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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 ...
Niel de Beaudrap's user avatar
19 votes
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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 ...
DaftWullie's user avatar
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10 votes
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Quantum phase estimation and HHL algorithm - knowledge of eigenvalues required?

You should know a bound on the eigenvalues (both upper and lower). As you say, you can then normalise $A$ by rescaling $t$. Indeed, you should do this to get the most accurate estimate possible, ...
DaftWullie's user avatar
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8 votes
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Quantum Circuit for $e^{iAt}$ Hamiltonian Simulation in HHL algorithm

As requested through the comment by the OP. Given a Hermitian matrix $H$, we can always write it as linear combination of Pauli strings. That is, $$ H = \sum_i \alpha_i P_i \hspace{1 cm} P_i \in \{I,...
KAJ226's user avatar
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8 votes
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What is the difference between amplitude amplification, amplitude estimation, and phase estimation?

To find the differences between them, you only need to know the aim of the problems. The aim of AA is to find the answers from unstructured data(or more directly, amplify the probability of the right ...
narip's user avatar
  • 2,989
7 votes
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Clarification on state prep for quantum phase estimation

Your supposition is spot on. Indeed, quantum phase estimation (QPE) can be applied even when the input state is not an eigenstate of the unitary $U$. The key point to note is that, by expanding the ...
bm442's user avatar
  • 1,127
6 votes
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Quantum algorithm for linear systems of equations (HHL09): Step 1 - Confusion regarding the usage of phase estimation algorithm

It depends on the papers but I saw 2 approaches: In most of the papers I read about the HHL algorithm and its implementation, the Hamiltonian evolution time $t$ is taken such that this factor ...
Adrien Suau's user avatar
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6 votes
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Are the squared absolute values of the eigenvalues of a unitary matrix always 1?

Good question. The answer turns out to be Yes. You don't even need the vector to be normalized. Watch: Start with the definition of eigenvalues and eigenvectors: $$ \begin{align} U|\psi\rangle &=...
user1271772 No more free time's user avatar
6 votes
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Why can I use the Sum of Eigenvectors for Phase Estimation in Shor

As you say, we would be able to use phase estimation if we knew the eigenvector $|u_s \rangle$ that depends on the order $r$ and the integer $s$ which is $0 \leq s \leq r - 1$. However, we don't know ...
Attila Kun's user avatar
6 votes
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Quantum Amplitude Estimation vs Quantum Phase Estimation

Ok let's break this down. Firstly the success probability for QPE and QAE are defined slightly differently. With QPE there are two error bounds $|\hat{x} - x| < \epsilon$ to consider $\epsilon < ...
Sam Palmer's user avatar
6 votes

In the quantum phase estimation algorithm, why can't we directly compute the eigenvalue from the known eigenvector?

There are two different issues at play here. First is the difficulty of the calculation. If you have an $n$-qubit unitary $U$, then to evaluate $Ux$, you have to multiply a $2^n\times 2^n$ matrix with ...
DaftWullie's user avatar
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6 votes
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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 ...
dylan7's user avatar
  • 324
6 votes
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Does phase kickback require the system to be in the eigenstate?

The first circuit equality fails when $|\psi_k\rangle$ is not an eigenstate of $U$. A simple way to see this is to set the control qubit to $|1\rangle$. In this case, the RHS circuit is equivalent to $...
Adam Zalcman's user avatar
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6 votes

$QFT^{-1}$ at the end of Shor's algorithm and $QFT$ at the end of Hidden Subgroup algorithm

$\text{QFT}\big(|0\rangle^{\otimes n}\big) = \text{QFT}^{-1}\big(|0\rangle^{\otimes n}\big) = |{+}\rangle^{\otimes n}$, so a QFT, inverse QFT, or a column of Hadamard gates are all equivalent at the ...
benrg's user avatar
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6 votes
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$QFT^{-1}$ at the end of Shor's algorithm and $QFT$ at the end of Hidden Subgroup algorithm

Note that $\text{QFT}^2$ is a permutation $|k\rangle \rightarrow |(-k) \bmod 2^n\rangle$. This is a classical operation. It can be applied in the post processing of the measurements, and in fact it ...
Craig Gidney's user avatar
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6 votes
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Requirement of vector 'b' in the definition of Phase Estimation Sampling (PES)

The discussion in question appears to be discussing usage of the Quantum Phase Estimation algorithm when we do not have access to an eigenstate $|\eta_j \rangle$ of the unitary matrix $U$ in question. ...
jsbaker's user avatar
  • 156
5 votes
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How to implement exponentiation of a gate without breaking complexity?

As a general rule, just because you can produce controlled-$U$, it does not mean that you can produce controlled-$U^{2^k}$ with the same complexity. Modular exponentiation is a very special case where ...
DaftWullie's user avatar
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5 votes
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Given a state $|\phi\rangle=\frac{1}{\sqrt{2}}(|0\rangle+e^{i\theta}|1\rangle)$, how do I know the angle $\theta$?

Answer to the first question: As mentioned in the comments of the question I assume that we can prepare $|\phi \rangle$ as many as we want. Let's calculate the relative phase for this one qubit pure ...
Davit Khachatryan's user avatar
5 votes

Are the squared absolute values of the eigenvalues of a unitary matrix always 1?

@user1271772's answer is excellent, and absolutely the right answer. I just wanted to add in some additional perspective, given recent questions regarding Hamiltonians. Many physicists start from the ...
DaftWullie's user avatar
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5 votes
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Which angle is estimated by the phase estimation algorithm?

Neither. Phase estimation algorithm does not estimate a property of a qubit state (and the angles $\theta$ and $\varphi$ in your question are exactly that - a property of a given qubit state). ...
Mariia Mykhailova's user avatar
5 votes
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How to get the relative phase of an entangled pair of qubits

Apply a CNOT gate with one of the qubits as control and the other as target. You'll get $$\frac{1}{\sqrt{2}}(|0\rangle+e^{i\theta}|1\rangle) \otimes |0\rangle$$ Use the methodology from How to get ...
Mariia Mykhailova's user avatar
5 votes

Why are $U$s raised to successive powers of two in quantum phase estimation when we use $n$ register qubits $|0\rangle|0\rangle|0\rangle$?

The objective of all those gates is to put the quantum state in a "nice form" to manipulate. Let me explain. Let's note $U |\psi \rangle = e^{2i\pi\theta}|\psi\rangle$ all the variables we ...
Lena's user avatar
  • 2,597
5 votes
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Can quantum search be performed without phase estimation?

I'm sure I'm not the first to notice this; does it appear somewhere? Section 6 of the 1996 paper "Tight Bounds on Quantum Searching" by Boyer et al uses this strategy of iteratively trying ...
Craig Gidney's user avatar
  • 37.8k
5 votes
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Phase estimation using $U_3$ gate

If you were to apply QPE to this unitary, what you will get, assuming you start with a proper eigenvector $|\Lambda\rangle$, is an estimation of $x$, if the associated eigenvalue $\Lambda$ is written ...
Tristan Nemoz's user avatar
  • 6,462
4 votes

Phase estimation error analysis

Let me augment the discussion by adding some insight into the derivation of the estimate provided. This will give you a good understanding of when the result is an approximation and when it is precise....
Alex East's user avatar
4 votes

Quantum algorithm for linear systems of equations (HHL09): Step 1 - Confusion regarding the usage of phase estimation algorithm

What am I missing here? Where did the factor of $\frac{t}{2\pi}$ vanish in their algorithm? Remember that in Dirac notation, whatever you write inside the ket is an arbitrary label referring to ...
DaftWullie's user avatar
  • 58.8k
4 votes

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 $t$-...
DaftWullie's user avatar
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4 votes
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Simulating QPE + Grover using Low-Rank Stabilizer Decomposition

I agree that the Bravyi et al. paper is not easy to understand and they should have made some reference implementation available. Without going into details, I don't think it is likely to get an ...
Markus Heinrich's user avatar
4 votes

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 ...
Sam Palmer's user avatar
4 votes
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How would Quantum Phase Estimation be solved classically?

Solving the (classical) problem of finding the eigenvalue corresponding to a given matrix and eigenvector (under the promise that the given vector is indeed an eigenvector) really isn't hard. But I ...
glS's user avatar
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