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Recently I've found the recursive Bernstein-Vazirani algorithm which is explained quite good in this paper CSE 599d - Quantum Computing The Recursive and Nonrecursive Bernstein-Vazirani Algorithm of Dave Bacon.

However, I still don't understand some points of the algorithm. In the paper it is stated that instead of trying to find some $s$ such that given a function $f$, we have $f(x) = x\cdot s \;\text{mod}\;2$ (this is the non-recursive B-V algorithm), we are interested to find some function $g:\{0,1\}^{n} \longrightarrow \{0,1\}$ on $s$, that is $g(s)$.

It is also stated that the oracle is now two $n$ strings, called respectively $x \in \{0,1\}^n$ and $y \in \{0,1\}^n$, so the function that must be queried first takes the input and then outputs $s_x \cdot y$, i.e. the function is $f:\{0,1\}^n \times \{0,1\}^n \longrightarrow \{0,1\}$ and is given by $f(x,y) = s_x \cdot y$.

The following part is where my doubts arise. When asking what exactly is $s_x$, Bacon says that the $s_x$ are "$2^n$ different bit strings, labeleed by $x \in \{0,1\}^n$, with the property that when computing $g(s_x)$ we require that these $2^n$ bits satisfy $g(s_x) = x\cdot s$ for some unknown $s$; thus the problem is to identify $g(s)$". (A)

But still, I don't understand the role of the different $s_x$ strings. Lastly, the paper states that when we are at the last point of the recursion, e.g. suppose we have the $k=2$ level problem, that is when we have $g(s_{x1}) = s\cdot x_1$, we finally get the hidden string $s$. (B)

So the question is: are we trying to find $g(s)$ (as in A), or the ultimate goal is still to find the hidden string $s$ (as in B)?

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So the intuition here is that the function $f$ encodes a tiny amount of information: a single $n$ bit string. Extracting this information is quantumly easy, but also classically easy. In order to make the problem harder, we might consider a strategy to encode much more information within a function. However, we can't just increase the size of the string, since we can't exactly query a $2^n$-bit string in polynomial time. We want to drastically increase the amount of information encoded in a clever way, without enlarging the queries by too much.

To make the problem of harder, we compose it with itself. The idea is instead of directly giving query access to a function $f_s(x) = x \cdot s$, we hide the function $f_s$ behind yet another secret string. We promise that $f_s = x \cdot s = g(s_x)$ for some secret string $s_x$. Here we are in the first level.

The ultimate goal is still to compute $s$. In order to compute $g(s)$ for any possible choice of $g$, you would need to know $s$. You can assume that $g(s)$ is a simple function to compute, but is tricky enough that you can't instantly guess the secret strings from any nice properties of $g$. In an extremely crude analogy, you can think of $g$ as a very lousy hash.

In order to compute $s$, we need to make queries to the function $f_s(x)$. We no longer have oracle access to $f_s(x)$, but we can compute $f_s(x)$ if we can find the string $s_x$, then compute $g(s_x)$.

The new oracle takes two strings $x, y$ and produces $s_x \cdot y$. If we fix $x$, this is the same as the basic problem. We have a single secret string $s_x$, and a single query string $y$. We can find $s_x$ in $n$ classical queries. Once we find $s_x$, we can make one "query" to $f_s(x) = s \cdot x = g(s_x)$. This "query" of course is done by just calculating $g(s_x)$.

Here is an illustrative example: suppose we want to query the string $10...0$, in order to find the first bit of $s$. To do so, we need to find the secret string corresponding to $10...0$. We make queries of the form $f(10...0, y)$ in order find all the bits of $s_x$, then compute $g(s_x)$ to find one bit of $s$.

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  • $\begingroup$ So, what could be a problem of this type? In the non recursive B-V algorithm, we have a function f(x) that does something and we implement its oracle in order to find the hidden string s. What would be an example of a problem of the recursive version? Will we have a function g or f? Could you give a formal example? In the paper Bacon says that a function g of this type would have to be |x|mod3, but I can't understand how to formulate the example in order to create the oracles $\endgroup$
    – aghin00
    Feb 25 at 14:37
  • $\begingroup$ You will need both $g$ and $f$. $g$ is known to the solver, so either we compute it ourselves or we have a second oracle for $g$. This is done so that $g$ does not show up in the runtime in any way. $f(x,y)$ defines the problem because it encodes the hidden strings, in the same way that $f_s(x)$ defines the non-recursive problem. Each unique $f(x, y)$ gives information about the strings $s, s_x$ through its output. In order to create an example of the problem, we: 1. choose a secret $s$, 2. choose string $s_x$ for every x, s.t. $g(s_x) = s \cdot x$, 3. define $f(x, y) = s_x \cdot y$ $\endgroup$
    – xzkxyz
    Feb 25 at 16:48
  • $\begingroup$ Thank you for your answer, but that's not what i meant. What I want is to create an example from scratch; suppose I choose the string s=11 as the solutio: what function g and f do i define, just for the purpose of an example? To claryfi, the non recursive version could implement the function f(x1x2) = x1, given x1,x2 $\in \{0,1\}^n$; but what function can I create for the recursive version (both g and f), that's easy enough to demonstrate the algorithm? $\endgroup$
    – aghin00
    Feb 25 at 17:41
  • $\begingroup$ Follow the previous comment. Lets choose $g(x) = |x| mod 3$ for the example. To define $f(x, y)$ compute all the bits $s \cdot x \forall x \in \{0, 1\}$. For each bit, choose a string $s_x$ such that $|s_x| mod 3 = s \cdot x$. From this list of $s_x$, you've defined $f(x, y)$. $f(x, y)$ is essentially defined by the mapping $x \to s_x$, and you are free to choose any list of strings $s_x$, so long as they meet the condition. $\endgroup$
    – xzkxyz
    Feb 27 at 20:21

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