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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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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What is the significance of the phase angle? [duplicate]

I've the following circuit which gives an output of 1 with a phase angle of 3π/4. When we measure the circuit all we get is the ...
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Iterative Phase Estimation with noise vs standard Quantum Phase Estimation with noise

I am doing Qiskit Lab 4 about Iterative Phase Estimation. I created a circuit implementing IPE for theta = 1/3 (phase of 2pi/3). Here's the circuit: It seems to do okay if I run it without noise in a ...
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If we can prepare a ground state efficiently, when can we prepare the second-lowest energy eigenstate?

I'd like to know if there's anything that can be said about whether and when we can efficiently prepare a state corresponding to the second-lowest eigenvalue of a given Hamiltonian, or in any other ...
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Question about the phase kickback in the phase estimation algorithm

I have an issue with the quantum phase estimation algorithm as explained Nielsen and Chuang. There was a question very similar to mine asked about this 2 years ago, but my question is different... ...
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Finding Eigen Values from Quantum Phase Estimation - Using qiskit

I am trying to use the quantum phase estimation(EigsQPE) of qiskit to find the eigen values of a matrix. As I am new to quantum computing so I am confused what to measure in the circuit to derive the ...
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Quantum Circuit for $e^{iAt}$ Hamiltonian Simulation in HHL algorithm

In HHL algorithm, there is a step in Quantum Phase Estimation where we have to apply powers of $e^{iAt}$ to the register (see pic). I am not able to understand how to find the quantum circuit ...
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3-qubit phase shift gate/circuit implementation without any Ancilla qubits

Hi, I need help me with figuring out the 3-qubit phase shift circuit without any ancillas similar to the 2-qubit circuit shown in below attached picture....... Please do let me know! Thanks in advance!...
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HHL algorithm, How to implement exp(iAt) gates?

From this paper Quantum Circuit Design for Solving Linear Systems of Equations, in figure 4 The paper shows what inside operator $e^{-iAt}$ but didn't shows how to connect the control qubit (register ...
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What is the meaning of approximating a phase to an accuracy of $2^{-n}$?

In quantum phase estimation we often see that approximating $\phi$ ito an accuracy of $2^{-n}$. Can anybody explain what is the meaning of that? Does that mean after decimal we can only believe the n ...
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How do I calculate the number of uses of a unitary $U$ in iterative phase estimation?

How would one go along to calculate the number of uses of an unitary $U$ in Iterative Phase Estimation (IPE) to compare it to the number of uses of $U$ in standard Phase Estimation (Qiskit QPE)?
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How to implement Majority Vote

I am trying to boost the success propability of standart phase estimation by repeating the procedure enought times and taking a majority vote that will be encoded in a quantum register. My problem is ...
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How to perform a phase operator on register that contains two or more qubits?

My problem is easy to understand, just how to calculate the matrix of phase operator(or phase gate) acts on multi-qubits so that i can perfrom it in quantum circuit on IBM Quantum Experience just like ...
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Simulating QPE + Grover using Low-Rank Stabilizer Decomposition

I want to simulate a 40-45 Qubit circuit that applies Grover + QPE. I've tried running a simulation on qiskit but can't really go past 18 qubits on my machine. As an alternative, I've been reading ...
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Why do the controlled unitary operations in quantum phase estimation have $2^n$ in their exponents?

Why do the unitary gates on the measurement qubits have $2^n$? Why do we need to apply the unitary gates for any power at all? What would happen if we applied the controlled-$U$ only once, for ...
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Efficient QFT-based QPEA complexity

The HHL algorithm lies on an implementation of the Quantum Phase Estimation algorithm. One popular implementation is based on the Quantum Fourier Transform which can be divided in three steps. Let $U$ ...
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In the quantum phase estimation algorithm, why can't we directly compute the eigenvalue from the known eigenvector?

The Quantum Phase Estimation algorithm wants to approximate the phase $\varphi$ of an eigenvalue $\lambda = e^{2\pi i \varphi}$ of a unitary operator $U$. Besides $U$ an eigenvector $x$ corresponding ...
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Shor's algorithm - modular exponentiation and Quantum Fourier transform and quantum phase estimation method

I have a question about Shor's algorithm with respect to the eigenvector representation of the second (lower) register. In the following I use the notation of Nielsen, M., Chuang, I., 2016, Quantum ...
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HHL algorithm, How can I get result from register $|b\rangle$?

From the paper A survey on HHL algorithm: From theory to application in quantum machine learning , I use qasm code from here. I try to follow the example in page 7. with Ax = b and the answer x ...
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Quantum Amplitude Estimation vs Quantum Phase Estimation

Quick question concerning the probability of success after a phase estimation algorithm vs an amplitude estimation algorithm. Given the calculation on the wikipedia page, the probability of measuring ...
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Output of Quantum Phase Estimation Algorithm

In section 5.2.1 of Nielsen Chuang, Performance and Requirements, there is an idea, that what happens if we can't prepare eigen state $|u\rangle$ and instead have a state $|\psi\rangle$ which is ...
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A question regarding quantum phase estimation algorithm

Why are $U$s raised to successive powers of two in quantum phase estimation circuit diagram when we use $n$ register qubits $|0\rangle|0\rangle|0\rangle$?
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Trying to perform Quantum Phase Estimation on T-gate

I'm trying to perform QPE on the T-gate in Quirk but I'm not getting the correct result. For the T-gate, I should be measuring (001) with 100% probability, but instead, I'm getting the following: I'...
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In Nielsen and Chuang, how can $\frac{1}{2(e-1)}$ result from $\frac12\int_{e-1}^{2^{t-1}-1}dl\frac{1}{l^2}$?

From Nielsen and Chuang's book: $\textit{Quantum computation and quantum information}$, how can (5.34) equal (5.33)? I.e. $$\dfrac{1}{2} \int_{e-1}^{2^{t-1}-1} dl \dfrac{1}{l^2} = \dfrac{1}{2(e-1)}.$$...
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Given a matrix, how do I proceed with the quantum phase estimation algorithm and choose $\theta$?

Given a matrix, say $\begin{bmatrix} 1.5 & 0.5\\ 0.5& 1.5 \end{bmatrix}$, with eigenvalues $1$ and $2$, how do I proceed with the quantum phase estimation algorithm? In particular, how do I ...
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Quantum Phase estimation with $2\pi$ replaced with $2e$

The QPE on IBM platform finds the eigenvalue of a unitary operator, i.e $$U|\phi\rangle=e^{2\pi i\theta}|\phi\rangle$$ and uses the rotation operators as $$U(\theta)=\begin{bmatrix}0 & 1\\ 1 &...
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Reducing cost of Phase Estimation for Trotterization

Even though Trotterized Hamiltonians have polynomial time scaling directly, the process of quantum phase estimation means that the controlled unitaries $ CU$ scale exponentially with number of ...