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.
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.
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.
Measurement-based quantum computation (MBQC) has emerged 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 takes unitary operations as fundamental. Among measurement-based quantum computation methods the recently introduced one-way quantum computer stands out as basic and fundamental. The key idea is to start from an entangled state and then use measurements and one-qubit unitaries, which may be dependent on the outcomes of measurements, to guide the computation. The main point is that one never has to perform unitaries on more than one qubit at a time after the initial preparation of an entangled state. The “programs” that one writes in this model are traditionally called “patterns.” In this chapter, we develop a rigorous mathematical model underlying measurement-based quantum computation. We give syntax, operational semantics, denotational semantics, and an algebra of programs derived from the denotational semantics. We also present a rewrite theory and prove a general standardization theorem that allows all programs 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. We use our general measurement calculus not just to formalize the one-way model but also several other measurement-based models, e.g., Teleportation, Phase, and Pauli models, and present compositional embeddings of them into and from the one-way model.
We develop a formal model for distributed measurement-based quantum computations, adopting an agent-based view, such that computations are described locally where possible. Because the network quantum state is in general entangled, we need to model it as a global structure, reminiscent of global memory in classical agent systems. Local quantum computations are described as measurement patterns. Since measurement-based quantum computation is inherently distributed, this allows us to extend naturally several concepts of the measurement calculus [V. Danos, E. Kashefi and P. Panangaden, The measurement calculus (2004), arXiv:quant-ph/0412135], a formal model for such computations. Our goal is to define an assembly language, i.e. we assume that computations are well-defined and we do not concern ourselves with verification techniques. The operational semantics for systems of agents is given by a probabilistic transition system, and we define operational equivalence in a way that it corresponds to the notion of bisimilarity. With this in place, we prove that teleportation is bisimilar to a direct quantum channel, and this also within the context of larger networks.
We develop a formal model for distributed measurement-based quantum computations, adopting an agent-based view, such that computations are described locally where possible. Because the network quantum state is in general entangled, we need to model it as a global structure, reminiscent of global memory in classical agent systems. Local quantum computations are described as measurement patterns. Since measurement-based quantum computation is inherently distributed, this allows us to extend naturally several concepts of the measurement calculus, a formal model for such computations. Our goal is to define an assembly language, i.e. we assume that computations are well-defined and we do not concern ourselves with verification techniques. The operational semantics for systems of agents is given by a probabilistic transition system, and we define operational equivalence in a way that it corresponds to the notion of bisimilarity. With this in place, we prove that teleportation is bisimilar to a direct quantum channel, and this also within the context of larger networks.
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.
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.