Universality, Invariance, and the Foundations of Computational Complexity in the Light of the Quantum Computer
Computational complexity theory is a branch of computer science dedicated to classifying computational problems in terms of their difficulty. While computability theory tells us what we can compute in principle, complexity theory informs us regarding what is feasible. In this chapter I argue that the science of quantum computing illuminates complexity theory by emphasising that its fundamental concepts are not model-independent, but that this does not, as some suggest, force us to radically revise the foundations of the theory. For model-independence never has been essential to those foundations. The fundamental aim of complexity theory is to describe what is achievable in practice under various models of computation for our various practical purposes. Reflecting on quantum computing illuminates complexity theory by reminding us of this, too often under-emphasised, fact.
This project was supported financially by the Foundational Questions Institute (FQXi) and by the Rotman Institute of Philosophy. I am grateful to Scott Aaronson, Walter Dean, William Demopoulos, Armond Duwell, Laura Felline, Sona Ghosh, Amit Hagar, Gregory Lavers, and Markus Müller for their comments on a previous draft of this chapter. This chapter also benefited from the comments and questions of audience members at the Montreal Inter-University Workshop on the History and Philosophy of Mathematics, and at the Perimeter Institute for Theoretical Physics.
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