Time required to compute the determinant of a $50$ $X$ $50$ matrix

Question

So I am asked to solve this question which says the following...

If I have a quantum computer which can make $1000^{10}$ numeric operations (addition and multiplication) per second, I am asked to estimate the time it takes a quantum computer to evaluate the determinant of a matrix of size $(50,50)$ by iterated cofactor expansion.

Some estimation steps I am apparently supposed to aim for here is, the number of operations is greater than $50!$, and $50!>10^{50}$ where $10^{50}=\sqrt{Googol}$.

I am also told that

$1$ $ns$= $1$ $nanosecond$ = $10^{-9}$

$1$ $\mu s$= $1$ $microsecond$ = $10^{-6}$

$1$ $ms$= $1$ $millisecond$ = $10^{-3}$

$1$ $Day$ $< 10^5$ $seconds$

$1$ $Year$ $< 10^8$ $seconds$

$Age$ $of$ $the$ $universe$: $<15 \cdot 10^9 < 10^{11}$ $years$

My approach was that I am already given the rate of calculations. It's $1000^{10}$ calculations per second. So I just need the number of calculations required. Then I can just multiply the two numbers to get the amount of time required.

The problem is though I can't figure out the number of calculations required. My assumption was to just use $50!$ which is close to $10^{50}$ or $\sqrt{Googol}$ but I don't think that's right since I have to account for all the multiplication of the cofactors of each minor and then all the additions.

How would I go about finding the number of calculations required for a $50$ X $50$ matrix?

Answer 1

In fact you are right. To calculate the determinant of a $50\times 50$ matrix directly, would require more than $50!$ operations, and $50! \approx 3\times 10^{64}$. Thus, using the computer that performs $1000^{10} = 10^{30}$ operations per second, it would take approximately $3\times 10^{34}$ seconds $\approx 10^{27}$ years.

Thus, it would require more than $10^{17}$ times the age of the universe! For more, see page 269, Algebra: A Modern Introduction by David Poole. Hope it helps.

Answer 2

So to calculate the determinant of an $n \times n$ matrix using cofactor expansion you expand along a row or column, lets say a row. So for each n terms you multiply by the determinant of a $(n-1)\times (n-1)$ matrix. Then you add or subtract all these terms. So an $n\times n$ matrix requires a number of calculations equal to $n\times f(n-1)+n+(n-1)$ where $f(n)$ is the number of calculations required for an $n\times n$ matrix. The $n\times f(n-1)+n$ comes from multiplying n terms by the determinant of a $(n-1)\times (n-1)$ matrix, then n-1 comes from the adding or subtracting of their terms. So $f(n)= n\cdot f(n-1)+ 2n -1$ now we have the number of calculations in terms of a recursive function, but this can be simplified further with the upper incomplete gamma function and be written as $f(n) = e\Gamma(n+1,1)-2$. This works out to be approximately $8\times 10^{64}$ calculations which would take $2.5\cdot10^{27}$ years. This in fact is longer then the current age of the universe.