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Let us assume we have used the HHL algorithm to approximately prepare $|x\rangle$, where $|x\rangle$ is a normalized quantum state corresponding to $\vec{x}$, the solution of a system of linear equations.

Is there any way we can recover information on the signs of $\vec{x}$?

So far, I can think of no way that doesn't include classically solving for $\vec{x}$ and extracting the relevant signs from $\vec{x}$, which obviously defeats the purpose of using HHL in the first place. Is there maybe some efficient classical way of determining the signs using $A$ and $b$ that takes at most $O(\log N)$ steps and therefore doesn't destroy the computational advantage of HHL?

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Let us denote: $$|x\rangle=\sum_ix_i|i\rangle$$ If it was possible to learn the sign of each $x_i$, then you would have a way to distinguish $|x\rangle$ and $-|x\rangle$. Since these states only differ by a global phase, this is not possible.

There may be an algorithm that allows you to learn the relative signs between given $x_i$ and $x_j$, but learning them all would of course take at least $O(N)$ operations.

HHL is more to be seen as a routine in a bigger quantum algorithm, like evaluating $\langle x|M|x\rangle$ for instance, rather than an actual solver for systems of equations.

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  • $\begingroup$ Thank you for the anwer! So would you say there's not just no way to learn the signs based on $|x\rangle$, but also no way to learn it from "other parts of the circuit" (I'm thinking of how the normalization constant can be estimated by measuring the Ancilla qubit)? $\endgroup$ Mar 4, 2023 at 21:26
  • $\begingroup$ @DominikVereno If I'm not mistaken, the ancilla will only give you information on the norm of $x$. If you think about it, there's no reason for the ancilla to contain anything else: the relative phases are already encoded in $|x\rangle$. I may be mistaken, but I don't think the ancilla can be of any use here $\endgroup$ Mar 4, 2023 at 22:40
  • $\begingroup$ I wasn't talking about the ancilla in particular, but rather as an example of a "source of information" other than the result qubits. To be honest, I was mainly hoping to have overlooked something, some source of retrieving sign information. $\endgroup$ Mar 7, 2023 at 21:03

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