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Quantum Algorithms and Complexity for Continuous Problems

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Encyclopedia of Complexity and Systems Science

Definition of the Subject

Most continuous mathematical formulations arising in science and engineeringcan only be solved numerically and therefore approximately. We shall alwaysassume that we are dealing with a numerical approximation to the solution.

There are two major motivations for studying quantum algorithms andcomplexity for continuous problems.

  1. 1.

    Are quantum computers more powerful than classical computers forimportant scientific problems? How much more powerful? This would answerthe question posed by Nielsen and Chuang (p. 47 in [48]).

  2. 2.

    Many important scientific and engineering problems have continuousformulations. These problems occur in fields such as physics, chemistry,engineering and finance.The continuous formulations include path integration, partialdifferential equations (in particular, the Schrödinger equation) andcontinuous optimization.

To answer the first question we must know the classical computationalcomplexity (for brevity, complexity)...

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Abbreviations

Black box model :

This model assumes we can collect knowledge about aninput f through queries without knowing how the answer to the query is computed.A synonym for black box is oracle.

Classical computer:

A computer which does not use the principles of quantum computing to carry out itscomputations.

Computational complexity :

In this article, complexity for brevity.The minimal cost of solving a problem by an algorithm. Some authors use theword complexity when cost would be preferable. An upper bound on thecomplexity is given by the cost of an algorithm. A lower bound is given by a theoremwhich states there cannot be an algorithm which does better.

Continuous problem :

A problem involving real or complex functions of realor complex variables. Examples of continuous problem are integrals, pathintegrals, and partial differential equations.

Cost of an algorithm :

The price of executing an algorithm. The cost depends on the model of computation.

Discrete problem:

A problem whose inputs are from a countable set. Examples of discrete problems are integer factorization, traveling salesman and satisfiability.

e-Approximation:

Most real-world continuous problems can only besolved numerically and therefore approximately, that is to within anerror threshold e. The definition of e-approximation depends onthe setting. See worst-case setting, randomized setting, quantum setting.

Information-based complexity :

The discipline that studies algorithms and complexity of continuous problems.

Model of computation :

The rules stating what is permitted in a computation and how much it costs. The modelof computation is an abstraction of a physical computer. Examples of modelsare Turing machines, real number model, quantum circuit model.

Optimal algorithm :

An algorithm whose cost equals the complexity of the problem.

Promise:

A statement of what is known about a problem a priori before anyqueries are made. An example in quantum computation is the promisethat an unknown 1-bit function is constant or balanced. Ininformation-based complexity a promise is also called global information.

Quantum computing speedup :

The amount by which a quantum computer can solve a problem faster than a classicalcomputer. To compute the speedup one must know the classical complexity and it isdesirable to also know the quantum complexity. Grover proved quadratic speedupfor search in an unstructured database. Its only conjectured that Shor's algorithm provides exponential speedup for integer factorization since the classical complexity is unknown.

Query:

One obtains knowledge about a particular input through queries. For example, if the problem is numerical approximation of \( { \int_0^1 f(x)\mskip2mu\mathrm{d} x } \) a query might be the evaluation of fat a point. In information-based complexity the same concept is called aninformation operation.

Quantum setting:

There are a number of quantum settings. An example is a guarantee of error at most e with probability greater than 1/2.

Qubit complexity :

The minimal number of qubits to solve a problem.

Query complexity :

The minimal number of queries required to solve the problem.

Randomized setting:

In this setting the expected error with respect tothe probability measure generating the random variables is at most e. Thecomputation is randomized. An important example of a randomized algorithm isthe Monte Carlo method.

Worst-case setting:

In this setting an error of at most e is guaranteed for all inputs satisfying the promise. The computation is deterministic.

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Acknowledgments

We are grateful to Erich Novak, University of Jena, and Henryk Wozniakowski,Columbia University and University of Warsaw, for their very helpful comments.We thank Jason Petras, Columbia University, for checking the complexityestimates appearing in the tables.

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Authors and Affiliations

  1. Department of Computer Science, Columbia University, New York, USA

    Anargyros Papageorgiou & Joseph F. Traub

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  1. Anargyros Papageorgiou
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Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Papageorgiou, A., Traub, J.F. (2009). Quantum Algorithms and Complexity for Continuous Problems. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_424

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