The amount of ROM needed to implement a 4 bit multiplier is

To implement a 4-bit multiplier using a ROM:

1. Inputs: A 4-bit multiplier multiplies two 4-bit numbers (say $A$ and $B$). Total number of input bits is $4 + 4 = 8$ bits. These 8 bits serve as the address lines of the ROM.

2. Number of memory locations (words): With $8$ address lines, the ROM has $2^8 = 256$ words.

3. Outputs: The product of two 4-bit numbers has a maximum value of $15 \times 15 = 225$, which requires $\lceil \log_2(225 + 1) \rceil = 8$ bits. So each word must store an 8-bit output.

4. Total ROM capacity:

Thus, the correct option is D.