Representation of Quantum Circuits with Clifford and $π/8$ Gates

Details Representation of Quantum Circuits with Clifford and $π/8$ Gates Representation of Quantum Circuits with Clifford and $π/8$ Gates

2008-06-24

arxiv.org/abs/0806.3834v1

Physics

Quantum Physics

17 pages, 3 figures

Scientific paper

In this paper, we introduce the notion of a normal form of one qubit quantum circuits over the basis $\{H, P, T\}$, where $H$, $P$ and $T$ denote the Hadamard, Phase and $\pi/8$ gates, respectively. This basis is known as the {\it standard set} and its universality has been shown by Boykin et al. [FOCS '99]. Our normal form has several nice properties: (i) Every circuit over this basis can easily be transformed into a normal form, and (ii) Every two normal form circuits compute same unitary matrix if and only if both circuits are identical. We also show that the number of unitary operations that can be represented by a circuit over this basis that contains at most $n$ $T$-gates is exactly $192 \cdot (3 \cdot 2^n - 2)$.

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