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Computação quântica baseada em medidas

Thiago de Souza Ferreira

Published Jan 1, 2020• 0 citations
OpenAlex

Abstract

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

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•Jul 1, 2015•1 citation
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
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 computation is a novel way of information processing that allows, for certain classes of problems, exponential speedups over classical computation. Various models of quantum computation exist, such as the adiabatic, circuit, and measurement-based models. They have been proven equivalent in their computational power, but operate very differently. As such, they may be suitable for realization in different physical systems, and also offer different perspectives on open questions such as the precise origin of the quantum speedup. Here, we give an introduction to the one-way quantum computer, a scheme of measurement-based quantum computation (MBQC). In this model, the computation is driven by local measurements on a carefully chosen, highly entangled state. We discuss various aspects of this computational scheme, such as the role of entanglement and quantum correlations. We also give examples for ground states of simple Hamiltonians that enable universal quantum computation by local measurements.
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.
The term quantum parallelism is commonly used to refer to a property of quantum computations where an algorithm can act simultaneously on a superposition of states. However, this is not the only aspect of parallelism in quantum computing. Analogously to the classical computing model, every algorithm consists of elementary quantum operations and the application of them could be parallelised itself. This kind of parallelism is explored in this thesis in the one way quantum computing (1WQC) and the quantum circuit model. In the quantum circuit model we explore arithmetic circuits and circuit complexity theory. Two new arithmetic circuits for quantum computers are introduced in this work: an adder and a multiply-adder. The latter is especially interesting because its depth (i.e. the number of parallel steps required to finish the computation) is smaller than for any known classical circuit when applied sequentially. From the complexity theoretical perspective we concentrate on the classes QAC0 and QAC0[2], the quantum counterparts of AC0 and AC0[2]. The class AC0 comprises of constant depth circuits with unbounded fan-in AND and OR gates and AC0[2] is obtained when unbounded fan-in parity gates are added to AC0 circuits. We prove that QAC0 circuits with two layers of multi-qubit gates cannot compute parity exactly. This is a step towards proving QAC0 6= QAC0[2], a relation known to hold for AC0 and AC0[2]. In 1WQC, computation is done through measurements on an entangled state called the resource state. Two well known parallelisation methods exist in this model: signal shifting and finding the maximally delayed general flow. The first one uses the measurement calculus formalism to rewrite the dependencies of an existing computation, whereas the second technique exploits the geometry of the resource state to find the optimal ordering of measurements. We prove that the aforementioned methods result in same depth computations when the input and output sizes are equal. Through showing this equivalence we reveal new properties of 1WQC computations and design a new algorithm for the above mentioned parallelisations.