CleveQuantumComplexityTheory - arXiv:quant-ph/9906111v1 28...

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Unformatted text preview: arXiv:quant-ph/9906111v1 28 Jun 1999 An Introduction to Quantum Complexity Theory Richard Cleve University of Calgary Abstract We give a basic overview of computational complexity, query com- plexity, and communication complexity, with quantum information in- corporated into each of these scenarios. The aim is to provide simple but clear definitions, and to highlight the interplay between the three scenarios and currently-known quantum algorithms. Complexity theory is concerned with the inherent cost required to solve in- formation processing problems, where the cost is measured in terms of various well-defined resources. In this context, a problem can usually be thought of as a function whose input is a problem instance and whose corresponding output is the solution to it. Sometimes the solution is not unique, in which case the problem can be thought of as a relation, rather than a function. Resources are usually measured in terms of: some designated elementary operations, mem- ory usage, or communication. We consider three specific complexity scenarios, which illustrate different advantages of working with quantum information: 1. Computational complexity 2. Query complexity 3. Communication complexity . Despite the differences between these models, there are some intimate rela- tionships among them. The usefulness of many currently-known quantum al- gorithms is ultimately best expressed in the computational complexity model; however, virtually all of these algorithms evolved from algorithms in the query complexity model. The query complexity model is a natural setting for dis- covering interesting quantum algorithms, which frequently have interesting counterparts in the computational complexity model. Quantum algorithms in the query complexity model can also be transformed into protocols in the * Department of Computer Science, University of Calgary, Calgary, Alberta, Canada T2N 1N4. Email: . 1 communication complexity model that use quantum information (and some- times these are more efficient than any classical protocol can be). Also, this latter relationship, taken in its contrapositive form, can be used to prove that some problems are inherently difficult in the query complexity model. 1 Computational complexity In the computational complexity scenario, an input is encoded as a binary string (say) and supplied to an algorithm, which must compute an output string corresponding to the input. For example, in the case of the factoring problem, for input 100011 (representing 35 in binary), the valid outputs might be 000101 or 000111 (representing the factors of 35). The algorithm must produce the required output by a series of local operations. By this, we do not necessarily mean local in space, but, rather, that each operation involves a small portion of the data. In other words, a local operation is a transformation that is confined to a small number of bits or qubits (such as two or three)....
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CleveQuantumComplexityTheory - arXiv:quant-ph/9906111v1 28...

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