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A promise BQP-complete string rewriting problem

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Janzing,  D
Max Planck Institute for Biological Cybernetics, Max Planck Society;
Department Empirical Inference, Max Planck Institute for Biological Cybernetics, Max Planck Society;

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Citation

Janzing, D., & Wocjan, P. (2010). A promise BQP-complete string rewriting problem. Quantum Information and Computation, 10(3), 234-257.


Cite as: http://hdl.handle.net/21.11116/0000-0002-7726-E
Abstract
We consider the following combinatorial problem. We are given three strings s, t, and t'of length L over some fixed finite alphabet and an integer m that is polylogarithmic inL. We have a symmetric relation on substrings of constant length that specifies whichsubstrings are allowed to be replaced with each other. Let Δ(n) denote the differencebetween the numbers of possibilities to obtain t from s, and t' from s after n ∈ Nreplacements. The problem is to determine the sign of Δ(m). As promises we have agap condition and a growth condition. The former states that |Δ(m)| ≥ εcm where ε isinverse polylogarithmic in L and c > 0 is a constant. The latter is given by Δ(n) ≤ cnfor all n. We show that this problem is PromiseBQP-complete, i.e., it represents theclass of problems that can be solved efficiently on a quantum computer.