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The chapter about solving linear systems in the qiskit textbook describes the last (6th) step of the HHL algorithm as follows

Apply an observable $M$ to calculate $F(x):=\langle x |M|x\rangle$.

How is this observable $M$ chosen? What are the considerations?

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The main limitation of the HHL algorithm is that the solution to the linear system $Ax = b$ is obtained as a quantum state $|x\rangle$. Thus, we cannot determine the full vector $x$ directly without running the algorithm an exponential number of times.

However, we can estimate $f(x)$ where $f$ is any function of the solution which can be expressed as $f(x) = \langle x|M|x \rangle$ for a Hermitian operator $M$ which we know how to measure. The HHL algorithm is useful in applications where we know how to cast the quantity of interest as such a function. Designing and implementing an observable $M$ appropriate for a given application is one of the key tasks in using the algorithm.


Example of what can be computed

A fairly general example of a quantity we know how to efficiently estimate from $|x\rangle$ is the overlap $f_\psi(x) = \langle x| \psi\rangle$ of $|x\rangle$ with any quantum state $|\psi\rangle$ for which we know the preparation, e.g. $|\psi\rangle = U|0\rangle$. We can accomplish this using the Hadamard test.

We start with an auxiliary qubit $A$ in the $|+\rangle$ state and a quantum register $R$ in the $|0\dots0\rangle$ state. Using controlled operations we apply $U$ to $R$ if $A$ is $|0\rangle$ and if $A$ is $|1\rangle$ we put $R$ into the state $|x\rangle$ using the HHL algorithm. These operations result in the entangled state

$$ \frac{1}{\sqrt{2}}(|0\rangle|\psi\rangle + |1\rangle|x\rangle).\tag1 $$

Now, in order to estimate the real part of $\langle x|\psi\rangle$ we first apply Hadamard to $A$, obtaining

$$ \frac{1}{2}|0\rangle(|\psi\rangle + |x\rangle) + \frac{1}{2}|1\rangle(|\psi\rangle - |x\rangle) $$

and then we measure $A$ in the computational basis. The output probabilities are

$$ p_0 = \langle\phi|0\rangle\langle 0|\phi\rangle = \frac{1}{4}(\langle\psi|\psi\rangle + \langle\psi|x\rangle + \langle x|\psi\rangle + \langle x|x\rangle) = \frac{1}{2} + \frac{\mathrm{Re} \, \langle x|\psi\rangle}{2} \\ p_1 = \langle\phi|1\rangle\langle 1|\phi\rangle = \frac{1}{4}(\langle\psi|\psi\rangle - \langle\psi|x\rangle - \langle x|\psi\rangle + \langle x|x\rangle) = \frac{1}{2} - \frac{\mathrm{Re} \, \langle x|\psi\rangle}{2}. $$

In order to estimate the imaginary part of $\langle x|\psi\rangle$ we first apply the $S$ gate and then Hadamard to $A$ in $(1)$, obtaining

$$ \frac{1}{2}|0\rangle(|\psi\rangle + i|x\rangle) + \frac{1}{2}|1\rangle(|\psi\rangle - i|x\rangle) $$

and then we measure $A$ in the computational basis. This time, the output probabilities are

$$ p_0 = \langle\phi|0\rangle\langle 0|\phi\rangle = \frac{1}{4}(\langle\psi|\psi\rangle + i\langle\psi|x\rangle - i\langle x|\psi\rangle + \langle x|x\rangle) = \frac{1}{2} + \frac{\mathrm{Im} \, \langle x|\psi\rangle}{2} \\ p_1 = \langle\phi|1\rangle\langle 1|\phi\rangle = \frac{1}{4}(\langle\psi|\psi\rangle - i\langle\psi|x\rangle + i\langle x|\psi\rangle + \langle x|x\rangle) = \frac{1}{2} - \frac{\mathrm{Im} \, \langle x|\psi\rangle}{2}. $$

A simple special case of the above is the efficient estimation of the $k$th component $x_k$ of $x$. Another is the determination whether solutions $x_1$ and $x_2$ to two linear systems $A_1x_1=b_1$ and $A_2x_2=b_2$ are orthogonal.

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