How to implement non-unitary operator in qiskit?

Question

I am trying to make a operator for S-Box for AES. The matrix is non-unitary. When I use Operator command for declaring it as a function. It gives error, the matrix is non-unitary. Thus, the qiskit does not allow non-unitary operators to be implemented. Is there any way to implement the non-unitary operator in qiskit or cirq etc?

Answer 1

There are several ways to apply a classical function in quantum computing. Here, I think you're interested in one of the following two.

The first one is the standard oracle: $$U_f|x,y\rangle=|x,y\oplus f(x)\rangle$$ where $\oplus$ is the bitwise-XOR. This is the most common, as it allows to apply quantum functions even if they're not bijective.

However, there's another type of oracle, called erasing oracle or minimal oracle: $$V_f|x\rangle=|f(x)\rangle$$ Because of the requirement of unitarity, in order to use this oracle the function must be injective (though bijective is easier).

The thing is, the AES S-BOX is bijective, since it is essentially a lookup table with unique values. Since it takes an 8-bit number and returns another 8-bit number, you can build a $2^8\times2^8$ matrix where the first column is all $0$ except in $S(0)=$0x63$=99$, the second column is all zeros except in $S(1)=$0x7c$=124$, etc...

You can then apply this matrix, which is unitary since it's a permutation matrix, which corresponds in this case to an erasing oracle.

The standard oracle is a bit different, but similar in the process. You take a $2^{16}\times2^{16}$ matrix, where the $256x+y$ column is all zeros except in $256x+(y\oplus S(x)$, for all $x$ and $y$ being between $0$ and $255$ included. Once again, it is unitary since it's a permutation matrix.

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As a side note, you probably don't want to implement the actual AES in Qiskit. AES' state being 128-bit long, you'd need a $2^{128}$ statevector to represent it with Qiskit, which clearly won't fit in your computer memory, not to mention the $2^{128}\times2^{128}$ matrices you'd have to apply. These numbers are for an erasing oracle, while you probably want a standard one in order to apply Grover's algorithm, in which case replace the $2^{128}$ by $2^{256}$.

Thus, you probably want to design an "AES" that acts on very small states, where a case of your state is only represented on 1 bit or two. This would still allow you to understand how you could implement such a function in Qiskit.