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Algebraic characterisation of one-way patterns

Vedran Dunjko, Elham Kashefi

Published Aug 15, 2016• 0 citations
OpenAlex

Abstract

We give a complete structural characterisation of the map the positive branch of a one-way pattern implements. We start with the representation of the positive branch in terms of the phase map de-composition [4], which is then further analysed to obtain the primary structure of the matrix M, representing the phase map decomposition in the computational basis. Using this approach we ob-tain some preliminary results on the connection between the columns structure of a given unitary and the angles of measurements in a pattern that implements it. We believe this work is a step forward towards a full characterisation of those unitaries with an efficient one-way model implementation. 1

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We give a complete structural characterisation of the map the positive branch of a one-way pattern implements. We start with the representation of the positive branch in terms of the phase map de-composition [4], which is then further analysed to obtain the primary structure of the matrix M, representing the phase map decomposition in the computational basis. Using this approach we ob-tain some preliminary results on the connection between the columns structure of a given unitary and the angles of measurements in a pattern that implements it. We believe this work is a step forward towards a full characterisation of those unitaries with an efficient one-way model implementation. 1
We give a complete structural characterisation of the map the positive branch of a one-way pattern implements. We start with the representation of the positive branch in terms of the phase map decomposition, which is then further analysed to obtain the primary structure of the matrix M, representing the phase map decomposition in the computational basis. Using this approach we obtain some preliminary results on the connection between the columns structure of a given unitary and the angles of measurements in a pattern that implements it. We believe this work is a step forward towards a full characterisation of those unitaries with an efficient one-way model implementation.
In the study of quantum computation, data is represented in terms of linear operators which form a generalized model of probability, and computations are most commonly described as products of unitary transformations, which are the transformations which preserve the quality of the data in a precise sense. This naturally leads to "unitary circuit models", which are models of computation in which unitary operators are expressed as a product of "elementary" unitary transformations. However, unitary transformations can also be effected as a composition of operations which are not all unitary themselves: the "one-way measurement model" is one such model of quantum computation. In this thesis, we examine the relationship between representations of unitary operators and decompositions of those operators in the one-way measurement model. In particular, we consider different circumstances under which a procedure in the one-way measurement model can be described as simulating a unitary circuit, by considering the combinatorial structures which are common to unitary circuits and two simple constructions of one-way based procedures. These structures lead to a characterization of the one-way measurement patterns which arise from these constructions, which can then be related to efficiently testable properties of graphs. We also consider how these characterizations provide automatic techniques for obtaining complete measurement-based decompositions, from unitary transformations which are specified by operator expressions bearing a formal resemblance to path integrals. These techniques are presented as a possible means to devise new algorithms in the one-way measurement model, independently of algorithms in the unitary circuit model.
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.
In this book chapter, we provide a tutorial introduction to one-way quantum computation and many of the techniques one can use to understand it. The techniques which are described include the stabilizer formalism and the logical Heisenberg picture. We highlight ways in which it is useful to understand one-way computation beyond simple equivalence with the quantum circuit model. We briefly review current proposals of implementations and experimental progress and summarize some recent related theoretical developments. Although the chapter is primarily didactic in focus, we include a number of new methods and observations. These include: a simpler and more compact formulation of one-way quantum computation in the stabilizer formalism; A new way of implementing unitaries diagonal in the computational basis; New results on the family of operations which may be implemented in a single round of measurements; A method for constructing compact one-way patterns by decomposing unitaries in terms of diagonal unitaries and Clifford group transformations.
In the study of quantum computation, data is represented in terms of linear operators which form a generalized of probability, and computations are most commonly described as products of transformations, which are the transformations which preserve the quality of the data in a precise sense. This naturally leads to unitary circuit models, which are models of computation in which operators are expressed as a product of elementary transformations. However, transformations can also be effected as a composition of operations which are not all themselves: the measurement model is one such of quantum computation. In this thesis, we examine the relationship between representations of operators and decompositions of those operators in the one-way measurement model. In particular, we consider different circumstances under which a procedure in the one-way measurement can be described as simulating a circuit, by considering the combinatorial structures which are common to circuits and two simple constructions of one-way based procedures. These structures lead to a characterization of the one-way measurement patterns which arise from these constructions, which can then be related to efficiently testable properties of graphs. We also consider how these characterizations provide automatic techniques for obtaining complete measurement-based decompositions, from transformations which are specified by operator expressions bearing a formal resemblance to path integrals. These techniques are presented as a possible means to devise new algorithms in the one-way measurement model, independently of algorithms in the circuit model.
We give a complete structural characterisation of the map implemented by the positive branch of a one-way pattern. Our approach is based on the phase map decomposition (de Beaudrap et al. 2006; de Beaudrap et al. 2008) and leads to some preliminary results on the connection between the column structure of a given unitary and the angles of measurements in a pattern that implements it. Our characterisation highlights the role of entanglement in the efficiency of the simulation of a one-way pattern, and it is a step forward towards a full characterisation of those unitaries that have an efficient one-way model implementation.