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In the 10th Anniversary Edition of Nielsen and Chuang Quantum Computation and Quantum Information textbook, Chapter 6.7 talks about Black Box algorithm limits.

It is given:

$f:\{0,1\}^n \rightarrow \{0,1\}$

$F:\{X_0,X_1,X_2,....,X_{N-1} \}\rightarrow \{0,1\}$ $\text{such that} \space F \space \text{is a boolean function,} \space X_k=f(k) \space and \space N=2^n-1$

It is then mentioned that:

We say that a polynomial $p: R^N\rightarrow R$ represents $F$ if $p(X)=F(X)$ for all $X \in \{0,1\}^N$ (where $R$ denotes the real numbers). Such a polynomial $p$ always exists, since we can explicitly construct a suitable candidate:
$$p(X)=\sum_{Y\in \{0,1\}^N} F(Y)\prod_{k=0}^{N-1}[1-(Y_k-X_k)^2]$$

Can someone explain this formula to me and whether the construction is a result of rigorous steps or by intuition? Will also be good if there are useful materials related to this for me to read.

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Plug in an arbitrary $X$ into the formula.

Look at each summand for each particular $Y \in \{0,1\}^N$

If $Y \neq X$, then there must be at least one index $i$ such that $X_k \neq Y_k$. But both $Y_k$ and $X_k$ are only either $0$ or $1$. So if they are not equal, then the difference must be either $+1$ or $-1$. Square that and you get $1$ if they are different. Then one of the terms in the product will be $1-1=0$.

So if $Y \neq X$, that particular summand is $0$.

The only summand that remains is when $Y=X$. In that case each of the $1-(Y_k-X_k)^2$ are equal to $1$. So the product of all of those still gives $1$ and the result for that summand is $F(Y)=F(X)$.

Add together the summands for all the $Y$, and you get the only nonzero term, $F(X)$. So $p(X)=F(X)$ for all $X \in \{0,1\}^N$.

As desired, $p$ represents $F$. However, there may be simpler $p$ that still do the same. This is just one of them. The one of smallest degree gives the notion of degree of a boolean function which is related to sensitivity and block sensitivity.

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  • $\begingroup$ Thanks AHusain. I'll have to progress further to understand your last paragraph. $\endgroup$ – C.C. Jul 14 at 4:59

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