An Application of the Deutsch-Jozsa Algorithm to Formal Languages and the Word Problem in Groups

We adapt the Deutsch-Jozsa algorithm to the context of formal language theory. Specifically, we use the algorithm to distinguish between trivial and nontrivial words in groups given by finite presentations, under the promise that a word is of a certain type. This is done by extending the original algorithm to functions of arbitrary length binary output, with the introduction of a more general concept of parity. We provide examples in which properties of the algorithm allow to reduce the number of oracle queries with respect to the deterministic classical case. This has some consequences for the word problem in groups with a particular kind of presentation. 1 The Deutsch-Jozsa Algorithm and Formal Languages We apply a direct generalization of the Deutsch-Jozsa algorithm to the context of formal language theory. More particularly, we extend the algorithm to distinguish between trivial and nontrivial words in groups given by finite presentations, under the promise that a word is of a certain type. For background information, we refer the reader to [1] and [2]. The Deutsch-Jozsa algorithm concerns maps f : {0, 1}n −→ {0, 1}, which we may think of as words of length 2 in a two-letter alphabet. Instead, let us consider words of length 2 in a four-letter alphabet A = {a, b, c, d}. We identify the letters with binary strings of length 2: a ↔ 00, b ↔ 01, c ↔ 10 and d ↔ 11. In this way we can look at words of length 2 as in one-to-one correspondence with maps f : {0, 1}n −→ {00, 01, 10, 11}. First, consider the case n = 1, that is when the words have length 2. As with the standard formulation of the Deutsch-Jozsa algorithm, the promise will be vacuous in this case.We use the quantum circuit represented below. The circuit essentially implements the Deutsch’s algorithm, but with input |11〉 rather than |1〉: Corresponding author. Y. Kawano and M. Mosca (Eds.): TQC 2008, LNCS 5106, pp. 57–69, 2008. c © Springer-Verlag Berlin Heidelberg 2008