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arxiv: 2606.30525 · v1 · pith:J4BAFHMEnew · submitted 2026-06-29 · 🪐 quant-ph · cs.DS

Working with measurement-based computations on qudits

Piotr Mitosek , Miriam Backens This is my paper

Pith reviewed 2026-06-30 06:02 UTC · model grok-4.3

classification 🪐 quant-ph cs.DS
keywords qudit flowmeasurement-based quantum computationgraph statesflow-finding algorithmadaptive correctionsprime dimension
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The pith

A simpler definition of qudit flow supports an O(n^3) finding algorithm for prime-dimensional graph states.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a streamlined definition of flow for measurement-based computations on prime-dimensional qudits. Flow indicates when adaptive measurement corrections can make the overall procedure deterministic. The new definition draws directly from the qubit case, which simplifies prior qudit versions and allows proofs of focusability and canonicity. It yields an algorithm that locates flows in cubic time relative to the number of qudits, plus several transformations that preserve flow. A reader would care because the change lowers the cost of handling adaptivity and supplies concrete tools for building and optimizing qudit computations.

Core claim

By defining qudit flow through a pair of functions on the vertices that satisfy the same correction and commutation relations used for qubits, the authors prove that focused flow is canonical, refine the algebraic description of focused flow, and obtain an O(n^3) algorithm that finds whether a flow exists; they further exhibit flow-preserving pivots, vertex insertions and removals, and reversibility.

What carries the argument

The simpler qudit flow, a pair of functions on graph vertices obeying correction and commutation conditions that guarantee a deterministic adaptive strategy exists for the measurements.

If this is right

  • Focused flow is the canonical representative for any given computation.
  • The cubic-time algorithm matches the best qubit result and improves the prior quartic bound for qudits.
  • Pivoting, insertion and removal of certain vertices, and flow reversal all preserve the existence of flow and can be used to rewrite computations.
  • An algorithmic generator can systematically produce large families of qudit computations that admit flow.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The transformations could be composed into a search procedure that finds resource-efficient graph states for a target computation.
  • The generation method supplies training data for machine-learning models that predict or enforce flow.
  • The cubic scaling makes exhaustive checks feasible for graphs an order of magnitude larger than before.
  • The same simplification might apply to other resource states beyond graph states once the prime-dimension assumption is relaxed.

Load-bearing premise

The definition, its properties, and the algorithm apply when the entangled resource is a prime-dimensional qudit graph state.

What would settle it

A concrete prime-dimensional qudit graph state and measurement pattern for which a valid deterministic correction strategy exists but the O(n^3) algorithm reports that no flow is present.

Figures

Figures reproduced from arXiv: 2606.30525 by Miriam Backens, Piotr Mitosek.

Figure 1
Figure 1. Figure 1: Running example of a labelled open F𝑑-graph with 𝑑 = 5. 1a: a labelled open F𝑑-graph with one input 𝑖 and two outputs 𝑜1 and 𝑜2. The measurement labels of non-outputs are given near the names of the vertices. Edge weights are denoted by numbers near the middles of the edges. 1b: the corresponding adjacency matrix. The dotted lines subdivide input, internal, and output vertices for improved readability but … view at source ↗
Figure 2
Figure 2. Figure 2: The algebraic formulation of the labelled open [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The labelled open F𝑑-graph from [PITH_FULL_IMAGE:figures/full_fig_p041_3.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript introduces a simpler definition of flow for measurement-based quantum computation on prime-dimensional qudit graph states. It establishes that this flow is focusable and that focused flow is canonical, improves an algebraic formulation to obtain an O(n^3) flow-finding algorithm, explores flow-preserving transformations (pivoting, vertex removal/insertion, reversibility), and proposes an algorithmic method for generating large qudit computations with flow.

Significance. If the results hold, the work lowers the barrier to applying flow in qudit MBQC, matches the best-known qubit complexity, and supplies concrete tools for optimization and systematic generation of computations. These contributions are directly useful for extending adaptive-correction techniques beyond qubits.

minor comments (2)
  1. Abstract: the statement that the new definition 'inherits focusability and canonicity from the qubit case' would be strengthened by a one-sentence pointer to the specific qubit result being invoked.
  2. The manuscript would benefit from an explicit statement, early in the development, of the precise conditions under which the O(n^3) bound holds (e.g., whether it requires the resource state to be a prime-dimensional graph state).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work on a simplified definition of flow for qudit MBQC, the O(n^3) algorithm, flow-preserving operations, and the generation method. We appreciate the recommendation for minor revision and the assessment that the contributions lower the barrier for applying flow in this setting.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central contributions—a simpler definition of qudit flow, its focusability and canonicity properties, and the O(n^3) flow-finding algorithm—are obtained by direct algebraic reformulation of the new definition into linear equations over GF(p) followed by standard Gaussian elimination. These steps rely on explicit mappings from the established qubit flow literature and generic complexity bounds for linear algebra, without any reduction of the claimed results to fitted parameters, self-citations, or definitional tautologies. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard mathematical properties of graph states and flow in the qubit case, extended to prime-dimensional qudits; no free parameters, new physical entities, or ad-hoc axioms are introduced.

axioms (1)
  • domain assumption Resource states are prime-dimensional qudit graph states whose flow properties extend from the qubit case
    Invoked when the new definition and algorithm are stated to apply to these states.

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Reference graph

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    Multiplying by𝐶𝑢,𝑤, we get: 𝑀𝑣,𝑢 𝐶𝑢,𝑤 =𝜆 𝑍 (𝑣) −1 𝐴𝑣,𝑢 𝐶𝑢,𝑤 and hence by linearity: (𝑀𝐶) 𝑣,𝑤 =𝜆 𝑍 (𝑣) −1 (𝐴𝐶) 𝑣,𝑤 (1)

    Suppose𝜆 𝑋 (𝑣)=0and thus𝑀 𝑣,𝑢 =𝜆 𝑍 (𝑣) −1 𝐴𝑣,𝑢. Multiplying by𝐶𝑢,𝑤, we get: 𝑀𝑣,𝑢 𝐶𝑢,𝑤 =𝜆 𝑍 (𝑣) −1 𝐴𝑣,𝑢 𝐶𝑢,𝑤 and hence by linearity: (𝑀𝐶) 𝑣,𝑤 =𝜆 𝑍 (𝑣) −1 (𝐴𝐶) 𝑣,𝑤 (1)

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    When𝜆 𝑋 (𝑣)≠0,thennecessarily𝑣∉𝐼andthus 𝑀has a𝑣-column

    Suppose𝜆 𝑋 (𝑣)≠0andthus𝑀 𝑣,𝑢 =𝜆 𝑋 (𝑣) −1𝛿𝑣,𝑢. When𝜆 𝑋 (𝑣)≠0,thennecessarily𝑣∉𝐼andthus 𝑀has a𝑣-column. Multiplying𝑀 𝑣,𝑢 be𝐶 𝑢,𝑤, we get: 𝑀𝑣,𝑢 𝐶𝑢,𝑤 =𝜆 𝑋 (𝑣) −1𝛿𝑣,𝑢 𝐶𝑢,𝑤 which equals0unless𝑣=𝑢when it equals𝜆 𝑋 (𝑣) −1𝐶𝑣,𝑤 . Using this fact, we find that: (𝑀𝐶) 𝑣,𝑤 = ∑︁ 𝑢∈ ¯𝐼 𝑀𝑣,𝑢 𝐶𝑢,𝑤 =𝜆 𝑋 (𝑣) −1𝐶𝑣,𝑤 (2) Now we can split the proof into two directions. (⇒):Supp...

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    Thus (𝐴𝐶) 𝑣,𝑣 =𝜆 𝑍 (𝑣), i.e

    For all𝑣such that𝜆 𝑋 (𝑣)=0, by Equation (1), we have1=(𝑀𝐶) 𝑣,𝑣 =𝜆 𝑍 (𝑣) −1 (𝐴𝐶) 𝑣,𝑣 . Thus (𝐴𝐶) 𝑣,𝑣 =𝜆 𝑍 (𝑣), i.e. (C1) holds

  47. [47]

    Since𝜆 𝑍 (𝑣)≠0(as𝜆(𝑣)≠(0,0)), this implies(𝐴𝐶) 𝑣,𝑤 =0 and thus (FX) holds

    Furthermore, for all𝑣such that𝜆 𝑋 (𝑣)=0and for all𝑤such that𝑣≠𝑤, by Equation (1) we have 0=(𝑀𝐶) 𝑣,𝑤 =𝜆 𝑍 (𝑣) −1 (𝐴𝐶) 𝑣,𝑤 . Since𝜆 𝑍 (𝑣)≠0(as𝜆(𝑣)≠(0,0)), this implies(𝐴𝐶) 𝑣,𝑤 =0 and thus (FX) holds

  48. [48]

    Thus 𝐶𝑣,𝑣 =𝜆 𝑋 (𝑣), i.e

    For all𝑣such that𝜆 𝑋 (𝑣)≠0, by Equation (2), we have1=(𝑀𝐶) 𝑣,𝑣 =𝜆 𝑋 (𝑣) −1𝐶𝑣,𝑣 . Thus 𝐶𝑣,𝑣 =𝜆 𝑋 (𝑣), i.e. (C2) holds. P . Mitosek & M. Backens23

  49. [49]

    This implies𝐶𝑣,𝑤 =0and thus (FZ) holds

    Finally, for all𝑣such that𝜆 𝑋 (𝑣)≠0and for all𝑤such that𝑣≠𝑤, by Equation 2 we have 0=(𝑀𝐶) 𝑣,𝑤 =𝜆 𝑋 (𝑣) −1𝐶𝑣,𝑤 . This implies𝐶𝑣,𝑤 =0and thus (FZ) holds. Hence the desired properties are all satisfied. (⇐):Suppose the four properties hold. We must show that(𝑀𝐶)𝑣,𝑣 =1and(𝑀𝐶) 𝑣,𝑤 =0for all 𝑣≠𝑤∈ ¯𝑂. Observethatallfourpointsinthe(⇒)partoftheproofarereversible. ...

  50. [50]

    Then(𝑁𝐶) 𝑣,𝑣 =−𝜆 𝑍 (𝑣)𝐶 𝑣,𝑣 which equals0if and only if𝐶𝑣,𝑣 =0

    Suppose𝜆 𝑋 (𝑣)=0. Then(𝑁𝐶) 𝑣,𝑣 =−𝜆 𝑍 (𝑣)𝐶 𝑣,𝑣 which equals0if and only if𝐶𝑣,𝑣 =0. Consid- ering all𝑣with𝜆 𝑋 (𝑣)=0, we obtain equivalence of(𝑁𝐶) 𝑣,𝑣 =0to (C3)

  51. [51]

    By the assumption that𝐶satisfies (C2), we have𝐶𝑣,𝑣 =𝜆 𝑋 (𝑣)

    Suppose𝜆 𝑋 (𝑣)≠0. By the assumption that𝐶satisfies (C2), we have𝐶𝑣,𝑣 =𝜆 𝑋 (𝑣). Then: (𝑁𝐶) 𝑣,𝑣 =𝜆 𝑋 (𝑣) (𝐴𝐶) 𝑣,𝑣 −𝜆 𝑍 (𝑣)𝐶 𝑣,𝑣 =𝜆 𝑋 (𝑣) (𝐴𝐶) 𝑣,𝑣 −𝜆 𝑍 (𝑣)𝜆 𝑋 (𝑣)=𝜆 𝑋 (𝑣) ((𝐴𝐶) 𝑣,𝑣 −𝜆 𝑍 (𝑣)) whichequals0ifandonlyif(𝐴𝐶) 𝑣,𝑣 =𝜆 𝑍 (𝑣). Consideringall𝑣with𝜆 𝑋 ≠0,weobtainequivalence of(𝑁𝐶) 𝑣,𝑣 =0to (C4). Thus, the above two cases imply that the main diagonal of𝑁𝐶...

  52. [52]

    By the assumption that𝐶satisfies (FX), we have(𝐴𝐶) 𝑣,𝑤 =0

    Suppose𝜆 𝑋 (𝑣)=0. By the assumption that𝐶satisfies (FX), we have(𝐴𝐶) 𝑣,𝑤 =0. Thus the condition of (O1) does not hold. Ergo the condition of (O2) necessarily applies, leading to 𝐶𝑣,𝑤 ≠0. Therefore (3) becomes: (𝑁𝐶) 𝑣,𝑤 =𝜆 𝑋 (𝑣) (𝐴𝐶) 𝑣,𝑤 −𝜆 𝑍 (𝑣)𝐶 𝑣,𝑤 =−𝜆 𝑍 (𝑣)𝐶 𝑣,𝑤 ≠0 24Working with measurement-based computations on qudits

  53. [53]

    Bytheassumptionthat𝐶satisfies(FZ),wehave𝐶 𝑣,𝑤 =0

    Suppose𝜆 𝑋 (𝑣)≠0. Bytheassumptionthat𝐶satisfies(FZ),wehave𝐶 𝑣,𝑤 =0. Thusthecondition of (O2) does not apply. Ergo the condition of (O1) necessarily applies, leading to(𝐴𝐶)𝑣,𝑤 ≠0. Therefore (3) becomes: (𝑁𝐶) 𝑣,𝑤 =𝜆 𝑋 (𝑣) (𝐴𝐶) 𝑣,𝑤 −𝜆 𝑍 (𝑣)𝐶 𝑣,𝑤 =𝜆 𝑋 (𝑣) (𝐴𝐶) 𝑣,𝑤 ≠0 In both cases,(𝑁𝐶)𝑣,𝑤 ≠0as desired. (⇐): Suppose that(𝑁𝐶) 𝑣,𝑤 ≠0, which means: 𝜆𝑋 (𝑣) (𝐴𝐶) 𝑣,...

  54. [54]

    This implies(𝐴®𝒶) 𝑣 =0, i.e.𝑣∉supp ( 𝐴®𝒶)

    Suppose𝜆 𝑋 (𝑣)=0, then: 0=(𝑀®𝒶) 𝑣 = ∑︁ 𝑢∈ ¯𝐼 𝑀𝑣,𝑢 ®𝒶𝑢 =𝜆 𝑍 (𝑣) −1 ∑︁ 𝑢∈ ¯𝐼 𝐴𝑣,𝑢 ®𝒶𝑢 =𝜆 𝑍 (𝑣) −1 (𝐴®𝒶)𝑣 Since𝜆 𝑋 (𝑣)=0, necessarily𝜆 𝑍 (𝑣)≠0. This implies(𝐴®𝒶) 𝑣 =0, i.e.𝑣∉supp ( 𝐴®𝒶). By contra- position: if𝑣∈supp ( 𝐴®𝒶), then𝜆𝑋 (𝑣)≠0, i.e. (Fs2) holds

  55. [55]

    Since𝑀 𝑣,𝑢 =0for𝑣≠𝑢, we have: 0=(𝑀®𝒶) 𝑣 = ∑︁ 𝑢∈ ¯𝐼 𝑀𝑣,𝑢 ®𝒶𝑢 =𝜆 𝑋 (𝑣) −1 ®𝒶𝑣 Then as𝜆 𝑋 (𝑣)≠0, we have®𝒶𝑣 =0, i.e.𝑣∉supp ( ®𝒶)

    Suppose𝜆 𝑋 (𝑣)≠0. Since𝑀 𝑣,𝑢 =0for𝑣≠𝑢, we have: 0=(𝑀®𝒶) 𝑣 = ∑︁ 𝑢∈ ¯𝐼 𝑀𝑣,𝑢 ®𝒶𝑢 =𝜆 𝑋 (𝑣) −1 ®𝒶𝑣 Then as𝜆 𝑋 (𝑣)≠0, we have®𝒶𝑣 =0, i.e.𝑣∉supp ( ®𝒶). Again by contraposition: if𝑣∈supp ( 𝐴), then𝜆 𝑋 (𝑣)=0, meaning (Fs1) holds. Hence®𝒶satisfies conditions (Fs1) and (Fs2) over the set{𝑣}as required. (⇐): Let®𝒶be focused over{𝑣}. Once again, we consider two cases ...

  56. [56]

    Following the same sequence of equalities as in the first case of the(⇒)part of the proof, we find(𝑀®𝒶)𝑣 =0

    Suppose𝜆 𝑋 (𝑣)=0, then by (Fs2):𝑣∉supp ( 𝐴®𝒶), i.e.(𝐴®𝒶)𝑣 =0. Following the same sequence of equalities as in the first case of the(⇒)part of the proof, we find(𝑀®𝒶)𝑣 =0

  57. [57]

    pivot simplification

    Suppose𝜆 𝑋 (𝑣)≠0, then by (Fs1):𝑣∉supp ( ®𝒶), i.e.( ®𝒶)𝑣 =0. Following the same sequence of equalities as in the second case of the(⇒)part of the proof, we again find(𝑀®𝒶)𝑣 =0. Therefore,(𝑀®𝒶)𝑣 =0, meaning𝑣∉supp (𝑀®𝒶), ergo𝑣∈𝐹 ®𝒶, which ends the proof.□ Using the above lemma, we find that the kernel of the flow-demand matrix forms is exactly the set of fo...