Quantum Arithmetic on Galois Fields

Details Quantum Arithmetic on Galois Fields Quantum Arithmetic on Galois Fields

2003-01-29

arxiv.org/abs/quant-ph/0301163v1

Physics

Quantum Physics

29 pages, 12 figures. This is the most recent version, dated 11 April 02. Paper only posted now in reply to quant-ph/0301141,

Scientific paper

In this paper we discuss the problem of performing elementary finite field arithmetic on a quantum computer. Of particular interest, is the controlled-multiplication operation, which is the only group-specific operation in Shor's algorithms for factoring and solving the Discrete Log Problem. We describe how to build quantum circuits for performing this operation on the generic Galois fields GF($p^k$), as well as the boundary cases GF($p$) and GF($2^k$). We give the detailed size, width and depth complexity of such circuits, which ultimately will allow us to obtain detailed upper bounds on the amount of quantum resources needed to solve instances of the DLP on such fields.

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