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2

You start with a polarisation filter. This does nothing to the path of your photon and, effectively, measures the polarisation of the photon, meaning that you prepare the "second" qubit in the fixed state determined by what polarisation the filter is detecting. So, at this point, you have $$|0\rangle|-\rangle$$ Then, you input to a beamsplitter. ...

1

Not exactly. You are correct, there is a single photon; qubit 1 is its path, qubit 2 is its polarization. The waveplates implement an oracle (one from 4 possible). You are wrong about the beam splitters; beam splitters do not affect polarization, so they act on the qubit 1 only as Hadamard gates. The $|-\rangle$ state of qubit 2 is created by the ...

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This corresponds to the problem of counting the number of 1's in some $n$-bit input string. It is well known that for exact counting there can be no significant speedup. This follows from the $\Omega(n)$ lower bound on the quantum query complexity of parity (see here). For approximate counting you can get a quantum speedup, as is described here.

0

We can construct quantum verified delay functions ( QVDF ) and delay authentication ( QDA ) circuits using single quibit quantum circuits. Like quantum randomness generators ( QRN ), these delay functions can be used for auction and lottery systems. We can possibly construct quantum ring structures (QRS) as building blocks for quibit storage, sensing ...

4

Let's say that the first bit of $s$ $s_0=1$ (the argument will be exactly the same for any bit, just for convenience). You can split the space of inputs $x \in \{0,1\}^n$ in two halves: one half where $x_0 = 0$ and the other half where $x_0 = 1$. For each bitstring $x$ from the first half you'll have a bitstring $\tilde{x}$ from the second half which will ...

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There aren't many examples! The main reason for advantages in quantum computers is the ability to constructively combine amplitudes - if you've only got 1 qubit, there aren't any amplitudes to combine! The best use case I can think of is randomness. A quantum computer (implemented with arbitrary error) could theoretically be a near perfect source of entropy,...

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