Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2606.30525

Working with measurement-based computations on qudits

Piotr Mitosek, Miriam Backens

quant-phcs.DSSubmitted Jun 29, 2026• Announced Jun 30, 2026• 0 citations
arXivHTMLPDF

Abstract

Measurement-based quantum computing is a universal model of quantum computation in which successive product measurements of an entangled resource state drive the computation. The non-deterministic nature of measurements necessitates adaptivity to ensure an overall deterministic computation. Flow structures characterise cases in which such an adaptive correction procedure is possible. Recently, flow has been defined in a setting where the resource states are prime-dimensional qudit graph states rather than the usual qubit graph states. Yet, this qudit flow definition is more burdensome to work with than analogous definitions for qubits. Here, we give a simpler definition of qudit flow and consider various useful properties of this flow, drawing on results for the qubit case. In particular, we show how to focus qudit flow and argue that focused flow is canonical. We improve the previous algebraic formulation to capture focused flow and use it to obtain an $O(n^3)$ flow-finding algorithm (where $n$ is the number of qudits), matching the best known complexity for qubit flows and improving on the previous $O(n^4)$ result for qudits. Furthermore, we explore multiple flow-preserving transformations, thus opening a pathway to using flow for optimisation. These transformations include pivoting, removal and insertion of certain types of vertices, and reversibility of flow. Lastly, we propose an algorithmic approach to generating large qudit computations with flow, for testing or machine learning.

Comments
Accepted for proceedings of QPL 2026

Related papers

Flow conditions for continuous variable measurement-based quantum computing

Robert I. Booth, Damian Markham

In measurement-based quantum computing (MBQC), computation is carried out by a sequence of measurements and corrections on an entangled state. Flow, and related concepts, are powerful techniques for characterising the dependence of the corrections on previous measurement results. We introduce flow-based methods for quantum computation with continuous-variable graph states, which we call CV-flow. These are inspired by, but not equivalent to, the notions of causal flow and g-flow for qubit MBQC. We also show that an MBQC with CV-flow approximates a unitary arbitrarily well in the infinite-squeezing limit, addressing issues of convergence which are unavoidable in the infinite-dimensional setting. In developing our proofs, we provide a method for converting a CV-MBQC computation into a circuit form, analogous to the circuit extraction method of Miyazaki et al, and an efficient algorithm for finding CV-flow when it exists based on the qubit version by Mhalla and Perdrix. Our results and techniques naturally extend to the cases of MBQC for quantum computation with qudits of prime local dimension.

arXiv:2104.00572•Quantum•Oct 19, 2023•3 citations

Outcome determinism in measurement-based quantum computation with qudits

Robert I. Booth, Aleks Kissinger, Damian Markham +2 more

Abstract In measurement-based quantum computing (MBQC), computation is carried out by a sequence of measurements and corrections on an entangled state. Flow, and related concepts, are powerful techniques for characterising the dependence of the corrections on previous measurement outcomes. We introduce flow-based methods for MBQC with qudit graph states, which we call Z d -flow, when the local dimension is an odd prime. Our main results are a proof that Z d -flow is a necessary and sufficient condition for a strong form of outcome determinism. Along the way, we find a suitable generalisation of the concept of measurement planes to this setting and characterise the allowed measurements in a qudit MBQC. We also provide a polynomial-time algorithm for finding an optimal Z d -flow whenever one exists.

arXiv:2109.13810•Journal of Physics A Mathematical and Theoretical•Feb 9, 2023•6 citations

Finding gflow on unlabelled open graphs

Miriam Backens, Victor Gasse, Piotr Mitosek +1 more

The one-way model of measurement-based quantum computing implements computations via successive adaptive single-qubit measurements on a resource graph state. This model has practical applications, particularly in photonics, and it is also useful as a theoretical tool e.g. for optimisation. Gflow is a necessary and sufficient condition for implementing certain one-way computations deterministically (in a suitable sense); it is also used in efficient translations from the one-way model to quantum circuits. For a computation on $n$ qubits, a gflow can be found in $\mathcal{O}(n^3)$ time. Here, we consider an incompletely specified computation given by an unlabelled open graph: the graph state as well as the input and output qubits are known, but the measurements have not yet been fixed. We give an algorithm that identifies a measurement labelling and a compatible gflow, and runs in $\mathcal{O}(n^3)$, strictly generalising the previous approach. The new algorithm can also handle restrictions on the order of the measurements and returns only solutions compatible with these constraints. We additionally prove that if an open graph has equal numbers of inputs and outputs, it has at most one labelling compatible with gflow; and show how to identify additional inputs for an open graph that does not yet have the maximal number, without breaking an existing gflow.

quant-pharXiv:2609.40099•Oct 1, 2026•0 citations

Inserting Planar-Measured Qubits into MBQC Patterns while Preserving Flow

Miriam Backens, Thomas Perez

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.

arXiv:2508.14671•Electronic Proceedings in Theoretical Computer Science•Aug 19, 2025•0 citations

Generating one-way computations with flow: flow-preserving rewriting that ignores the interpretation

Miriam Backens

The one-way model is a universal model of quantum computation, driven by successive adaptive single-qubit measurements on an entangled resource state. Measurements are non-deterministic, yet if the computation satisfies one of several related families of conditions known as 'flows', the computation can be made deterministic overall by modifying later measurements depending on the outcomes of earlier ones. Flow properties also enable efficient translation from one-way computations to circuits, motivating research into rewriting one-way computations while preserving the existence of flow. Existing approaches to flow-preserving rewriting are used for compilation or optimisation and preserve both the interpretation and the existence of flow. Here, we broaden our perspective to consider flow-preserving rewriting that does not necessarily preserve the interpretation, with applications to creating test instances for software that works with flow, as well as to generating ansätze for quantum machine learning. We show that a family of just three flow-preserving rewrite rules suffices to generate any diagram with flow from a trivial diagram with the desired number of inputs and outputs. This rule set is nearly the same as the complete set of flow- and interpretation-preserving rewrite rules for one-way computations in which all measurements are Pauli; and just a small subset of the flow- and interpretation-preserving rewrite rules for arbitrary measurements.

quant-pharXiv:2607.03250•Jul 6, 2026•0 citations

Pauli Flow on Open Graphs with Unknown Measurement Labels

Piotr Mitosek

One-way quantum computation, or measurement-based quantum computation, is a universal model of quantum computation alternative to the circuit model. The computation progresses by measurements of a pre-prepared resource state together with corrections of undesired outcomes via applications of Pauli gates to yet unmeasured qubits. The fundamental question of this model is determining whether computation can be implemented deterministically. Pauli flow is one of the most general structures guaranteeing determinism. It is also essential for polynomial time ancilla-free circuit extraction. It is known how to efficiently determine the existence of Pauli flow given an open graph together with a measurement labelling (a choice of measurements to be performed). In this work, we focus on the problem of deciding the existence of Pauli flow for a given open graph when the measurement labelling is unknown. We show that this problem is in RP by providing a random polynomial time algorithm. To do it, we extend previous algebraic interpretations of Pauli flow, by showing that, in the case of X and Z measurements only, flow existence corresponds to the right-invertibility of a matrix derived from the adjacency matrix. We also use this interpretation to show that the number of output qubits can always be reduced to match the number of input qubits while preserving the existence of flow.

arXiv:2408.06059•Electronic Proceedings in Theoretical Computer Science•Aug 9, 2024•0 citations
Measurement-based quantum computing is a universal model of quantum computation in which successive product measurements of an entangled resource state drive the computation. The non-deterministic nature of measurements necessitates adaptivity to ensure an overall deterministic computation. Flow structures characterise cases in which such an adaptive correction procedure is possible. Recently, flow has been defined in a setting where the resource states are prime-dimensional qudit graph states rather than the usual qubit graph states. Yet, this qudit flow definition is more burdensome to work with than analogous definitions for qubits.
Here, we give a simpler definition of qudit flow and consider various useful properties of this flow, drawing on results for the qubit case. In particular, we show how to focus qudit flow and argue that focused flow is canonical. We improve the previous algebraic formulation to capture focused flow and use it to obtain an O(n3)O(n^3)O(n3) flow-finding algorithm (where nnn is the number of qudits), matching the best known complexity for qubit flows and improving on the previous O(n4)O(n^4)O(n4) result for qudits. Furthermore, we explore multiple flow-preserving transformations, thus opening a pathway to using flow for optimisation. These transformations include pivoting, removal and insertion of certain types of vertices, and reversibility of flow. Lastly, we propose an algorithmic approach to generating large qudit computations with flow, for testing or machine learning.
In measurement-based quantum computing (MBQC), computation is carried out by a sequence of measurements and corrections on an entangled state. Flow, and related concepts, are powerful techniques for characterising the dependence of the corrections on previous measurement results. We introduce flow-based methods for quantum computation with continuous-variable graph states, which we call CV-flow. These are inspired by, but not equivalent to, the notions of causal flow and g-flow for qubit MBQC. We also show that an MBQC with CV-flow approximates a unitary arbitrarily well in the infinite-squeezing limit, addressing issues of convergence which are unavoidable in the infinite-dimensional setting. In developing our proofs, we provide a method for converting a CV-MBQC computation into a circuit form, analogous to the circuit extraction method of Miyazaki et al, and an efficient algorithm for finding CV-flow when it exists based on the qubit version by Mhalla and Perdrix. Our results and techniques naturally extend to the cases of MBQC for quantum computation with qudits of prime local dimension.
Abstract In measurement-based quantum computing (MBQC), computation is carried out by a sequence of measurements and corrections on an entangled state. Flow, and related concepts, are powerful techniques for characterising the dependence of the corrections on previous measurement outcomes. We introduce flow-based methods for MBQC with qudit graph states, which we call Z d -flow, when the local dimension is an odd prime. Our main results are a proof that Z d -flow is a necessary and sufficient condition for a strong form of outcome determinism. Along the way, we find a suitable generalisation of the concept of measurement planes to this setting and characterise the allowed measurements in a qudit MBQC. We also provide a polynomial-time algorithm for finding an optimal Z d -flow whenever one exists.
The one-way model of measurement-based quantum computing implements computations via successive adaptive single-qubit measurements on a resource graph state. This model has practical applications, particularly in photonics, and it is also useful as a theoretical tool e.g. for optimisation. Gflow is a necessary and sufficient condition for implementing certain one-way computations deterministically (in a suitable sense); it is also used in efficient translations from the one-way model to quantum circuits. For a computation on nnn qubits, a gflow can be found in O(n3)\mathcal{O}(n^3)O(n3) time. Here, we consider an incompletely specified computation given by an unlabelled open graph: the graph state as well as the input and output qubits are known, but the measurements have not yet been fixed. We give an algorithm that identifies a measurement labelling and a compatible gflow, and runs in O(n3)\mathcal{O}(n^3)O(n3), strictly generalising the previous approach. The new algorithm can also handle restrictions on the order of the measurements and returns only solutions compatible with these constraints. We additionally prove that if an open graph has equal numbers of inputs and outputs, it has at most one labelling compatible with gflow; and show how to identify additional inputs for an open graph that does not yet have the maximal number, without breaking an existing gflow.
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.
The one-way model is a universal model of quantum computation, driven by successive adaptive single-qubit measurements on an entangled resource state. Measurements are non-deterministic, yet if the computation satisfies one of several related families of conditions known as 'flows', the computation can be made deterministic overall by modifying later measurements depending on the outcomes of earlier ones. Flow properties also enable efficient translation from one-way computations to circuits, motivating research into rewriting one-way computations while preserving the existence of flow. Existing approaches to flow-preserving rewriting are used for compilation or optimisation and preserve both the interpretation and the existence of flow. Here, we broaden our perspective to consider flow-preserving rewriting that does not necessarily preserve the interpretation, with applications to creating test instances for software that works with flow, as well as to generating ansätze for quantum machine learning. We show that a family of just three flow-preserving rewrite rules suffices to generate any diagram with flow from a trivial diagram with the desired number of inputs and outputs. This rule set is nearly the same as the complete set of flow- and interpretation-preserving rewrite rules for one-way computations in which all measurements are Pauli; and just a small subset of the flow- and interpretation-preserving rewrite rules for arbitrary measurements.
One-way quantum computation, or measurement-based quantum computation, is a universal model of quantum computation alternative to the circuit model. The computation progresses by measurements of a pre-prepared resource state together with corrections of undesired outcomes via applications of Pauli gates to yet unmeasured qubits. The fundamental question of this model is determining whether computation can be implemented deterministically. Pauli flow is one of the most general structures guaranteeing determinism. It is also essential for polynomial time ancilla-free circuit extraction. It is known how to efficiently determine the existence of Pauli flow given an open graph together with a measurement labelling (a choice of measurements to be performed). In this work, we focus on the problem of deciding the existence of Pauli flow for a given open graph when the measurement labelling is unknown. We show that this problem is in RP by providing a random polynomial time algorithm. To do it, we extend previous algebraic interpretations of Pauli flow, by showing that, in the case of X and Z measurements only, flow existence corresponds to the right-invertibility of a matrix derived from the adjacency matrix. We also use this interpretation to show that the number of output qubits can always be reduced to match the number of input qubits while preserving the existence of flow.