ExamplePaper_MeasurementBasedCS290_S

ExamplePaper_MeasurementBasedCS290_S - Review...

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Review: Measurement-Based Quantum Computation Mark Howard (Dated: June 16, 2007) We review the key results and unifying ideas behind measurement-based quantum computing (MQC). Specifically, we discuss a measurement-based scheme derived from an adaption of the fa- miliar teleportation circuit, called TQC. We then discuss a seemingly dissimilar scheme, QC C , which relies on single qubit measurements of a highly entangled initial state called a cluster (hence the subscript C ). For each scheme we will motivate their ability to perform universal quantum computa- tion by explicitly showing how the universal gate set can be implemented. These two measurement schemes can be shown to be equivalent by constructing a systematic mapping from one to the other. A third, unifying picture of MQC is provided by the valence-bond model (or matrix product model), which we mention briefly. This latter model has proved fruitful for answering fundamental questions about which classes of states lead to classically simulable evolution, and which classes of states provide full quantum computational power. The review concludes with a recent result on fault-tolerance and thresholds in MQC. PACS numbers: Introduction and Timeline The new paradigm of measurement-based quantum computing (MQC) is somewhat counterintuitive for those familiar with the standard circuit model. The standard model consists of preparing an initially unentangled state ( | 0 n ) and then applying a sequence of one and two qubit gates which amount to some desired unitary trans- formation. Measurement is usually performed only as a final step (note that we will consistently use the word measurement in the sense of a projective measurement only, and ignore the possibility of using POVMs). In con- trast, measurement (and appropriate use of the measure- ment outcome) is the computational primitive in MQC. In e ff ect one can steer forward the desired computation (unitary operation) by an appropriate choice of measure- ments. For TQC, a number of joint measurements on at least two qubits is required , in order to e ff ect arbitrary one and two qubit unitary transformations. QC C , on the other hand, requires only single qubit measurements to achieve the same result, but has a very specific requirement on the initial state of the system. A 1999 paper by Gottesman and Chuang [1] showed how an adaptation of the teleportation procedure allowed for universal quantum computation without explicit use of 2-qubit gates like the CNOT. This result still required the ability to perform single qubit operations. This mo- tivated a 2001 paper by Nielsen [2] in which it was shown that 4-qubit measurements alone were universal (i.e. the universal gate set could be accomplished). Later the same year this construction was refined to one requiring 3-qubit measurements by Leung and Nielsen [3] and Fen- ner and Zhang[4], and finally Leung [5] provided a con- struction showing that 2-qubit measurements were nec- essary and su ffi cient.
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