The Quantum Challenge to Structural Complexity

This is a non-technical survey paper of recent quantum-mechanical discoveries that challenge generally accepted complexity-theoretic versions of the Church{Turing thesis. In particular, building on pi-onering work of David Deutsch and Richard Jozsa, we construct an oracle relative to which there exists a set that can be recognized in Quantum Polynomial Time (QP), yet any Turing machine that recognizes it would require exponential time even if allowed to be probabilistic, provided that errors are not tolerated. In particular, QP 6 6 ZPP relative to this oracle. Furthermore, there are cryptographic tasks that are demonstrably impossible to implement with unlimited computing power probabilistic interactive Turing machines , yet they can be implemented even in practice by quantum mechanical apparatus. 1 Deutsch's Quantum Computer In a bold paper published in the Proceedings of the Royal Society, David Deutsch put forth in 1985 the quantum computer 7] (see also 8]). Even though this may come one day to be recognized as one of the most important papers of that decade in theoretical computer science, it was written primarily for the beneet of physicists, in a language somewhat foreign to computer scientists. It is not our purpose here to describe the principles of quantum computing; rather, we wish to explore some of its implications on structural complexity theory. Nevertheless, a short review of quantum computing is in order. What makes the quantum computer so diierent from a Turing machine is the possibility it ooers for Supported in part by an nserc postgraduate fellowship. y Supported in part by the E. W. R. Steacie Memorial Fellowship (nserc). massive parallelism within a single piece of hardware. Let f be a computable function. If you have to compute it on two diierent inputs x and y, you may have to compute f(x) and then start afresh to compute f(y), thus requiring about twice as long as if you had needed only one of these values. The quantum computer allows you to prepare an input that encodes both x and y in a so-called quantum superposition. If you run your program that computes f on that input, it produces the quantum superposition of f(x) and f(y) in the time needed to compute the function once only. Even better, you can prepare in linear time the quantum superposition of an exponential number of inputs, after which a single call to the program computes the quantum superposition of all the corresponding …