# Understand $U|\psi\rangle = e^{2\pi i\phi}|\psi\rangle$ in phase estimation algorithms

I'm trying to understand the motivations behind $$U|\psi\rangle = e^{2\pi i\phi}|\psi\rangle$$ in quantum phase estimation. In my interpretation, since $$U$$ is the unitary operator, this equation wraps the spectrum of a system in a complex unit circle, and that's where the factor $$2\pi$$ comes from. However, I wonder is there another way I can understand the factor $$2\pi$$? Can I say this equation allows us to learn the static information on a system (energy spectrum) from dynamics (time-evolution)?

Also, if I know there's an energy $$E_1$$ in the system, should I set $$e^{2\pi i\phi} = e^{iE_1t}$$, or $$e^{-iE_1t}$$?

Thanks!

You are right. The main purpose of using QPE is to examine the eigenstate of Hamiltonian (energies) when applied to some system.

$$e^{−iE_1t}$$ should be set, just as physics tells you, and qubits do not limit you to positive rotations (as long as you are consistent). You can read about it in Elucidating Reaction Mechanisms on Quantum Computers

What's special about QPE, is that it is able to find a variable that is transparent to quantum computation - which is the global phase. The global phase is such an $$e^{2πiϕ}$$ multiplication on both $$|0\rangle$$ and $$|1\rangle$$ and not only on one of them. Since QC can't find a global phase, it must use the QPE which is using an ancilla qubits, to convert a global phase, to a phase that is global only on the sub system, what ends in a relative phase (between 0 and 1).

Another use case for QPE is in Shor's algorithm, you can read more about it here.

The reason for $$2π$$ is a basic mathematical concept, where $$e^{2πiϕ}$$ represents a complex number on the unit circle, where the units of $$ϕ$$ is the fraction of the circle ($$ϕ=1/4$$ is $$90$$ degress or $$π/2$$) it is just convention, and it can be done with any other units.

• I think this is misleading. Global phases are by definition non-physical in any scenario. They're just artifacts of the mathematical formalism used. QPE doesn't "find a global phase", it allows to estimate a specific property of a unitary, that is, one of its eigenvalues. At no point QPE can be used to retrieve the global phase of a state, that just wouldn't make any sense.
– glS
Commented Jun 19, 2022 at 9:44
• This is not true that "Global phases are by definition non-physical in any scenario". global phases are physical. They are just meaningless when you talk about qubits, since they do not affect the probabilities of measurement at the end of the algorithm. A physical operation of Hamiltonian, when applied on and eigensate of the system, is creating a global phase on the physical system. The QPE is an algorithm that is converting the global phase to a relative phase, in order to be able to measure it after some proccess Commented Jun 19, 2022 at 10:20
• it is not. I think the issue here is that you're conflating global and relative phases. Changing the phase of a subsystem can change the relative phase of the overall state, and thus be an observable effect, but it's not accurate to say that this is changing a global phase in any way. If you have $|00\rangle+|11\rangle$ and apply a control-phase operation on the second qubit, you're not "changing the global phase of the second qubit", you're changing the relative phase of the overall state.
– glS
Commented Jun 19, 2022 at 10:23
• this has nothing to do with qubits by the way. It is wrong to think that global phases are somehow physical but unobservable in some circumstances. They outright do not exist if one takes a bit more care in handling the formalism. Or even just if you work with density matrices rather than bra-kets
– glS
Commented Jun 19, 2022 at 10:27
• Now we agree, I really had to say instead of global phase, that it is a global phase of the sub-system. But I like to call it global phase, because the whole idea of QPE, is to find global phase, using adding more qubits, what makes it not a global phase, but a phase that is common to the sub system Commented Jun 19, 2022 at 11:06

The goal of QPE is to estimate an eigenvalue of $$U$$. Being $$U$$ unitary, its eigenvalues have unit modulus. Any complex number with unit modulus can be written as $$e^{i\phi}$$ with $$\phi\in[0,2\pi]$$, or equivalently, as $$e^{2\pi i \phi}$$ with $$\phi\in[0,1]$$. It is just a matter of notational convenience which notation is used.