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arXiv:quant-ph/0702229 · Quantum Information and Computation

An extremal result for geometries in the one-way measurement model

Niel de Beaudrap, Martin Pei

Published May 1, 2008• 8 citations
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Abstract

We present an extremal result for the class of graphs $G$ which (together with some specified sets of input and output vertices, $I$ and $O$) have a certain ``flow'' property introduced by Danos and Kashefi for the one-way measurement model of quantum computation. The existence of a flow for a triple $(G,I,O)$ allows a unitary embedding to be derived from any choice of measurement bases allowed in the one-way measurement model. We prove an upper bound on the number of edges that a graph $G$ may have, in order for a triple $(G,I,O)$ to have a flow for some $I, O \subseteq V(G)$, in terms of the number of vertices in $G$ and $O$. This implies that finding a flow for a triple $(G,I,O)$ when $\lvert I \rvert = \lvert O \rvert = k$ (corresponding to unitary transformations in the measurement model) and $\lvert V(G) \rvert = n$ can be performed in time $O(k^2 n)$, improving the earlier known bound of $O(km)$ given in \cite{B06a}, where $m = \lvert E(G) \rvert$.

DOI
10.26421/qic8.5-5

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We present an extremal result for the class of graphs GGG which (together with some specified sets of input and output vertices, III and OOO) have a certain ``flow'' property introduced by Danos and Kashefi for the one-way measurement model of quantum computation. The existence of a flow for a triple (G,I,O)(G,I,O)(G,I,O) allows a unitary embedding to be derived from any choice of measurement bases allowed in the one-way measurement model. We prove an upper bound on the number of edges that a graph GGG may have, in order for a triple (G,I,O)(G,I,O)(G,I,O) to have a flow for some I,O⊆V(G)I, O \subseteq V(G)I,O⊆V(G), in terms of the number of vertices in GGG and OOO. This implies that finding a flow for a triple (G,I,O)(G,I,O)(G,I,O) when ∣I∣=∣O∣=k\lvert I \rvert = \lvert O \rvert = k∣I∣=∣O∣=k (corresponding to unitary transformations in the measurement model) and ∣V(G)∣=n\lvert V(G) \rvert = n∣V(G)∣=n can be performed in time O(k2n)O(k^2 n)O(k2n), improving the earlier known bound of O(km)O(km)O(km) given in \cite{B06a}, where m=∣E(G)∣m = \lvert E(G) \rvertm=∣E(G)∣.
The one-way measurement model is a framework for universal quantum computation in which algorithms are partially described by a graph GGG of entanglement relations on a collection of qubits. A sufficient condition for an algorithm to perform a unitary embedding between two Hilbert spaces is for the graph GGG, together with input and output I\mathit{I}I, O\mathit{O}O vertices $I,O\ensuremath{\subseteq}V(G)$, to have a flow in the sense introduced by Danos and Kashefi [Phys. Rev. A 74, 052310 (2006)]. For the special case of ∣I∣=∣O∣|I|=|O|∣I∣=∣O∣, using a graph-theoretic characterization, I show that such flows are unique when they exist. This leads to an efficient algorithm for finding flows by a reduction to solved problems in graph theory.
We extend the notion of quantum information flow defined by Danos and Kashefi (2006 Phys. Rev. A 74 052310) for the one-way model (Raussendorf and Briegel 2001 Phys. Rev. Lett. 86 910) and present a necessary and sufficient condition for the stepwise uniformly deterministic computation in this model. The generalized flow also applied in the extended model with measurements in the ( X , Y ), ( X , Z ) and ( Y , Z ) planes. We apply both measurement calculus and the stabiliser formalism to derive our main theorem which for the first time gives a full characterization of the stepwise uniformly deterministic computation in the one-way model. We present several examples to show how our result improves over the traditional notion of flow, such as geometries (entanglement graph with input and output) with no flow but having generalized flow and we discuss how they lead to an optimal implementation of the unitaries. More importantly one can also obtain a better quantum computation depth with the generalized flow rather than with flow. We believe our characterization result is particularly valuable for the study of the algorithms and complexity in the one-way model.
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.
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.
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.
We introduce a flow condition on open graph states (graph states with inputs and outputs) which guarantees globally deterministic behavior of a class of measurement patterns defined over them. Dependent Pauli corrections are derived for all such patterns, which equalize all computation branches, and only depend on the underlying entanglement graph and its choice of inputs and outputs. The class of patterns having flow is stable under composition and tensorization, and has unitary embeddings as realizations. The restricted class of patterns having both flow and reverse flow supports an operation of adjunction, and has all and only unitaries as realizations.