Questions tagged [phase-estimation]

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

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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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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 ...
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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 ...
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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 $U$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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)...
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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 ...
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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 |+&...
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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 ...
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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, ...
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Exponential Grover iterations in Quantum Counting

In a quantum counting circuit such as the one below: ...
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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 \...
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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? ...
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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 ...
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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-...
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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 ...
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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 $...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $...
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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: ...
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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 ...
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Can we use quantum phase estimation to estimate the phase of an arbitrary single-qubit state?

Can we use quantum phase estimation (or any other) algorithm to estimate the phase of an arbitrary single-qubit state, without measuring it? That is: estimate the relative phase 𝜙 of the qubit 𝑎|0⟩ +...
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Qiskit textbook: Shor's algorithm

The Qiskit textbook shows the following circuit for implementing the phase $(\frac{s}{r})$ estimation stage of Shor's algorithm for factorizing 15: My understanding is that the register of qubits 8-...
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Why doesn't Z-gate change phase of |0⟩

Since the Pauli Z gate equate to a rotation around z axes of the Bloch sphere by $\pi$ radians, the phase of anything that lies on z axes is expected to change by $\pi$ by applying z-gate. As $|0⟩$ ...
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How can I understand the notation for the final state of QPE algorithm?

I've been reading about the standard phase estimation from the Qiskit tutorial and got stuck in interpreting the final state representation. What does the state $|2^t\theta\rangle$ mean? Is that the ...
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Does phase kickback require the system to be in the eigenstate?

I've been watching this video for the introduction to phase kickback. And here's a diagram: I got confused if we really need $|\psi_k\rangle$ to be an eigenstate to make the kickback work. It seems ...
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Understand $U|\psi\rangle = e^{2\pi i\phi}|\psi\rangle$ in phase estimation algorithms

I'm trying to understand the motivations behind $U|\psi\rangle = e^{2\pi i\phi}|\psi\rangle$ in quantum phase estimation. In my interpretation, since $U$ is the unitary operator, this equation wraps ...
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Why do we start from the least significant bit in phase estimation algorithms?

I've been watching some videos and tutorials for quantum phase estimation. Here's a video I found helpful, which explains the phase estimation in general. I also learned the iterative phase estimation ...
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What phase should I expect from pauli X?

Here's an example I found that implements the phase estimation algorithm: Here the eigenvector is initialized to the $|+\rangle$ state, which is an eigenvector of Pauli X with eigenvalue 1. The ...
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Writing a Phase Estimation function in Cirq

I am trying to write a phase estimation algorithm using Cirq. The algorithm works for different inputs but I receive a few errors in the estimate_phi(mystery) function. ...
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Expectation value of Pauli string for VQE

One approach to get the expectation value $\langle\psi|P\psi\rangle$ of a pauli string $P\in \{I, X, Y, X\}^{\otimes n}$ is the following. Let $(a_i, |\lambda_i\rangle)$ be eigenvalue-eigenvector ...
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The phase of the eigenvalue for the LCU method and randomized product formula

If we implement the linear combination of unitary(LCU), $\tilde{U}=\sum_i p_i U_i$, onto its eigenstate, $|\psi \rangle$, we can express its eigenvalue as $re^{i\theta}$, where $\sum_ip_i=1$, $0\leq ...
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Cost of Modular Exponentiation in Shor's algorithm

In the Shor's algorithm, we need to compute the sequence of controlled $U^{2^j}$ operations used by the phase estimation procedure, where $U$ is defined as $$ U|y\rangle=|xy\;(\mod N)\rangle\text{ ...
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