We present a translation from the standard network model of quantum computation to the one-way model of quantum computation. The translation is compositional, i.e., it preserves the structure of computations, which allows us to abstract the concrete realizations in a monadic abstraction layer. We briefly review the By-Product and Measurement Calculus approaches to combining circuits in the one-way model, and show how the procedures can be expressed in the exact same notation, bearing a direct relationship with the representation of circuits in the standard network model, and using monads. Discussions on improving the abstraction yield us to introducing an alternative approach to combining circuits in the one-way model: a graphical one.
We present a translation from the standard network model of quantum computation to the one-way model of quantum computation. The translation is compositional, i.e., it preserves the structure of computations, which allows us to abstract the concrete realizations in a monadic abstraction layer. We briefly review the By-Product and Measurement Calculus approaches to combining circuits in the one-way model, and show how the procedures can be expressed in the exact same notation, bearing a direct relationship with the representation of circuits in the standard network model, and using monads. Discussions on improving the abstraction yield us to introducing an alternative approach to combining circuits in the one-way model: a graphical one.
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 present elements of quantum circuits trans- lations from the (standard) network or circuit model to the one-way one. We present a translation scheme, give an account of currently existing tools to apply the scheme, and propose an extension of those tools into a complete translation calculus. We analyse the set of difficulties incurred from such work, and show an engendered opening to new sets of problems and ideas. Among others, this paper introduces the notions of graphical concatenation, graph state reduction (GSR) and graph state extension (GSE) passes, as well as an algorithm for running the (extended) measurement calculus with acceptable efficiency.
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.
Measurement-based quantum computation has emerged from the physics community as a new approach to quantum computation where the notion of measurement is the main driving force of computation. This is in contrast with the more traditional circuit model which is based on unitary operations. Among measurement-based quantum computation methods, the recently introduced one-way quantum computer stands out as fundamental. We develop a rigorous mathematical model underlying the one-way quantum computer and present a concrete syntax and operational semantics for programs, which we call patterns, and an algebra of these patterns derived from a denotational semantics. More importantly, we present a calculus for reasoning locally and compositionally about these patterns. We present a rewrite theory and prove a general standardization theorem which allows all patterns to be put in a semantically equivalent standard form. Standardization has far-reaching consequences: a new physical architecture based on performing all the entanglement in the beginning, parallelization by exposing the dependency structure of measurements and expressiveness theorems. Furthermore we formalize several other measurement-based models: Teleportation, Phase and Pauli models and present compositional embeddings of them into and from the one-way model. This allows us to transfer all the theory we develop for the one-way model to these models. This shows that the framework we have developed has a general impact on measurement-based computation and is not just particular to the one-way quantum computer.
Complete sets of graphical rewrite rules enable fully graphical reasoning about quantum computations and have been an area of active research for more than a decade. Many recent applications of the ZX-calculus have made use of the close correspondence between ZX-diagrams and computations in the one-way model of measurement-based quantum computation. In this model, various kinds of flow properties ensure deterministic implementability; for ZX-diagrams, these same properties allow efficient translation into quantum circuits (a problem that is known to be #P-hard in general). Therefore, flow-preserving ZX-calculus rewrite rules are of strong interest.
Here, we extend the set of flow-preserving rules appearing in the literature with a few new rules and extensions of existing rules. We then show that the resulting rule set is complete for all flow-preserving translations between ZX-diagrams of appropriate form. The proof employs a manifestly flow-preserving equivalent of circuit extraction, where a diagram with gflow is transformed, using only flow-preserving rewrite rules, into a diagram with causal flow.
Measurement-based quantum computation has emerged from the physics community as a new approach to quantum computation where the notion of measurement is the main driving force of computation. This is in contrast with the more traditional circuit model that is based on unitary operations. Among measurement-based quantum computation methods, the recently introduced one-way quantum computer [Raussendorf and Briegel 2001] stands out as fundamental. We develop a rigorous mathematical model underlying the one-way quantum computer and present a concrete syntax and operational semantics for programs, which we call patterns , and an algebra of these patterns derived from a denotational semantics. More importantly, we present a calculus for reasoning locally and compositionally about these patterns. We present a rewrite theory and prove a general standardization theorem which allows all patterns to be put in a semantically equivalent standard form. Standardization has far-reaching consequences: a new physical architecture based on performing all the entanglement in the beginning, parallelization by exposing the dependency structure of measurements and expressiveness theorems. Furthermore we formalize several other measurement-based models, for example, Teleportation, Phase and Pauli models and present compositional embeddings of them into and from the one-way model. This allows us to transfer all the theory we develop for the one-way model to these models. This shows that the framework we have developed has a general impact on measurement-based computation and is not just particular to the one-way quantum computer.