Questions tagged [phase-estimation]

Refers to the quantum algorithm used to estimate the eigenvalue corresponding to an eigenvector of a unitary operator. (Wikipedia)

Filter by
Sorted by
Tagged with
2
votes
1answer
42 views

Why can I use the Sum of Eigenvectors for Phase Estimation in Shor

In phase estimation, we start by using an eigenvector $\newcommand{\ket}[1]{\lvert#1\rangle}\ket u$ to find the corresponding eigenvalue lambda. So far so good. In the order finding algorithm, we also ...
3
votes
2answers
491 views

Given a state $|\phi\rangle=\frac{1}{\sqrt{2}}(|0\rangle+e^{i\theta}|1\rangle)$, how do I know the angle $\theta$?

Question1. If there is a state $|\phi\rangle=\frac{1}{\sqrt{2}}(|0\rangle+e^{i\theta}|1\rangle)$, and I want to know the angle $\theta$. What kind of measurement should I do? Could somebody give me ...
3
votes
1answer
59 views

How to get the relative phase of an entangled pair of qubits

I have an extension to the following question: How to get the relative phase of a qubit? How do I get the relative phase of a pair of entangled qubits such as $$\frac{1}{\sqrt{2}}(|00\rangle+e^{i\...
2
votes
1answer
40 views

Indexing an “unknown” quantum state

Assuming I have a state $$|x\rangle = \frac{1}{\sqrt{n}}\sum_n |x_n\rangle$$ where $|x_n\rangle$ are quantum state vectors $$|x_n\rangle = \frac{1}{\|x_n\|}\sum_i x_{in}|i\rangle$$ and that I have a ...
2
votes
1answer
73 views

Is it possible to detect the phase $\pi$ or 0 for the single qubit circuit X H P?

I found an answer that shows how to detect the phase in cases like $0$, $\pi/8$, $\pi/2$, $\pi/4$ or $\pi$ for circuit to prepare state as H P, where P is a phase gate like $I$, $U1(\pi/8)$, $S$, $T$ ...
3
votes
1answer
41 views

Finding phase angle in Q#

I've trying to measure the phase angle from X axis of a qubit, but unable to find any function in Q# documentation, can anyone help me with this?
0
votes
0answers
111 views

Is it possible to express u1(λ) through the gates rx, ry, rz while maintaining the phase? In Qiskit for example

Is it possible to express gate u1(λ) through the gates rx, ry, rz while maintaining the phase? Both in principle and in practice (in Qiskit for example)? The single gate rz(λ) is not suitable, because ...
1
vote
1answer
59 views

Quantum Phase Estimation Circuit and Modular Exponentiaton

In Nielsen and Chuang, it is stated that the effect of phase estimation circuit is mapping state $|j\rangle |u\rangle$ to $|j\rangle U^j |u\rangle$. Here is my solution: Consider the first $CU^{2^0}...
2
votes
0answers
110 views

What is the usefulness of the Suzuki-Trotter formula?

I can't seem to wrap my head around the suzuki-trotter formula. I have seen This answer but I am still confused of the applicability of the formula. Let me explain: As I understand it Trotterization ...
1
vote
0answers
38 views

Quantum Phase Estimation on a superposed state

Is it possible, using the QPE algorithm, to map the state $\sum_j\alpha_j\,\left|v_j\right\rangle\,|0\rangle^{\otimes n}$ to the state $\sum_j\alpha_j\,\left|v_j\right\rangle\,\left|\theta_j\right\...
1
vote
1answer
67 views

How does Inverse QFT work in Quantum Phase Estimation?

I'm trying to implement Quantum Phase Estimation from qiskit textbook. Below is the implementation circuit taken from the above-mentioned site: The output at position 2 will be as follows: $$|\psi ...
1
vote
1answer
23 views

Why is the number of qubits linear in the inverse of epsilon in Quantum Phase Estimation?

On Wikipedia, one can read the following about Quantum Phase Estimation: the algorithm estimates the value of $\theta$ with high probability within additive error $\varepsilon$, using $O(1/\...
1
vote
0answers
40 views

Repeating Quantum Phase Estimation algorithm

In the Quantum Recommendation Systems paper, the authors use the Quantum Phase Estimation algorithm in a way that is slightly different from what I've seen so far. This is how it is described: The ...
3
votes
2answers
55 views

How to implement exponentiation of a gate without breaking complexity?

In the application of QFT for quantum phase estimation (QPE) of a unitary $\mathbf{U}$, one has to perform successive controlled operations using powers of $\mathbf{U}$. In order not to break the ...
4
votes
1answer
253 views

How to decompose a multi-target controlled gate?

I'm trying to replicate with qiskit the results of this paper in which basically they implement a quantum version of the Principal Component Analysis applying Quantum Phase Estimation algorithm in ...
1
vote
1answer
158 views

Is it possible to demonstrate a quadratic speed-up of a quantum algorithm on a classical computer?

In article Quantum computational finance: Monte Carlo pricing of financial derivatives the authors said that: Firstly: While a practical quantum computer has yet to become a reality, we can ...
3
votes
2answers
122 views

What is Quantum Phase Estimation in Shor's Algorithm?

While I'm studying Algorithm, I couldn't understand what Quantum Phase Estimation is. And I heard there is relation between Phase-Kickback and Quantum Phase Estimation. I wonder what it is. Also, I'm ...
1
vote
1answer
152 views

Prepending initial state to a quantum circuit in Qiskit

I am trying to generate a Quantum Phase Estimation (QPE) circuit in QISKIT the following way. 1 - First, I generate a QPE circuit with the following code: ...
0
votes
0answers
51 views

How to properly make phase estimation with qiskit's modules

I have been making several experiments with qiskits' Quantum Phase Estimation modules, such as with QPE and EigsQPE. For example,...
5
votes
1answer
373 views

Travelling salesman problem on quantum computer

Recently a pre-print of article Efficient quantum algorithm for solving travelling salesman problem: An IBM quantum experience appeared. The authors use a phase estimation as a core for their ...
0
votes
2answers
150 views

Right way to use Quantum Phase Estimation using aqua

I have been experimenting with Qiskit lately, but I have found the implementation of algorithms in Aqua extremely confusing. Currently I am trying to implement a very simple circuit that will return ...
3
votes
1answer
274 views

Cannot replicate results in article on pricing financial derivatives on IBM Q

I am trying to implement a circuit for searching for the largest eigenvalue and respective eigenvector of an operator, i.e. phase estimation, introduced in article Towards Pricing Financial ...
2
votes
1answer
181 views

Quantum Phase Estimation implementation

I tried implementing quantum phase estimation in qiskit, however, I'm not getting the expected results. I choose a controlled $U1$ gate. First of, I implemented inverse QFT operation (basically a ...
4
votes
2answers
218 views

Phase estimation error analysis

This question is about Lemma $7.1.2$ in Kaye, Laflamme, and Mosca's textbook: Let $\omega = \frac{x}{2^n} = 0.x_1x_2\ldots x_n$ be some fixed number. The phase estimation algorithm applied to the ...
0
votes
1answer
57 views

Which angle is estimated by the phase estimation algorithm?

Is it $\theta$ or $\varphi$ as usually depicted on the Bloch sphere? In other words, is it the angle projected on the $xy$-plane or is it the one on a plane that intersects the $z$-axis of the Bloch ...
2
votes
0answers
58 views

Why must we take multiple measurements for different values of $M$?

The circuit for Kitaev phase estimation is given as: By varying $\theta$, we are able to determine $\sin(2 \pi M \phi_k)$ and $\cos (2 \pi M \phi_k)$ from sampling the circuit and calculating the ...
2
votes
1answer
91 views

Sketch the quantum logic gates correctly and give a proof for the identity

If I denote by $U^c$ the controlled version of the quantum operation $U$ $$U^c=|0\rangle \langle 0|\otimes \mathbb{1}+|1\rangle \langle 1|\otimes U$$ I can first apply $U^c$ and afterward measure ...
3
votes
1answer
26 views

Obtaining phases of all basis states

I’m wondering that is there a way to separate phases of basis states from the magnitudes (namely putting phases on some new basis states with the same probability magnitude) if the exact values of ...
2
votes
1answer
66 views

Method to find $r$ in the case when $r'$ returned by the continued fractions procedure is a factor of $r$

In the Quantum Computation and Quantum Information (10th ed.) textbook by Nielsen and Chuang, section 5.3.1 (titled "Application: order-finding") describes how phase estimation can be used to find the ...
2
votes
1answer
85 views

Eigenstate of unitary operator used for Order-Finding

In the "Quantum Computation and Quantum Information 10th Anniversary textbook by Nielsen and Chuang", chapter 5.3.1 introduces the concept of solving the Order-Finding Problem. (Eqn 5.36) states ...
6
votes
1answer
157 views

Phase estimation algorithm: probability bound of obtaining $m$

Note: Cross-posted on Physics SE. Hi, I'm studying the quantum phase estimation algorithm from this book: M.A. Nielsen, I.L. Chuang, "Quantum Computation and Quantum Information", Cambridge Univ. ...
6
votes
0answers
271 views

What is the complexity of the quantum phase estimation in Grover's algorithm?

Suppose we are using GA (Grover's algorithm) such that we are given it has 2 or more solutions. The search space is of size $N$. We all know Grover's algorithm has, at worst, a time complexity ...
13
votes
3answers
2k views

Why does the “Phase Kickback” mechanism work in the Quantum phase estimation algorithm?

I've probably read the chapter The quantum Fourier transform and its applications from Nielsen and Chuang (10 th anniversary edition) a couple of times before and this took this thing for granted, but ...
11
votes
2answers
1k views

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

I have been trying to get my head around the famous(?) paper Quantum algorithm for linear systems of equations (Harrow, Hassidim & Lloyd, 2009) (more popularly known as the HHL09 algorithm paper) ...
4
votes
1answer
154 views

How to analyze highly entangled quantum circuits?

I came across a quantum circuit very similar to the phase estimation circuit, which is shown below: In the ...
4
votes
2answers
187 views

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

I'm going through the phase estimation algorithm, and wanted to sanity-check my calculations by making sure the state I'd calculated was still normalized. It is, assuming the square of the absolute ...
12
votes
1answer
866 views

Quantum phase estimation and HHL algorithm - knowledge of eigenvalues required?

The quantum phase estimation algorithm (QPE) computes an approximation of the eigenvalue associated to a given eigenvector of a quantum gate $U$. Formally, let $\left|\psi\right>$ be an ...
7
votes
1answer
388 views

SWAP gate(s) in the $R(\lambda^{-1})$ step of the HHL circuit for $4\times 4$ systems

Context: On the 5th page of the paper Quantum circuit design for solving linear systems of equations (Cao et al, 2012) there's this circuit: Schematic: A brief schematic of what's actually ...