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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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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Mathematical reasoning of Repeating the Phase Estimation in the Order Finding Algorithm

Repeating the order-finding algorithm of quantum computing twice obtains $\dfrac{s_1'}{r_1'}$ the first time, and $\dfrac{s_2'}{r_2'}$ the second time, where $\dfrac{s_i'}{r_i'}$ is $\dfrac{s_i}{r}$ ...
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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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How to define the phase obtained from the quantum phase estimation when using the randomized product formula as the Hamiltonian simulation?

Several randomized Hamiltonian simulation methods are developed in recent years. For example, the qDRIFT method is developed by Earl Campbell https://arxiv.org/pdf/1811.08017.pdf, or the randomized ...
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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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Theta value passed as input to quantum phase estimation: qiskit textbook

I'm trying to understand the Quantum Phase Estimation in the qiskit textbook https://qiskit.org/textbook/ch-algorithms/quantum-phase-estimation.html. I know QPE is used to estimate the $\theta$ given ...
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specificity of quantum phase estimation in Childs et al.'s NAND formula evaluation algorithm

I have some uncertainties regarding the quantum algorithm for NAND formula evaluation presented in this paper : Every NAND formula of size N can be evaluated in time N^1/2 +o(1) on a quantum computer. ...
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Chronology of discovery of quantum phase estimation algorithm

I'm a bit confused about exactly when the phase estimation algorithm was discovered. The Wiki article, as well as various textbooks and papers, says that it was introduced in 1995 by Alexei Kitaev, ...
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Prove that applying the QFT twice is equivalent to classical multiplication by $-1$ modulo $2^n$

While going through the answer given on this post, I came across the sentence: If you apply the $QFT$ twice, it is equivalent to a classical multiplication by -1 modulo $2^n$ where $n$ is the size of ...
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Can I combine controlled unitaries in IPE?

Suppose I have two controlled unitary ($U_3$) gates which implement controlled $e^{-iHt_1}$ and controlled $e^{-iHt_2}$ (they share the same control qubits), where $H$ is the same Hamiltonian. My ...
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Applying controlled unitary operations during quantum phase estimation

I am trying to understand Shor's algorithm for a personal research project. I am currently going through quantum phase estimation, and have came accross something I'm struggling to understand in the ...
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How to determine appropriate time evolution for phase estimation algorithm?

In phase estimation algorithms, we have $U|\psi\rangle = e^{2\pi i\theta}|\psi\rangle$, where $|\psi\rangle$ is an eigenvector and $ e^{2\pi i\theta}$ is the corresponding eigenvalue. Since $U$ ...
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Did we entangle the qubits in phase estimation algorithm?

This is a follow-up question to my earlier post about the phase estimation algorithms. From the Qiskit tutorials of QPE and IPE, the qubit $q_1$ represents the physical system on which $U$ operates ...
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Calculating the Inner Product using Quantum Phase Estimation

I'm following the method laid out in https://arxiv.org/abs/2011.03429 (Page 23 Equations 13-23) to calculate the inner product of two amplitude embedded vectors using Quantum Phase Estimation. I'm ...
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State evolution and phase correction for iterative phase estimation

I'm learning about the iterative phase estimation (IPE) algorithm from the qiskit textbook. Here's a circuit I generated to implement this algorithm on a random single-qubit Hamiltonian. Instead of ...
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How much information we can get about eigenstate from phase estimation?

I'm learning about the standard phase estimation algorithm. Here's a diagram from Qiskit tutorial. Suppose we use $\mathcal{C}$ to represent the entire circuit, and $|\psi\rangle$ is an eigenstate of ...
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Why does QAA need QPE to estimate the probability of the good state?

I have some trouble understanding some aspects of the Quantum Amplification Algorithm (QAA). I am using this reference for the following discussion (section 4). The QAA applies a certain operator $Q :=...
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Why For loop is not appending the circuit in the first iteration causing qubit argument mismatch with the gate expectation?

Why For loop is not appending the circuit in the first iteration, it only starts executing on the second iteration, due to which the qubit argument is mismatching with the gate expectation? ...
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Implementing controlled unitary gate in QPE

I'm reading the Qiskit tutorial on quantum phase estimation. In this tutorial, the controlled unitaries on the diagram are denoted as $U^{2^{t-1}}, ..., U^{2^{0}}$: In the actual quantum circuit, ...
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Interpreting the phase in QPE

Given a unitary operator $U$, the quantum phase estimation algorithm estimates the phase $\theta$ in $U|\psi\rangle = e^{2\pi i\theta}|\psi\rangle$. From this Qiskit tutorial, the phase is estimated ...
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Phase estimation using $U_3$ gate

I'm trying to understand how to implement quantum phase estimation (QPE) for a generic single-qubit Hamiltonian. The general time-evolution could be simulated using $U_3$ gate, in Qiskit documentation,...
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Paradox on the evolution direction in controlled Hamiltonian simulation for Quantum Phase Estimation

Suppose we want to perform Quantum Phase Estimation over a Linear Combination of Unitaries Hamiltonian. One of the most efficient ways to do so is to use qubitization: \begin{equation} Q=(2|0\rangle\...
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Why does quantum phase estimation complexity scale with maximum representable energy?

In Quantum simulation of chemistry with sublinear scaling in basis size Ryan Babbush and other authors from Google Quantum team argue, when talking about performing Quantum Phase Estimation in 1st ...
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Applying QPE on a large matrix on amazon-braket

I'm running a QPE algorithm on the amazon-braket but it can only apply on a 22 or 44 matrix, when I want to expand it into a 5*5 or more, it will come an error. As I know, there is no theoretical ...
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Definition(s) of $\delta$ in quantum phase estimation

I read the chapter on QPE (quantum phase estimation) in Nielsen and noticed that $\delta$ is defined there as follows: $0 \leq \delta \leq 2^{-t}$, see: 5.2.1 Performance and requirements The above ...
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Register size in factoring 15 using Shor's algorithm

In Nielsen and Chuang's book: Quantum computation and quantum information (2016), there is an example in Box 5.4 which shows how to factor $15$ using Shor's algorithm. I am confused about a ...
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Derivation of efficiency of Phase Estimation Algorithm

In the section Performance and requirements of the phase estimation algorithm of Page 224, Quantum Computation and Quantum Information by Nielsen and Chuang In order to obtain Eq. 5.27 we have ...
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Equating the state of the Phase Estimation algorithm to $\frac{1}{2^{t/2}}\sum_{k=0}^{2^t-1} e^{2\pi i\phi k}|k\rangle$

It is stated in the Phase Estimation algorithm in Page 222, Quantum Computation and Quantum Information by Nielsen and Chuang that It seems to say that taking the inverse Quantum Fourier transform of ...
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Can we use Hadamard test to estimate phases?

There have been some questions discussing the Hadamard test and quantum phase estimation (QPE), but I did not find the answer to the following question. Suppose we are given $|\psi\rangle$, which is ...
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Implementation of quantum phase estimation in Quirk

after reading the chapter of QPE (Quantum phase estimation) in Nielsen, I wanted to try an implementation in Quirk. My idea was to apply the T-gate, from which I know the following relation $T|1\...
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How would Quantum Phase Estimation be solved classically?

I would be interested to know how Quantum Phase Estimation (QPE) would be solved classically. So suppose we have a matrix and a vector description of $U$ and $|\psi\rangle$. I would present here what ...
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In the HHL algorithm, does the controlled unitary depend on the Hermitian matrix coefficients?

In HHL algorithm, does the controlled unitary (Hamiltonian simulation part of Quantum phase estimation) depend on Hermitian matrix coefficients and how?
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Phase estimation algorithm: Bounding of probability in Nielsen and Chuang

I am currently studying the Quantum Phase Estimation (QPE) algorithm as described in Nielsen and Chuang, pages 223-224. We have the following situation there, we have the state: $$\frac{1}{2^t} \sum\...
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Formulation of quantum phase estimation in Nielsen and Chuang is different then from other sources?

In a chapter of Quantum Computation and Quantum Information by Nielsen and Chuang (10th edition) about quantum phase estimation I get a little confused. Namely: Before applying inverse QFT our quantum ...
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Phase estimation algorithm: Modulo part in Nielsen and Chuang

In Nielsen and Chuang the explanation of phase estimation states: We have the following state: $$\frac{1}{2^{t/2}} \sum\limits_{k=0}^{2^t-1} e^{2 \pi i \varphi k}|k\rangle$$ Now we apply the inverse ...
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Why does Hamiltonian simulation seek to find the energy minimum, if eigenvalues of unitaries are always unimodular?

I know I am wrong here and trying to find out where I am making a logical mistake. I'd appreciate it if you can help me untangle. A. We know that the eigenvalues of Unitaries are all unimodular (...
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Can we use quantum phase estimation to learn anything about the dynamics of puzzles like the Rubik's cube?

Introduction Consider a state $\vert\psi\rangle$ such as below, which is in a superposition of a difference between a Rubik's cube in a solved state and a Rubik's cube in the "superflip" ...
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Can quantum search be performed without phase estimation?

In the quantum search algorithm with a known number of answers $M$ out of a search space of size $N$, it seems that the algorithm works pretty well (though not with pinpoint accuracy) for a fairly ...
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Can I use the Lie product formula to simulate the Hamiltonian of an adjacency matrix by using the QPE to take Nth roots of permutation matrices?

I have gotten some great help recently on Hamiltonian simulation, and am interested in using Hamiltonian simulation to explore (classical) random walks on large graphs, but I'm running up against ...
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What are quantum algorithms with only one possible outcome with probability equal to one?

I would like to study circuits with only one possible outcome. Quantum phase estimation, Bernstein-Vazirani, and in part Deutsch-Jozsa (for constant functions) come to mind - do you know any other ...
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"Classical" phase estimation versus iterative phase estimation

In the article Arbitrary accuracy iterative phase estimation algorithm as a two qubit benchmark, the authors introduced implementation of phase estimation with two qubits only. The trick that bits ...
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Implementation of the Phase Estimation algorithm

I've been working on implementing quantum phase estimation in Qiskit for a $2^n \times 2^n$ Hamiltonian as part of my bachelor project, I'm using Trotterization as my Hamiltonian simulation of choice ...
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What is the difference between amplitude amplification, amplitude estimation, and phase estimation?

I'm confused about the difference among Amplitude amplification (AA) , phase estimation (PE), and Amplitude Estimation. I thought I understood AA and PE somewhat but when I heard the amplitude ...
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QPE Circuit test on Quantum Computer ('ibmq_16_melbourne')

After several atempts, I cannot mitigate the error when running the code on a NISQ, via the qiskit library (more specifically on the 'ibmq_16_melbourne'). I've already mapped the connected qubits and ...
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