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.
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 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 I, 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.
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⊆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 ∣I∣=∣O∣=k (corresponding to unitary transformations in the measurement model) and ∣V(G)∣=n can be performed in time O(k2n), improving the earlier known bound of O(km) given in \cite{B06a}, where m=∣E(G)∣.
In the one-way quantum computing model, the information processing is driven by measurements performed on an entangled state, called the resource state. In order to achieve parallelism, one aims to increase the number of simultaneous measurements, such that the effects of decoherence on the resource state are minimised due to the reduction of the time required to run the computation. At the heart of this question is the notion of quantum information flow, which specifies the dependency relations between the measurements in the computations. There exist two well-known techniques for reducing the time required to perform a one-way quantum computation without changing its semantics. The first one, called signal-shifting, transforms the flow of a graph (representing the resource state) within the measurement calculus formalism, whereas the second one, namely finding the maximally-delayed generalised flow, explores the geometry of the graph to increase the number of operations that can be performed simultaneously. In this paper, we show for the first time how these two techniques relate to each other. We prove that the application of the signal-shifting rules to a measurement pattern with flow results in a generalised flow for the pattern. Then, we prove that in the particular case when the input size equals the output size, the gflow obtained using signal-shifting has the lowest possible depth. As a side result, we construct an O(n3)-algorithm for finding maximally delayed gflows on graphs with flow. For those graphs, our algorithm is more efficient than the best previously known algorithm for the same task, which takes O(n4) operations to complete.
We introduce a new characterisation of determinism in Measurement-Based Quantum Computing (MBQC). The one-way model consists in performing local measurements over a large entangled state represented by a graph. The ability to perform an overall deterministic computation requires a correction strategy because of the non-determinism of each measurement. The existence of such a correction strategy depends on the underlying open graph, which is a description of the resource state together with the basis of the performed measurements. GFlow is a well-known graphical characterisation of robust determinism in MBQC when every measurement is performed in some specific planes of the Bloch sphere. While Pauli measurements are ubiquitous in MBQC, GFlow fails to be necessary for determinism when a measurement-based quantum computation involves Pauli measurements. Pauli Flow was designed as a generalisation of GFlow to handle MBQC with Pauli measurements, and guarantees robust determinism, however, it has been shown more recently that it fails to be a necessary condition. Our contribution is twofold. First, we demonstrate that Pauli flow is actually necessary for robust determinism in a weaker sense: given an open graph, i.e. a resource state, a deterministic computation can be driven iff it has a Pauli flow. However, the Pauli flows do not reflect all the possible correction strategies over a particular resource state, and properties like measurement order or computational depth are not necessarily reflected by a Pauli flow. Thus, to characterise determinism in full generality, we introduce a further extension called Shadow Pauli Flow that we prove necessary and sufficient for robust determinism: An MBQC is robustly deterministic if and only if its correction strategy is consistent with a Shadow Pauli flow. Furthermore, we show that Shadow Pauli flow can be computed in polynomial time.
Graph states are an elegant and powerful quantum resource for measurement based quantum computation (MBQC). They are also used for many quantum protocols (error correction, secret sharing, etc.). The main focus of this paper is to provide a structural characterization of the graph states that can be used for quantum information processing. The existence of a gflow (generalized flow) is known to be a requirement for open graphs (graph, input set and output set) to perform uniformly and strongly deterministic computations. We weaken the gflow conditions to define two new more general types of MBQC: uniform equiprobability and constant probability. These classes can be useful from a cryptographic and information point of view because even though we can not do a deterministic computation in general we can preserve the information and transfer it perfectly from the inputs to the outputs. We derive simple graph characterizations for these classes and prove that the deterministic and uniform equiprobability classes collapse when the cardinalities of inputs and outputs are the same. We also prove the reversibility of gflow in that case. The new graphical characterizations allow us to go from open graphs to graphs in general and to consider this question: given a graph with no inputs or outputs fixed, which vertices can be chosen as input and output for quantum information processing? We present a characterization of the sets of possible inputs and ouputs for the equiprobability class, which is also valid for deterministic computations with inputs and ouputs of the same cardinality.
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.