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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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 ...
glS♦
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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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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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