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arXiv:quant-ph/0611284 · Physical Review A

Finding flows in the one-way measurement model

Niel de Beaudrap

Published Feb 29, 2008• 25 citations
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Abstract

The one-way measurement model is a framework for universal quantum computation in which algorithms are partially described by a graph $G$ 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 $G$, together with input and output $\mathit{I}$, $\mathit{O}$ 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 $|I|=|O|$, 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.

DOI
10.1103/physreva.77.022328

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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⊆V(G)I,O\ensuremath{\subseteq}V(G)I,O\ensuremath⊆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 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 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.
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.
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.
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.
We propose a universal decomposition of unitary maps over a tensorial power of C^2, introducing the key concept of "phase maps", and investigate how this decomposition can be used to implement unitary maps directly in the measurement-based model for quantum computing. Specifically, we show how to extract from such a decomposition a matching entangled graph state (with inputs), and a set of measurements angles, when there is one. Next, we check whether the obtained graph state verifies a "flow" condition, which guarantees an execution order such that the dependent measurements and corrections of the pattern yield deterministic results. Using a graph theoretic characterization of flows, we can determine whether a flow can be constructed for a graph state in polynomial time. This approach yields an algorithmic procedure which, when it succeeds, may produce an efficient pattern for a given unitary.