Even if the matrix only required one byte per entry (e.g., uint8), a matrix that big would require a processor able to handle a 220 bit address space. The largest address space available with commercial processors is 48 bit. You do not have any hope of using a matrix that big.
Is it possible that you want something that is (10^6)X(10^6) instead of (10^60)X(10^6) ?? At one byte per entry that would require less than a terabyte of memory, a 40 bit address space. It is possible to get systems with that much memory... but it is likely to be quite expensive.
We cannot really discuss reasonable solutions without more knowledge of what kind of calculations you are trying to do. Sometimes it is quite reasonable to "block" the calculations (apply the calculations to subsets of the data, and then work on the cross-block terms.)
By the way, how long were you anticipating that a single run of your algorithm should take?
Have you considered using parallel processing?
