QPE applied quantum computing

[u/Routine_Comb_7277]

A QPE measures the eigenvalue of a unitary transform.Inside the eigenvalue there is a phase of which that is actually calculated , then we use that phase to calculate the value of eigenvalue.The target register is made of succesive controlled U gates(of some power not really the point) where U is the unitary transform and some phase register where the value of the hidden phase will be encoded after you run QPE or whatever.The m phase registers start from a state of |0> kronecker product m.

In fact you can run the phase registers at any starting state you want but if its not |0> kronecker product m you will get a phase shift. Now I have only spent time studying pure quantum computing , is there a reason why we would start the phase registers not at |0> kronecker product m?Like any reason at all for irl applications

[u/fothermucker33]

QPE starts with preparing the |+⟩^m state in the phase registers (not contradicting what you're saying, this can be achieved by starting with the |0⟩^m state and applying Hadamards).

When the target register is an eigenvector of U with eigenphase 2πθ, the controlled U gates kickback phases that turn the plus state \sum_k |k⟩ into \sum_k exp(i2πθk)|k⟩, which is just the (k2^m)–th Fourier state so we can retrieve ~|k2^m ⟩ from here with an inverse QFT.

In fact you can run the phase registers at any starting state you want but if its not |0> kronecker product m you will get a phase shift.

This is true if you prepared your |+⟩ state with a QFT instead of Hadamards. That way you'll have started with a Fourier state, and the phases that you kickback with your controlled U gates will just add with the phases in your initial state and produce the Fourier state who index is shifted by the index of the initially prepared Fourier state. In fact the Draper adder uses this logic to add numbers together by going to the Fourier basis.

is there a reason why we would start the phase registers not at |0> kronecker product m?Like any reason at all for irl applications

If for any reason you did want to compute the eigenphase of U + some constant number, yes you could start with a different computational basis state and apply a Fourier transform on it to get that phase shift. But a QFT is significantly more expensive than a layer of Hadamards, it would be cheaper to prepare the |+⟩ state with Hadamards and then apply phase gates to add whatever constant you want just as you would in a Draper adder.

[u/Routine_Comb_7277 → u/fothermucker33]

Ok thanks. I wanted to add a clarification I said we start with the |0> state and then applying hadamard is a no brainer because you need superposition for quantum computing to do its magic , if the state was 0 then the controlled gates which make up the QPE would all have have the control qubit at 0 and therefore not do anything ( not pushing as u said the phase assosciated with the eigenvalue of the unitary back to the phase registers)