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Boolean Functions, Quantum Gates, Hamilton Operators, Spin Systems and Computer Algebra

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 Publication date 2014
and research's language is English




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We describe the construction of quantum gates (unitary operators) from boolean functions and give a number of applications. Both non-reversible and reversible boolean functions are considered. The construction of the Hamilton operator for a quantum gate is also described with the Hamilton operator expressed as spin system. Computer algebra implementations are provided.



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We introduce two new packages, Nemo and Hecke, written in the Julia programming language for computer algebra and number theory. We demonstrate that high performance generic algorithms can be implemented in Julia, without the need to resort to a low-level C implementation. For specialised algorithms, we use Julias efficient native C interface to wrap existing C/C++ libraries such as Flint, Arb, Antic and Singular. We give examples of how to use Hecke and Nemo and discuss some algorithms that we have implemented to provide high performance basic arithmetic.
It has been proved that almost all $n$-bit Boolean functions have exact classical query complexity $n$. However, the situation seemed to be very different when we deal with exact quantum query complexity. In this paper, we prove that almost all $n$-bit Boolean functions can be computed by an exact quantum algorithm with less than $n$ queries. More exactly, we prove that ${AND}_n$ is the only $n$-bit Boolean function, up to isomorphism, that requires $n$ queries.
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