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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“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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HHL algorithm Qiskit version

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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 ...
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How to use the measurement in quantum phase estimation?

Reading the quantum phase estimation algorithm on Wikipedia, I am wondering how exactly the measurements are used to obtain the phase $\delta$. I understand that the value of phase is encoded into the ...
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In Shor's algorithm, how can we guarantee that each controlled-U will kickback to the same eigenvalue?

I'm studying the Shor algorithm as part of my thesis and have a question about the "measured" phases after the QPE. So, I take the controlled-U operations on the second register and in cause ...
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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 ...
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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\...
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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 $I$, $U1(\pi/8)$, $S$, $T$ ...
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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?
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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 ...
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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 ...
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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\...
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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}...
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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 ...
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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 ...
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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/\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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: ...