We present a comprehensive and universal quantum computing model based on 2-qubit measurements (TQC), which embeds trivially in a model based on 1-qubit measurements known as the one-way model (1WQC). One inherits through this embedding a calculus whereby each TQC computing pattern can be put in a standard form where entanglement is done first, then measurement, then local corrections.
We present a comprehensive and universal quantum computing model based on 2-qubit measurements (TQC), which embeds trivially in a model based on 1-qubit measurements known as the one-way model (1WQC). One inherits through this embedding a calculus whereby each TQC computing pattern can be put in a standard form where entanglement is done first, then measurement, then local corrections.
In the formalism of measurement based quantum computation we start with a given fixed entangled state of many qubits and perform computation by applying a sequence of measurements to designated qubits in designated bases. The choice of basis for later measurements may depend on earlier measurement outcomes and the final result of the computation is determined from the classical data of all the measurement outcomes. This is in contrast to the more familiar gate array model in which computational steps are unitary operations, developing a large entangled state prior to some final measurements for the output. Two principal schemes of measurement based computation are teleportation quantum computation (TQC) and the so-called cluster model or one-way quantum computer (1WQC). We will describe these schemes and show how they are able to perform universal quantum computation. We will outline various possible relationships between the models which serve to clarify their workings. We will also discuss possible novel computational benefits of the measurement based models compared to the gate array model, especially issues of parallelisability of algorithms.
Quantum computing allows the realization of certain tasks more efficiently than its classical counterpart. This monograph seeks to present a type of quantum computation that is different from the standard form: Measurement-Based Quantum Computation, more specifically a model known as One-Way Quantum Computation (1WQC), developed by R. Raussendorf and H. Briegel. To understand it, this work starts from the circuit model, the most common type of quantum computing, describing its main components. The 1WQC model is then introduced, with an attempt to didactically present its operation. A way of applying one- and two-qubit quantum logic gates is also shown, thus explaining how universal quantum computing is possible in this model. As an example, the translation of Deutsch’s algorithm to this model is also discussed. The result of this operation raises, however, open questions concerning the translation of Oracle-type algorithms, such as Deutsch’s, to the 1WQC 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.
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.
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.
In one-way quantum computation (1WQC) model, universal quantum computations are performed using measurements to designated qubits in a highly entangled state. The choices of bases for these measurements as well as the structure of the entanglements specify a quantum algorithm. As scalable and reliable quantum computers have not been implemented yet, quantum computation simulators are the only widely available tools to design and test quantum algorithms. However, simulating the quantum computations on a standard classical computer in most cases requires exponential memory and time. In this paper, a general direct simulator for 1WQC, called OWQS, is presented. Some techniques such as qubit elimination, pattern reordering and implicit simulation of actions are used to considerably reduce the time and memory needed for the simulations. Moreover, our simulator is adjusted to simulate the measurement patterns with a generalized flow without calculating the measurement probabilities which is called extended one-way quantum computation simulator (EOWQS). Experimental results validate the feasibility of the proposed simulators and that OWQS and EOWQS are faster as compared with the well-known quantum circuit simulators, i.e., QuIDDPro and libquantum for simulating 1WQC model.