If you don't supply a $|u\rangle$ as an input, there are two possible things you might want to get out:
The $\varphi$ for a randomly chosen (but unknown) eigenstate $|u\rangle$;
Both $\varphi$ and $|u\rangle$ for one or more eigenstates.
Let's first look at 1. Since eigenstates form a complete basis, any input state you use can be interpreted as a superposition of the eigenstates of $U$. Due to the linearity of quantum mechanics, the algorithm would then run for all these states at once. Then at the end, when you measure the output, it will randomly collapse to a given instance. This means that you'll be given a $\varphi$ for a randomly chosen eigenstate, but you won't know which it is. The existing phase estimation algorithm therefore can supply us with the first possible application.
The second application is something we can't do with standard phase estimation, but let's consider it hypothetically. Any algorithm that could do this would be giving us an eigenstate $|u\rangle$ as part of the output. So if you want to actually know what $|u\rangle$ is, you'd have to do tomography on the output. Since tomography is inefficient, there would probably be better ways to go about doing this.
If you don't want to do tomography, but just want the state, note that $|u\rangle$ would also be an eigenstate of a Hamiltonian $H$ for which $U=e^{iH}$. For the particular case of the ground state of $H$, this can be done via adiabatic QC. I'm not so sure about other eigenstates, though.
So there is indeed questions that phase estimation alone cannot answer. But it is good for what it is good at, and can be supplemented by other algorithms for other applications.
If you don't supply a $|u\rangle$ as an input, there are two possible things you might want to get out:
The $\varphi$ for a randomly chosen (but unknown) eigenstate $|u\rangle$;
Both $\varphi$ and $|u\rangle$ for one or more eigenstates.
Let's first look at 1. Since eigenstates form a complete basis, any input state you use can be interpreted as a superposition of the eigenstates of $U$. Due to the linearity of quantum mechanics, the algorithm would then run for all these states at once. Then at the end, when you measure the output, it will randomly collapse to a given instance. This means that you'll be given a $\varphi$ for a randomly chosen eigenstate, but you won't know which it is. The existing phase estimation algorithm therefore can supply us with the first possible application.
The second application is something we can't do with standard phase estimation, but let's consider it hypothetically. Any algorithm that could do this would be giving us an eigenstate $|u\rangle$ as part of the output. So if you want to actually know what $|u\rangle$ is, you'd have to do tomography on the output. Since tomography is inefficient, there would probably be better ways to go about doing this.
If you don't want to do tomography, but just want the state, note that $|u\rangle$ would also be an eigenstate of a Hamiltonian $H$ for which $U=e^{iH}$. For the particular case of the ground state of $H$, this can be done via adiabatic QC. I'm not so sure about other eigenstates, though.
So there is indeed questions that phase estimation alone cannot answer. But it is good for what it is good at, and can be supplemented by other algorithms for other applications.
If you don't supply a $|u\rangle$ as an input, there are two possible things you might want to get out:
The $\varphi$ for a randomly chosen (but unknown) eigenstate $|u\rangle$;
Both $\varphi$ and $|u\rangle$ for one or more eigenstates.
Let's first look at 1. Since eigenstates form a complete basis, any input state you use can be interpreted as a superposition of the eigenstates of $U$. Due to the linearity of quantum mechanics, the algorithm would then run for all these states at once. Then at the end, when you measure the output, it will randomly collapse to a given instance. This means that you'll be given a $\varphi$ for a randomly chosen eigenstate, but you won't know which it is. The existing phase estimation algorithm therefore can supply us with the first possible application.
The second application is something we can't do with standard phase estimation, but let's consider it hypothetically. Any algorithm that could do this would be giving us an eigenstate $|u\rangle$ as part of the output. So if you want to actually know what $|u\rangle$ is, you'd have to do tomography on the output. Since tomography is inefficient, there would probably be better ways to go about doing this.
