Continuous quantum computation: Difference between revisions

Revision as of 19:26, 8 March 2007

Revision as of 15:38, 7 July 2007

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'''Continuous Quantumquantum Computationcomputation''' Two major motivations for studying continuous quantum computation are:

* Many scientific problems have continuous mathematical formulations. Examples of such formulations are

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* In their standard monograph Nielsen and Chuang state "Of particular interest is a decisive answer to the problem whether quantum computers are more powerful than classical computers." To answer this question one must know the classical and quantum computational complexities

We discuss the second motivation. By computational complexity (complexity for brevity) is meant the '''minimal''' computational resources needed to solve a problem. Two of the most important resources for quantum computing are qubits[[qubit]]s and queries. Classical complexity has been extensively studied in [[Information-Based Complexity| Information-Based Complexity]]. The classical complexity of many continuous problems is known. Therefore, when the quantum complexity of these problems is obtained, the question as to whether quantum computers are more powerful than classical can be answered. Furthermore, it can be established how much more powerful. In contrast, the complexity of discrete problems is typically unknown; one has to settle for the complexity hierarchy. For example, the classical complexity of integrer factorization is unknown.

==An Exampleexample: Pathpath Integrationintegration==

Path integration has numerous applications including quantum mechanics, quantum chemistry, statistical mechanics, and computational finance. We want to compute an approximation to within error at most <math>\epsilonscripstyle\varepsilon</math> with probability, say, at least 3/4. Then the following was [http://arXiv.org/quant-ph/0109113 shown] by Traub and Woźniakowski:

* A quantum computer enjoys exponential speedup over the classical worst case and quadratic speedup over the classical randomized case.

* The query complexity is of order <math>\epsilonscriptstyle\varepsilon^{-1}</math>.

* The qubit complexity is of order <math>\epsilonscriptstyle\varepsilon^{-2}\log\epsilonvarepsilon^{-1}</math>.

Thus the qubit complexity of path integration is a second degree polynomial in <math>\epsilonscriptstyle\varepsilon^{-1}</math>. That seems pretty good but we probably won't have enough qubits for a long time to do new science especially with error correction. Since this is a complexity result we can't do better by inventing a clever new algorithm. But perhaps we can do better by slightly modifying the queries.

In the standard model of quantum computation the probabilistic nature of quantum computation enters only through measurement; the queries are deterministic. In analogy with classical Monte Carlo Woźniakowski introduced the idea of a quantum setting with [http://arXiv.org/quant-ph/0601196 randomized queries]. He showed that in this setting the qubit complexity is of order <math>\scripstyle \log\epsilonvarepsilon^{-1}</math>, thus achieving an exponential improvement over the qubit complexity in the standard quantum computing setting.

==Applications==