# Tag Info

### Why does the "Phase Kickback" mechanism work in the Quantum phase estimation algorithm?

A first remark This same phenomenon of 'control' qubits changing states in some circumstances also occurs with controlled-NOT gates; in fact, this is the entire basis of eigenvalue estimation. So not ...
Accepted

### Who discovered the phase kickback trick?

The phase kickback trick appears in this paper: Richard Cleve, Artur Ekert, Chiara Macchiavello, Michele Mosca. Quantum Algorithms Revisited. Proceedings of the Royal Society of London A, 454(1969):...
Accepted

### Why does the "Phase Kickback" mechanism work in the Quantum phase estimation algorithm?

Imagine you have an eigenvector $|u\rangle$ of $U$. If you have a state such as $|1\rangle|u\rangle$ and you apply controlled-$U$ to it, you get out $e^{i\phi}|1\rangle|u\rangle$. The phase isn't ...
Accepted

### Can we use Hadamard test to estimate phases?

So QPE using $\mathcal{O}(1/\epsilon)$ queries to $U$ outputs an estimate of the eigenphase $\theta$ given a corresponding eigenvector with additive error and $\Omega(1)$ probability. The method using ...

### What is Quantum Phase Estimation in Shor's Algorithm?

Phase estimation is the process by which you are given a controlled-$U$ unitary, and a state that you are promised is an eigenvector of $U$ with eigenvalue $e^{2\pi ix/2^t}$, then you can use a $t$-...

### Question about the phase kickback in the phase estimation algorithm

The phase is applied to the overall wave function $|\phi\rangle$, therefore you can factor the phase to any individual qubit. For example if we have a wave function as a result of a controlled ...
Accepted

### In the Bernstein-Vazirani circuit, if the secret is all 0s, would the oracle just be nothing?

Yes, this is because the initial state will be starting as $|\psi_{init} \rangle = |000\cdots0\rangle = |0\rangle^{\otimes 8}$. Then you apply a layer of Hadamard gates, follow by the Oracle ...

### Unitary Operator impact on both - the Control Qubit and the Target Register in Shor's Algorithm

The key idea in understanding phase kickback is that phase factors do not belong to one register or the other, but instead belong to terms in superposition and are shared by the registers. For example,...

### How to understand intuitively the quantum gate phase kickback?

Part of the problem people usually have here is a sort-of-classical intuition. Because you're trying to describe the action of a gate such as controlled-$U$, we divide it up as "if the control ...

### Is the $|-\rangle$ state the only one that can do the trick for Grover's algorithm?

Remember that the states $|+\rangle$ and $|-\rangle$ form a basis. That means that any state $|\psi\rangle$ that you use can be written as $$|\psi\rangle=\alpha|+\rangle+\beta|-\rangle.$$ By ...
Accepted

### Is the $|-\rangle$ state the only one that can do the trick for Grover's algorithm?

It would still work if you rotated the $|−\rangle$ state, you refer to, by an angle with cosine less than $\frac{1}{L}$ where $L$ is the square root of the size of the search space (which is a power ...

### Show that these two expressions for the oracle transformation are equivalent

You are given a quantum circuit for $U$ compiled into the H/CNOT/T gateset. Derive a controlled version of $U$ by adding a control qubit $q$, replacing every H with a controlled H, every CNOT with a ...