Background
I'm reviewing basis encoding in this paper. In their example, they state:
$(-.07, 0.1, 0.2)^T \in \mathbb{R}^3$ is encoded as $|x\rangle = |11011\;01011\;00011\rangle$
Unpacking this, it seems clear that the first bit representing each value is the sign bit and this appears to be corroborated in the text:
The sign of the real number is encoded by an additional leading binary number, e.g. '1' for '−' and '0' for '+'.
Removing the sign bit, we thus have:
$0.7 \mapsto 1011$
$0.1 \mapsto 1011$
$0.2 \mapsto 0011$
Given we have 4 bits, that gives us 16 possible states. Alright. Assuming we're representing values in the interval $(-1.0, 1.0)$, $1011$ gives us $11$ in decimal and $11/16 = 0.6875$, where $\lceil 0.6875 \rceil = 0.7$. By similar logic, $0011$ gives us $3$ in decimal and $3/16 = 0.1875$, where $\lceil 0.1875 \rceil = 0.2$. This all appears to make sense.
But then we have:
$0.1 \mapsto 1011$
Okay, this doesn't make sense. $1011$ is the binary encoding we had for $0.7$ and, by the prior logic, I'd expect:
$0.1 \mapsto 0001$,
which would give us $1$ in decimal and $1/16 = 0.0625$, where $\lceil 0.0625 \rceil = 0.1$.
Question
What am I missing here? No alternative logic appears to be clearly correct. Maybe there's an error in the text but it's unclear to me where.
Can someone please explain the exact manipulations being done to go from the data $(-0.7, 0.1, 0.2)$ to the state $|x\rangle = |11011\;01011\;00011\rangle$? Given this isn’t a pre-print, I’m assuming I’m the one who’s made an error or, worst case, this represents a fundamental conceptual gap in my knowledge of very basic QIS concepts.
