Problem adding two numbers in a quantum computer with Drapper's algorithm

Question

I would like to add numbers. In this simple example a = 1 and b = 1. The circuit I created looks like this:

We have two registers a and b. Both have a size of 3 qubits. I'm using Big Endian notation, so for example the bit array [1, 1, 0] = 6. I start with encoding both numbers by adding a X-Gate to qubit 2 and 5. After that I follow Drapper's algorithm for integer addition: QFT on a, Evolve a by b. QFT-Inverted on a.

However the result is wrong that I'm getting at the end: The state 17 (= [0, 1, 0, 0, 0, 1]) has probability 1. I would like to see state 2 with probability 1 obviously. What am I doing wrong?

Answer 1

You are misunderstanding your output. All quantum calculations must be reversible. Having two 3-bit inputs yield a six-bit output is not reversible.

The result you're actually seeing is the three-bit result $[0, 1, 0]$ corresponding to the sum and the three-bit result $[0, 0, 1]$ which is the second argument. This is reversible, and you could rederive your original $a$ and $b$.

Answer 2

You are just interpreting your output the wrong way. The result of the addition will only be in register a as a 3-qubit result. You are evolving b over into a. If you only look at your top 3 qubits, they are in the state [0, 1, 0] which corresponds to decimal 2, which is the correct output.

You can verify this by modifying the input values in a and b in [0, 3], but please account for the carry (in essence, since you reserved 3 qubits for a, in effect you only have 2 qubits available to specify values 0 to 3 inclusive).

For reference, I have the corresponding code here.