arXiv subjects

Piotr Mitosek

Publications and source records attributed to Piotr Mitosek.

4 recordsLinked to original sources

arXiv2026

Working with measurement-based computations on qudits

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.

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arXiv2024

An algebraic interpretation of Pauli flow, leading to faster flow-finding algorithms

The one-way model of quantum computation is an alternative to the circuit model. A one-way computation is driven entirely by successive adaptive measurements of a pre-prepared entangled resource state. For each measurement, only one outcome is desired; hence a fundamental question is whether some intended measurement scheme can be performed in a robustly deterministic way. So-called flow structures witness robust determinism by providing instructions for correcting undesired outcomes. Pauli flow is one of the broadest of these structures and has been studied extensively. It is known how to find flow structures in polynomial time when they exist; nevertheless, their lengthy and complex definitions often hinder working with them. We simplify these definitions by providing a new algebraic interpretation of Pauli flow. This involves defining two matrices arising from the adjacency matrix of the underlying graph: the flow-demand matrix $M$ and the order-demand matrix $N$. We show that Pauli flow exists if and only if there is a right inverse $C$ of $M$ such that the product $NC$ forms the adjacency matrix of a directed acyclic graph. From the newly defined algebraic interpretation, we obtain $\mathcal{O}(n^3)$ algorithms for finding Pauli flow, improving on the previous $\mathcal{O}(n^4)$ bound for finding generalised flow, a weaker variant of flow, and $\mathcal{O}(n^5)$ bound for finding Pauli flow. We also introduce a first lower bound for the Pauli flow-finding problem, by linking it to the matrix invertibility and multiplication problems over $\mathbb{F}_2$.

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arXiv2024

Pauli Flow on Open Graphs with Unknown Measurement Labels

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.

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arXiv2024

Constructing $\mathrm{NP}^{\mathord{\#}\mathrm P}$-complete problems and ${\mathord{\#}\mathrm P}$-hardness of circuit extraction in phase-free ZH

The ZH calculus is a graphical language for quantum computation reasoning. The phase-free variant offers a simple set of generators that guarantee universality. ZH calculus is effective in MBQC and analysis of quantum circuits constructed with the universal gate set Toffoli+H. While circuits naturally translate to ZH diagrams, finding an ancilla-free circuit equivalent to a given diagram is hard. Here, we show that circuit extraction for phase-free ZH calculus is ${\mathord{\#}\mathrm P}$-hard, extending the existing result for ZX calculus. Another problem believed to be hard is comparing whether two diagrams represent the same process. We show that two closely related problems are $\mathrm{NP}^{\mathord{\#}\mathrm P}$-complete. The first problem is: given two processes represented as diagrams, determine the existence of a computational basis state on which they equalize. The second problem is checking whether the matrix representation of a given diagram contains an entry equal to a given number. Our proof adapts the proof of Cook-Levin theorem to a reduction from a non-deterministic Turing Machine with access to ${\mathord{\#}\mathrm P}$ oracle.

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