How to implement linear combination of two unitary gates in a quantum circuit?

Question

I wanted to implement a non-unitary operation. I came to know that I can do it as a linear combination of unitaries from this paper (published version). Let us say I want to implement an operation like $\alpha_1 A_1 + i \alpha_2 A_2$ in a quantum circuit, where $A_1$ and $A_2$ and unitary operators. Coeffient $\alpha$ are real. How do you do that

Answer 1

The place where I came to know about this technique was here, which will give more details than I'm about to reproduce. In overview, you want to make a unitary $B$ such that $$ B|0\rangle=(\sqrt{\alpha_1}|0\rangle+\sqrt{\alpha_2}|1\rangle)/\sqrt{\alpha_1+\alpha_2}, $$ (I'm assuming the $\alpha_i$ are positive) and a second unitary $$ U=|0\rangle\langle 0|\otimes A_1+|1\rangle\langle 1|\otimes A_2 $$

A very crude way of implementing the operation would then be to start with $$ |0\rangle|\psi\rangle, $$ where $|\psi\rangle$ is the state that you want to apply the superposition of unitaries. You apply $B$ to the first qubit, $U$ across both, then $S$ (phase gate) and $B^\dagger$ on the first qubit. Measure the first qubit, and if it is $|0\rangle$, you have succeeded.

If your amplitudes were negative, you can compensate for that by changing the phase rotation at the point where I applied $S$ in that sequence.

To see this, the evolution sequence is \begin{align*} |0\rangle|\psi\rangle &\xrightarrow{B} (\sqrt{\alpha_1}|0\rangle+\sqrt{\alpha_2}|1\rangle)|\psi\rangle/\sqrt{\alpha_1+\alpha_2} \\ &\xrightarrow{U} (\sqrt{\alpha_1}|0\rangle(A_1|\psi\rangle)+\sqrt{\alpha_2}|1\rangle(A_2|\psi\rangle))/\sqrt{\alpha_1+\alpha_2} \\ &\xrightarrow{S} (\sqrt{\alpha_1}|0\rangle(A_1|\psi\rangle)+\sqrt{\alpha_2}i|1\rangle(A_2|\psi\rangle))/\sqrt{\alpha_1+\alpha_2} \end{align*} This gets you up to just before applying $B^\dagger$ and measuring. This is equivalent to projecting the first qubit onto $(\sqrt{\alpha_1}\langle 0|+\sqrt{\alpha_2}\langle 1|)/\sqrt{\alpha_1+\alpha_2}$. This measuremnt result leaves the second qubit in $$ \frac{\alpha_1A_1+i\alpha_2A_2}{\alpha_1+\alpha_2}|\psi\rangle, $$ which you can use to assess the success probability.

For such a small number of terms in superposition, this is likely to be highly successful. However, if the probability of success is low, then you can improve your changes by getting rid of the measurement and instead performing amplitude amplification (i.e. essentially Grover's search, searching for when that first qubit is $|0\rangle$).