for Z measurements can be dropped while maintaining universality. We answer this question in the affirmative by showing that universality is possible in this scenario.