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Questions tagged [quantum-phase-estimation]

For questions about the quantum phase estimation algorithm.

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Difficulty in understanding Quantum Phase Estimation [closed]

I am trying to understand Quantum Phase Algorithm. I see circuit of Hadamard Test is very short. It contains 2 Hadamard gate, 1 controlled Unitary gate and finally measurement. Also notice that IBM ...
Sanu's user avatar
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quantum phase estimation overperiod value estimation

Basacally, QPE allows to find eigenvalue within a period $[0$ to $2\pi)$. Is there an extention that allows to calculate eigenvalue out of these limits, e.g. greater or equal to $2 \pi$?
Марина Лисниченко's user avatar
3 votes
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38 views

Alternative algorithm for quantum phase estimation problem

The Quantum Phase estimation problem is stated below: Input: Given $U$ as a unitary operator acting on an m-qubit register. If $| \psi \rangle$ is an eigenvector of $U$, then U$| \psi\rangle$ = $e^{ ...
Manish Kumar's user avatar
1 vote
1 answer
216 views

Exponential Quantum Speedup for the Traveling Salesman Problem - where is the catch?

Such an article claims that an NP-complete problem can be solved efficiently. Is it real? I noticed that they prepare a state $|0\rangle\langle0|+|1\rangle\langle1|$ on an ancilla, which is impossible ...
Ron Cohen's user avatar
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2 votes
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Is BQP contained in BPP with Quantum Phase Estimation (QPE) oracle?

I am trying to see if the below proposition holds: Proposition-1: $BQP\subseteq BPP^{QPE}$. Here, QPE is the Quantum Phase estimation algorithm. QPE takes an eigenstate and the unitary matrix as ...
Manish Kumar's user avatar
2 votes
1 answer
48 views

Complexity of controlled-$U^j$ operations in QPE applied to Hamiltonian simulation

One method to obtain the eigenvalues of a Hamiltonian $H$ is by applying quantum phase estimation to its time-evolution operator $U(t) = e^{-iHt}$. If I want to obtain an eigenvalue to $k$ bits in ...
Solarflare0's user avatar
4 votes
2 answers
72 views

Generalizing error propagation formula to multi-parameters

For single parameter phase estimation we have the Cramer-Rao bound $$(\Delta \theta)^2 \geq \frac{1}{F_{Q}[\rho, \hat{A}]},$$where $F_{Q}$ is the quantum Fisher information and where instead of an ...
John Doe's user avatar
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3 votes
1 answer
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In what limit does the estimator sample variance converge to the Cramer-Rao bound?

In the context of a single phase estimation problem of a quantum photonics experiment (related post). For example consider a 3-photon quantum circuit (such as the Mach-Zehnder which depends on some ...
John Doe's user avatar
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3 votes
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Why FACTORING is in second level of Fourier hierarchy?

As per comlexityzoo web, the definition of the k-th level of Fourier Hierarchy (FH) is: $FH_k$ is the class of problems solvable by a uniform family of polynomial-size quantum circuits, with k levels ...
Manish Kumar's user avatar
4 votes
1 answer
154 views

Requirement of vector 'b' in the definition of Phase Estimation Sampling (PES)

In this paper (last paragraph, page 3) by Wocjan and Zhang, the definition of PES requires vector/bit string b. The phase estimation problem (PE) very much inspires the definition. I cannot ...
Manish Kumar's user avatar
2 votes
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Modelling Mach-Zehnder and saturating Cramer-Rao bound

I am simulating (using Mathematica) a Mach-Zehnder interferometer, with photon counting measurements at the end (based on the setup described in the recent post) for the input state $|\psi\rangle:=|0,...
John Doe's user avatar
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3 answers
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Shor: Modular exponentiation vs modular multiplication

In his original article Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer, Peter Shor constructed an algorithm for finding a period $r$ of the modular ...
Martin Vesely's user avatar
1 vote
1 answer
134 views

Quantum Phase Estimation answers distribution

Suppose I have a random unitary matrix, known eigenvectors and eigenvalues. I know that exact eigenvalue for the given matrix is $0.5491617699847768+0.835716070437315j$. From here, if I'm not mistaken ...
Марина Лисниченко's user avatar
3 votes
0 answers
32 views

How to have seperate registers from scrach?

I am very new to quantum computing, so this might be a really simple question. I am coding a quantum computer simulator in python from scrach and I'm not sure how to make registers work. I would just ...
Gytis Vejelis's user avatar
4 votes
1 answer
90 views

How does the complexity of extracting eigenvalues via quantum phase estimation compare with the classical one?

Suppose, I have ideal quantum computer that allows me to find exact eigenvalues with QPE algorithm under perfect matrix, eigenvectors and eigenvalues conditions. How the complexity of this algorithm ...
Марина Лисниченко's user avatar
1 vote
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79 views

Deriving circuit templates for Hamiltonian simulation

Background I've been reading the paper entitled Some improvements to product formula circuits for Hamiltonian simulation. The authors propose three improvements motivated by phase estimation type ...
Callum's user avatar
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How to determine the single-qubit phase?

In some works it is suggested to find the qubit phase by the following method: Apply $X/2$ (or $Y/2$) gate - preparing the qubit in superposition. Detune the qubit by flux pulse, for example here ...
Curious's user avatar
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Can I get valid solution with HHL algorithm even if the QPE is not completely correct?

Can I get the valid solution of linear problem using HHL algorithm even if the QPE is not completely correct? My example is $A=diag(0.5, 0.2, 0.3, 0.6)$, so the solution is $[0.5, 0.2, 0.3, 0.6]$ ...
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2 answers
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How to estimate the negative amplitude of multiple qubits?

The probability of measurement is the square of amplitude. After measurement, how to guess the original amplitude of state?? For example, in linear problem, we would like to know the exact solution, ...
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Accuracy of QPE using histogram fit

In regular QPE, accuracy is $1/2^n$, where $n$ is estimation register size, and it is done with few shots and depth as $O(2^n)$ (total cost of $O(2^n*NumShots))=2^n=1/resolution$). I thought of the ...
Ron Cohen's user avatar
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Problem with the mathematical definition of the eigenvalue algorithm on a specific exercise

I think I understand well how the eigenvalue algorithm works but when I try to define it mathematically I have problems. Specifically I have the matrix U: $$ U = \begin{pmatrix} 0 & i \\ i & 0 ...
Francescov20's user avatar
2 votes
1 answer
160 views

Problem with eigenvalue evaluation algorithm application on matrix $U$

Once I get to the end of the algorithm, I can't understand how to calculate the eigenvalue using formulas. Bear in mind that it is an exercise to be carried out with pen and paper. the matrix of $U$ ...
Francescov20's user avatar
2 votes
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Does this measurement for quantum phase estimation look correct?

I have implemented Shor's algorithm for $N=15$ from this tutorial. I understand the algorithm pretty well, but I'm a little confused at the output I'm getting from running the circuit. It appears to ...
Jackson Walters's user avatar
3 votes
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100 views

Adiabatic state preparation for quantum phase estimation

I'm trying to understand the problem of state preparation for quantum phase estimation (QPE). Specifically how states are prepared adiabatically. I have a couple of questions: 1). Typically when one ...
Benjamin's user avatar
4 votes
2 answers
179 views

How to derive the expression for the probability in quantum phase estimation? ((5.27) Nielsen & Chuang)

I'm trying to understand the QPE algorithm that is presented in the Nielsen and Chuang textbook. More precisely, I do not understand Equation $(5.27)$. Context: In the following, let $b$ be a natural ...
Marcus's user avatar
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1 answer
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Constructing a controlled phase gate from given gates

As part of a project in a quantum computing course we were asked to classically simulate the quantum phase estimation algorithm, which has inverse QFT as one of its components. On the Wikipedia page ...
Ziv's user avatar
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78 views

Is the QFT optimal in the quantum phase estimation algorithm?

We can concisely summarise the quantum phase estimation (QPE) algorithm as follows: Generate the state $\sum_{k=0}^{2^n-1} \lambda^k |k\rangle$ efficiently using a series of controlled-unitary ...
glS's user avatar
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4 votes
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56 views

We take the reciprocal $\lambda^{-1}$ of eigenvalues in HHL - but what's stopping us from raising them to a positive exponent $\lambda^m$?

The HHL algorithm generally can be thought of as diagonalizing our matrix $A$ with the quantum phase estimation algorithm, and applying a specific function $f(\lambda)=\lambda^{-1}$ to the eigenvalues ...
Mark Spinelli's user avatar
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1 answer
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Amplitude Estimation/Counting - unsatisfiability

The Amplitude Amplification paper states in Theorem 13: For any positive integers $M$ and $k$, and any Boolean function $f: \{0,1,\ldots,N-1\}\rightarrow\{0,1\}$, the algorithm Count $\left(f,M\right)...
inq's user avatar
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1 answer
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How would you draw the phase-estimation circuit for the eigenvalues of $U = \mathrm{diag}(1,1,\exp(\pi i/4),\exp(\pi i/8)) $?

How would you draw the phase-estimation circuit for the eigenvalues of: $U = \mathrm{diag}(1,1,e^{(\pi i)/ 4}, e^{(\pi i)/8}) $ corresponding to the eigenstates $|10\rangle$ and $|11\rangle$? What is ...
Charlie Plath's user avatar
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1 answer
76 views

Phase estimation of the Pauli-Y matrix

I'm trying to use the phase estimation algorithm to extract the eigen value for both eigen vectors of the Pauli-Y matrix using the ibm quantum experiance. So far I have this for the possitive state |+&...
Charlie Plath's user avatar
1 vote
1 answer
144 views

Quantum Phase Estimation on the X gate

So I'm trying to perform QPE on the X-gate in IBM quantum but I'm not 100% sure that my implementation is correct. I've been able to do this for the T, S, and Z gates by using P gates with different ...
Conejo11's user avatar
0 votes
1 answer
153 views

Qiskit - Approximation of Hamiltonian energy via QPE

I'm trying to study QPE with the motivation of obtaining eigenvalues of Hamiltonian, i.e. energies of a system. My problem is, that while np.linalg.eig and VQE are agreeing on the lowest energy, ...
Eenoku's user avatar
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0 answers
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Exponential Grover iterations in Quantum Counting

In a quantum counting circuit such as the one below: ...
Lucas's user avatar
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1 vote
1 answer
65 views

Sequence of controlled gates on non-eigenstates

Quantum phase estimation predicts the eigenvalues of a unitary operator given an eigenstate, using a sequence of controlled versions of that operator. The math relies on the fact that $ |0, \psi \...
Loic Stoic's user avatar
1 vote
0 answers
114 views

HHL phase estimation step

I have got an HHL circuit that looks as follows: In the phase estimation part we are trying to find the eigenvalues of the matrix A. But what is the role of the piece of circuit hightlighted below? ...
Марина Лисниченко's user avatar
0 votes
1 answer
230 views

Quantum counting with ancillary qubits

I have been trying to implement quantum counting using my own oracle, however I've been unsuccessful getting results that make sense. The circuit I'm using looks like this (I'm only showing the ...
Lucas's user avatar
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4 votes
1 answer
146 views

Is the phase-estimation a specific case of the Hidden Subgroup Problem?

I read Nielsen & Chuang and I have difficulties understanding the links between the Hidden Subgroup Problem and the Phase Estimation. In Exercise 5.14 (Section 5.3.1 "Application: order-...
user8622655's user avatar
1 vote
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67 views

Reduced density matrix accuracy in amplitude estimation

I am implementing QAE (Quantum Amplitude Estimation), which is very similar to QPE (Quantum Phase Estimation) with a Grover Operator as the U matrix of QPE. I want to check my results, in the outputs ...
Ron Cohen's user avatar
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4 votes
1 answer
346 views

Question regarding Quantum Phase Estimation (Nielsen and Chuang exercise 5.8)

I was working through Nielsen and Chuang's book on quantum computing and they state the following result regarding the performance of the Quantum Phase Estimation algorithm, "... given the input $...
Bikrant Bhattacharyya's user avatar
5 votes
2 answers
858 views

Clarification on state prep for quantum phase estimation

I have a question about how to prepare a state $|\psi\rangle$ for quantum phase estimation (QPE). My question is about whether the state prepared in QPE has to be the exact eigenstate of the operator ...
Callum's user avatar
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2 votes
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Quantum phase estimation error on most likely phase

I have found an error formula for quantum phase estimation in terms of the total number of qubits, $t$, necessary to to measure the closest phase to an accuracy of $n$ bits with a probability of at ...
kηives's user avatar
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2 answers
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$QFT^{-1}$ at the end of Shor's algorithm and $QFT$ at the end of Hidden Subgroup algorithm

In the usual presentations (e.g. Nielsen and Chuang) Shor's algorithm (in its quantum part) is presented as a special case of phase estimation, meaning it uses a circuit of the form "generate ...
Gadi A's user avatar
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1 answer
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Is Shor demonstration wrong?

in Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer by Peter W. Shor (also in Algorithms for quantum computation: discrete logarithms and factoring). In ...
Philip.q.c's user avatar
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0 answers
48 views

How does QPE work if target register is in superposition of eigenstates?

In regular Quantum Phase Estimation algorithm the target register shall be in the eigenstate of the investigated operator. If it's the case, then applying controlled operator $U$ we can get its phase ...
EugeneB's user avatar
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2 votes
0 answers
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Is there a code version for the Quantum Mean Approximation algorithm?

I found this paper https://arxiv.org/pdf/1106.4267.pdf which describes a method to find the mean of all numbers in a list using a quantum algorithm. However, all it gives is the math behind it. Is ...
AadiTiwari's user avatar
1 vote
0 answers
53 views

Quantum algortihm for SVD and eigendecompostion

In this paper by Rebentrost, Steffens, and Lloyd, it is stated that: Such tasks [eigen- and singular value decomposition of a matrix] could be performed efficiently via phase estimation on a ...
atman's user avatar
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1 vote
1 answer
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Kaye Exercise 7.1.3, Quantum Phase Estimation

Prove that $O(\log_2(r))$ phase estimations with $n = m$ and taking the outcome that occurs most often provides an estimate $\tilde \omega$ of the phase $\omega$ which will with probability at least $...
Giorgos Giapitzakis's user avatar
0 votes
1 answer
563 views

Accuracy of Quantum Phase estimation; Finding the max difference integer, e

Working through Lab 5 in the Qiskit text, I have been attempting to complete Part 1, Step B. I implemented the following code as it seemed, at the time, to be what the question was asking for: ...
PGibbon's user avatar
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2 votes
1 answer
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Confusion about Rodeo algorithm "spectral weight suppression" argument

In this first paper on the Rodeo algorithm, there is an argument on the second page about the suppression of "spectral weights" that I don't really understand. In short, the algorithm is ...
tomdodd4598's user avatar