On quantum one-way permutations

Authors:
Elham Kashefi;Harumichi Nishimura;Vlatko Vedral
Affiliations:
Centre for Quantum Computation, Clarendon Laboratory, University of Oxford, Oxford, England and The Blacket Laboratory, Imperial College, London, England;Centre for Quantum Computation, Clarendon Laboratory, University of Oxford, Oxford, England and CREST, Japan Science and Technology;The Blacket Laboratory, Imperial College, London, England
Venue:
Quantum Information & Computation
Year:
2002

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Abstract

We discuss the question of the existence of quantum one-way permutations. First, we consider the question: if a state is difficult to prepare, is the reflection operator about that state difficult to construct? By revisiting Grover's algorithm, we present the relationship between this question and the existence of quantum one-way permutations. Next, we prove the equivalence between inverting a permutation and that of constructing polynomial size quantum networks for reflection operators about a class of quantum states. We will consider both the worst case and the average case complexity scenarios for this problem. Moreover, we compare our method to Grover's algorithm and discuss possible applications of our results.