### M Theory Lesson 116

In quantum computation [1] the Fourier transform for $2^{n}$ basis states on $n$ qubits is implemented with Hadamard gates and unitary gates $G_{k}$ of the form

$1$ $0$

$0$ $\textrm{exp} \frac{2 \pi i}{2^{k}}$

which add a phase factor to $| 1 >$. A ternary analogue of such gates would be a $3 \times 3$ diagonal matrix with entries $1$, $\textrm{exp} \frac{2 \pi i}{3^{k}}$ and $\textrm{exp} \frac{4 \pi i}{3^{k}}$, responsible for adding phases to each of the three basis states. For example, for two ternary objects ($n = 2$) the central phase factor in $G_{2}$ is $\textrm{exp} \frac{2 \pi i}{9}$. Any number from 1 to 9 is expressable in base 3 as a combination of 1, 3 and 9, so the product representation of the qubit transform has a ternary analogue. This ability to switch from an additive expression to a product expansion in superpositions of qutrits is quite reminiscent of Euler's relation for zeta functions. It is no surprise then that the quantum qubit transform is used in the algorithm for integer factorization.

[1] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge (2000)

$1$ $0$

$0$ $\textrm{exp} \frac{2 \pi i}{2^{k}}$

which add a phase factor to $| 1 >$. A ternary analogue of such gates would be a $3 \times 3$ diagonal matrix with entries $1$, $\textrm{exp} \frac{2 \pi i}{3^{k}}$ and $\textrm{exp} \frac{4 \pi i}{3^{k}}$, responsible for adding phases to each of the three basis states. For example, for two ternary objects ($n = 2$) the central phase factor in $G_{2}$ is $\textrm{exp} \frac{2 \pi i}{9}$. Any number from 1 to 9 is expressable in base 3 as a combination of 1, 3 and 9, so the product representation of the qubit transform has a ternary analogue. This ability to switch from an additive expression to a product expansion in superpositions of qutrits is quite reminiscent of Euler's relation for zeta functions. It is no surprise then that the quantum qubit transform is used in the algorithm for integer factorization.

[1] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge (2000)

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